\(\int \frac {g+h x}{(a+b x)^5 (c+d x)^2 (e+f x)^{3/2}} \, dx\) [185]

Optimal result

Integrand size = 29, antiderivative size = 1991 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Output:

-1/64*f*(64*a^5*d^4*f^4*(-c*f*h-2*d*e*h+3*d*f*g)-b^5*(320*d^5*e^5*g-c^3*d^ 
2*e^2*f^2*(-104*e*h+85*f*g)-16*c^2*d^3*e^3*f*(-10*e*h+7*f*g)-10*c^4*d*e*f^ 
3*(-8*e*h+7*f*g)+35*c^5*f^4*(-8*e*h+9*f*g)-16*c*d^4*e^4*(16*e*h+11*f*g))+a 
*b^4*(35*c^5*f^5*h-c^3*d^2*e*f^3*(-547*e*h+450*f*g)-3*c^2*d^3*e^2*f^2*(-28 
0*e*h+197*f*g)+5*c^4*d*f^4*(-262*e*h+301*f*g)+16*d^5*e^4*(4*e*h+89*f*g)-16 
*c*d^4*e^3*f*(71*e*h+58*f*g))-5*a^4*b*d^3*f^3*(103*c^2*f^2*h+2*c*d*f*(-199 
*e*h+64*f*g)+d^2*e*(103*e*h+64*f*g))-a^2*b^3*d*f*(185*c^4*f^4*h+c^3*d*f^3* 
(-2298*e*h+2785*f*g)-3*c^2*d^2*e*f^2*(-595*e*h+422*f*g)+16*d^4*e^3*(19*e*h 
+149*f*g)-3*c*d^3*e^2*f*(632*e*h+661*f*g))+a^3*b^2*d^2*f^2*(409*c^3*f^3*h+ 
17*c^2*d*f^2*(-90*e*h+139*f*g)+d^3*e^2*(568*e*h+1723*f*g)-c*d^2*e*f*(1367* 
e*h+2166*f*g)))/(-a*d+b*c)^5/(-a*f+b*e)^5/(-c*f+d*e)^2/(f*x+e)^(1/2)+1/192 
*d*(a^4*d^3*f^3*(-1219*c*f*h+1027*d*e*h+192*d*f*g)+b^4*(960*d^4*e^4*g-5*c^ 
3*d*e*f^2*(-40*e*h+49*f*g)-35*c^4*f^3*(-8*e*h+9*f*g)-32*c^2*d^2*e^2*f*(-3* 
e*h+5*f*g)-48*c*d^3*e^3*(16*e*h+f*g))+a*b^3*(35*c^4*f^4*h+c^2*d^2*e*f^2*(- 
728*e*h+1055*f*g)+5*c^3*d*f^3*(-255*e*h+301*f*g)-48*d^4*e^3*(4*e*h+79*f*g) 
+16*c*d^3*e^2*f*(183*e*h+29*f*g))+a^3*b*d^2*f^2*(409*c^2*f^2*h-d^2*e*(1336 
*e*h+3131*f*g)+c*d*f*(1695*e*h+2363*f*g))-a^2*b^2*d*f*(185*c^3*f^3*h+c^2*d 
*f^2*(-2113*e*h+2785*f*g)-16*d^3*e^2*(51*e*h+341*f*g)+c*d^2*e*f*(3896*e*h+ 
1519*f*g)))/(-a*d+b*c)^5/(-a*f+b*e)^4/(-c*f+d*e)/(d*x+c)/(f*x+e)^(1/2)-1/4 
*(-a*h+b*g)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^4/(d*x+c)/(f*x+e)^(1/2)+1/24*...
 

Mathematica [C] (verified)

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

Time = 13.79 (sec) , antiderivative size = 1410, normalized size of antiderivative = 0.71 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(48*(-(b*g) + a*h) - (8*(-11*a^2*d*f*h + b^2*(-10*d*e*g - 9*c*f*g + 8*c*e* 
h) + a*b*(19*d*f*g + 2*d*e*h + c*f*h))*(a + b*x))/((b*c - a*d)*(b*e - a*f) 
) + ((a + b*x)^2*(2*(b*c - a*d)^3*(b*e - a*f)^2*(99*a^3*d^2*f^2*h - a^2*b* 
d*f*(219*d*f*g + 52*d*e*h + 26*c*f*h) + a*b^2*(7*c^2*f^2*h + 2*c*d*f*(101* 
f*g - 82*e*h) + 4*d^2*e*(59*f*g + 4*e*h)) + b^3*(-80*d^2*e^2*g + 7*c^2*f*( 
-9*f*g + 8*e*h) + 4*c*d*e*(-19*f*g + 16*e*h))) - ((a + b*x)*((b*c - a*d)^2 
*(b*e - a*f)*(d*e - c*f)^2*(-693*a^4*d^3*f^3*h + a^3*b*d^2*f^2*(1725*d*f*g 
 + 698*d*e*h + 349*c*f*h) + a*b^3*(35*c^3*f^3*h + c^2*d*f^2*(1379*f*g - 11 
42*e*h) + 12*c*d^2*e*f*(155*f*g - 116*e*h) + 16*d^3*e^2*(121*f*g + 6*e*h)) 
 + b^4*(-480*d^3*e^3*g + 35*c^3*f^2*(-9*f*g + 8*e*h) + 16*c*d^2*e^2*(-31*f 
*g + 24*e*h) + 2*c^2*d*e*f*(-217*f*g + 184*e*h)) - a^2*b^2*d*f*(171*c^2*f^ 
2*h + c*d*f*(2309*f*g - 1604*e*h) + 2*d^2*e*(1433*f*g + 208*e*h))) + (a + 
b*x)*(d*(b*c - a*d)*(b*e - a*f)*(-(d*e) + c*f)*(a^4*d^3*f^3*(192*d*f*g + 1 
027*d*e*h - 1219*c*f*h) + b^4*(960*d^4*e^4*g + 32*c^2*d^2*e^2*f*(-5*f*g + 
3*e*h) + 35*c^4*f^3*(-9*f*g + 8*e*h) - 48*c*d^3*e^3*(f*g + 16*e*h) + 5*c^3 
*d*e*f^2*(-49*f*g + 40*e*h)) + a*b^3*(35*c^4*f^4*h + c^2*d^2*e*f^2*(1055*f 
*g - 728*e*h) + 5*c^3*d*f^3*(301*f*g - 255*e*h) - 48*d^4*e^3*(79*f*g + 4*e 
*h) + 16*c*d^3*e^2*f*(29*f*g + 183*e*h)) + a^3*b*d^2*f^2*(409*c^2*f^2*h - 
d^2*e*(3131*f*g + 1336*e*h) + c*d*f*(2363*f*g + 1695*e*h)) + a^2*b^2*d*f*( 
-185*c^3*f^3*h + 16*d^3*e^2*(341*f*g + 51*e*h) + c^2*d*f^2*(-2785*f*g +...
 

Rubi [A] (verified)

Time = 6.05 (sec) , antiderivative size = 2129, normalized size of antiderivative = 1.07, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {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.

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\int \frac {2 b \left (5 d e g+\frac {9 c f g}{2}-4 c e h\right )-a (8 d f g+2 d e h+c f h)+11 d f (b g-a h) x}{2 (a+b x)^4 (c+d x)^2 (e+f x)^{3/2}}dx}{4 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {b (10 d e g+9 c f g-8 c e h)-a (8 d f g+2 d e h+c f h)+11 d f (b g-a h) x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}}dx}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {d f (48 d f g+34 d e h+17 c f h) a^2-b \left (2 e (73 f g+8 e h) d^2+c f (121 f g-92 e h) d+7 c^2 f^2 h\right ) a+b^2 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )+9 d f \left (11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 d e g+9 c f g-8 c e h)\right ) x}{2 (a+b x)^3 (c+d x)^2 (e+f x)^{3/2}}dx}{3 (b c-a d) (b e-a f)}-\frac {11 a^2 d f h-a b (c f h+2 d e h+19 d f g)+b^2 (-8 c e h+9 c f g+10 d e g)}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {d f (48 d f g+34 d e h+17 c f h) a^2-b \left (2 e (73 f g+8 e h) d^2+c f (121 f g-92 e h) d+7 c^2 f^2 h\right ) a+b^2 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )+9 d f \left (11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 d e g+9 c f g-8 c e h)\right ) x}{(a+b x)^3 (c+d x)^2 (e+f x)^{3/2}}dx}{6 (b c-a d) (b e-a f)}-\frac {11 a^2 d f h-a b (c f h+2 d e h+19 d f g)+b^2 (-8 c e h+9 c f g+10 d e g)}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {99 a^3 d^2 f^2 h-a^2 b d f (26 c f h+52 d e h+219 d f g)+a b^2 \left (7 c^2 f^2 h+2 c d f (101 f g-82 e h)+4 d^2 e (4 e h+59 f g)\right )-b^3 \left (7 c^2 f (9 f g-8 e h)+4 c d e (19 f g-16 e h)+80 d^2 e^2 g\right )}{2 (a+b x)^2 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}-\frac {\int -\frac {d^2 f^2 (192 d f g+334 d e h+167 c f h) a^3-b d f \left (2 e (607 f g+152 e h) d^2+c f (895 f g-456 e h) d+122 c^2 f^2 h\right ) a^2+b^2 \left (32 e^2 (43 f g+3 e h) d^3+16 c e f (83 f g-59 e h) d^2+2 c^2 f^2 (469 f g-375 e h) d+35 c^3 f^3 h\right ) a-b^3 \left (35 f^2 (9 f g-8 e h) c^3+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )+7 d f \left (99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )\right ) x}{2 (a+b x)^2 (c+d x)^2 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {11 a^2 d f h-a b (c f h+2 d e h+19 d f g)+b^2 (-8 c e h+9 c f g+10 d e g)}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\int \frac {d^2 f^2 (192 d f g+334 d e h+167 c f h) a^3-b d f \left (2 e (607 f g+152 e h) d^2+c f (895 f g-456 e h) d+122 c^2 f^2 h\right ) a^2+b^2 \left (32 e^2 (43 f g+3 e h) d^3+16 c e f (83 f g-59 e h) d^2+2 c^2 f^2 (469 f g-375 e h) d+35 c^3 f^3 h\right ) a-b^3 \left (35 f^2 (9 f g-8 e h) c^3+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )+7 d f \left (99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )\right ) x}{(a+b x)^2 (c+d x)^2 (e+f x)^{3/2}}dx}{4 (b c-a d) (b e-a f)}+\frac {99 a^3 d^2 f^2 h-a^2 b d f (26 c f h+52 d e h+219 d f g)+a b^2 \left (7 c^2 f^2 h+2 c d f (101 f g-82 e h)+4 d^2 e (4 e h+59 f g)\right )-b^3 \left (7 c^2 f (9 f g-8 e h)+4 c d e (19 f g-16 e h)+80 d^2 e^2 g\right )}{2 (a+b x)^2 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {11 a^2 d f h-a b (c f h+2 d e h+19 d f g)+b^2 (-8 c e h+9 c f g+10 d e g)}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {b g-a h}{4 (a+b x)^4 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x) \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 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) \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {\frac {693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}-\frac {\int -\frac {d^3 f^3 (384 d f g+2054 d e h+1027 c f h) a^4-b d^2 f^2 \left (2 e (3131 f g+1336 e h) d^2+c f (3899 f g+100 e h) d+927 c^2 f^2 h\right ) a^3+b^2 d f \left (32 e^2 (341 f g+51 e h) d^3+12 c e f (941 f g-476 e h) d^2+c^2 f^2 (5975 f g-3794 e h) d+485 c^3 f^3 h\right ) a^2-b^3 \left (96 e^3 (79 f g+4 e h) d^4+16 c e^2 f (547 f g-336 e h) d^3+2 c^2 e f^2 (3595 f g-2752 e h) d^2+5 c^3 f^3 (777 f g-632 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+80 d e f^2 (7 f g-6 e h) c^3+144 d^2 e^2 f (5 f g-4 e h) c^2+256 d^3 e^3 (3 f g-2 e h) c+640 d^4 e^4 g\right )+5 d f \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )\right ) x}{2 (a+b x) (c+d x)^2 (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) \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 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) \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {\frac {693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\int \frac {d^3 f^3 (384 d f g+2054 d e h+1027 c f h) a^4-b d^2 f^2 \left (2 e (3131 f g+1336 e h) d^2+c f (3899 f g+100 e h) d+927 c^2 f^2 h\right ) a^3+b^2 d f \left (32 e^2 (341 f g+51 e h) d^3+12 c e f (941 f g-476 e h) d^2+c^2 f^2 (5975 f g-3794 e h) d+485 c^3 f^3 h\right ) a^2-b^3 \left (96 e^3 (79 f g+4 e h) d^4+16 c e^2 f (547 f g-336 e h) d^3+2 c^2 e f^2 (3595 f g-2752 e h) d^2+5 c^3 f^3 (777 f g-632 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+80 d e f^2 (7 f g-6 e h) c^3+144 d^2 e^2 f (5 f g-4 e h) c^2+256 d^3 e^3 (3 f g-2 e h) c+640 d^4 e^4 g\right )+5 d f \left (693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x)^2 (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) \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 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) \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {\frac {693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\frac {2 d \left (d^3 f^3 (192 d f g+1027 d e h-1219 c f h) a^4+b d^2 f^2 \left (-e (3131 f g+1336 e h) d^2+c f (2363 f g+1695 e h) d+409 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (341 f g+51 e h) d^3+c e f (1519 f g+3896 e h) d^2+c^2 f^2 (2785 f g-2113 e h) d+185 c^3 f^3 h\right ) a^2+b^3 \left (-48 e^3 (79 f g+4 e h) d^4+16 c e^2 f (29 f g+183 e h) d^3+c^2 e f^2 (1055 f g-728 e h) d^2+5 c^3 f^3 (301 f g-255 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-5 d e f^2 (49 f g-40 e h) c^3-32 d^2 e^2 f (5 f g-3 e h) c^2-48 d^3 e^3 (f g+16 e h) c+960 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {\int \frac {3 \left (64 d^4 f^4 (3 d f g-2 d e h-c f h) a^5-b d^3 f^3 \left (128 e (f g-4 e h) d^2+c f (640 f g-771 e h) d+515 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-128 e^2 (11 f g+6 e h) d^3+c e f (197 f g+328 e h) d^2+c^2 f^2 (2363 f g-1121 e h) d+409 c^3 f^3 h\right ) a^3-b^3 d f \left (-512 e^3 (6 f g+e h) d^4-16 c e^2 f (29 f g-125 e h) d^3+c^2 e f^2 (1519 f g-328 e h) d^2+c^3 f^3 (2785 f g-2113 e h) d+185 c^4 f^4 h\right ) a^2+b^4 \left (-64 e^4 (37 f g+2 e h) d^5-16 c e^3 f (29 f g-112 e h) d^4+16 c^2 e^2 f^2 (29 f g+7 e h) d^3+c^3 e f^3 (1055 f g-728 e h) d^2+5 c^4 f^4 (301 f g-255 e h) d+35 c^5 f^5 h\right ) a+b^5 (d e-c f) \left (35 f^3 (9 f g-8 e h) c^4+80 d e f^2 (7 f g-6 e h) c^3+144 d^2 e^2 f (5 f g-4 e h) c^2+256 d^3 e^3 (3 f g-2 e h) c+640 d^4 e^4 g\right )+b d f \left (d^3 f^3 (192 d f g+1027 d e h-1219 c f h) a^4+b d^2 f^2 \left (-e (3131 f g+1336 e h) d^2+c f (2363 f g+1695 e h) d+409 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (341 f g+51 e h) d^3+c e f (1519 f g+3896 e h) d^2+c^2 f^2 (2785 f g-2113 e h) d+185 c^3 f^3 h\right ) a^2+b^3 \left (-48 e^3 (79 f g+4 e h) d^4+16 c e^2 f (29 f g+183 e h) d^3+c^2 e f^2 (1055 f g-728 e h) d^2+5 c^3 f^3 (301 f g-255 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-5 d e f^2 (49 f g-40 e h) c^3-32 d^2 e^2 f (5 f g-3 e h) c^2-48 d^3 e^3 (f g+16 e h) c+960 d^4 e^4 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)}}{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) \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 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) \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {\frac {693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\frac {2 d \left (d^3 f^3 (192 d f g+1027 d e h-1219 c f h) a^4+b d^2 f^2 \left (-e (3131 f g+1336 e h) d^2+c f (2363 f g+1695 e h) d+409 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (341 f g+51 e h) d^3+c e f (1519 f g+3896 e h) d^2+c^2 f^2 (2785 f g-2113 e h) d+185 c^3 f^3 h\right ) a^2+b^3 \left (-48 e^3 (79 f g+4 e h) d^4+16 c e^2 f (29 f g+183 e h) d^3+c^2 e f^2 (1055 f g-728 e h) d^2+5 c^3 f^3 (301 f g-255 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-5 d e f^2 (49 f g-40 e h) c^3-32 d^2 e^2 f (5 f g-3 e h) c^2-48 d^3 e^3 (f g+16 e h) c+960 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \int \frac {64 d^4 f^4 (3 d f g-2 d e h-c f h) a^5-b d^3 f^3 \left (128 e (f g-4 e h) d^2+c f (640 f g-771 e h) d+515 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (-128 e^2 (11 f g+6 e h) d^3+c e f (197 f g+328 e h) d^2+c^2 f^2 (2363 f g-1121 e h) d+409 c^3 f^3 h\right ) a^3-b^3 d f \left (-512 e^3 (6 f g+e h) d^4-16 c e^2 f (29 f g-125 e h) d^3+c^2 e f^2 (1519 f g-328 e h) d^2+c^3 f^3 (2785 f g-2113 e h) d+185 c^4 f^4 h\right ) a^2+b^4 \left (-64 e^4 (37 f g+2 e h) d^5-16 c e^3 f (29 f g-112 e h) d^4+16 c^2 e^2 f^2 (29 f g+7 e h) d^3+c^3 e f^3 (1055 f g-728 e h) d^2+5 c^4 f^4 (301 f g-255 e h) d+35 c^5 f^5 h\right ) a+b^5 (d e-c f) \left (35 f^3 (9 f g-8 e h) c^4+80 d e f^2 (7 f g-6 e h) c^3+144 d^2 e^2 f (5 f g-4 e h) c^2+256 d^3 e^3 (3 f g-2 e h) c+640 d^4 e^4 g\right )+b d f \left (d^3 f^3 (192 d f g+1027 d e h-1219 c f h) a^4+b d^2 f^2 \left (-e (3131 f g+1336 e h) d^2+c f (2363 f g+1695 e h) d+409 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (341 f g+51 e h) d^3+c e f (1519 f g+3896 e h) d^2+c^2 f^2 (2785 f g-2113 e h) d+185 c^3 f^3 h\right ) a^2+b^3 \left (-48 e^3 (79 f g+4 e h) d^4+16 c e^2 f (29 f g+183 e h) d^3+c^2 e f^2 (1055 f g-728 e h) d^2+5 c^3 f^3 (301 f g-255 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-5 d e f^2 (49 f g-40 e h) c^3-32 d^2 e^2 f (5 f g-3 e h) c^2-48 d^3 e^3 (f g+16 e h) c+960 d^4 e^4 g\right )\right ) x}{(a+b x) (c+d x) (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)}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {b g-a h}{4 (b c-a d) (b e-a f) (a+b x)^4 (c+d x) \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 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) \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {\frac {693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\frac {2 d \left (d^3 f^3 (192 d f g+1027 d e h-1219 c f h) a^4+b d^2 f^2 \left (-e (3131 f g+1336 e h) d^2+c f (2363 f g+1695 e h) d+409 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (341 f g+51 e h) d^3+c e f (1519 f g+3896 e h) d^2+c^2 f^2 (2785 f g-2113 e h) d+185 c^3 f^3 h\right ) a^2+b^3 \left (-48 e^3 (79 f g+4 e h) d^4+16 c e^2 f (29 f g+183 e h) d^3+c^2 e f^2 (1055 f g-728 e h) d^2+5 c^3 f^3 (301 f g-255 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-5 d e f^2 (49 f g-40 e h) c^3-32 d^2 e^2 f (5 f g-3 e h) c^2-48 d^3 e^3 (f g+16 e h) c+960 d^4 e^4 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 (64 d^4 f^4 (3 d f g-2 d e h-c f h) a^5-5 b d^3 f^3 \left (e (64 f g+103 e h) d^2+2 c f (64 f g-199 e h) d+103 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (e^2 (1723 f g+568 e h) d^3-c e f (2166 f g+1367 e h) d^2+17 c^2 f^2 (139 f g-90 e h) d+409 c^3 f^3 h\right ) a^3-b^3 d f \left (16 e^3 (149 f g+19 e h) d^4-3 c e^2 f (661 f g+632 e h) d^3-3 c^2 e f^2 (422 f g-595 e h) d^2+c^3 f^3 (2785 f g-2298 e h) d+185 c^4 f^4 h\right ) a^2+b^4 \left (16 e^4 (89 f g+4 e h) d^5-16 c e^3 f (58 f g+71 e h) d^4-3 c^2 e^2 f^2 (197 f g-280 e h) d^3-c^3 e f^3 (450 f g-547 e h) d^2+5 c^4 f^4 (301 f g-262 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 (7 f g-8 e h) c^4-d^2 e^2 f^2 (85 f g-104 e h) c^3-16 d^3 e^3 f (7 f g-10 e h) c^2-16 d^4 e^4 (11 f g+16 e h) c+320 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 \int \frac {64 d^5 f^5 (3 d f g-2 d e h-c f h) a^6+320 b d^4 f^4 \left (-e (f g-2 e h) d^2-c f (2 f g+e h) d+2 c^2 f^2 h\right ) a^5-5 b^2 d^3 f^3 \left (256 e^2 (f g+e h) d^3-c e f (768 f g+281 e h) d^2+2 c^2 f^2 (64 f g+153 e h) d+103 c^3 f^3 h\right ) a^4+b^3 d^2 f^2 \left (640 e^3 (7 f g+2 e h) d^4-3 c e^2 f (2199 f g+1304 e h) d^3-3 c^2 e f^2 (722 f g-1891 e h) d^2+17 c^3 f^3 (139 f g-90 e h) d+409 c^4 f^4 h\right ) a^3-b^4 d f \left (320 e^4 (17 f g+2 e h) d^5-16 c e^3 f (371 f g+281 e h) d^4-c^2 e^2 f^2 (1983 f g-5144 e h) d^3-3 c^3 e f^3 (422 f g-595 e h) d^2+c^4 f^4 (2785 f g-2298 e h) d+185 c^5 f^5 h\right ) a^2+b^5 \left (64 e^5 (47 f g+2 e h) d^6-304 c e^4 f (9 f g+8 e h) d^5-16 c^2 e^3 f^2 (58 f g-149 e h) d^4-3 c^3 e^2 f^3 (197 f g-280 e h) d^3-c^4 e f^4 (450 f g-547 e h) d^2+5 c^5 f^5 (301 f g-262 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+80 d e f^2 (7 f g-6 e h) c^3+144 d^2 e^2 f (5 f g-4 e h) c^2+256 d^3 e^3 (3 f g-2 e h) c+640 d^4 e^4 g\right )+b d f \left (64 d^4 f^4 (3 d f g-2 d e h-c f h) a^5-5 b d^3 f^3 \left (e (64 f g+103 e h) d^2+2 c f (64 f g-199 e h) d+103 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (e^2 (1723 f g+568 e h) d^3-c e f (2166 f g+1367 e h) d^2+17 c^2 f^2 (139 f g-90 e h) d+409 c^3 f^3 h\right ) a^3-b^3 d f \left (16 e^3 (149 f g+19 e h) d^4-3 c e^2 f (661 f g+632 e h) d^3-3 c^2 e f^2 (422 f g-595 e h) d^2+c^3 f^3 (2785 f g-2298 e h) d+185 c^4 f^4 h\right ) a^2+b^4 \left (16 e^4 (89 f g+4 e h) d^5-16 c e^3 f (58 f g+71 e h) d^4-3 c^2 e^2 f^2 (197 f g-280 e h) d^3-c^3 e f^3 (450 f g-547 e h) d^2+5 c^4 f^4 (301 f g-262 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 (7 f g-8 e h) c^4-d^2 e^2 f^2 (85 f g-104 e h) c^3-16 d^3 e^3 f (7 f g-10 e h) c^2-16 d^4 e^4 (11 f g+16 e h) c+320 d^5 e^5 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)}}{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) \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 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) \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {\frac {693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\frac {2 d \left (d^3 f^3 (192 d f g+1027 d e h-1219 c f h) a^4+b d^2 f^2 \left (-e (3131 f g+1336 e h) d^2+c f (2363 f g+1695 e h) d+409 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (341 f g+51 e h) d^3+c e f (1519 f g+3896 e h) d^2+c^2 f^2 (2785 f g-2113 e h) d+185 c^3 f^3 h\right ) a^2+b^3 \left (-48 e^3 (79 f g+4 e h) d^4+16 c e^2 f (29 f g+183 e h) d^3+c^2 e f^2 (1055 f g-728 e h) d^2+5 c^3 f^3 (301 f g-255 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-5 d e f^2 (49 f g-40 e h) c^3-32 d^2 e^2 f (5 f g-3 e h) c^2-48 d^3 e^3 (f g+16 e h) c+960 d^4 e^4 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 (64 d^4 f^4 (3 d f g-2 d e h-c f h) a^5-5 b d^3 f^3 \left (e (64 f g+103 e h) d^2+2 c f (64 f g-199 e h) d+103 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (e^2 (1723 f g+568 e h) d^3-c e f (2166 f g+1367 e h) d^2+17 c^2 f^2 (139 f g-90 e h) d+409 c^3 f^3 h\right ) a^3-b^3 d f \left (16 e^3 (149 f g+19 e h) d^4-3 c e^2 f (661 f g+632 e h) d^3-3 c^2 e f^2 (422 f g-595 e h) d^2+c^3 f^3 (2785 f g-2298 e h) d+185 c^4 f^4 h\right ) a^2+b^4 \left (16 e^4 (89 f g+4 e h) d^5-16 c e^3 f (58 f g+71 e h) d^4-3 c^2 e^2 f^2 (197 f g-280 e h) d^3-c^3 e f^3 (450 f g-547 e h) d^2+5 c^4 f^4 (301 f g-262 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 (7 f g-8 e h) c^4-d^2 e^2 f^2 (85 f g-104 e h) c^3-16 d^3 e^3 f (7 f g-10 e h) c^2-16 d^4 e^4 (11 f g+16 e h) c+320 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\int \frac {64 d^5 f^5 (3 d f g-2 d e h-c f h) a^6+320 b d^4 f^4 \left (-e (f g-2 e h) d^2-c f (2 f g+e h) d+2 c^2 f^2 h\right ) a^5-5 b^2 d^3 f^3 \left (256 e^2 (f g+e h) d^3-c e f (768 f g+281 e h) d^2+2 c^2 f^2 (64 f g+153 e h) d+103 c^3 f^3 h\right ) a^4+b^3 d^2 f^2 \left (640 e^3 (7 f g+2 e h) d^4-3 c e^2 f (2199 f g+1304 e h) d^3-3 c^2 e f^2 (722 f g-1891 e h) d^2+17 c^3 f^3 (139 f g-90 e h) d+409 c^4 f^4 h\right ) a^3-b^4 d f \left (320 e^4 (17 f g+2 e h) d^5-16 c e^3 f (371 f g+281 e h) d^4-c^2 e^2 f^2 (1983 f g-5144 e h) d^3-3 c^3 e f^3 (422 f g-595 e h) d^2+c^4 f^4 (2785 f g-2298 e h) d+185 c^5 f^5 h\right ) a^2+b^5 \left (64 e^5 (47 f g+2 e h) d^6-304 c e^4 f (9 f g+8 e h) d^5-16 c^2 e^3 f^2 (58 f g-149 e h) d^4-3 c^3 e^2 f^3 (197 f g-280 e h) d^3-c^4 e f^4 (450 f g-547 e h) d^2+5 c^5 f^5 (301 f g-262 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+80 d e f^2 (7 f g-6 e h) c^3+144 d^2 e^2 f (5 f g-4 e h) c^2+256 d^3 e^3 (3 f g-2 e h) c+640 d^4 e^4 g\right )+b d f \left (64 d^4 f^4 (3 d f g-2 d e h-c f h) a^5-5 b d^3 f^3 \left (e (64 f g+103 e h) d^2+2 c f (64 f g-199 e h) d+103 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (e^2 (1723 f g+568 e h) d^3-c e f (2166 f g+1367 e h) d^2+17 c^2 f^2 (139 f g-90 e h) d+409 c^3 f^3 h\right ) a^3-b^3 d f \left (16 e^3 (149 f g+19 e h) d^4-3 c e^2 f (661 f g+632 e h) d^3-3 c^2 e f^2 (422 f g-595 e h) d^2+c^3 f^3 (2785 f g-2298 e h) d+185 c^4 f^4 h\right ) a^2+b^4 \left (16 e^4 (89 f g+4 e h) d^5-16 c e^3 f (58 f g+71 e h) d^4-3 c^2 e^2 f^2 (197 f g-280 e h) d^3-c^3 e f^3 (450 f g-547 e h) d^2+5 c^4 f^4 (301 f g-262 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 (7 f g-8 e h) c^4-d^2 e^2 f^2 (85 f g-104 e h) c^3-16 d^3 e^3 f (7 f g-10 e h) c^2-16 d^4 e^4 (11 f g+16 e h) c+320 d^5 e^5 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)}}{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) \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 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) \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {\frac {693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\frac {2 d \left (d^3 f^3 (192 d f g+1027 d e h-1219 c f h) a^4+b d^2 f^2 \left (-e (3131 f g+1336 e h) d^2+c f (2363 f g+1695 e h) d+409 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (341 f g+51 e h) d^3+c e f (1519 f g+3896 e h) d^2+c^2 f^2 (2785 f g-2113 e h) d+185 c^3 f^3 h\right ) a^2+b^3 \left (-48 e^3 (79 f g+4 e h) d^4+16 c e^2 f (29 f g+183 e h) d^3+c^2 e f^2 (1055 f g-728 e h) d^2+5 c^3 f^3 (301 f g-255 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-5 d e f^2 (49 f g-40 e h) c^3-32 d^2 e^2 f (5 f g-3 e h) c^2-48 d^3 e^3 (f g+16 e h) c+960 d^4 e^4 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 (64 d^4 f^4 (3 d f g-2 d e h-c f h) a^5-5 b d^3 f^3 \left (e (64 f g+103 e h) d^2+2 c f (64 f g-199 e h) d+103 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (e^2 (1723 f g+568 e h) d^3-c e f (2166 f g+1367 e h) d^2+17 c^2 f^2 (139 f g-90 e h) d+409 c^3 f^3 h\right ) a^3-b^3 d f \left (16 e^3 (149 f g+19 e h) d^4-3 c e^2 f (661 f g+632 e h) d^3-3 c^2 e f^2 (422 f g-595 e h) d^2+c^3 f^3 (2785 f g-2298 e h) d+185 c^4 f^4 h\right ) a^2+b^4 \left (16 e^4 (89 f g+4 e h) d^5-16 c e^3 f (58 f g+71 e h) d^4-3 c^2 e^2 f^2 (197 f g-280 e h) d^3-c^3 e f^3 (450 f g-547 e h) d^2+5 c^4 f^4 (301 f g-262 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 (7 f g-8 e h) c^4-d^2 e^2 f^2 (85 f g-104 e h) c^3-16 d^3 e^3 f (7 f g-10 e h) c^2-16 d^4 e^4 (11 f g+16 e h) c+320 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {64 d^5 \left (a d (3 d f g-2 d e h-c f h)+b \left (11 f h c^2-d (13 f g+8 e h) c+10 d^2 e g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx (b e-a f)^5}{b c-a d}+\frac {b^2 (d e-c f)^2 \left (1155 d^4 f^4 h a^5-231 b d^3 f^3 (13 d f g+8 d e h+4 c f h) a^4+66 b^2 d^2 f^2 \left (8 e (13 f g+3 e h) d^2+2 c f (39 f g-20 e h) d+9 c^2 f^2 h\right ) a^3-22 b^3 d f \left (8 e^2 (39 f g+4 e h) d^3+24 c e f (13 f g-8 e h) d^2+3 c^2 f^2 (65 f g-48 e h) d+10 c^3 f^3 h\right ) a^2+b^4 \left (128 e^3 (26 f g+e h) d^4+32 c e^2 f (117 f g-76 e h) d^3+48 c^2 e f^2 (65 f g-51 e h) d^2+20 c^3 f^3 (91 f g-76 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+80 d e f^2 (7 f g-6 e h) c^3+144 d^2 e^2 f (5 f g-4 e h) c^2+256 d^3 e^3 (3 f g-2 e h) c+640 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)}}{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) \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 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) \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {\frac {693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\frac {2 d \left (d^3 f^3 (192 d f g+1027 d e h-1219 c f h) a^4+b d^2 f^2 \left (-e (3131 f g+1336 e h) d^2+c f (2363 f g+1695 e h) d+409 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (341 f g+51 e h) d^3+c e f (1519 f g+3896 e h) d^2+c^2 f^2 (2785 f g-2113 e h) d+185 c^3 f^3 h\right ) a^2+b^3 \left (-48 e^3 (79 f g+4 e h) d^4+16 c e^2 f (29 f g+183 e h) d^3+c^2 e f^2 (1055 f g-728 e h) d^2+5 c^3 f^3 (301 f g-255 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-5 d e f^2 (49 f g-40 e h) c^3-32 d^2 e^2 f (5 f g-3 e h) c^2-48 d^3 e^3 (f g+16 e h) c+960 d^4 e^4 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 (64 d^4 f^4 (3 d f g-2 d e h-c f h) a^5-5 b d^3 f^3 \left (e (64 f g+103 e h) d^2+2 c f (64 f g-199 e h) d+103 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (e^2 (1723 f g+568 e h) d^3-c e f (2166 f g+1367 e h) d^2+17 c^2 f^2 (139 f g-90 e h) d+409 c^3 f^3 h\right ) a^3-b^3 d f \left (16 e^3 (149 f g+19 e h) d^4-3 c e^2 f (661 f g+632 e h) d^3-3 c^2 e f^2 (422 f g-595 e h) d^2+c^3 f^3 (2785 f g-2298 e h) d+185 c^4 f^4 h\right ) a^2+b^4 \left (16 e^4 (89 f g+4 e h) d^5-16 c e^3 f (58 f g+71 e h) d^4-3 c^2 e^2 f^2 (197 f g-280 e h) d^3-c^3 e f^3 (450 f g-547 e h) d^2+5 c^4 f^4 (301 f g-262 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 (7 f g-8 e h) c^4-d^2 e^2 f^2 (85 f g-104 e h) c^3-16 d^3 e^3 f (7 f g-10 e h) c^2-16 d^4 e^4 (11 f g+16 e h) c+320 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {128 d^5 \left (a d (3 d f g-2 d e h-c f h)+b \left (11 f h c^2-d (13 f g+8 e h) c+10 d^2 e g\right )\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^2 (d e-c f)^2 \left (1155 d^4 f^4 h a^5-231 b d^3 f^3 (13 d f g+8 d e h+4 c f h) a^4+66 b^2 d^2 f^2 \left (8 e (13 f g+3 e h) d^2+2 c f (39 f g-20 e h) d+9 c^2 f^2 h\right ) a^3-22 b^3 d f \left (8 e^2 (39 f g+4 e h) d^3+24 c e f (13 f g-8 e h) d^2+3 c^2 f^2 (65 f g-48 e h) d+10 c^3 f^3 h\right ) a^2+b^4 \left (128 e^3 (26 f g+e h) d^4+32 c e^2 f (117 f g-76 e h) d^3+48 c^2 e f^2 (65 f g-51 e h) d^2+20 c^3 f^3 (91 f g-76 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+80 d e f^2 (7 f g-6 e h) c^3+144 d^2 e^2 f (5 f g-4 e h) c^2+256 d^3 e^3 (3 f g-2 e h) c+640 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)}}{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) \sqrt {e+f x}}-\frac {-\frac {11 d f h a^2-b (19 d f g+2 d e h+c f h) a+b^2 (10 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) \sqrt {e+f x}}-\frac {\frac {99 d^2 f^2 h a^3-b d f (219 d f g+52 d e h+26 c f h) a^2+b^2 \left (4 e (59 f g+4 e h) d^2+2 c f (101 f g-82 e h) d+7 c^2 f^2 h\right ) a-b^3 \left (7 f (9 f g-8 e h) c^2+4 d e (19 f g-16 e h) c+80 d^2 e^2 g\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {\frac {693 d^3 f^3 h a^4-b d^2 f^2 (1725 d f g+698 d e h+349 c f h) a^3+b^2 d f \left (2 e (1433 f g+208 e h) d^2+c f (2309 f g-1604 e h) d+171 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (121 f g+6 e h) d^3+12 c e f (155 f g-116 e h) d^2+c^2 f^2 (1379 f g-1142 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+2 d e f (217 f g-184 e h) c^2+16 d^2 e^2 (31 f g-24 e h) c+480 d^3 e^3 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\frac {2 d \left (d^3 f^3 (192 d f g+1027 d e h-1219 c f h) a^4+b d^2 f^2 \left (-e (3131 f g+1336 e h) d^2+c f (2363 f g+1695 e h) d+409 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (341 f g+51 e h) d^3+c e f (1519 f g+3896 e h) d^2+c^2 f^2 (2785 f g-2113 e h) d+185 c^3 f^3 h\right ) a^2+b^3 \left (-48 e^3 (79 f g+4 e h) d^4+16 c e^2 f (29 f g+183 e h) d^3+c^2 e f^2 (1055 f g-728 e h) d^2+5 c^3 f^3 (301 f g-255 e h) d+35 c^4 f^4 h\right ) a+b^4 \left (-35 f^3 (9 f g-8 e h) c^4-5 d e f^2 (49 f g-40 e h) c^3-32 d^2 e^2 f (5 f g-3 e h) c^2-48 d^3 e^3 (f g+16 e h) c+960 d^4 e^4 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 (64 d^4 f^4 (3 d f g-2 d e h-c f h) a^5-5 b d^3 f^3 \left (e (64 f g+103 e h) d^2+2 c f (64 f g-199 e h) d+103 c^2 f^2 h\right ) a^4+b^2 d^2 f^2 \left (e^2 (1723 f g+568 e h) d^3-c e f (2166 f g+1367 e h) d^2+17 c^2 f^2 (139 f g-90 e h) d+409 c^3 f^3 h\right ) a^3-b^3 d f \left (16 e^3 (149 f g+19 e h) d^4-3 c e^2 f (661 f g+632 e h) d^3-3 c^2 e f^2 (422 f g-595 e h) d^2+c^3 f^3 (2785 f g-2298 e h) d+185 c^4 f^4 h\right ) a^2+b^4 \left (16 e^4 (89 f g+4 e h) d^5-16 c e^3 f (58 f g+71 e h) d^4-3 c^2 e^2 f^2 (197 f g-280 e h) d^3-c^3 e f^3 (450 f g-547 e h) d^2+5 c^4 f^4 (301 f g-262 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 (7 f g-8 e h) c^4-d^2 e^2 f^2 (85 f g-104 e h) c^3-16 d^3 e^3 f (7 f g-10 e h) c^2-16 d^4 e^4 (11 f g+16 e h) c+320 d^5 e^5 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {-\frac {128 d^{9/2} \left (a d (3 d f g-2 d e h-c f h)+b \left (11 f h c^2-d (13 f g+8 e h) c+10 d^2 e g\right )\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^{3/2} (d e-c f)^2 \left (1155 d^4 f^4 h a^5-231 b d^3 f^3 (13 d f g+8 d e h+4 c f h) a^4+66 b^2 d^2 f^2 \left (8 e (13 f g+3 e h) d^2+2 c f (39 f g-20 e h) d+9 c^2 f^2 h\right ) a^3-22 b^3 d f \left (8 e^2 (39 f g+4 e h) d^3+24 c e f (13 f g-8 e h) d^2+3 c^2 f^2 (65 f g-48 e h) d+10 c^3 f^3 h\right ) a^2+b^4 \left (128 e^3 (26 f g+e h) d^4+32 c e^2 f (117 f g-76 e h) d^3+48 c^2 e f^2 (65 f g-51 e h) d^2+20 c^3 f^3 (91 f g-76 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+80 d e f^2 (7 f g-6 e h) c^3+144 d^2 e^2 f (5 f g-4 e h) c^2+256 d^3 e^3 (3 f g-2 e h) c+640 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)}}{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)^2*(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)*Sqrt[e + f 
*x]) - (-1/3*(11*a^2*d*f*h + b^2*(10*d*e*g + 9*c*f*g - 8*c*e*h) - a*b*(19* 
d*f*g + 2*d*e*h + c*f*h))/((b*c - a*d)*(b*e - a*f)*(a + b*x)^3*(c + d*x)*S 
qrt[e + f*x]) - ((99*a^3*d^2*f^2*h - a^2*b*d*f*(219*d*f*g + 52*d*e*h + 26* 
c*f*h) - b^3*(80*d^2*e^2*g + 4*c*d*e*(19*f*g - 16*e*h) + 7*c^2*f*(9*f*g - 
8*e*h)) + a*b^2*(7*c^2*f^2*h + 2*c*d*f*(101*f*g - 82*e*h) + 4*d^2*e*(59*f* 
g + 4*e*h)))/(2*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)*Sqrt[e + f*x 
]) + ((693*a^4*d^3*f^3*h - a^3*b*d^2*f^2*(1725*d*f*g + 698*d*e*h + 349*c*f 
*h) + b^4*(480*d^3*e^3*g + 2*c^2*d*e*f*(217*f*g - 184*e*h) + 16*c*d^2*e^2* 
(31*f*g - 24*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*(1379*f*g - 1142*e*h) + 12*c*d^2*e*f*(155*f*g - 116*e*h) + 16*d^3* 
e^2*(121*f*g + 6*e*h)) + a^2*b^2*d*f*(171*c^2*f^2*h + c*d*f*(2309*f*g - 16 
04*e*h) + 2*d^2*e*(1433*f*g + 208*e*h)))/((b*c - a*d)*(b*e - a*f)*(a + b*x 
)*(c + d*x)*Sqrt[e + f*x]) + ((2*d*(a^4*d^3*f^3*(192*d*f*g + 1027*d*e*h - 
1219*c*f*h) + b^4*(960*d^4*e^4*g - 5*c^3*d*e*f^2*(49*f*g - 40*e*h) - 35*c^ 
4*f^3*(9*f*g - 8*e*h) - 32*c^2*d^2*e^2*f*(5*f*g - 3*e*h) - 48*c*d^3*e^3*(f 
*g + 16*e*h)) + a*b^3*(35*c^4*f^4*h + c^2*d^2*e*f^2*(1055*f*g - 728*e*h) + 
 5*c^3*d*f^3*(301*f*g - 255*e*h) - 48*d^4*e^3*(79*f*g + 4*e*h) + 16*c*d^3* 
e^2*f*(29*f*g + 183*e*h)) + a^3*b*d^2*f^2*(409*c^2*f^2*h - d^2*e*(3131*f*g 
 + 1336*e*h) + c*d*f*(2363*f*g + 1695*e*h)) - a^2*b^2*d*f*(185*c^3*f^3*...
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3943\) vs. \(2(1945)=3890\).

Time = 0.08 (sec) , antiderivative size = 3944, normalized size of antiderivative = 1.98

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

Input:

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

Output:

2*f^5*(-(-e*h+f*g)/(c*f-d*e)^2/(a*f-b*e)^5/(f*x+e)^(1/2)+d^5/(a*d-b*c)^6/f 
^5/(c*f-d*e)^2*((1/2*a*c*d*f*h-1/2*a*d^2*f*g-1/2*b*c^2*f*h+1/2*b*c*d*f*g)* 
(f*x+e)^(1/2)/((f*x+e)*d+c*f-d*e)+1/2*(a*c*d*f*h+2*a*d^2*e*h-3*a*d^2*f*g-1 
1*b*c^2*f*h+8*b*c*d*e*h+13*b*c*d*f*g-10*b*d^2*e*g)/((c*f-d*e)*d)^(1/2)*arc 
tan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))+b^2/(a*d-b*c)^6/f^5/(a*f-b*e)^5* 
(((297/64*b^5*a^3*c^2*d^2*f^4*h-55/32*b^6*a^2*c^3*d*f^4*h-2*b^8*c*d^3*e^3* 
f*g-231/32*b^4*a^4*c*d^3*f^4*h-5/8*a^3*b^5*c*d^3*e*f^3*h+39/4*a^2*b^6*c^2* 
d^2*e*f^3*h+3*a^2*b^6*c*d^3*e^2*f^2*h-189/8*a^2*b^6*c*d^3*e*f^3*g-47/8*a*b 
^7*c^3*d*e*f^3*h-57/8*a*b^7*c^2*d^2*e^2*f^2*h+99/8*a*b^7*c^2*d^2*e*f^3*g-a 
*b^7*c*d^3*e^3*f*h+45/4*a*b^7*c*d^3*e^2*f^2*g-187/128*b^8*c^4*f^4*g+647/32 
*a^3*b^5*c*d^3*f^4*g+109/8*a^3*b^5*d^4*e*f^3*g-1185/64*a^2*b^6*c^2*d^2*f^4 
*g-69/8*a^2*b^6*d^4*e^2*f^2*g+263/32*a*b^7*c^3*d*f^4*g+2*a*b^7*d^4*e^3*f*g 
+7/4*b^8*c^3*d*e^2*f^2*h-19/8*b^8*c^3*d*e*f^3*g+3/2*b^8*c^2*d^2*e^3*f*h-21 
/8*b^8*c^2*d^2*e^2*f^2*g+19/8*a^3*b^5*d^4*e^2*f^2*h-1/2*a^2*b^6*d^4*e^3*f* 
h-71/16*a^4*b^4*d^4*e*f^3*h+515/128*a^5*b^3*d^4*f^4*h+19/16*b^8*c^4*e*f^3* 
h+35/128*b^7*a*c^4*f^4*h-1083/128*a^4*d^4*f^4*g*b^4)*(f*x+e)^(7/2)+1/384*f 
*b^2*(5153*a^6*d^4*f^4*h-9652*a^5*b*c*d^3*f^4*h-10633*a^5*b*d^4*e*f^3*h-10 
633*a^5*b*d^4*f^4*g+6534*a^4*b^2*c^2*d^2*f^4*h+9284*a^4*b^2*c*d^3*e*f^3*h+ 
25908*a^4*b^2*c*d^3*f^4*g+8312*a^4*b^2*d^4*e^2*f^2*h+27257*a^4*b^2*d^4*e*f 
^3*g-2420*a^3*b^3*c^3*d*f^4*h+5274*a^3*b^3*c^2*d^2*e*f^3*h-24150*a^3*b^...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((h*x+g)/(b*x+a)^5/(d*x+c)^2/(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. 7412 vs. \(2 (1945) = 3890\).

Time = 0.85 (sec) , antiderivative size = 7412, normalized size of antiderivative = 3.72 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/64*(640*b^7*d^4*e^4*g + 768*b^7*c*d^3*e^3*f*g - 3328*a*b^6*d^4*e^3*f*g + 
 720*b^7*c^2*d^2*e^2*f^2*g - 3744*a*b^6*c*d^3*e^2*f^2*g + 6864*a^2*b^5*d^4 
*e^2*f^2*g + 560*b^7*c^3*d*e*f^3*g - 3120*a*b^6*c^2*d^2*e*f^3*g + 6864*a^2 
*b^5*c*d^3*e*f^3*g - 6864*a^3*b^4*d^4*e*f^3*g + 315*b^7*c^4*f^4*g - 1820*a 
*b^6*c^3*d*f^4*g + 4290*a^2*b^5*c^2*d^2*f^4*g - 5148*a^3*b^4*c*d^3*f^4*g + 
 3003*a^4*b^3*d^4*f^4*g - 512*b^7*c*d^3*e^4*h - 128*a*b^6*d^4*e^4*h - 576* 
b^7*c^2*d^2*e^3*f*h + 2432*a*b^6*c*d^3*e^3*f*h + 704*a^2*b^5*d^4*e^3*f*h - 
 480*b^7*c^3*d*e^2*f^2*h + 2448*a*b^6*c^2*d^2*e^2*f^2*h - 4224*a^2*b^5*c*d 
^3*e^2*f^2*h - 1584*a^3*b^4*d^4*e^2*f^2*h - 280*b^7*c^4*e*f^3*h + 1520*a*b 
^6*c^3*d*e*f^3*h - 3168*a^2*b^5*c^2*d^2*e*f^3*h + 2640*a^3*b^4*c*d^3*e*f^3 
*h + 1848*a^4*b^3*d^4*e*f^3*h - 35*a*b^6*c^4*f^4*h + 220*a^2*b^5*c^3*d*f^4 
*h - 594*a^3*b^4*c^2*d^2*f^4*h + 924*a^4*b^3*c*d^3*f^4*h - 1155*a^5*b^2*d^ 
4*f^4*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^11*c^6*e^5 - 6*a 
*b^10*c^5*d*e^5 + 15*a^2*b^9*c^4*d^2*e^5 - 20*a^3*b^8*c^3*d^3*e^5 + 15*a^4 
*b^7*c^2*d^4*e^5 - 6*a^5*b^6*c*d^5*e^5 + a^6*b^5*d^6*e^5 - 5*a*b^10*c^6*e^ 
4*f + 30*a^2*b^9*c^5*d*e^4*f - 75*a^3*b^8*c^4*d^2*e^4*f + 100*a^4*b^7*c^3* 
d^3*e^4*f - 75*a^5*b^6*c^2*d^4*e^4*f + 30*a^6*b^5*c*d^5*e^4*f - 5*a^7*b^4* 
d^6*e^4*f + 10*a^2*b^9*c^6*e^3*f^2 - 60*a^3*b^8*c^5*d*e^3*f^2 + 150*a^4*b^ 
7*c^4*d^2*e^3*f^2 - 200*a^5*b^6*c^3*d^3*e^3*f^2 + 150*a^6*b^5*c^2*d^4*e^3* 
f^2 - 60*a^7*b^4*c*d^5*e^3*f^2 + 10*a^8*b^3*d^6*e^3*f^2 - 10*a^3*b^8*c^...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 5.79 (sec) , antiderivative size = 72551, normalized size of antiderivative = 36.44 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3465*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b 
)*sqrt(a*f - b*e)))*a**9*b*c**4*d**4*f**7*h - 10395*sqrt(b)*sqrt(e + f*x)* 
sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c 
**3*d**5*e*f**6*h + 3465*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt( 
e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c**3*d**5*f**7*h*x + 10395*s 
qrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( 
a*f - b*e)))*a**9*b*c**2*d**6*e**2*f**5*h - 10395*sqrt(b)*sqrt(e + f*x)*sq 
rt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c** 
2*d**6*e*f**6*h*x - 3465*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt( 
e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*c*d**7*e**3*f**4*h + 10395*s 
qrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( 
a*f - b*e)))*a**9*b*c*d**7*e**2*f**5*h*x - 3465*sqrt(b)*sqrt(e + f*x)*sqrt 
(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*b*d**8* 
e**3*f**4*h*x - 2772*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + 
f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**8*b**2*c**5*d**3*f**7*h + 2772*sqrt( 
b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
- b*e)))*a**8*b**2*c**4*d**4*e*f**6*h - 9009*sqrt(b)*sqrt(e + f*x)*sqrt(a* 
f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**8*b**2*c**4* 
d**4*f**7*g + 11088*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f 
*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**8*b**2*c**4*d**4*f**7*h*x + 8316*s...