\(\int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx\) [184]

Optimal result

Integrand size = 29, antiderivative size = 1340 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Output:

-1/8*f*(8*a^4*d^3*f^3*(-c*f*h-2*d*e*h+3*d*f*g)+b^4*(32*d^4*e^4*g-c^2*d^2*e 
^2*f*(-18*e*h+13*f*g)-2*c^3*d*e*f^2*(-6*e*h+5*f*g)+5*c^4*f^3*(-6*e*h+7*f*g 
)-4*c*d^3*e^3*(6*e*h+5*f*g))-a*b^3*(5*c^4*f^4*h+2*c^3*d*f^3*(-53*e*h+65*f* 
g)-7*c^2*d^2*e*f^2*(-11*e*h+8*f*g)+4*d^4*e^3*(2*e*h+27*f*g)-2*c*d^3*e^2*f* 
(40*e*h+43*f*g))-a^3*b*d^2*f^2*(41*c^2*f^2*h+2*c*d*f*(-89*e*h+32*f*g)+d^2* 
e*(41*e*h+32*f*g))+a^2*b^2*d*f*(22*c^3*f^3*h+c^2*d*f^2*(-110*e*h+167*f*g)+ 
d^3*e^2*(30*e*h+119*f*g)-2*c*d^2*e*f*(43*e*h+71*f*g)))/(-a*d+b*c)^4/(-a*f+ 
b*e)^4/(-c*f+d*e)^2/(f*x+e)^(1/2)+1/24*d*(a^3*d^2*f^2*(-113*c*f*h+89*d*e*h 
+24*d*f*g)-b^3*(96*d^3*e^3*g-c^2*d*e*f*(-18*e*h+25*f*g)-5*c^3*f^2*(-6*e*h+ 
7*f*g)-12*c*d^2*e^2*(6*e*h+f*g))-a*b^2*(5*c^3*f^3*h+c^2*d*f^2*(-101*e*h+13 
0*f*g)-12*d^3*e^2*(2*e*h+23*f*g)+2*c*d^2*e*f*(96*e*h+37*f*g))+a^2*b*d*f*(2 
2*c^2*f^2*h-d^2*e*(78*e*h+239*f*g)+c*d*f*(128*e*h+167*f*g)))/(-a*d+b*c)^4/ 
(-a*f+b*e)^3/(-c*f+d*e)/(d*x+c)/(f*x+e)^(1/2)-1/3*(-a*h+b*g)/(-a*d+b*c)/(- 
a*f+b*e)/(b*x+a)^3/(d*x+c)/(f*x+e)^(1/2)+1/12*(9*a^2*d*f*h+b^2*(-6*c*e*h+7 
*c*f*g+8*d*e*g)-a*b*(c*f*h+2*d*e*h+15*d*f*g))/(-a*d+b*c)^2/(-a*f+b*e)^2/(b 
*x+a)^2/(d*x+c)/(f*x+e)^(1/2)+1/24*(63*a^3*d^2*f^2*h-a^2*b*d*f*(20*c*f*h+4 
0*d*e*h+129*d*f*g)-b^3*(48*d^2*e^2*g+2*c*d*e*(-18*e*h+23*f*g)+5*c^2*f*(-6* 
e*h+7*f*g))+a*b^2*(5*c^2*f^2*h+2*c*d*f*(-43*e*h+58*f*g)+2*d^2*e*(6*e*h+71* 
f*g)))/(-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)/(d*x+c)/(f*x+e)^(1/2)+1/8*b^(3/2) 
*(105*a^4*d^3*f^3*h-21*a^3*b*d^2*f^2*(3*c*f*h+6*d*e*h+11*d*f*g)+b^4*(64...
 

Mathematica [C] (verified)

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

Time = 11.65 (sec) , antiderivative size = 902, normalized size of antiderivative = 0.67 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {-\frac {8 (b c-a d) (b e-a f) (b g-a h)}{(a+b x)^3}+\frac {2 \left (9 a^2 d f h+b^2 (8 d e g+7 c f g-6 c e h)-a b (15 d f g+2 d e h+c f h)\right )}{(a+b x)^2}+\frac {63 a^3 d^2 f^2 h-a^2 b d f (129 d f g+40 d e h+20 c f h)+a b^2 \left (5 c^2 f^2 h+2 c d f (58 f g-43 e h)+2 d^2 e (71 f g+6 e h)\right )+b^3 \left (-48 d^2 e^2 g+5 c^2 f (-7 f g+6 e h)+2 c d e (-23 f g+18 e h)\right )}{(b c-a d) (b e-a f) (a+b x)}+\frac {d (b c-a d) (b e-a f) (-d e+c f) \left (a^3 d^2 f^2 (-24 d f g-89 d e h+113 c f h)+b^3 \left (96 d^3 e^3 g+5 c^3 f^2 (-7 f g+6 e h)-12 c d^2 e^2 (f g+6 e h)+c^2 d e f (-25 f g+18 e h)\right )+a b^2 \left (5 c^3 f^3 h+c^2 d f^2 (130 f g-101 e h)-12 d^3 e^2 (23 f g+2 e h)+2 c d^2 e f (37 f g+96 e h)\right )+a^2 b d f \left (-22 c^2 f^2 h+d^2 e (239 f g+78 e h)-c d f (167 f g+128 e h)\right )\right )-3 (c+d x) \left (b (d e-c f)^2 \left (105 a^4 d^3 f^3 h-21 a^3 b d^2 f^2 (11 d f g+6 d e h+3 c f h)+b^4 \left (64 d^3 e^3 g+5 c^3 f^2 (7 f g-6 e h)+12 c^2 d e f (5 f g-4 e h)+24 c d^2 e^2 (3 f g-2 e h)\right )+9 a^2 b^2 d f \left (3 c^2 f^2 h+3 c d f (11 f g-6 e h)+4 d^2 e (11 f g+2 e h)\right )-a b^3 \left (5 c^3 f^3 h+3 c^2 d f^2 (55 f g-42 e h)+8 d^3 e^2 (33 f g+2 e h)-24 c d^2 e f (-11 f g+7 e h)\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {b (e+f x)}{b e-a f}\right )-8 d^3 (b e-a f)^4 \left (a d (3 d f g-2 d e h-c f h)+b \left (8 d^2 e g+9 c^2 f h-c d (11 f g+6 e h)\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {d (e+f x)}{d e-c f}\right )\right )}{(b c-a d)^3 (b e-a f)^2 (d e-c f)^2}}{24 (b c-a d)^2 (b e-a f)^2 (c+d x) \sqrt {e+f x}} \] Input:

Integrate[(g + h*x)/((a + b*x)^4*(c + d*x)^2*(e + f*x)^(3/2)),x]
 

Output:

((-8*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))/(a + b*x)^3 + (2*(9*a^2*d*f*h + 
b^2*(8*d*e*g + 7*c*f*g - 6*c*e*h) - a*b*(15*d*f*g + 2*d*e*h + c*f*h)))/(a 
+ b*x)^2 + (63*a^3*d^2*f^2*h - a^2*b*d*f*(129*d*f*g + 40*d*e*h + 20*c*f*h) 
 + a*b^2*(5*c^2*f^2*h + 2*c*d*f*(58*f*g - 43*e*h) + 2*d^2*e*(71*f*g + 6*e* 
h)) + b^3*(-48*d^2*e^2*g + 5*c^2*f*(-7*f*g + 6*e*h) + 2*c*d*e*(-23*f*g + 1 
8*e*h)))/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + (d*(b*c - a*d)*(b*e - a*f)* 
(-(d*e) + c*f)*(a^3*d^2*f^2*(-24*d*f*g - 89*d*e*h + 113*c*f*h) + b^3*(96*d 
^3*e^3*g + 5*c^3*f^2*(-7*f*g + 6*e*h) - 12*c*d^2*e^2*(f*g + 6*e*h) + c^2*d 
*e*f*(-25*f*g + 18*e*h)) + a*b^2*(5*c^3*f^3*h + c^2*d*f^2*(130*f*g - 101*e 
*h) - 12*d^3*e^2*(23*f*g + 2*e*h) + 2*c*d^2*e*f*(37*f*g + 96*e*h)) + a^2*b 
*d*f*(-22*c^2*f^2*h + d^2*e*(239*f*g + 78*e*h) - c*d*f*(167*f*g + 128*e*h) 
)) - 3*(c + d*x)*(b*(d*e - c*f)^2*(105*a^4*d^3*f^3*h - 21*a^3*b*d^2*f^2*(1 
1*d*f*g + 6*d*e*h + 3*c*f*h) + b^4*(64*d^3*e^3*g + 5*c^3*f^2*(7*f*g - 6*e* 
h) + 12*c^2*d*e*f*(5*f*g - 4*e*h) + 24*c*d^2*e^2*(3*f*g - 2*e*h)) + 9*a^2* 
b^2*d*f*(3*c^2*f^2*h + 3*c*d*f*(11*f*g - 6*e*h) + 4*d^2*e*(11*f*g + 2*e*h) 
) - a*b^3*(5*c^3*f^3*h + 3*c^2*d*f^2*(55*f*g - 42*e*h) + 8*d^3*e^2*(33*f*g 
 + 2*e*h) - 24*c*d^2*e*f*(-11*f*g + 7*e*h)))*Hypergeometric2F1[-1/2, 1, 1/ 
2, (b*(e + f*x))/(b*e - a*f)] - 8*d^3*(b*e - a*f)^4*(a*d*(3*d*f*g - 2*d*e* 
h - c*f*h) + b*(8*d^2*e*g + 9*c^2*f*h - c*d*(11*f*g + 6*e*h)))*Hypergeomet 
ric2F1[-1/2, 1, 1/2, (d*(e + f*x))/(d*e - c*f)]))/((b*c - a*d)^3*(b*e -...
 

Rubi [A] (verified)

Time = 3.78 (sec) , antiderivative size = 1452, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {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.

\(\displaystyle \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\int \frac {b (8 d e g+7 c f g-6 c e h)-a (6 d f g+2 d e h+c f h)+9 d f (b g-a h) x}{2 (a+b x)^3 (c+d x)^2 (e+f x)^{3/2}}dx}{3 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {b (8 d e g+7 c f g-6 c e h)-a (6 d f g+2 d e h+c f h)+9 d f (b g-a h) x}{(a+b x)^3 (c+d x)^2 (e+f x)^{3/2}}dx}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {d f (24 d f g+26 d e h+13 c f h) a^2-b \left (2 e (43 f g+6 e h) d^2+c f (67 f g-44 e h) d+5 c^2 f^2 h\right ) a+b^2 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 g\right )+7 d f \left (9 d f h a^2-b (15 d f g+2 d e h+c f h) a+b^2 (8 d e g+7 c f g-6 c e h)\right ) x}{2 (a+b x)^2 (c+d x)^2 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}-\frac {9 a^2 d f h-a b (c f h+2 d e h+15 d f g)+b^2 (-6 c e h+7 c f g+8 d e g)}{2 (a+b x)^2 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {d f (24 d f g+26 d e h+13 c f h) a^2-b \left (2 e (43 f g+6 e h) d^2+c f (67 f g-44 e h) d+5 c^2 f^2 h\right ) a+b^2 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 g\right )+7 d f \left (9 d f h a^2-b (15 d f g+2 d e h+c f h) a+b^2 (8 d e g+7 c f g-6 c e h)\right ) x}{(a+b x)^2 (c+d x)^2 (e+f x)^{3/2}}dx}{4 (b c-a d) (b e-a f)}-\frac {9 a^2 d f h-a b (c f h+2 d e h+15 d f g)+b^2 (-6 c e h+7 c f g+8 d e g)}{2 (a+b x)^2 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {63 a^3 d^2 f^2 h-a^2 b d f (20 c f h+40 d e h+129 d f g)+a b^2 \left (5 c^2 f^2 h+2 c d f (58 f g-43 e h)+2 d^2 e (6 e h+71 f g)\right )-b^3 \left (5 c^2 f (7 f g-6 e h)+2 c d e (23 f g-18 e h)+48 d^2 e^2 g\right )}{(a+b x) (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}-\frac {\int -\frac {d^2 f^2 (48 d f g+178 d e h+89 c f h) a^3-b d f \left (2 e (239 f g+78 e h) d^2+c f (311 f g-56 e h) d+56 c^2 f^2 h\right ) a^2+b^2 \left (24 e^2 (23 f g+2 e h) d^3+2 c e f (281 f g-162 e h) d^2+4 c^2 f^2 (80 f g-57 e h) d+15 c^3 f^3 h\right ) a-3 b^3 \left (5 f^2 (7 f g-6 e h) c^3+12 d e f (5 f g-4 e h) c^2+24 d^2 e^2 (3 f g-2 e h) c+64 d^3 e^3 g\right )+5 d f \left (63 d^2 f^2 h a^3-b d f (129 d f g+40 d e h+20 c f h) a^2+b^2 \left (2 e (71 f g+6 e h) d^2+2 c f (58 f g-43 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 g\right )\right ) x}{2 (a+b x) (c+d x)^2 (e+f x)^{3/2}}dx}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {9 a^2 d f h-a b (c f h+2 d e h+15 d f g)+b^2 (-6 c e h+7 c f g+8 d e g)}{2 (a+b x)^2 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\int \frac {d^2 f^2 (48 d f g+178 d e h+89 c f h) a^3-b d f \left (2 e (239 f g+78 e h) d^2+c f (311 f g-56 e h) d+56 c^2 f^2 h\right ) a^2+b^2 \left (24 e^2 (23 f g+2 e h) d^3+2 c e f (281 f g-162 e h) d^2+4 c^2 f^2 (80 f g-57 e h) d+15 c^3 f^3 h\right ) a-3 b^3 \left (5 f^2 (7 f g-6 e h) c^3+12 d e f (5 f g-4 e h) c^2+24 d^2 e^2 (3 f g-2 e h) c+64 d^3 e^3 g\right )+5 d f \left (63 d^2 f^2 h a^3-b d f (129 d f g+40 d e h+20 c f h) a^2+b^2 \left (2 e (71 f g+6 e h) d^2+2 c f (58 f g-43 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^2 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}+\frac {63 a^3 d^2 f^2 h-a^2 b d f (20 c f h+40 d e h+129 d f g)+a b^2 \left (5 c^2 f^2 h+2 c d f (58 f g-43 e h)+2 d^2 e (6 e h+71 f g)\right )-b^3 \left (5 c^2 f (7 f g-6 e h)+2 c d e (23 f g-18 e h)+48 d^2 e^2 g\right )}{(a+b x) (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {9 a^2 d f h-a b (c f h+2 d e h+15 d f g)+b^2 (-6 c e h+7 c f g+8 d e g)}{2 (a+b x)^2 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x) \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (15 d f g+2 d e h+c f h) a+b^2 (8 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}-\frac {\frac {63 d^2 f^2 h a^3-b d f (129 d f g+40 d e h+20 c f h) a^2+b^2 \left (2 e (71 f g+6 e h) d^2+2 c f (58 f g-43 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\frac {2 d \left (d^2 f^2 (24 d f g+89 d e h-113 c f h) a^3+b d f \left (-e (239 f g+78 e h) d^2+c f (167 f g+128 e h) d+22 c^2 f^2 h\right ) a^2-b^2 \left (-12 e^2 (23 f g+2 e h) d^3+2 c e f (37 f g+96 e h) d^2+c^2 f^2 (130 f g-101 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (25 f g-18 e h) c^2-12 d^2 e^2 (f g+6 e h) c+96 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {\int \frac {3 \left (8 d^3 f^3 (3 d f g-2 d e h-c f h) a^4-b d^2 f^2 \left (8 e (f g-6 e h) d^2+c f (64 f g-65 e h) d+41 c^2 f^2 h\right ) a^3+b^2 d f \left (-24 e^2 (5 f g+2 e h) d^3+c e f (25 f g+42 e h) d^2+c^2 f^2 (167 f g-88 e h) d+22 c^3 f^3 h\right ) a^2-b^3 \left (-8 e^3 (21 f g+2 e h) d^4-4 c e^2 f (3 f g-28 e h) d^3+2 c^2 e f^2 (37 f g-12 e h) d^2+c^3 f^3 (130 f g-101 e h) d+5 c^4 f^4 h\right ) a-b^4 (d e-c f) \left (5 f^2 (7 f g-6 e h) c^3+12 d e f (5 f g-4 e h) c^2+24 d^2 e^2 (3 f g-2 e h) c+64 d^3 e^3 g\right )+b d f \left (d^2 f^2 (24 d f g+89 d e h-113 c f h) a^3+b d f \left (-e (239 f g+78 e h) d^2+c f (167 f g+128 e h) d+22 c^2 f^2 h\right ) a^2-b^2 \left (-12 e^2 (23 f g+2 e h) d^3+2 c e f (37 f g+96 e h) d^2+c^2 f^2 (130 f g-101 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (25 f g-18 e h) c^2-12 d^2 e^2 (f g+6 e h) c+96 d^3 e^3 g\right )\right ) x\right )}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x) \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (15 d f g+2 d e h+c f h) a+b^2 (8 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}-\frac {\frac {63 d^2 f^2 h a^3-b d f (129 d f g+40 d e h+20 c f h) a^2+b^2 \left (2 e (71 f g+6 e h) d^2+2 c f (58 f g-43 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}+\frac {\frac {2 d \left (d^2 f^2 (24 d f g+89 d e h-113 c f h) a^3+b d f \left (-e (239 f g+78 e h) d^2+c f (167 f g+128 e h) d+22 c^2 f^2 h\right ) a^2-b^2 \left (-12 e^2 (23 f g+2 e h) d^3+2 c e f (37 f g+96 e h) d^2+c^2 f^2 (130 f g-101 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (25 f g-18 e h) c^2-12 d^2 e^2 (f g+6 e h) c+96 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \int \frac {8 d^3 f^3 (3 d f g-2 d e h-c f h) a^4-b d^2 f^2 \left (8 e (f g-6 e h) d^2+c f (64 f g-65 e h) d+41 c^2 f^2 h\right ) a^3+b^2 d f \left (-24 e^2 (5 f g+2 e h) d^3+c e f (25 f g+42 e h) d^2+c^2 f^2 (167 f g-88 e h) d+22 c^3 f^3 h\right ) a^2-b^3 \left (-8 e^3 (21 f g+2 e h) d^4-4 c e^2 f (3 f g-28 e h) d^3+2 c^2 e f^2 (37 f g-12 e h) d^2+c^3 f^3 (130 f g-101 e h) d+5 c^4 f^4 h\right ) a-b^4 (d e-c f) \left (5 f^2 (7 f g-6 e h) c^3+12 d e f (5 f g-4 e h) c^2+24 d^2 e^2 (3 f g-2 e h) c+64 d^3 e^3 g\right )+b d f \left (d^2 f^2 (24 d f g+89 d e h-113 c f h) a^3+b d f \left (-e (239 f g+78 e h) d^2+c f (167 f g+128 e h) d+22 c^2 f^2 h\right ) a^2-b^2 \left (-12 e^2 (23 f g+2 e h) d^3+2 c e f (37 f g+96 e h) d^2+c^2 f^2 (130 f g-101 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (25 f g-18 e h) c^2-12 d^2 e^2 (f g+6 e h) c+96 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x) \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (15 d f g+2 d e h+c f h) a+b^2 (8 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}-\frac {\frac {63 d^2 f^2 h a^3-b d f (129 d f g+40 d e h+20 c f h) a^2+b^2 \left (2 e (71 f g+6 e h) d^2+2 c f (58 f g-43 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 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^2 f^2 (24 d f g+89 d e h-113 c f h) a^3+b d f \left (-e (239 f g+78 e h) d^2+c f (167 f g+128 e h) d+22 c^2 f^2 h\right ) a^2-b^2 \left (-12 e^2 (23 f g+2 e h) d^3+2 c e f (37 f g+96 e h) d^2+c^2 f^2 (130 f g-101 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (25 f g-18 e h) c^2-12 d^2 e^2 (f g+6 e h) c+96 d^3 e^3 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 (8 d^3 f^3 (3 d f g-2 d e h-c f h) a^4-b d^2 f^2 \left (e (32 f g+41 e h) d^2+2 c f (32 f g-89 e h) d+41 c^2 f^2 h\right ) a^3+b^2 d f \left (e^2 (119 f g+30 e h) d^3-2 c e f (71 f g+43 e h) d^2+c^2 f^2 (167 f g-110 e h) d+22 c^3 f^3 h\right ) a^2-b^3 \left (4 e^3 (27 f g+2 e h) d^4-2 c e^2 f (43 f g+40 e h) d^3-7 c^2 e f^2 (8 f g-11 e h) d^2+2 c^3 f^3 (65 f g-53 e h) d+5 c^4 f^4 h\right ) a+b^4 \left (5 f^3 (7 f g-6 e h) c^4-2 d e f^2 (5 f g-6 e h) c^3-d^2 e^2 f (13 f g-18 e h) c^2-4 d^3 e^3 (5 f g+6 e h) c+32 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 \int \frac {8 d^4 f^4 (3 d f g-2 d e h-c f h) a^5+32 b d^3 f^3 \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^4-b^2 d^2 f^2 \left (16 e^2 (7 f g+6 e h) d^3-c e f (320 f g+103 e h) d^2+2 c^2 f^2 (32 f g+55 e h) d+41 c^3 f^3 h\right ) a^3+b^3 d f \left (32 e^3 (9 f g+2 e h) d^4-c e^2 f (409 f g+226 e h) d^3-2 c^2 e f^2 (71 f g-173 e h) d^2+c^3 f^3 (167 f g-110 e h) d+22 c^4 f^4 h\right ) a^2-b^4 \left (8 e^4 (29 f g+2 e h) d^5-4 c e^3 f (61 f g+44 e h) d^4-2 c^2 e^2 f^2 (43 f g-104 e h) d^3-7 c^3 e f^3 (8 f g-11 e h) d^2+2 c^4 f^4 (65 f g-53 e h) d+5 c^5 f^5 h\right ) a+b^5 (d e-c f)^2 \left (5 f^2 (7 f g-6 e h) c^3+12 d e f (5 f g-4 e h) c^2+24 d^2 e^2 (3 f g-2 e h) c+64 d^3 e^3 g\right )+b d f \left (8 d^3 f^3 (3 d f g-2 d e h-c f h) a^4-b d^2 f^2 \left (e (32 f g+41 e h) d^2+2 c f (32 f g-89 e h) d+41 c^2 f^2 h\right ) a^3+b^2 d f \left (e^2 (119 f g+30 e h) d^3-2 c e f (71 f g+43 e h) d^2+c^2 f^2 (167 f g-110 e h) d+22 c^3 f^3 h\right ) a^2-b^3 \left (4 e^3 (27 f g+2 e h) d^4-2 c e^2 f (43 f g+40 e h) d^3-7 c^2 e f^2 (8 f g-11 e h) d^2+2 c^3 f^3 (65 f g-53 e h) d+5 c^4 f^4 h\right ) a+b^4 \left (5 f^3 (7 f g-6 e h) c^4-2 d e f^2 (5 f g-6 e h) c^3-d^2 e^2 f (13 f g-18 e h) c^2-4 d^3 e^3 (5 f g+6 e h) c+32 d^4 e^4 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)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x) \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (15 d f g+2 d e h+c f h) a+b^2 (8 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}-\frac {\frac {63 d^2 f^2 h a^3-b d f (129 d f g+40 d e h+20 c f h) a^2+b^2 \left (2 e (71 f g+6 e h) d^2+2 c f (58 f g-43 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 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^2 f^2 (24 d f g+89 d e h-113 c f h) a^3+b d f \left (-e (239 f g+78 e h) d^2+c f (167 f g+128 e h) d+22 c^2 f^2 h\right ) a^2-b^2 \left (-12 e^2 (23 f g+2 e h) d^3+2 c e f (37 f g+96 e h) d^2+c^2 f^2 (130 f g-101 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (25 f g-18 e h) c^2-12 d^2 e^2 (f g+6 e h) c+96 d^3 e^3 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 (8 d^3 f^3 (3 d f g-2 d e h-c f h) a^4-b d^2 f^2 \left (e (32 f g+41 e h) d^2+2 c f (32 f g-89 e h) d+41 c^2 f^2 h\right ) a^3+b^2 d f \left (e^2 (119 f g+30 e h) d^3-2 c e f (71 f g+43 e h) d^2+c^2 f^2 (167 f g-110 e h) d+22 c^3 f^3 h\right ) a^2-b^3 \left (4 e^3 (27 f g+2 e h) d^4-2 c e^2 f (43 f g+40 e h) d^3-7 c^2 e f^2 (8 f g-11 e h) d^2+2 c^3 f^3 (65 f g-53 e h) d+5 c^4 f^4 h\right ) a+b^4 \left (5 f^3 (7 f g-6 e h) c^4-2 d e f^2 (5 f g-6 e h) c^3-d^2 e^2 f (13 f g-18 e h) c^2-4 d^3 e^3 (5 f g+6 e h) c+32 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\int \frac {8 d^4 f^4 (3 d f g-2 d e h-c f h) a^5+32 b d^3 f^3 \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^4-b^2 d^2 f^2 \left (16 e^2 (7 f g+6 e h) d^3-c e f (320 f g+103 e h) d^2+2 c^2 f^2 (32 f g+55 e h) d+41 c^3 f^3 h\right ) a^3+b^3 d f \left (32 e^3 (9 f g+2 e h) d^4-c e^2 f (409 f g+226 e h) d^3-2 c^2 e f^2 (71 f g-173 e h) d^2+c^3 f^3 (167 f g-110 e h) d+22 c^4 f^4 h\right ) a^2-b^4 \left (8 e^4 (29 f g+2 e h) d^5-4 c e^3 f (61 f g+44 e h) d^4-2 c^2 e^2 f^2 (43 f g-104 e h) d^3-7 c^3 e f^3 (8 f g-11 e h) d^2+2 c^4 f^4 (65 f g-53 e h) d+5 c^5 f^5 h\right ) a+b^5 (d e-c f)^2 \left (5 f^2 (7 f g-6 e h) c^3+12 d e f (5 f g-4 e h) c^2+24 d^2 e^2 (3 f g-2 e h) c+64 d^3 e^3 g\right )+b d f \left (8 d^3 f^3 (3 d f g-2 d e h-c f h) a^4-b d^2 f^2 \left (e (32 f g+41 e h) d^2+2 c f (32 f g-89 e h) d+41 c^2 f^2 h\right ) a^3+b^2 d f \left (e^2 (119 f g+30 e h) d^3-2 c e f (71 f g+43 e h) d^2+c^2 f^2 (167 f g-110 e h) d+22 c^3 f^3 h\right ) a^2-b^3 \left (4 e^3 (27 f g+2 e h) d^4-2 c e^2 f (43 f g+40 e h) d^3-7 c^2 e f^2 (8 f g-11 e h) d^2+2 c^3 f^3 (65 f g-53 e h) d+5 c^4 f^4 h\right ) a+b^4 \left (5 f^3 (7 f g-6 e h) c^4-2 d e f^2 (5 f g-6 e h) c^3-d^2 e^2 f (13 f g-18 e h) c^2-4 d^3 e^3 (5 f g+6 e h) c+32 d^4 e^4 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)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x) \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (15 d f g+2 d e h+c f h) a+b^2 (8 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}-\frac {\frac {63 d^2 f^2 h a^3-b d f (129 d f g+40 d e h+20 c f h) a^2+b^2 \left (2 e (71 f g+6 e h) d^2+2 c f (58 f g-43 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 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^2 f^2 (24 d f g+89 d e h-113 c f h) a^3+b d f \left (-e (239 f g+78 e h) d^2+c f (167 f g+128 e h) d+22 c^2 f^2 h\right ) a^2-b^2 \left (-12 e^2 (23 f g+2 e h) d^3+2 c e f (37 f g+96 e h) d^2+c^2 f^2 (130 f g-101 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (25 f g-18 e h) c^2-12 d^2 e^2 (f g+6 e h) c+96 d^3 e^3 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 (8 d^3 f^3 (3 d f g-2 d e h-c f h) a^4-b d^2 f^2 \left (e (32 f g+41 e h) d^2+2 c f (32 f g-89 e h) d+41 c^2 f^2 h\right ) a^3+b^2 d f \left (e^2 (119 f g+30 e h) d^3-2 c e f (71 f g+43 e h) d^2+c^2 f^2 (167 f g-110 e h) d+22 c^3 f^3 h\right ) a^2-b^3 \left (4 e^3 (27 f g+2 e h) d^4-2 c e^2 f (43 f g+40 e h) d^3-7 c^2 e f^2 (8 f g-11 e h) d^2+2 c^3 f^3 (65 f g-53 e h) d+5 c^4 f^4 h\right ) a+b^4 \left (5 f^3 (7 f g-6 e h) c^4-2 d e f^2 (5 f g-6 e h) c^3-d^2 e^2 f (13 f g-18 e h) c^2-4 d^3 e^3 (5 f g+6 e h) c+32 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {b^2 (d e-c f)^2 \left (105 d^3 f^3 h a^4-21 b d^2 f^2 (11 d f g+6 d e h+3 c f h) a^3+9 b^2 d f \left (4 e (11 f g+2 e h) d^2+3 c f (11 f g-6 e h) d+3 c^2 f^2 h\right ) a^2-b^3 \left (8 e^2 (33 f g+2 e h) d^3+24 c e f (11 f g-7 e h) d^2+3 c^2 f^2 (55 f g-42 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (5 f^2 (7 f g-6 e h) c^3+12 d e f (5 f g-4 e h) c^2+24 d^2 e^2 (3 f g-2 e h) c+64 d^3 e^3 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {8 d^4 (b e-a f)^4 \left (a d (3 d f g-2 d e h-c f h)+b \left (9 f h c^2-d (11 f g+6 e h) c+8 d^2 e g\right )\right ) \int \frac {1}{(c+d 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)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x) \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (15 d f g+2 d e h+c f h) a+b^2 (8 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}-\frac {\frac {63 d^2 f^2 h a^3-b d f (129 d f g+40 d e h+20 c f h) a^2+b^2 \left (2 e (71 f g+6 e h) d^2+2 c f (58 f g-43 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 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^2 f^2 (24 d f g+89 d e h-113 c f h) a^3+b d f \left (-e (239 f g+78 e h) d^2+c f (167 f g+128 e h) d+22 c^2 f^2 h\right ) a^2-b^2 \left (-12 e^2 (23 f g+2 e h) d^3+2 c e f (37 f g+96 e h) d^2+c^2 f^2 (130 f g-101 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (25 f g-18 e h) c^2-12 d^2 e^2 (f g+6 e h) c+96 d^3 e^3 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 (8 d^3 f^3 (3 d f g-2 d e h-c f h) a^4-b d^2 f^2 \left (e (32 f g+41 e h) d^2+2 c f (32 f g-89 e h) d+41 c^2 f^2 h\right ) a^3+b^2 d f \left (e^2 (119 f g+30 e h) d^3-2 c e f (71 f g+43 e h) d^2+c^2 f^2 (167 f g-110 e h) d+22 c^3 f^3 h\right ) a^2-b^3 \left (4 e^3 (27 f g+2 e h) d^4-2 c e^2 f (43 f g+40 e h) d^3-7 c^2 e f^2 (8 f g-11 e h) d^2+2 c^3 f^3 (65 f g-53 e h) d+5 c^4 f^4 h\right ) a+b^4 \left (5 f^3 (7 f g-6 e h) c^4-2 d e f^2 (5 f g-6 e h) c^3-d^2 e^2 f (13 f g-18 e h) c^2-4 d^3 e^3 (5 f g+6 e h) c+32 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {2 b^2 (d e-c f)^2 \left (105 d^3 f^3 h a^4-21 b d^2 f^2 (11 d f g+6 d e h+3 c f h) a^3+9 b^2 d f \left (4 e (11 f g+2 e h) d^2+3 c f (11 f g-6 e h) d+3 c^2 f^2 h\right ) a^2-b^3 \left (8 e^2 (33 f g+2 e h) d^3+24 c e f (11 f g-7 e h) d^2+3 c^2 f^2 (55 f g-42 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (5 f^2 (7 f g-6 e h) c^3+12 d e f (5 f g-4 e h) c^2+24 d^2 e^2 (3 f g-2 e h) c+64 d^3 e^3 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}-\frac {16 d^4 (b e-a f)^4 \left (a d (3 d f g-2 d e h-c f h)+b \left (9 f h c^2-d (11 f g+6 e h) c+8 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 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)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x) \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (15 d f g+2 d e h+c f h) a+b^2 (8 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}-\frac {\frac {63 d^2 f^2 h a^3-b d f (129 d f g+40 d e h+20 c f h) a^2+b^2 \left (2 e (71 f g+6 e h) d^2+2 c f (58 f g-43 e h) d+5 c^2 f^2 h\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+2 d e (23 f g-18 e h) c+48 d^2 e^2 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^2 f^2 (24 d f g+89 d e h-113 c f h) a^3+b d f \left (-e (239 f g+78 e h) d^2+c f (167 f g+128 e h) d+22 c^2 f^2 h\right ) a^2-b^2 \left (-12 e^2 (23 f g+2 e h) d^3+2 c e f (37 f g+96 e h) d^2+c^2 f^2 (130 f g-101 e h) d+5 c^3 f^3 h\right ) a-b^3 \left (-5 f^2 (7 f g-6 e h) c^3-d e f (25 f g-18 e h) c^2-12 d^2 e^2 (f g+6 e h) c+96 d^3 e^3 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 (8 d^3 f^3 (3 d f g-2 d e h-c f h) a^4-b d^2 f^2 \left (e (32 f g+41 e h) d^2+2 c f (32 f g-89 e h) d+41 c^2 f^2 h\right ) a^3+b^2 d f \left (e^2 (119 f g+30 e h) d^3-2 c e f (71 f g+43 e h) d^2+c^2 f^2 (167 f g-110 e h) d+22 c^3 f^3 h\right ) a^2-b^3 \left (4 e^3 (27 f g+2 e h) d^4-2 c e^2 f (43 f g+40 e h) d^3-7 c^2 e f^2 (8 f g-11 e h) d^2+2 c^3 f^3 (65 f g-53 e h) d+5 c^4 f^4 h\right ) a+b^4 \left (5 f^3 (7 f g-6 e h) c^4-2 d e f^2 (5 f g-6 e h) c^3-d^2 e^2 f (13 f g-18 e h) c^2-4 d^3 e^3 (5 f g+6 e h) c+32 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {16 d^{7/2} (b e-a f)^4 \left (a d (3 d f g-2 d e h-c f h)+b \left (9 f h c^2-d (11 f g+6 e h) c+8 d^2 e g\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d) \sqrt {d e-c f}}-\frac {2 b^{3/2} (d e-c f)^2 \left (105 d^3 f^3 h a^4-21 b d^2 f^2 (11 d f g+6 d e h+3 c f h) a^3+9 b^2 d f \left (4 e (11 f g+2 e h) d^2+3 c f (11 f g-6 e h) d+3 c^2 f^2 h\right ) a^2-b^3 \left (8 e^2 (33 f g+2 e h) d^3+24 c e f (11 f g-7 e h) d^2+3 c^2 f^2 (55 f g-42 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (5 f^2 (7 f g-6 e h) c^3+12 d e f (5 f g-4 e h) c^2+24 d^2 e^2 (3 f g-2 e h) c+64 d^3 e^3 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)}\)

Input:

Int[(g + h*x)/((a + b*x)^4*(c + d*x)^2*(e + f*x)^(3/2)),x]
 

Output:

-1/3*(b*g - a*h)/((b*c - a*d)*(b*e - a*f)*(a + b*x)^3*(c + d*x)*Sqrt[e + f 
*x]) - (-1/2*(9*a^2*d*f*h + b^2*(8*d*e*g + 7*c*f*g - 6*c*e*h) - a*b*(15*d* 
f*g + 2*d*e*h + c*f*h))/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)*Sqr 
t[e + f*x]) - ((63*a^3*d^2*f^2*h - a^2*b*d*f*(129*d*f*g + 40*d*e*h + 20*c* 
f*h) - b^3*(48*d^2*e^2*g + 2*c*d*e*(23*f*g - 18*e*h) + 5*c^2*f*(7*f*g - 6* 
e*h)) + a*b^2*(5*c^2*f^2*h + 2*c*d*f*(58*f*g - 43*e*h) + 2*d^2*e*(71*f*g + 
 6*e*h)))/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)*Sqrt[e + f*x]) + (( 
2*d*(a^3*d^2*f^2*(24*d*f*g + 89*d*e*h - 113*c*f*h) - b^3*(96*d^3*e^3*g - c 
^2*d*e*f*(25*f*g - 18*e*h) - 5*c^3*f^2*(7*f*g - 6*e*h) - 12*c*d^2*e^2*(f*g 
 + 6*e*h)) - a*b^2*(5*c^3*f^3*h + c^2*d*f^2*(130*f*g - 101*e*h) - 12*d^3*e 
^2*(23*f*g + 2*e*h) + 2*c*d^2*e*f*(37*f*g + 96*e*h)) + a^2*b*d*f*(22*c^2*f 
^2*h - d^2*e*(239*f*g + 78*e*h) + c*d*f*(167*f*g + 128*e*h))))/((b*c - a*d 
)*(d*e - c*f)*(c + d*x)*Sqrt[e + f*x]) + (3*((-2*f*(8*a^4*d^3*f^3*(3*d*f*g 
 - 2*d*e*h - c*f*h) + b^4*(32*d^4*e^4*g - c^2*d^2*e^2*f*(13*f*g - 18*e*h) 
- 2*c^3*d*e*f^2*(5*f*g - 6*e*h) + 5*c^4*f^3*(7*f*g - 6*e*h) - 4*c*d^3*e^3* 
(5*f*g + 6*e*h)) - a*b^3*(5*c^4*f^4*h + 2*c^3*d*f^3*(65*f*g - 53*e*h) - 7* 
c^2*d^2*e*f^2*(8*f*g - 11*e*h) + 4*d^4*e^3*(27*f*g + 2*e*h) - 2*c*d^3*e^2* 
f*(43*f*g + 40*e*h)) - a^3*b*d^2*f^2*(41*c^2*f^2*h + 2*c*d*f*(32*f*g - 89* 
e*h) + d^2*e*(32*f*g + 41*e*h)) + a^2*b^2*d*f*(22*c^3*f^3*h + c^2*d*f^2*(1 
67*f*g - 110*e*h) + d^3*e^2*(119*f*g + 30*e*h) - 2*c*d^2*e*f*(71*f*g + ...
 

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 = 398.57 (sec) , antiderivative size = 1943, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1943\)
default	\(\text {Expression too large to display}\)	\(1943\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(2422\)

Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)^2/(f*x+e)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2*f^4*(-(-e*h+f*g)/(c*f-d*e)^2/(a*f-b*e)^4/(f*x+e)^(1/2)+b^2/(a*d-b*c)^5/f 
^4/(a*f-b*e)^4*(((19/16*b^6*c^3*f^3*g-15/8*a^3*b^3*d^3*e*f^2*h+137/16*a^2* 
b^4*c*d^2*f^3*g+1/2*a^2*b^4*d^3*e^2*f*h+19/4*a^2*b^4*d^3*e*f^2*g-85/16*a*b 
^5*c^2*d*f^3*g-3/2*a*b^5*d^3*e^2*f*g-b^6*c^2*d*e^2*f*h+7/4*b^6*c^2*d*e*f^2 
*g+3/2*b^6*c*d^2*e^2*f*g-63/16*b^3*a^3*c*d^2*f^3*h+27/16*b^4*a^2*c^2*d*f^3 
*h-1/8*a^2*b^4*c*d^2*e*f^2*h+23/8*a*b^5*c^2*d*e*f^2*h+1/2*a*b^5*c*d^2*e^2* 
f*h-13/2*a*b^5*c*d^2*e*f^2*g+41/16*a^4*b^2*d^3*f^3*h-71/16*a^3*d^3*f^3*g*b 
^3-7/8*b^6*c^3*e*f^2*h-5/16*b^5*a*c^3*f^3*h)*(f*x+e)^(5/2)+1/6*b*f*(35*a^5 
*d^3*f^3*h-57*a^4*b*c*d^2*f^3*h-59*a^4*b*d^3*e*f^2*h-59*a^4*b*d^3*f^3*g+27 
*a^3*b^2*c^2*d*f^3*h+57*a^3*b^2*c*d^2*e*f^2*h+117*a^3*b^2*c*d^2*f^3*g+30*a 
^3*b^2*d^3*e^2*f*h+119*a^3*b^2*d^3*e*f^2*g-5*a^2*b^3*c^3*f^3*h+9*a^2*b^3*c 
^2*d*e*f^2*h-75*a^2*b^3*c^2*d*f^3*g+6*a^2*b^3*c*d^2*e^2*f*h-201*a^2*b^3*c* 
d^2*e*f^2*g-6*a^2*b^3*d^3*e^3*h-78*a^2*b^3*d^3*e^2*f*g-7*a*b^4*c^3*e*f^2*h 
+17*a*b^4*c^3*f^3*g-48*a*b^4*c^2*d*e^2*f*h+99*a*b^4*c^2*d*e*f^2*g-6*a*b^4* 
c*d^2*e^3*h+102*a*b^4*c*d^2*e^2*f*g+18*a*b^4*d^3*e^3*g+12*b^5*c^3*e^2*f*h- 
17*b^5*c^3*e*f^2*g+12*b^5*c^2*d*e^3*h-24*b^5*c^2*d*e^2*f*g-18*b^5*c*d^2*e^ 
3*g)*(f*x+e)^(3/2)+(-97/16*a^5*b*c*d^2*f^5*h-9*a^5*b*d^3*e*f^4*h+53/16*a^4 
*b^2*c^2*d*f^5*h+183/16*a^4*b^2*c*d^2*f^5*g+131/16*a^4*b^2*d^3*e^2*f^3*h+1 
31/8*a^4*b^2*d^3*e*f^4*g-123/16*a^3*b^3*c^2*d*f^5*g-281/16*a^3*b^3*d^3*e^2 
*f^3*g+1/4*a^2*b^4*c^3*e*f^4*h+33/4*a^2*b^4*d^3*e^3*f^2*g+25/16*a*b^5*c...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)^2/(f*x+e)^(3/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)**4/(d*x+c)**2/(f*x+e)**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)/(b*x+a)^4/(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. 4339 vs. \(2 (1298) = 2596\).

Time = 0.49 (sec) , antiderivative size = 4339, normalized size of antiderivative = 3.24 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)^2/(f*x+e)^(3/2),x, algorithm="giac")
 

Output:

-1/8*(64*b^6*d^3*e^3*g + 72*b^6*c*d^2*e^2*f*g - 264*a*b^5*d^3*e^2*f*g + 60 
*b^6*c^2*d*e*f^2*g - 264*a*b^5*c*d^2*e*f^2*g + 396*a^2*b^4*d^3*e*f^2*g + 3 
5*b^6*c^3*f^3*g - 165*a*b^5*c^2*d*f^3*g + 297*a^2*b^4*c*d^2*f^3*g - 231*a^ 
3*b^3*d^3*f^3*g - 48*b^6*c*d^2*e^3*h - 16*a*b^5*d^3*e^3*h - 48*b^6*c^2*d*e 
^2*f*h + 168*a*b^5*c*d^2*e^2*f*h + 72*a^2*b^4*d^3*e^2*f*h - 30*b^6*c^3*e*f 
^2*h + 126*a*b^5*c^2*d*e*f^2*h - 162*a^2*b^4*c*d^2*e*f^2*h - 126*a^3*b^3*d 
^3*e*f^2*h - 5*a*b^5*c^3*f^3*h + 27*a^2*b^4*c^2*d*f^3*h - 63*a^3*b^3*c*d^2 
*f^3*h + 105*a^4*b^2*d^3*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f 
))/((b^9*c^5*e^4 - 5*a*b^8*c^4*d*e^4 + 10*a^2*b^7*c^3*d^2*e^4 - 10*a^3*b^6 
*c^2*d^3*e^4 + 5*a^4*b^5*c*d^4*e^4 - a^5*b^4*d^5*e^4 - 4*a*b^8*c^5*e^3*f + 
 20*a^2*b^7*c^4*d*e^3*f - 40*a^3*b^6*c^3*d^2*e^3*f + 40*a^4*b^5*c^2*d^3*e^ 
3*f - 20*a^5*b^4*c*d^4*e^3*f + 4*a^6*b^3*d^5*e^3*f + 6*a^2*b^7*c^5*e^2*f^2 
 - 30*a^3*b^6*c^4*d*e^2*f^2 + 60*a^4*b^5*c^3*d^2*e^2*f^2 - 60*a^5*b^4*c^2* 
d^3*e^2*f^2 + 30*a^6*b^3*c*d^4*e^2*f^2 - 6*a^7*b^2*d^5*e^2*f^2 - 4*a^3*b^6 
*c^5*e*f^3 + 20*a^4*b^5*c^4*d*e*f^3 - 40*a^5*b^4*c^3*d^2*e*f^3 + 40*a^6*b^ 
3*c^2*d^3*e*f^3 - 20*a^7*b^2*c*d^4*e*f^3 + 4*a^8*b*d^5*e*f^3 + a^4*b^5*c^5 
*f^4 - 5*a^5*b^4*c^4*d*f^4 + 10*a^6*b^3*c^3*d^2*f^4 - 10*a^7*b^2*c^2*d^3*f 
^4 + 5*a^8*b*c*d^4*f^4 - a^9*d^5*f^4)*sqrt(-b^2*e + a*b*f)) + (8*b*d^6*e*g 
 - 11*b*c*d^5*f*g + 3*a*d^6*f*g - 6*b*c*d^5*e*h - 2*a*d^6*e*h + 9*b*c^2*d^ 
4*f*h - a*c*d^5*f*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^5...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Hanged} \] Input:

int((g + h*x)/((e + f*x)^(3/2)*(a + b*x)^4*(c + d*x)^2),x)
 

Output:

\text{Hanged}
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 1.00 (sec) , antiderivative size = 46641, normalized size of antiderivative = 34.81 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)^2/(f*x+e)^(3/2),x)
 

Output:

(315*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**7*b*c**4*d**3*f**6*h - 945*sqrt(b)*sqrt(e + f*x)*sqr 
t(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b*c**3 
*d**4*e*f**5*h + 315*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**7*b*c**3*d**4*f**6*h*x + 945*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**7*b*c**2*d**5*e**2*f**4*h - 945*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**7*b*c**2*d**5*e 
*f**5*h*x - 315*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**7*b*c*d**6*e**3*f**3*h + 945*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**7*b*c*d**6*e**2*f**4*h*x - 315*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**7*b*d**7*e**3*f**3*h* 
x - 189*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**6*b**2*c**5*d**2*f**6*h + 189*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**6* 
b**2*c**4*d**3*e*f**5*h - 693*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**6*b**2*c**4*d**3*f**6*g + 7 
56*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*s 
qrt(a*f - b*e)))*a**6*b**2*c**4*d**3*f**6*h*x + 567*sqrt(b)*sqrt(e + f*...