\(\int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx\) [194]

Optimal result

Integrand size = 29, antiderivative size = 2754 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Output:

-1/64*f*(48*a^6*d^5*f^5*(-c*f*h-4*d*e*h+5*d*f*g)+16*a^5*b*d^4*f^4*(23*c^2* 
f^2*h-c*d*f*(-47*e*h+75*f*g)-5*d^2*e*(-4*e*h+3*f*g))-b^6*(960*d^6*e^6*g+c^ 
3*d^3*e^3*f^2*(-296*e*h+195*f*g)+3*c^4*d^2*e^2*f^3*(-56*e*h+45*f*g)+15*c^5 
*d*e*f^4*(-8*e*h+7*f*g)-35*c^6*f^5*(-8*e*h+9*f*g)-80*c*d^5*e^5*(8*e*h+21*f 
*g)+8*c^2*d^4*e^4*f*(148*e*h+45*f*g))+a^2*b^4*d*f*(225*c^5*f^5*h-3*c^3*d^2 
*e*f^3*(-1289*e*h+905*f*g)+3*c^4*d*f^4*(-1097*e*h+1355*f*g)-8*d^5*e^4*(178 
*e*h+795*f*g)+7*c*d^4*e^3*f*(904*e*h+1905*f*g)-5*c^2*d^3*e^2*f^2*(1861*e*h 
+945*f*g))+a^4*b^2*d^3*f^3*(1083*c^3*f^3*h+c^2*d*f^2*(-7009*e*h+1920*f*g)- 
d^3*e^2*(1723*e*h+480*f*g)+c*d^2*e*f*(4049*e*h+2160*f*g))-a*b^5*(35*c^6*f^ 
6*h-c^3*d^3*e^2*f^3*(-1661*e*h+1125*f*g)-3*c^4*d^2*e*f^4*(-323*e*h+265*f*g 
)+15*c^5*d*f^5*(-103*e*h+119*f*g)-80*d^6*e^5*(4*e*h+51*f*g)+16*c*d^5*e^4*f 
*(197*e*h+480*f*g)-c^2*d^4*e^3*f^2*(5392*e*h+2025*f*g))-a^3*b^3*d^2*f^2*(6 
33*c^4*f^4*h+15*c^3*d*f^3*(-181*e*h+301*f*g)-d^4*e^3*(2384*e*h+4035*f*g)+c 
*d^3*e^2*f*(6679*e*h+10185*f*g)-c^2*d^2*e*f^2*(7013*e*h+5865*f*g)))/(-a*d+ 
b*c)^6/(-a*f+b*e)^5/(-c*f+d*e)^3/(f*x+e)^(1/2)+1/192*d*(3*a^4*d^3*f^3*(-60 
7*c*f*h+575*d*e*h+32*d*f*g)+b^4*(1440*d^4*e^4*g-c^3*d*e*f^2*(-328*e*h+399* 
f*g)-2*c^2*d^2*e^2*f*(-128*e*h+195*f*g)-35*c^4*f^3*(-8*e*h+9*f*g)-240*c*d^ 
3*e^3*(4*e*h+f*g))+a*b^3*(35*c^4*f^4*h+3*c^3*d*f^3*(-459*e*h+553*f*g)+3*c^ 
2*d^2*e*f^2*(-454*e*h+659*f*g)-240*d^4*e^3*(2*e*h+23*f*g)+4*c*d^3*e^2*f*(8 
92*e*h+375*f*g))-a^2*b^2*d*f*(211*c^3*f^3*h+3*c^2*d*f^2*(-813*e*h+1159*...
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 16.57 (sec) , antiderivative size = 14143, normalized size of antiderivative = 5.14 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(g + h*x)/((a + b*x)^5*(c + d*x)^3*(e + f*x)^(3/2)),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 8.83 (sec) , antiderivative size = 2899, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {168, 27, 168, 27, 168, 27, 168, 27, 168, 27, 168, 27, 169, 27, 174, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\int \frac {2 b \left (6 d e g+\frac {9 c f g}{2}-4 c e h\right )-a (8 d f g+4 d e h+c f h)+13 d f (b g-a h) x}{2 (a+b x)^4 (c+d x)^3 (e+f x)^{3/2}}dx}{4 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {b (12 d e g+9 c f g-8 c e h)-a (8 d f g+4 d e h+c f h)+13 d f (b g-a h) x}{(a+b x)^4 (c+d x)^3 (e+f x)^{3/2}}dx}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {d f (48 d f g+76 d e h+19 c f h) a^2-b \left (4 e (51 f g+10 e h) d^2+c f (123 f g-88 e h) d+7 c^2 f^2 h\right ) a+b^2 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )+11 d f \left (13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)\right ) x}{2 (a+b x)^3 (c+d x)^3 (e+f x)^{3/2}}dx}{3 (b c-a d) (b e-a f)}-\frac {13 a^2 d f h-a b (c f h+4 d e h+21 d f g)+b^2 (-8 c e h+9 c f g+12 d e g)}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {d f (48 d f g+76 d e h+19 c f h) a^2-b \left (4 e (51 f g+10 e h) d^2+c f (123 f g-88 e h) d+7 c^2 f^2 h\right ) a+b^2 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )+11 d f \left (13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)\right ) x}{(a+b x)^3 (c+d x)^3 (e+f x)^{3/2}}dx}{6 (b c-a d) (b e-a f)}-\frac {13 a^2 d f h-a b (c f h+4 d e h+21 d f g)+b^2 (-8 c e h+9 c f g+12 d e g)}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {143 a^3 d^2 f^2 h-3 a^2 b d f (10 c f h+40 d e h+93 d f g)+a b^2 \left (7 c^2 f^2 h+2 c d f (111 f g-88 e h)+8 d^2 e (5 e h+42 f g)\right )-b^3 \left (7 c^2 f (9 f g-8 e h)+16 c d e (6 f g-5 e h)+120 d^2 e^2 g\right )}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}-\frac {\int -\frac {3 d^2 f^2 (64 d f g+292 d e h+73 c f h) a^3-b d f \left (236 e (9 f g+4 e h) d^2+c f (963 f g-304 e h) d+134 c^2 f^2 h\right ) a^2+b^2 \left (80 e^2 (33 f g+4 e h) d^3+8 c e f (249 f g-169 e h) d^2+2 c^2 f^2 (483 f g-370 e h) d+35 c^3 f^3 h\right ) a-b^3 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )+9 d f \left (143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )\right ) x}{2 (a+b x)^2 (c+d x)^3 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {13 a^2 d f h-a b (c f h+4 d e h+21 d f g)+b^2 (-8 c e h+9 c f g+12 d e g)}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\int \frac {3 d^2 f^2 (64 d f g+292 d e h+73 c f h) a^3-b d f \left (236 e (9 f g+4 e h) d^2+c f (963 f g-304 e h) d+134 c^2 f^2 h\right ) a^2+b^2 \left (80 e^2 (33 f g+4 e h) d^3+8 c e f (249 f g-169 e h) d^2+2 c^2 f^2 (483 f g-370 e h) d+35 c^3 f^3 h\right ) a-b^3 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )+9 d f \left (143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )\right ) x}{(a+b x)^2 (c+d x)^3 (e+f x)^{3/2}}dx}{4 (b c-a d) (b e-a f)}+\frac {143 a^3 d^2 f^2 h-3 a^2 b d f (10 c f h+40 d e h+93 d f g)+a b^2 \left (7 c^2 f^2 h+2 c d f (111 f g-88 e h)+8 d^2 e (5 e h+42 f g)\right )-b^3 \left (7 c^2 f (9 f g-8 e h)+16 c d e (6 f g-5 e h)+120 d^2 e^2 g\right )}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {13 a^2 d f h-a b (c f h+4 d e h+21 d f g)+b^2 (-8 c e h+9 c f g+12 d e g)}{3 (a+b x)^3 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\int -\frac {3 d^3 f^3 (575 c f h+4 d (32 f g+575 e h)) a^4-b d^2 f^2 \left (4 e (3861 f g+2866 e h) d^2+c f (5013 f g+2920 e h) d+1195 c^2 f^2 h\right ) a^3+b^2 d f \left (8 e^2 (3765 f g+968 e h) d^3+8 c e f (2766 f g-1229 e h) d^2+c^2 f^2 (6819 f g-3460 e h) d+535 c^3 f^3 h\right ) a^2-b^3 \left (960 e^3 (23 f g+2 e h) d^4+8 c e^2 f (2505 f g-1504 e h) d^3+4 c^2 e f^2 (3021 f g-2264 e h) d^2+5 c^3 f^3 (819 f g-640 e h) d+105 c^4 f^4 h\right ) a+3 b^4 \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )+7 d f \left (1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )\right ) x}{2 (a+b x) (c+d x)^3 (e+f x)^{3/2}}dx}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\int \frac {3 d^3 f^3 (575 c f h+4 d (32 f g+575 e h)) a^4-b d^2 f^2 \left (4 e (3861 f g+2866 e h) d^2+c f (5013 f g+2920 e h) d+1195 c^2 f^2 h\right ) a^3+b^2 d f \left (8 e^2 (3765 f g+968 e h) d^3+8 c e f (2766 f g-1229 e h) d^2+c^2 f^2 (6819 f g-3460 e h) d+535 c^3 f^3 h\right ) a^2-b^3 \left (960 e^3 (23 f g+2 e h) d^4+8 c e^2 f (2505 f g-1504 e h) d^3+4 c^2 e f^2 (3021 f g-2264 e h) d^2+5 c^3 f^3 (819 f g-640 e h) d+105 c^4 f^4 h\right ) a+3 b^4 \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )+7 d f \left (1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\int \frac {2 \left (96 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (6262 d^2 h e^2-2363 c^2 f^2 h-c d f (1920 f g+1979 e h)\right ) a^4+b^2 d^2 f^2 \left (-2 e^2 (6915 f g+5456 e h) d^3+3 c e f (2785 f g+2396 e h) d^2+15 c^2 f^2 (557 f g-45 e h) d+1519 c^3 f^3 h\right ) a^3-b^3 d f \left (-48 e^3 (595 f g+158 e h) d^4+20 c e^2 f (327 f g+824 e h) d^3+3 c^2 e f^2 (4895 f g-1786 e h) d^2+3 c^3 f^3 (3085 f g-2021 e h) d+605 c^4 f^4 h\right ) a^2+b^4 \left (-480 e^4 (45 f g+4 e h) d^5+48 c e^3 f (35 f g+284 e h) d^4+2 c^2 e^2 f^2 (3885 f g-1352 e h) d^3+c^3 e f^3 (7905 f g-5728 e h) d^2+5 c^4 f^4 (945 f g-773 e h) d+105 c^5 f^5 h\right ) a+3 b^5 (d e-c f) \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )+5 b d f \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right ) x\right )}{(a+b x) (c+d x)^2 (e+f x)^{3/2}}dx}{2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\int \frac {96 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (6262 d^2 h e^2-2363 c^2 f^2 h-c d f (1920 f g+1979 e h)\right ) a^4+b^2 d^2 f^2 \left (-2 e^2 (6915 f g+5456 e h) d^3+3 c e f (2785 f g+2396 e h) d^2+15 c^2 f^2 (557 f g-45 e h) d+1519 c^3 f^3 h\right ) a^3-b^3 d f \left (-48 e^3 (595 f g+158 e h) d^4+20 c e^2 f (327 f g+824 e h) d^3+3 c^2 e f^2 (4895 f g-1786 e h) d^2+3 c^3 f^3 (3085 f g-2021 e h) d+605 c^4 f^4 h\right ) a^2+b^4 \left (-480 e^4 (45 f g+4 e h) d^5+48 c e^3 f (35 f g+284 e h) d^4+2 c^2 e^2 f^2 (3885 f g-1352 e h) d^3+c^3 e f^3 (7905 f g-5728 e h) d^2+5 c^4 f^4 (945 f g-773 e h) d+105 c^5 f^5 h\right ) a+3 b^5 (d e-c f) \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )+5 b d f \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right ) x}{(a+b x) (c+d x)^2 (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (48 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (3131 d^2 h e^2+3371 c^2 f^2 h-2 c d f (600 f g+2651 e h)\right ) a^4-b^2 d^2 f^2 \left (e^2 (6915 f g+5456 e h) d^3-c e f (13830 f g+10759 e h) d^2+35 c^2 f^2 (129 f g+202 e h) d+633 c^3 f^3 h\right ) a^3+3 b^3 d f \left (8 e^3 (595 f g+158 e h) d^4-5 c e^2 f (1473 f g+872 e h) d^3+c^2 e f^2 (450 f g+4843 e h) d^2+c^3 f^3 (1355 f g-1022 e h) d+75 c^4 f^4 h\right ) a^2-b^4 \left (240 e^4 (45 f g+4 e h) d^5-48 c e^3 f (305 f g+167 e h) d^4-3 c^2 e^2 f^2 (45 f g-3424 e h) d^3+c^3 e f^3 (990 f g-541 e h) d^2+5 c^4 f^4 (357 f g-302 e h) d+35 c^5 f^5 h\right ) a+b^5 \left (35 f^4 (9 f g-8 e h) c^5+10 d e f^3 (21 f g-16 e h) c^4+d^2 e^2 f^2 (75 f g+8 e h) c^3-24 d^3 e^3 f (5 f g-108 e h) c^2-240 d^4 e^4 (15 f g+8 e h) c+2880 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {\int \frac {3 \left (48 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+16 b d^4 f^4 \left (8 d^2 h e^2+23 c^2 f^2 h-c d f (75 f g-44 e h)\right ) a^5+b^2 d^3 f^3 \left (-32 e^2 (15 f g-44 e h) d^3+c e f (960 f g-1253 e h) d^2+2 c^2 f^2 (960 f g-1819 e h) d+1083 c^3 f^3 h\right ) a^4-3 b^3 d^2 f^2 \left (64 e^3 (15 f g+16 e h) d^4-5 c e^2 f (243 f g+272 e h) d^3-c^2 e f^2 (450 f g-19 e h) d^2+c^3 f^3 (1505 f g-694 e h) d+211 c^4 f^4 h\right ) a^3+b^4 d f \left (16 e^4 (495 f g+148 e h) d^5-8 c e^3 f (1095 f g+844 e h) d^4-c^2 e^2 f^2 (3375 f g-5224 e h) d^3+9 c^3 e f^3 (150 f g+89 e h) d^2+3 c^4 f^4 (1355 f g-1022 e h) d+225 c^5 f^5 h\right ) a^2-b^5 \left (320 e^5 (21 f g+2 e h) d^6-16 c e^4 f (435 f g+304 e h) d^5-80 c^2 e^3 f^2 (27 f g-61 e h) d^4-5 c^3 e^2 f^3 (27 f g-224 e h) d^3+c^4 e f^4 (990 f g-541 e h) d^2+5 c^5 f^5 (357 f g-302 e h) d+35 c^6 f^6 h\right ) a+b^6 (d e-c f)^2 \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )+b d f \left (48 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (3131 d^2 h e^2+3371 c^2 f^2 h-2 c d f (600 f g+2651 e h)\right ) a^4-b^2 d^2 f^2 \left (e^2 (6915 f g+5456 e h) d^3-c e f (13830 f g+10759 e h) d^2+35 c^2 f^2 (129 f g+202 e h) d+633 c^3 f^3 h\right ) a^3+3 b^3 d f \left (8 e^3 (595 f g+158 e h) d^4-5 c e^2 f (1473 f g+872 e h) d^3+c^2 e f^2 (450 f g+4843 e h) d^2+c^3 f^3 (1355 f g-1022 e h) d+75 c^4 f^4 h\right ) a^2-b^4 \left (240 e^4 (45 f g+4 e h) d^5-48 c e^3 f (305 f g+167 e h) d^4-3 c^2 e^2 f^2 (45 f g-3424 e h) d^3+c^3 e f^3 (990 f g-541 e h) d^2+5 c^4 f^4 (357 f g-302 e h) d+35 c^5 f^5 h\right ) a+b^5 \left (35 f^4 (9 f g-8 e h) c^5+10 d e f^3 (21 f g-16 e h) c^4+d^2 e^2 f^2 (75 f g+8 e h) c^3-24 d^3 e^3 f (5 f g-108 e h) c^2-240 d^4 e^4 (15 f g+8 e h) c+2880 d^5 e^5 g\right )\right ) x\right )}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (48 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (3131 d^2 h e^2+3371 c^2 f^2 h-2 c d f (600 f g+2651 e h)\right ) a^4-b^2 d^2 f^2 \left (e^2 (6915 f g+5456 e h) d^3-c e f (13830 f g+10759 e h) d^2+35 c^2 f^2 (129 f g+202 e h) d+633 c^3 f^3 h\right ) a^3+3 b^3 d f \left (8 e^3 (595 f g+158 e h) d^4-5 c e^2 f (1473 f g+872 e h) d^3+c^2 e f^2 (450 f g+4843 e h) d^2+c^3 f^3 (1355 f g-1022 e h) d+75 c^4 f^4 h\right ) a^2-b^4 \left (240 e^4 (45 f g+4 e h) d^5-48 c e^3 f (305 f g+167 e h) d^4-3 c^2 e^2 f^2 (45 f g-3424 e h) d^3+c^3 e f^3 (990 f g-541 e h) d^2+5 c^4 f^4 (357 f g-302 e h) d+35 c^5 f^5 h\right ) a+b^5 \left (35 f^4 (9 f g-8 e h) c^5+10 d e f^3 (21 f g-16 e h) c^4+d^2 e^2 f^2 (75 f g+8 e h) c^3-24 d^3 e^3 f (5 f g-108 e h) c^2-240 d^4 e^4 (15 f g+8 e h) c+2880 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \int \frac {48 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+16 b d^4 f^4 \left (8 d^2 h e^2+23 c^2 f^2 h-c d f (75 f g-44 e h)\right ) a^5+b^2 d^3 f^3 \left (-32 e^2 (15 f g-44 e h) d^3+c e f (960 f g-1253 e h) d^2+2 c^2 f^2 (960 f g-1819 e h) d+1083 c^3 f^3 h\right ) a^4-3 b^3 d^2 f^2 \left (64 e^3 (15 f g+16 e h) d^4-5 c e^2 f (243 f g+272 e h) d^3-c^2 e f^2 (450 f g-19 e h) d^2+c^3 f^3 (1505 f g-694 e h) d+211 c^4 f^4 h\right ) a^3+b^4 d f \left (16 e^4 (495 f g+148 e h) d^5-8 c e^3 f (1095 f g+844 e h) d^4-c^2 e^2 f^2 (3375 f g-5224 e h) d^3+9 c^3 e f^3 (150 f g+89 e h) d^2+3 c^4 f^4 (1355 f g-1022 e h) d+225 c^5 f^5 h\right ) a^2-b^5 \left (320 e^5 (21 f g+2 e h) d^6-16 c e^4 f (435 f g+304 e h) d^5-80 c^2 e^3 f^2 (27 f g-61 e h) d^4-5 c^3 e^2 f^3 (27 f g-224 e h) d^3+c^4 e f^4 (990 f g-541 e h) d^2+5 c^5 f^5 (357 f g-302 e h) d+35 c^6 f^6 h\right ) a+b^6 (d e-c f)^2 \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )+b d f \left (48 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (3131 d^2 h e^2+3371 c^2 f^2 h-2 c d f (600 f g+2651 e h)\right ) a^4-b^2 d^2 f^2 \left (e^2 (6915 f g+5456 e h) d^3-c e f (13830 f g+10759 e h) d^2+35 c^2 f^2 (129 f g+202 e h) d+633 c^3 f^3 h\right ) a^3+3 b^3 d f \left (8 e^3 (595 f g+158 e h) d^4-5 c e^2 f (1473 f g+872 e h) d^3+c^2 e f^2 (450 f g+4843 e h) d^2+c^3 f^3 (1355 f g-1022 e h) d+75 c^4 f^4 h\right ) a^2-b^4 \left (240 e^4 (45 f g+4 e h) d^5-48 c e^3 f (305 f g+167 e h) d^4-3 c^2 e^2 f^2 (45 f g-3424 e h) d^3+c^3 e f^3 (990 f g-541 e h) d^2+5 c^4 f^4 (357 f g-302 e h) d+35 c^5 f^5 h\right ) a+b^5 \left (35 f^4 (9 f g-8 e h) c^5+10 d e f^3 (21 f g-16 e h) c^4+d^2 e^2 f^2 (75 f g+8 e h) c^3-24 d^3 e^3 f (5 f g-108 e h) c^2-240 d^4 e^4 (15 f g+8 e h) c+2880 d^5 e^5 g\right )\right ) x}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (48 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (3131 d^2 h e^2+3371 c^2 f^2 h-2 c d f (600 f g+2651 e h)\right ) a^4-b^2 d^2 f^2 \left (e^2 (6915 f g+5456 e h) d^3-c e f (13830 f g+10759 e h) d^2+35 c^2 f^2 (129 f g+202 e h) d+633 c^3 f^3 h\right ) a^3+3 b^3 d f \left (8 e^3 (595 f g+158 e h) d^4-5 c e^2 f (1473 f g+872 e h) d^3+c^2 e f^2 (450 f g+4843 e h) d^2+c^3 f^3 (1355 f g-1022 e h) d+75 c^4 f^4 h\right ) a^2-b^4 \left (240 e^4 (45 f g+4 e h) d^5-48 c e^3 f (305 f g+167 e h) d^4-3 c^2 e^2 f^2 (45 f g-3424 e h) d^3+c^3 e f^3 (990 f g-541 e h) d^2+5 c^4 f^4 (357 f g-302 e h) d+35 c^5 f^5 h\right ) a+b^5 \left (35 f^4 (9 f g-8 e h) c^5+10 d e f^3 (21 f g-16 e h) c^4+d^2 e^2 f^2 (75 f g+8 e h) c^3-24 d^3 e^3 f (5 f g-108 e h) c^2-240 d^4 e^4 (15 f g+8 e h) c+2880 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (48 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+16 b d^4 f^4 \left (-5 e (3 f g-4 e h) d^2-c f (75 f g-47 e h) d+23 c^2 f^2 h\right ) a^5+b^2 d^3 f^3 \left (-e^2 (480 f g+1723 e h) d^3+c e f (2160 f g+4049 e h) d^2+c^2 f^2 (1920 f g-7009 e h) d+1083 c^3 f^3 h\right ) a^4-b^3 d^2 f^2 \left (-e^3 (4035 f g+2384 e h) d^4+c e^2 f (10185 f g+6679 e h) d^3-c^2 e f^2 (5865 f g+7013 e h) d^2+15 c^3 f^3 (301 f g-181 e h) d+633 c^4 f^4 h\right ) a^3+b^4 d f \left (-8 e^4 (795 f g+178 e h) d^5+7 c e^3 f (1905 f g+904 e h) d^4-5 c^2 e^2 f^2 (945 f g+1861 e h) d^3-3 c^3 e f^3 (905 f g-1289 e h) d^2+3 c^4 f^4 (1355 f g-1097 e h) d+225 c^5 f^5 h\right ) a^2-b^5 \left (-80 e^5 (51 f g+4 e h) d^6+16 c e^4 f (480 f g+197 e h) d^5-c^2 e^3 f^2 (2025 f g+5392 e h) d^4-c^3 e^2 f^3 (1125 f g-1661 e h) d^3-3 c^4 e f^4 (265 f g-323 e h) d^2+15 c^5 f^5 (119 f g-103 e h) d+35 c^6 f^6 h\right ) a-b^6 \left (-35 f^5 (9 f g-8 e h) c^6+15 d e f^4 (7 f g-8 e h) c^5+3 d^2 e^2 f^3 (45 f g-56 e h) c^4+d^3 e^3 f^2 (195 f g-296 e h) c^3+8 d^4 e^4 f (45 f g+148 e h) c^2-80 d^5 e^5 (21 f g+8 e h) c+960 d^6 e^6 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 \int \frac {48 d^6 f^6 (5 d f g-4 d e h-c f h) a^7+16 b d^5 f^5 \left (-5 e (3 f g-4 e h) d^2-c f (75 f g-47 e h) d+23 c^2 f^2 h\right ) a^6-80 b^2 d^4 f^4 \left (2 e^2 (3 f g-8 e h) d^3-c e f (27 f g-62 e h) d^2-c^2 f^2 (24 f g+25 e h) d+24 c^3 f^3 h\right ) a^5+b^3 d^3 f^3 \left (-160 e^3 (15 f g+28 e h) d^4+c e^2 f (9120 f g+12197 e h) d^3-c^2 e f^2 (13440 f g+8431 e h) d^2+3 c^3 f^3 (640 f g+1477 e h) d+1083 c^4 f^4 h\right ) a^4-b^4 d^2 f^2 \left (-80 e^4 (135 f g+68 e h) d^5+c e^3 f (29565 f g+17696 e h) d^4-c^2 e^2 f^2 (21015 f g+22441 e h) d^3-3 c^3 e f^3 (1955 f g-5289 e h) d^2+15 c^4 f^4 (301 f g-181 e h) d+633 c^5 f^5 h\right ) a^3+b^5 d f \left (-16 e^5 (915 f g+188 e h) d^6+8 c e^4 f (4305 f g+1868 e h) d^5-c^2 e^3 f^2 (17865 f g+24872 e h) d^4-75 c^3 e^2 f^3 (63 f g-181 e h) d^3-3 c^4 e f^4 (905 f g-1289 e h) d^2+3 c^5 f^5 (1355 f g-1097 e h) d+225 c^6 f^6 h\right ) a^2-b^6 \left (-320 e^6 (27 f g+2 e h) d^7+16 c e^5 f (1155 f g+424 e h) d^6-16 c^2 e^4 f^2 (495 f g+817 e h) d^5-27 c^3 e^3 f^3 (75 f g-224 e h) d^4-c^4 e^2 f^4 (1125 f g-1661 e h) d^3-3 c^5 e f^5 (265 f g-323 e h) d^2+15 c^6 f^6 (119 f g-103 e h) d+35 c^7 f^7 h\right ) a-b^7 (d e-c f)^3 \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )+b d f \left (48 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+16 b d^4 f^4 \left (-5 e (3 f g-4 e h) d^2-c f (75 f g-47 e h) d+23 c^2 f^2 h\right ) a^5+b^2 d^3 f^3 \left (-e^2 (480 f g+1723 e h) d^3+c e f (2160 f g+4049 e h) d^2+c^2 f^2 (1920 f g-7009 e h) d+1083 c^3 f^3 h\right ) a^4-b^3 d^2 f^2 \left (-e^3 (4035 f g+2384 e h) d^4+c e^2 f (10185 f g+6679 e h) d^3-c^2 e f^2 (5865 f g+7013 e h) d^2+15 c^3 f^3 (301 f g-181 e h) d+633 c^4 f^4 h\right ) a^3+b^4 d f \left (-8 e^4 (795 f g+178 e h) d^5+7 c e^3 f (1905 f g+904 e h) d^4-5 c^2 e^2 f^2 (945 f g+1861 e h) d^3-3 c^3 e f^3 (905 f g-1289 e h) d^2+3 c^4 f^4 (1355 f g-1097 e h) d+225 c^5 f^5 h\right ) a^2-b^5 \left (-80 e^5 (51 f g+4 e h) d^6+16 c e^4 f (480 f g+197 e h) d^5-c^2 e^3 f^2 (2025 f g+5392 e h) d^4-c^3 e^2 f^3 (1125 f g-1661 e h) d^3-3 c^4 e f^4 (265 f g-323 e h) d^2+15 c^5 f^5 (119 f g-103 e h) d+35 c^6 f^6 h\right ) a-b^6 \left (-35 f^5 (9 f g-8 e h) c^6+15 d e f^4 (7 f g-8 e h) c^5+3 d^2 e^2 f^3 (45 f g-56 e h) c^4+d^3 e^3 f^2 (195 f g-296 e h) c^3+8 d^4 e^4 f (45 f g+148 e h) c^2-80 d^5 e^5 (21 f g+8 e h) c+960 d^6 e^6 g\right )\right ) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (48 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (3131 d^2 h e^2+3371 c^2 f^2 h-2 c d f (600 f g+2651 e h)\right ) a^4-b^2 d^2 f^2 \left (e^2 (6915 f g+5456 e h) d^3-c e f (13830 f g+10759 e h) d^2+35 c^2 f^2 (129 f g+202 e h) d+633 c^3 f^3 h\right ) a^3+3 b^3 d f \left (8 e^3 (595 f g+158 e h) d^4-5 c e^2 f (1473 f g+872 e h) d^3+c^2 e f^2 (450 f g+4843 e h) d^2+c^3 f^3 (1355 f g-1022 e h) d+75 c^4 f^4 h\right ) a^2-b^4 \left (240 e^4 (45 f g+4 e h) d^5-48 c e^3 f (305 f g+167 e h) d^4-3 c^2 e^2 f^2 (45 f g-3424 e h) d^3+c^3 e f^3 (990 f g-541 e h) d^2+5 c^4 f^4 (357 f g-302 e h) d+35 c^5 f^5 h\right ) a+b^5 \left (35 f^4 (9 f g-8 e h) c^5+10 d e f^3 (21 f g-16 e h) c^4+d^2 e^2 f^2 (75 f g+8 e h) c^3-24 d^3 e^3 f (5 f g-108 e h) c^2-240 d^4 e^4 (15 f g+8 e h) c+2880 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (48 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+16 b d^4 f^4 \left (-5 e (3 f g-4 e h) d^2-c f (75 f g-47 e h) d+23 c^2 f^2 h\right ) a^5+b^2 d^3 f^3 \left (-e^2 (480 f g+1723 e h) d^3+c e f (2160 f g+4049 e h) d^2+c^2 f^2 (1920 f g-7009 e h) d+1083 c^3 f^3 h\right ) a^4-b^3 d^2 f^2 \left (-e^3 (4035 f g+2384 e h) d^4+c e^2 f (10185 f g+6679 e h) d^3-c^2 e f^2 (5865 f g+7013 e h) d^2+15 c^3 f^3 (301 f g-181 e h) d+633 c^4 f^4 h\right ) a^3+b^4 d f \left (-8 e^4 (795 f g+178 e h) d^5+7 c e^3 f (1905 f g+904 e h) d^4-5 c^2 e^2 f^2 (945 f g+1861 e h) d^3-3 c^3 e f^3 (905 f g-1289 e h) d^2+3 c^4 f^4 (1355 f g-1097 e h) d+225 c^5 f^5 h\right ) a^2-b^5 \left (-80 e^5 (51 f g+4 e h) d^6+16 c e^4 f (480 f g+197 e h) d^5-c^2 e^3 f^2 (2025 f g+5392 e h) d^4-c^3 e^2 f^3 (1125 f g-1661 e h) d^3-3 c^4 e f^4 (265 f g-323 e h) d^2+15 c^5 f^5 (119 f g-103 e h) d+35 c^6 f^6 h\right ) a-b^6 \left (-35 f^5 (9 f g-8 e h) c^6+15 d e f^4 (7 f g-8 e h) c^5+3 d^2 e^2 f^3 (45 f g-56 e h) c^4+d^3 e^3 f^2 (195 f g-296 e h) c^3+8 d^4 e^4 f (45 f g+148 e h) c^2-80 d^5 e^5 (21 f g+8 e h) c+960 d^6 e^6 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\int \frac {48 d^6 f^6 (5 d f g-4 d e h-c f h) a^7+16 b d^5 f^5 \left (-5 e (3 f g-4 e h) d^2-c f (75 f g-47 e h) d+23 c^2 f^2 h\right ) a^6-80 b^2 d^4 f^4 \left (2 e^2 (3 f g-8 e h) d^3-c e f (27 f g-62 e h) d^2-c^2 f^2 (24 f g+25 e h) d+24 c^3 f^3 h\right ) a^5+b^3 d^3 f^3 \left (-160 e^3 (15 f g+28 e h) d^4+c e^2 f (9120 f g+12197 e h) d^3-c^2 e f^2 (13440 f g+8431 e h) d^2+3 c^3 f^3 (640 f g+1477 e h) d+1083 c^4 f^4 h\right ) a^4-b^4 d^2 f^2 \left (-80 e^4 (135 f g+68 e h) d^5+c e^3 f (29565 f g+17696 e h) d^4-c^2 e^2 f^2 (21015 f g+22441 e h) d^3-3 c^3 e f^3 (1955 f g-5289 e h) d^2+15 c^4 f^4 (301 f g-181 e h) d+633 c^5 f^5 h\right ) a^3+b^5 d f \left (-16 e^5 (915 f g+188 e h) d^6+8 c e^4 f (4305 f g+1868 e h) d^5-c^2 e^3 f^2 (17865 f g+24872 e h) d^4-75 c^3 e^2 f^3 (63 f g-181 e h) d^3-3 c^4 e f^4 (905 f g-1289 e h) d^2+3 c^5 f^5 (1355 f g-1097 e h) d+225 c^6 f^6 h\right ) a^2-b^6 \left (-320 e^6 (27 f g+2 e h) d^7+16 c e^5 f (1155 f g+424 e h) d^6-16 c^2 e^4 f^2 (495 f g+817 e h) d^5-27 c^3 e^3 f^3 (75 f g-224 e h) d^4-c^4 e^2 f^4 (1125 f g-1661 e h) d^3-3 c^5 e f^5 (265 f g-323 e h) d^2+15 c^6 f^6 (119 f g-103 e h) d+35 c^7 f^7 h\right ) a-b^7 (d e-c f)^3 \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )+b d f \left (48 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+16 b d^4 f^4 \left (-5 e (3 f g-4 e h) d^2-c f (75 f g-47 e h) d+23 c^2 f^2 h\right ) a^5+b^2 d^3 f^3 \left (-e^2 (480 f g+1723 e h) d^3+c e f (2160 f g+4049 e h) d^2+c^2 f^2 (1920 f g-7009 e h) d+1083 c^3 f^3 h\right ) a^4-b^3 d^2 f^2 \left (-e^3 (4035 f g+2384 e h) d^4+c e^2 f (10185 f g+6679 e h) d^3-c^2 e f^2 (5865 f g+7013 e h) d^2+15 c^3 f^3 (301 f g-181 e h) d+633 c^4 f^4 h\right ) a^3+b^4 d f \left (-8 e^4 (795 f g+178 e h) d^5+7 c e^3 f (1905 f g+904 e h) d^4-5 c^2 e^2 f^2 (945 f g+1861 e h) d^3-3 c^3 e f^3 (905 f g-1289 e h) d^2+3 c^4 f^4 (1355 f g-1097 e h) d+225 c^5 f^5 h\right ) a^2-b^5 \left (-80 e^5 (51 f g+4 e h) d^6+16 c e^4 f (480 f g+197 e h) d^5-c^2 e^3 f^2 (2025 f g+5392 e h) d^4-c^3 e^2 f^3 (1125 f g-1661 e h) d^3-3 c^4 e f^4 (265 f g-323 e h) d^2+15 c^5 f^5 (119 f g-103 e h) d+35 c^6 f^6 h\right ) a-b^6 \left (-35 f^5 (9 f g-8 e h) c^6+15 d e f^4 (7 f g-8 e h) c^5+3 d^2 e^2 f^3 (45 f g-56 e h) c^4+d^3 e^3 f^2 (195 f g-296 e h) c^3+8 d^4 e^4 f (45 f g+148 e h) c^2-80 d^5 e^5 (21 f g+8 e h) c+960 d^6 e^6 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (48 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (3131 d^2 h e^2+3371 c^2 f^2 h-2 c d f (600 f g+2651 e h)\right ) a^4-b^2 d^2 f^2 \left (e^2 (6915 f g+5456 e h) d^3-c e f (13830 f g+10759 e h) d^2+35 c^2 f^2 (129 f g+202 e h) d+633 c^3 f^3 h\right ) a^3+3 b^3 d f \left (8 e^3 (595 f g+158 e h) d^4-5 c e^2 f (1473 f g+872 e h) d^3+c^2 e f^2 (450 f g+4843 e h) d^2+c^3 f^3 (1355 f g-1022 e h) d+75 c^4 f^4 h\right ) a^2-b^4 \left (240 e^4 (45 f g+4 e h) d^5-48 c e^3 f (305 f g+167 e h) d^4-3 c^2 e^2 f^2 (45 f g-3424 e h) d^3+c^3 e f^3 (990 f g-541 e h) d^2+5 c^4 f^4 (357 f g-302 e h) d+35 c^5 f^5 h\right ) a+b^5 \left (35 f^4 (9 f g-8 e h) c^5+10 d e f^3 (21 f g-16 e h) c^4+d^2 e^2 f^2 (75 f g+8 e h) c^3-24 d^3 e^3 f (5 f g-108 e h) c^2-240 d^4 e^4 (15 f g+8 e h) c+2880 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (48 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+16 b d^4 f^4 \left (-5 e (3 f g-4 e h) d^2-c f (75 f g-47 e h) d+23 c^2 f^2 h\right ) a^5+b^2 d^3 f^3 \left (-e^2 (480 f g+1723 e h) d^3+c e f (2160 f g+4049 e h) d^2+c^2 f^2 (1920 f g-7009 e h) d+1083 c^3 f^3 h\right ) a^4-b^3 d^2 f^2 \left (-e^3 (4035 f g+2384 e h) d^4+c e^2 f (10185 f g+6679 e h) d^3-c^2 e f^2 (5865 f g+7013 e h) d^2+15 c^3 f^3 (301 f g-181 e h) d+633 c^4 f^4 h\right ) a^3+b^4 d f \left (-8 e^4 (795 f g+178 e h) d^5+7 c e^3 f (1905 f g+904 e h) d^4-5 c^2 e^2 f^2 (945 f g+1861 e h) d^3-3 c^3 e f^3 (905 f g-1289 e h) d^2+3 c^4 f^4 (1355 f g-1097 e h) d+225 c^5 f^5 h\right ) a^2-b^5 \left (-80 e^5 (51 f g+4 e h) d^6+16 c e^4 f (480 f g+197 e h) d^5-c^2 e^3 f^2 (2025 f g+5392 e h) d^4-c^3 e^2 f^3 (1125 f g-1661 e h) d^3-3 c^4 e f^4 (265 f g-323 e h) d^2+15 c^5 f^5 (119 f g-103 e h) d+35 c^6 f^6 h\right ) a-b^6 \left (-35 f^5 (9 f g-8 e h) c^6+15 d e f^4 (7 f g-8 e h) c^5+3 d^2 e^2 f^3 (45 f g-56 e h) c^4+d^3 e^3 f^2 (195 f g-296 e h) c^3+8 d^4 e^4 f (45 f g+148 e h) c^2-80 d^5 e^5 (21 f g+8 e h) c+960 d^6 e^6 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {16 d^5 \left (\left (-143 f^2 h c^3+13 d f (15 f g+16 e h) c^2-20 d^2 e (15 f g+4 e h) c+120 d^3 e^2 g\right ) b^2+2 a d \left (10 e (3 f g-2 e h) d^2-c f (45 f g-22 e h) d+13 c^2 f^2 h\right ) b+3 a^2 d^2 f (5 d f g-4 d e h-c f h)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx (b e-a f)^5}{b c-a d}+\frac {b^3 (d e-c f)^3 \left (3003 d^4 f^4 h a^5-429 b d^3 f^3 (15 d f g+16 d e h+4 c f h) a^4+858 b^2 d^2 f^2 \left (4 e (5 f g+2 e h) d^2+2 c f (5 f g-2 e h) d+c^2 f^2 h\right ) a^3-26 b^3 d f \left (16 e^2 (45 f g+8 e h) d^3+12 c e f (45 f g-26 e h) d^2+3 c^2 f^2 (75 f g-52 e h) d+10 c^3 f^3 h\right ) a^2+b^4 \left (640 e^3 (15 f g+e h) d^4+64 c e^2 f (135 f g-86 e h) d^3+72 c^2 e f^2 (75 f g-58 e h) d^2+20 c^3 f^3 (105 f g-86 e h) d+35 c^4 f^4 h\right ) a-b^5 \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (48 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (3131 d^2 h e^2+3371 c^2 f^2 h-2 c d f (600 f g+2651 e h)\right ) a^4-b^2 d^2 f^2 \left (e^2 (6915 f g+5456 e h) d^3-c e f (13830 f g+10759 e h) d^2+35 c^2 f^2 (129 f g+202 e h) d+633 c^3 f^3 h\right ) a^3+3 b^3 d f \left (8 e^3 (595 f g+158 e h) d^4-5 c e^2 f (1473 f g+872 e h) d^3+c^2 e f^2 (450 f g+4843 e h) d^2+c^3 f^3 (1355 f g-1022 e h) d+75 c^4 f^4 h\right ) a^2-b^4 \left (240 e^4 (45 f g+4 e h) d^5-48 c e^3 f (305 f g+167 e h) d^4-3 c^2 e^2 f^2 (45 f g-3424 e h) d^3+c^3 e f^3 (990 f g-541 e h) d^2+5 c^4 f^4 (357 f g-302 e h) d+35 c^5 f^5 h\right ) a+b^5 \left (35 f^4 (9 f g-8 e h) c^5+10 d e f^3 (21 f g-16 e h) c^4+d^2 e^2 f^2 (75 f g+8 e h) c^3-24 d^3 e^3 f (5 f g-108 e h) c^2-240 d^4 e^4 (15 f g+8 e h) c+2880 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (48 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+16 b d^4 f^4 \left (-5 e (3 f g-4 e h) d^2-c f (75 f g-47 e h) d+23 c^2 f^2 h\right ) a^5+b^2 d^3 f^3 \left (-e^2 (480 f g+1723 e h) d^3+c e f (2160 f g+4049 e h) d^2+c^2 f^2 (1920 f g-7009 e h) d+1083 c^3 f^3 h\right ) a^4-b^3 d^2 f^2 \left (-e^3 (4035 f g+2384 e h) d^4+c e^2 f (10185 f g+6679 e h) d^3-c^2 e f^2 (5865 f g+7013 e h) d^2+15 c^3 f^3 (301 f g-181 e h) d+633 c^4 f^4 h\right ) a^3+b^4 d f \left (-8 e^4 (795 f g+178 e h) d^5+7 c e^3 f (1905 f g+904 e h) d^4-5 c^2 e^2 f^2 (945 f g+1861 e h) d^3-3 c^3 e f^3 (905 f g-1289 e h) d^2+3 c^4 f^4 (1355 f g-1097 e h) d+225 c^5 f^5 h\right ) a^2-b^5 \left (-80 e^5 (51 f g+4 e h) d^6+16 c e^4 f (480 f g+197 e h) d^5-c^2 e^3 f^2 (2025 f g+5392 e h) d^4-c^3 e^2 f^3 (1125 f g-1661 e h) d^3-3 c^4 e f^4 (265 f g-323 e h) d^2+15 c^5 f^5 (119 f g-103 e h) d+35 c^6 f^6 h\right ) a-b^6 \left (-35 f^5 (9 f g-8 e h) c^6+15 d e f^4 (7 f g-8 e h) c^5+3 d^2 e^2 f^3 (45 f g-56 e h) c^4+d^3 e^3 f^2 (195 f g-296 e h) c^3+8 d^4 e^4 f (45 f g+148 e h) c^2-80 d^5 e^5 (21 f g+8 e h) c+960 d^6 e^6 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {32 d^5 \left (\left (-143 f^2 h c^3+13 d f (15 f g+16 e h) c^2-20 d^2 e (15 f g+4 e h) c+120 d^3 e^2 g\right ) b^2+2 a d \left (10 e (3 f g-2 e h) d^2-c f (45 f g-22 e h) d+13 c^2 f^2 h\right ) b+3 a^2 d^2 f (5 d f g-4 d e h-c f h)\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x} (b e-a f)^5}{(b c-a d) f}+\frac {2 b^3 (d e-c f)^3 \left (3003 d^4 f^4 h a^5-429 b d^3 f^3 (15 d f g+16 d e h+4 c f h) a^4+858 b^2 d^2 f^2 \left (4 e (5 f g+2 e h) d^2+2 c f (5 f g-2 e h) d+c^2 f^2 h\right ) a^3-26 b^3 d f \left (16 e^2 (45 f g+8 e h) d^3+12 c e f (45 f g-26 e h) d^2+3 c^2 f^2 (75 f g-52 e h) d+10 c^3 f^3 h\right ) a^2+b^4 \left (640 e^3 (15 f g+e h) d^4+64 c e^2 f (135 f g-86 e h) d^3+72 c^2 e f^2 (75 f g-58 e h) d^2+20 c^3 f^3 (105 f g-86 e h) d+35 c^4 f^4 h\right ) a-b^5 \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{(b c-a d) f}}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {13 d f h a^2-b (21 d f g+4 d e h+c f h) a+b^2 (12 d e g+9 c f g-8 c e h)}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {143 d^2 f^2 h a^3-3 b d f (93 d f g+40 d e h+10 c f h) a^2+b^2 \left (8 e (42 f g+5 e h) d^2+2 c f (111 f g-88 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+16 d e (6 f g-5 e h) c+120 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {1287 d^3 f^3 h a^4-3 b d^2 f^2 (901 d f g+652 d e h+163 c f h) a^3+b^2 d f \left (4 e (1287 f g+326 e h) d^2+c f (2961 f g-1888 e h) d+197 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (93 f g+8 e h) d^3+56 c e f (51 f g-37 e h) d^2+c^2 f^2 (1533 f g-1244 e h) d+35 c^3 f^3 h\right ) a+b^4 \left (35 f^2 (9 f g-8 e h) c^3+4 d e f (147 f g-124 e h) c^2+40 d^2 e^2 (21 f g-16 e h) c+960 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (3 d^3 f^3 (32 d f g+575 d e h-607 c f h) a^4+b d^2 f^2 \left (-e (3861 f g+2866 e h) d^2+c f (3477 f g+2693 e h) d+557 c^2 f^2 h\right ) a^3-b^2 d f \left (-2 e^2 (3765 f g+968 e h) d^3+3 c e f (1159 f g+1580 e h) d^2+3 c^2 f^2 (1159 f g-813 e h) d+211 c^3 f^3 h\right ) a^2+b^3 \left (-240 e^3 (23 f g+2 e h) d^4+4 c e^2 f (375 f g+892 e h) d^3+3 c^2 e f^2 (659 f g-454 e h) d^2+3 c^3 f^3 (553 f g-459 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-d e f^2 (399 f g-328 e h) c^3-2 d^2 e^2 f (195 f g-128 e h) c^2-240 d^3 e^3 (f g+4 e h) c+1440 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (48 d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (3131 d^2 h e^2+3371 c^2 f^2 h-2 c d f (600 f g+2651 e h)\right ) a^4-b^2 d^2 f^2 \left (e^2 (6915 f g+5456 e h) d^3-c e f (13830 f g+10759 e h) d^2+35 c^2 f^2 (129 f g+202 e h) d+633 c^3 f^3 h\right ) a^3+3 b^3 d f \left (8 e^3 (595 f g+158 e h) d^4-5 c e^2 f (1473 f g+872 e h) d^3+c^2 e f^2 (450 f g+4843 e h) d^2+c^3 f^3 (1355 f g-1022 e h) d+75 c^4 f^4 h\right ) a^2-b^4 \left (240 e^4 (45 f g+4 e h) d^5-48 c e^3 f (305 f g+167 e h) d^4-3 c^2 e^2 f^2 (45 f g-3424 e h) d^3+c^3 e f^3 (990 f g-541 e h) d^2+5 c^4 f^4 (357 f g-302 e h) d+35 c^5 f^5 h\right ) a+b^5 \left (35 f^4 (9 f g-8 e h) c^5+10 d e f^3 (21 f g-16 e h) c^4+d^2 e^2 f^2 (75 f g+8 e h) c^3-24 d^3 e^3 f (5 f g-108 e h) c^2-240 d^4 e^4 (15 f g+8 e h) c+2880 d^5 e^5 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (48 d^5 f^5 (5 d f g-4 d e h-c f h) a^6+16 b d^4 f^4 \left (-5 e (3 f g-4 e h) d^2-c f (75 f g-47 e h) d+23 c^2 f^2 h\right ) a^5+b^2 d^3 f^3 \left (-e^2 (480 f g+1723 e h) d^3+c e f (2160 f g+4049 e h) d^2+c^2 f^2 (1920 f g-7009 e h) d+1083 c^3 f^3 h\right ) a^4-b^3 d^2 f^2 \left (-e^3 (4035 f g+2384 e h) d^4+c e^2 f (10185 f g+6679 e h) d^3-c^2 e f^2 (5865 f g+7013 e h) d^2+15 c^3 f^3 (301 f g-181 e h) d+633 c^4 f^4 h\right ) a^3+b^4 d f \left (-8 e^4 (795 f g+178 e h) d^5+7 c e^3 f (1905 f g+904 e h) d^4-5 c^2 e^2 f^2 (945 f g+1861 e h) d^3-3 c^3 e f^3 (905 f g-1289 e h) d^2+3 c^4 f^4 (1355 f g-1097 e h) d+225 c^5 f^5 h\right ) a^2-b^5 \left (-80 e^5 (51 f g+4 e h) d^6+16 c e^4 f (480 f g+197 e h) d^5-c^2 e^3 f^2 (2025 f g+5392 e h) d^4-c^3 e^2 f^3 (1125 f g-1661 e h) d^3-3 c^4 e f^4 (265 f g-323 e h) d^2+15 c^5 f^5 (119 f g-103 e h) d+35 c^6 f^6 h\right ) a-b^6 \left (-35 f^5 (9 f g-8 e h) c^6+15 d e f^4 (7 f g-8 e h) c^5+3 d^2 e^2 f^3 (45 f g-56 e h) c^4+d^3 e^3 f^2 (195 f g-296 e h) c^3+8 d^4 e^4 f (45 f g+148 e h) c^2-80 d^5 e^5 (21 f g+8 e h) c+960 d^6 e^6 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {-\frac {32 d^{9/2} \left (\left (-143 f^2 h c^3+13 d f (15 f g+16 e h) c^2-20 d^2 e (15 f g+4 e h) c+120 d^3 e^2 g\right ) b^2+2 a d \left (10 e (3 f g-2 e h) d^2-c f (45 f g-22 e h) d+13 c^2 f^2 h\right ) b+3 a^2 d^2 f (5 d f g-4 d e h-c f h)\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) (b e-a f)^5}{(b c-a d) \sqrt {d e-c f}}-\frac {2 b^{5/2} (d e-c f)^3 \left (3003 d^4 f^4 h a^5-429 b d^3 f^3 (15 d f g+16 d e h+4 c f h) a^4+858 b^2 d^2 f^2 \left (4 e (5 f g+2 e h) d^2+2 c f (5 f g-2 e h) d+c^2 f^2 h\right ) a^3-26 b^3 d f \left (16 e^2 (45 f g+8 e h) d^3+12 c e f (45 f g-26 e h) d^2+3 c^2 f^2 (75 f g-52 e h) d+10 c^3 f^3 h\right ) a^2+b^4 \left (640 e^3 (15 f g+e h) d^4+64 c e^2 f (135 f g-86 e h) d^3+72 c^2 e f^2 (75 f g-58 e h) d^2+20 c^3 f^3 (105 f g-86 e h) d+35 c^4 f^4 h\right ) a-b^5 \left (35 f^3 (9 f g-8 e h) c^4+120 d e f^2 (7 f g-6 e h) c^3+288 d^2 e^2 f (5 f g-4 e h) c^2+640 d^3 e^3 (3 f g-2 e h) c+1920 d^4 e^4 g\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) \sqrt {b e-a f}}}{(b e-a f) (d e-c f)}\right )}{(b c-a d) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}\)

Input:

Int[(g + h*x)/((a + b*x)^5*(c + d*x)^3*(e + f*x)^(3/2)),x]
 

Output:

-1/4*(b*g - a*h)/((b*c - a*d)*(b*e - a*f)*(a + b*x)^4*(c + d*x)^2*Sqrt[e + 
 f*x]) - (-1/3*(13*a^2*d*f*h + b^2*(12*d*e*g + 9*c*f*g - 8*c*e*h) - a*b*(2 
1*d*f*g + 4*d*e*h + c*f*h))/((b*c - a*d)*(b*e - a*f)*(a + b*x)^3*(c + d*x) 
^2*Sqrt[e + f*x]) - ((143*a^3*d^2*f^2*h - 3*a^2*b*d*f*(93*d*f*g + 40*d*e*h 
 + 10*c*f*h) - b^3*(120*d^2*e^2*g + 7*c^2*f*(9*f*g - 8*e*h) + 16*c*d*e*(6* 
f*g - 5*e*h)) + a*b^2*(7*c^2*f^2*h + 2*c*d*f*(111*f*g - 88*e*h) + 8*d^2*e* 
(42*f*g + 5*e*h)))/(2*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)^2*Sqrt 
[e + f*x]) + ((1287*a^4*d^3*f^3*h - 3*a^3*b*d^2*f^2*(901*d*f*g + 652*d*e*h 
 + 163*c*f*h) + b^4*(960*d^3*e^3*g + 4*c^2*d*e*f*(147*f*g - 124*e*h) + 40* 
c*d^2*e^2*(21*f*g - 16*e*h) + 35*c^3*f^2*(9*f*g - 8*e*h)) - a*b^3*(35*c^3* 
f^3*h + c^2*d*f^2*(1533*f*g - 1244*e*h) + 56*c*d^2*e*f*(51*f*g - 37*e*h) + 
 40*d^3*e^2*(93*f*g + 8*e*h)) + a^2*b^2*d*f*(197*c^2*f^2*h + c*d*f*(2961*f 
*g - 1888*e*h) + 4*d^2*e*(1287*f*g + 326*e*h)))/((b*c - a*d)*(b*e - a*f)*( 
a + b*x)*(c + d*x)^2*Sqrt[e + f*x]) + ((2*d*(3*a^4*d^3*f^3*(32*d*f*g + 575 
*d*e*h - 607*c*f*h) + b^4*(1440*d^4*e^4*g - c^3*d*e*f^2*(399*f*g - 328*e*h 
) - 2*c^2*d^2*e^2*f*(195*f*g - 128*e*h) - 35*c^4*f^3*(9*f*g - 8*e*h) - 240 
*c*d^3*e^3*(f*g + 4*e*h)) + a*b^3*(35*c^4*f^4*h + 3*c^3*d*f^3*(553*f*g - 4 
59*e*h) + 3*c^2*d^2*e*f^2*(659*f*g - 454*e*h) - 240*d^4*e^3*(23*f*g + 2*e* 
h) + 4*c*d^3*e^2*f*(375*f*g + 892*e*h)) - a^2*b^2*d*f*(211*c^3*f^3*h + 3*c 
^2*d*f^2*(1159*f*g - 813*e*h) - 2*d^3*e^2*(3765*f*g + 968*e*h) + 3*c*d^...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 4394, normalized size of antiderivative = 1.60

\[\text {output too large to display}\]

Input:

int((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(3/2),x)
 

Output:

2*f^6*(d^5/(a*d-b*c)^7/f^6/(c*f-d*e)^3*(((3/8*a^2*c*d^3*f^2*h+1/2*a^2*d^4* 
e*f*h-7/8*a^2*d^4*f^2*g-13/4*a*b*c^2*d^2*f^2*h+3/2*a*b*c*d^3*e*f*h+17/4*a* 
b*c*d^3*f^2*g-5/2*a*b*d^4*e*f*g+23/8*b^2*c^3*d*f^2*h-2*b^2*c^2*d^2*e*f*h-2 
7/8*b^2*c^2*d^2*f^2*g+5/2*b^2*c*d^3*e*f*g)*(f*x+e)^(3/2)+1/8*f*(5*a^2*c^2* 
d^2*f^2*h-a^2*c*d^3*e*f*h-9*a^2*c*d^3*f^2*g-4*a^2*d^4*e^2*h+9*a^2*d^4*e*f* 
g-30*a*b*c^3*d*f^2*h+42*a*b*c^2*d^2*e*f*h+38*a*b*c^2*d^2*f^2*g-12*a*b*c*d^ 
3*e^2*h-58*a*b*c*d^3*e*f*g+20*a*b*d^4*e^2*g+25*b^2*c^4*f^2*h-41*b^2*c^3*d* 
e*f*h-29*b^2*c^3*d*f^2*g+16*b^2*c^2*d^2*e^2*h+49*b^2*c^2*d^2*e*f*g-20*b^2* 
c*d^3*e^2*g)*(f*x+e)^(1/2))/((f*x+e)*d+c*f-d*e)^2+1/8*(3*a^2*c*d^2*f^2*h+1 
2*a^2*d^3*e*f*h-15*a^2*d^3*f^2*g-26*a*b*c^2*d*f^2*h-44*a*b*c*d^2*e*f*h+90* 
a*b*c*d^2*f^2*g+40*a*b*d^3*e^2*h-60*a*b*d^3*e*f*g+143*b^2*c^3*f^2*h-208*b^ 
2*c^2*d*e*f*h-195*b^2*c^2*d*f^2*g+80*b^2*c*d^2*e^2*h+300*b^2*c*d^2*e*f*g-1 
20*b^2*d^3*e^2*g)/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d) 
^(1/2)))-(-e*h+f*g)/(c*f-d*e)^3/(a*f-b*e)^5/(f*x+e)^(1/2)-b^3/(a*d-b*c)^7/ 
f^6/(a*f-b*e)^5*(((429/64*b^5*a^3*c^2*d^2*f^4*h-65/32*b^6*a^2*c^3*d*f^4*h- 
5*b^8*c*d^3*e^3*f*g-429/32*b^4*a^4*c*d^3*f^4*h+131/16*a^3*b^5*c*d^3*e*f^3* 
h+171/16*a^2*b^6*c^2*d^2*e*f^3*h+3/8*a^2*b^6*c*d^3*e^2*f^2*h-747/16*a^2*b^ 
6*c*d^3*e*f^3*g-103/16*a*b^7*c^3*d*e*f^3*h-93/8*a*b^7*c^2*d^2*e^2*f^2*h+33 
9/16*a*b^7*c^2*d^2*e*f^3*g-a*b^7*c*d^3*e^3*f*h+51/2*a*b^7*c*d^3*e^2*f^2*g- 
187/128*b^8*c^4*f^4*g+1025/32*a^3*b^5*c*d^3*f^4*g+465/16*a^3*b^5*d^4*e*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(3/2),x, algorithm="fricas")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)**5/(d*x+c)**3/(f*x+e)**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(3/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7241 vs. \(2 (2702) = 5404\).

Time = 1.54 (sec) , antiderivative size = 7241, normalized size of antiderivative = 2.63 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(3/2),x, algorithm="giac")
 

Output:

1/64*(1920*b^8*d^4*e^4*g + 1920*b^8*c*d^3*e^3*f*g - 9600*a*b^7*d^4*e^3*f*g 
 + 1440*b^8*c^2*d^2*e^2*f^2*g - 8640*a*b^7*c*d^3*e^2*f^2*g + 18720*a^2*b^6 
*d^4*e^2*f^2*g + 840*b^8*c^3*d*e*f^3*g - 5400*a*b^7*c^2*d^2*e*f^3*g + 1404 
0*a^2*b^6*c*d^3*e*f^3*g - 17160*a^3*b^5*d^4*e*f^3*g + 315*b^8*c^4*f^4*g - 
2100*a*b^7*c^3*d*f^4*g + 5850*a^2*b^6*c^2*d^2*f^4*g - 8580*a^3*b^5*c*d^3*f 
^4*g + 6435*a^4*b^4*d^4*f^4*g - 1280*b^8*c*d^3*e^4*h - 640*a*b^7*d^4*e^4*h 
 - 1152*b^8*c^2*d^2*e^3*f*h + 5504*a*b^7*c*d^3*e^3*f*h + 3328*a^2*b^6*d^4* 
e^3*f*h - 720*b^8*c^3*d*e^2*f^2*h + 4176*a*b^7*c^2*d^2*e^2*f^2*h - 8112*a^ 
2*b^6*c*d^3*e^2*f^2*h - 6864*a^3*b^5*d^4*e^2*f^2*h - 280*b^8*c^4*e*f^3*h + 
 1720*a*b^7*c^3*d*e*f^3*h - 4056*a^2*b^6*c^2*d^2*e*f^3*h + 3432*a^3*b^5*c* 
d^3*e*f^3*h + 6864*a^4*b^4*d^4*e*f^3*h - 35*a*b^7*c^4*f^4*h + 260*a^2*b^6* 
c^3*d*f^4*h - 858*a^3*b^5*c^2*d^2*f^4*h + 1716*a^4*b^4*c*d^3*f^4*h - 3003* 
a^5*b^3*d^4*f^4*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^12*c^7 
*e^5 - 7*a*b^11*c^6*d*e^5 + 21*a^2*b^10*c^5*d^2*e^5 - 35*a^3*b^9*c^4*d^3*e 
^5 + 35*a^4*b^8*c^3*d^4*e^5 - 21*a^5*b^7*c^2*d^5*e^5 + 7*a^6*b^6*c*d^6*e^5 
 - a^7*b^5*d^7*e^5 - 5*a*b^11*c^7*e^4*f + 35*a^2*b^10*c^6*d*e^4*f - 105*a^ 
3*b^9*c^5*d^2*e^4*f + 175*a^4*b^8*c^4*d^3*e^4*f - 175*a^5*b^7*c^3*d^4*e^4* 
f + 105*a^6*b^6*c^2*d^5*e^4*f - 35*a^7*b^5*c*d^6*e^4*f + 5*a^8*b^4*d^7*e^4 
*f + 10*a^2*b^10*c^7*e^3*f^2 - 70*a^3*b^9*c^6*d*e^3*f^2 + 210*a^4*b^8*c^5* 
d^2*e^3*f^2 - 350*a^5*b^7*c^4*d^3*e^3*f^2 + 350*a^6*b^6*c^3*d^4*e^3*f^2...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Hanged} \] Input:

int((g + h*x)/((e + f*x)^(3/2)*(a + b*x)^5*(c + d*x)^3),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^3 (e+f x)^{3/2}} \, dx=\int \frac {h x +g}{\left (b x +a \right )^{5} \left (d x +c \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}d x \] Input:

int((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(3/2),x)
 

Output:

int((h*x+g)/(b*x+a)^5/(d*x+c)^3/(f*x+e)^(3/2),x)