\(\int \frac {g+h x}{(a+b x)^3 (c+d x)^3 (e+f x)^{3/2}} \, dx\) [192]

Optimal result

Integrand size = 29, antiderivative size = 1292 \[ \int \frac {g+h x}{(a+b x)^3 (c+d x)^3 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Output:

-3/4*f*(a^4*d^3*f^3*(-c*f*h-4*d*e*h+5*d*f*g)+a^3*b*d^2*f^2*(5*c^2*f^2*h-c* 
d*f*(-11*e*h+17*f*g)-d^2*e*(-4*e*h+3*f*g))-b^4*(8*d^4*e^4*g+c^3*d*e*f^2*(- 
4*e*h+3*f*g)-c^4*f^3*(-4*e*h+5*f*g)-4*c*d^3*e^3*(e*h+4*f*g)+c^2*d^2*e^2*f* 
(9*e*h+5*f*g))+a^2*b^2*d*f*(5*c^3*f^3*h+2*c^2*d*f^2*(-23*e*h+8*f*g)-d^3*e^ 
2*(9*e*h+5*f*g)+c*d^2*e*f*(20*e*h+19*f*g))-a*b^3*(c^4*f^4*h+c^3*d*f^3*(-11 
*e*h+17*f*g)-4*d^4*e^3*(e*h+4*f*g)+2*c*d^3*e^2*f*(7*e*h+19*f*g)-c^2*d^2*e* 
f^2*(20*e*h+19*f*g)))/(-a*d+b*c)^4/(-a*f+b*e)^3/(-c*f+d*e)^3/(f*x+e)^(1/2) 
+1/4*d*(a^2*d*f*(-13*c*f*h+11*d*e*h+2*d*f*g)+b^2*(12*d^2*e^2*g-c^2*f*(-4*e 
*h+5*f*g)-c*d*e*(6*e*h+5*f*g))+a*b*(c^2*f^2*h+3*c*d*f*(3*e*h+5*f*g)-d^2*e* 
(6*e*h+19*f*g)))/(-a*d+b*c)^3/(-a*f+b*e)^2/(-c*f+d*e)/(d*x+c)^2/(f*x+e)^(1 
/2)-1/2*(-a*h+b*g)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^2/(d*x+c)^2/(f*x+e)^(1/2) 
+1/4*(9*a^2*d*f*h+b^2*(-4*c*e*h+5*c*f*g+8*d*e*g)-a*b*(c*f*h+4*d*e*h+13*d*f 
*g))/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)/(d*x+c)^2/(f*x+e)^(1/2)+1/4*d*(a^3* 
d^2*f^2*(-c*f*h-4*d*e*h+5*d*f*g)+b^3*(24*d^3*e^3*g+c^3*f^2*(-4*e*h+5*f*g)- 
12*c*d^2*e^2*(e*h+3*f*g)+c^2*d*e*f*(21*e*h+2*f*g))+a^2*b*d*f*(26*c^2*f^2*h 
+d^2*e*(21*e*h+2*f*g)-c*d*f*(32*e*h+17*f*g))-a*b^2*(c^3*f^3*h+12*d^3*e^2*( 
e*h+3*f*g)-2*c*d^2*e*f*(15*e*h+34*f*g)+c^2*d*f^2*(32*e*h+17*f*g)))/(-a*d+b 
*c)^4/(-a*f+b*e)^2/(-c*f+d*e)^2/(d*x+c)/(f*x+e)^(1/2)+3/4*b^(5/2)*(21*a^3* 
d^2*f^2*h-3*a^2*b*d*f*(2*c*f*h+8*d*e*h+11*d*f*g)-b^3*(16*d^2*e^2*g+c^2*f*( 
-4*e*h+5*f*g)+4*c*d*e*(-2*e*h+3*f*g))+a*b^2*(c^2*f^2*h+2*c*d*f*(-6*e*h+...
 

Mathematica [C] (verified)

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

Time = 11.72 (sec) , antiderivative size = 871, normalized size of antiderivative = 0.67 \[ \int \frac {g+h x}{(a+b x)^3 (c+d x)^3 (e+f x)^{3/2}} \, dx=\frac {\frac {-2 b g+2 a h}{(a+b x)^2}+\frac {9 a^2 d f h+b^2 (8 d e g+5 c f g-4 c e h)-a b (13 d f g+4 d e h+c f h)}{(b c-a d) (b e-a f) (a+b x)}-\frac {-d (b c-a d)^2 (b e-a f) (d e-c f)^2 \left (a^2 d f (2 d f g+11 d e h-13 c f h)+b^2 \left (12 d^2 e^2 g+c^2 f (-5 f g+4 e h)-c d e (5 f g+6 e h)\right )+a b \left (c^2 f^2 h+3 c d f (5 f g+3 e h)-d^2 e (19 f g+6 e h)\right )\right )+(c+d x) \left (-d (b c-a d) (b e-a f) (-d e+c f) \left (a^3 d^2 f^2 (-5 d f g+4 d e h+c f h)+b^3 \left (-24 d^3 e^3 g+12 c d^2 e^2 (3 f g+e h)+c^3 f^2 (-5 f g+4 e h)-c^2 d e f (2 f g+21 e h)\right )+a^2 b d f \left (-26 c^2 f^2 h-d^2 e (2 f g+21 e h)+c d f (17 f g+32 e h)\right )+a b^2 \left (c^3 f^3 h+12 d^3 e^2 (3 f g+e h)-2 c d^2 e f (34 f g+15 e h)+c^2 d f^2 (17 f g+32 e h)\right )\right )-3 (c+d x) \left (b^2 (d e-c f)^3 \left (-21 a^3 d^2 f^2 h+3 a^2 b d f (11 d f g+8 d e h+2 c f h)+b^3 \left (16 d^2 e^2 g+c^2 f (5 f g-4 e h)+4 c d e (3 f g-2 e h)\right )-a b^2 \left (c^2 f^2 h+2 c d f (11 f g-6 e h)+4 d^2 e (11 f g+2 e h)\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {b (e+f x)}{b e-a f}\right )-d^2 (b e-a f)^3 \left (a^2 d^2 f (5 d f g-4 d e h-c f h)+b^2 \left (16 d^3 e^2 g-21 c^3 f^2 h-4 c d^2 e (11 f g+2 e h)+3 c^2 d f (11 f g+8 e h)\right )+2 a b d \left (3 c^2 f^2 h+c d f (-11 f g+6 e h)+d^2 \left (6 e f g-4 e^2 h\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {d (e+f x)}{d e-c f}\right )\right )\right )}{(b c-a d)^4 (b e-a f)^2 (d e-c f)^3}}{4 (b c-a d) (b e-a f) (c+d x)^2 \sqrt {e+f x}} \] Input:

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

Output:

((-2*b*g + 2*a*h)/(a + b*x)^2 + (9*a^2*d*f*h + b^2*(8*d*e*g + 5*c*f*g - 4* 
c*e*h) - a*b*(13*d*f*g + 4*d*e*h + c*f*h))/((b*c - a*d)*(b*e - a*f)*(a + b 
*x)) - (-(d*(b*c - a*d)^2*(b*e - a*f)*(d*e - c*f)^2*(a^2*d*f*(2*d*f*g + 11 
*d*e*h - 13*c*f*h) + b^2*(12*d^2*e^2*g + c^2*f*(-5*f*g + 4*e*h) - c*d*e*(5 
*f*g + 6*e*h)) + a*b*(c^2*f^2*h + 3*c*d*f*(5*f*g + 3*e*h) - d^2*e*(19*f*g 
+ 6*e*h)))) + (c + d*x)*(-(d*(b*c - a*d)*(b*e - a*f)*(-(d*e) + c*f)*(a^3*d 
^2*f^2*(-5*d*f*g + 4*d*e*h + c*f*h) + b^3*(-24*d^3*e^3*g + 12*c*d^2*e^2*(3 
*f*g + e*h) + c^3*f^2*(-5*f*g + 4*e*h) - c^2*d*e*f*(2*f*g + 21*e*h)) + a^2 
*b*d*f*(-26*c^2*f^2*h - d^2*e*(2*f*g + 21*e*h) + c*d*f*(17*f*g + 32*e*h)) 
+ a*b^2*(c^3*f^3*h + 12*d^3*e^2*(3*f*g + e*h) - 2*c*d^2*e*f*(34*f*g + 15*e 
*h) + c^2*d*f^2*(17*f*g + 32*e*h)))) - 3*(c + d*x)*(b^2*(d*e - c*f)^3*(-21 
*a^3*d^2*f^2*h + 3*a^2*b*d*f*(11*d*f*g + 8*d*e*h + 2*c*f*h) + b^3*(16*d^2* 
e^2*g + c^2*f*(5*f*g - 4*e*h) + 4*c*d*e*(3*f*g - 2*e*h)) - a*b^2*(c^2*f^2* 
h + 2*c*d*f*(11*f*g - 6*e*h) + 4*d^2*e*(11*f*g + 2*e*h)))*Hypergeometric2F 
1[-1/2, 1, 1/2, (b*(e + f*x))/(b*e - a*f)] - d^2*(b*e - a*f)^3*(a^2*d^2*f* 
(5*d*f*g - 4*d*e*h - c*f*h) + b^2*(16*d^3*e^2*g - 21*c^3*f^2*h - 4*c*d^2*e 
*(11*f*g + 2*e*h) + 3*c^2*d*f*(11*f*g + 8*e*h)) + 2*a*b*d*(3*c^2*f^2*h + c 
*d*f*(-11*f*g + 6*e*h) + d^2*(6*e*f*g - 4*e^2*h)))*Hypergeometric2F1[-1/2, 
 1, 1/2, (d*(e + f*x))/(d*e - c*f)])))/((b*c - a*d)^4*(b*e - a*f)^2*(d*e - 
 c*f)^3))/(4*(b*c - a*d)*(b*e - a*f)*(c + d*x)^2*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 3.62 (sec) , antiderivative size = 1388, normalized size of antiderivative = 1.07, 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.

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {d f (8 d f g+44 d e h+11 c f h) a^2-b \left (4 e (19 f g+6 e h) d^2+c f (31 f g-8 e h) d+3 c^2 f^2 h\right ) a+3 b^2 \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 g\right )+7 d f \left (9 d f h a^2-b (13 d f g+4 d e h+c f h) a+b^2 (8 d e g+5 c f g-4 c e h)\right ) x}{2 (a+b x) (c+d x)^3 (e+f x)^{3/2}}dx}{(b c-a d) (b e-a f)}-\frac {9 a^2 d f h-a b (c f h+4 d e h+13 d f g)+b^2 (-4 c e h+5 c f g+8 d e g)}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {b g-a h}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {d f (8 d f g+44 d e h+11 c f h) a^2-b \left (4 e (19 f g+6 e h) d^2+c f (31 f g-8 e h) d+3 c^2 f^2 h\right ) a+3 b^2 \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 g\right )+7 d f \left (9 d f h a^2-b (13 d f g+4 d e h+c f h) a+b^2 (8 d e g+5 c f g-4 c e h)\right ) x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}-\frac {9 a^2 d f h-a b (c f h+4 d e h+13 d f g)+b^2 (-4 c e h+5 c f g+8 d e g)}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {b g-a h}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {\int \frac {2 \left (2 d^2 f^2 (5 d f g-4 d e h-c f h) a^3-b d f \left (-2 e (2 f g+21 e h) d^2+3 c f (8 f g+3 e h) d+13 c^2 f^2 h\right ) a^2+b^2 \left (-24 e^2 (3 f g+e h) d^3+c e f (41 f g+30 e h) d^2+c^2 f^2 (41 f g-19 e h) d+3 c^3 f^3 h\right ) a+3 b^3 (d e-c f) \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 g\right )+5 b d f \left (d f (2 d f g+11 d e h-13 c f h) a^2+b \left (-e (19 f g+6 e h) d^2+3 c f (5 f g+3 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (5 f g+6 e h) c+12 d^2 e^2 g\right )\right ) x\right )}{(a+b x) (c+d x)^2 (e+f x)^{3/2}}dx}{2 (b c-a d) (d e-c f)}+\frac {2 d \left (a^2 d f (-13 c f h+11 d e h+2 d f g)+a b \left (c^2 f^2 h+3 c d f (3 e h+5 f g)+d^2 (-e) (6 e h+19 f g)\right )+b^2 \left (c^2 (-f) (5 f g-4 e h)-c d e (6 e h+5 f g)+12 d^2 e^2 g\right )\right )}{(c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {9 a^2 d f h-a b (c f h+4 d e h+13 d f g)+b^2 (-4 c e h+5 c f g+8 d e g)}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {b g-a h}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\int \frac {2 d^2 f^2 (5 d f g-4 d e h-c f h) a^3-b d f \left (-2 e (2 f g+21 e h) d^2+3 c f (8 f g+3 e h) d+13 c^2 f^2 h\right ) a^2+b^2 \left (-24 e^2 (3 f g+e h) d^3+c e f (41 f g+30 e h) d^2+c^2 f^2 (41 f g-19 e h) d+3 c^3 f^3 h\right ) a+3 b^3 (d e-c f) \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 g\right )+5 b d f \left (d f (2 d f g+11 d e h-13 c f h) a^2+b \left (-e (19 f g+6 e h) d^2+3 c f (5 f g+3 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (5 f g+6 e h) c+12 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^2 (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}+\frac {2 d \left (a^2 d f (-13 c f h+11 d e h+2 d f g)+a b \left (c^2 f^2 h+3 c d f (3 e h+5 f g)+d^2 (-e) (6 e h+19 f g)\right )+b^2 \left (c^2 (-f) (5 f g-4 e h)-c d e (6 e h+5 f g)+12 d^2 e^2 g\right )\right )}{(c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {9 a^2 d f h-a b (c f h+4 d e h+13 d f g)+b^2 (-4 c e h+5 c f g+8 d e g)}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {b g-a h}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (13 d f g+4 d e h+c f h) a+b^2 (8 d e g+5 c f g-4 c e h)}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (d f (2 d f g+11 d e h-13 c f h) a^2+b \left (-e (19 f g+6 e h) d^2+3 c f (5 f g+3 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (5 f g+6 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (e (2 f g+21 e h) d^2-c f (17 f g+32 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (12 e^2 (3 f g+e h) d^3-2 c e f (34 f g+15 e h) d^2+c^2 f^2 (17 f g+32 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-4 e h) c^3+d e f (2 f g+21 e h) c^2-12 d^2 e^2 (3 f g+e h) c+24 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {\int \frac {3 \left (d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^3 \left (5 f h c^2-17 d f g c+10 d e h c+2 d^2 e g\right ) a^3+b^2 d f \left (-3 e^2 (f g-4 e h) d^3+2 c e f (f g-6 e h) d^2+4 c^2 f^2 (4 f g-5 e h) d+5 c^3 f^3 h\right ) a^2-b^3 \left (4 e^3 (5 f g+2 e h) d^4-2 c e^2 f (15 f g+8 e h) d^3-2 c^2 e f^2 (f g-6 e h) d^2+c^3 f^3 (17 f g-10 e h) d+c^4 f^4 h\right ) a+b^4 (d e-c f)^2 \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 g\right )+b d f \left (d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (e (2 f g+21 e h) d^2-c f (17 f g+32 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (12 e^2 (3 f g+e h) d^3-2 c e f (34 f g+15 e h) d^2+c^2 f^2 (17 f g+32 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-4 e h) c^3+d e f (2 f g+21 e h) c^2-12 d^2 e^2 (3 f g+e h) c+24 d^3 e^3 g\right )\right ) x\right )}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (13 d f g+4 d e h+c f h) a+b^2 (8 d e g+5 c f g-4 c e h)}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (d f (2 d f g+11 d e h-13 c f h) a^2+b \left (-e (19 f g+6 e h) d^2+3 c f (5 f g+3 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (5 f g+6 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (e (2 f g+21 e h) d^2-c f (17 f g+32 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (12 e^2 (3 f g+e h) d^3-2 c e f (34 f g+15 e h) d^2+c^2 f^2 (17 f g+32 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-4 e h) c^3+d e f (2 f g+21 e h) c^2-12 d^2 e^2 (3 f g+e h) c+24 d^3 e^3 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {3 \int \frac {d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^3 \left (5 f h c^2-17 d f g c+10 d e h c+2 d^2 e g\right ) a^3+b^2 d f \left (-3 e^2 (f g-4 e h) d^3+2 c e f (f g-6 e h) d^2+4 c^2 f^2 (4 f g-5 e h) d+5 c^3 f^3 h\right ) a^2-b^3 \left (4 e^3 (5 f g+2 e h) d^4-2 c e^2 f (15 f g+8 e h) d^3-2 c^2 e f^2 (f g-6 e h) d^2+c^3 f^3 (17 f g-10 e h) d+c^4 f^4 h\right ) a+b^4 (d e-c f)^2 \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 g\right )+b d f \left (d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (e (2 f g+21 e h) d^2-c f (17 f g+32 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (12 e^2 (3 f g+e h) d^3-2 c e f (34 f g+15 e h) d^2+c^2 f^2 (17 f g+32 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-4 e h) c^3+d e f (2 f g+21 e h) c^2-12 d^2 e^2 (3 f g+e h) c+24 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (13 d f g+4 d e h+c f h) a+b^2 (8 d e g+5 c f g-4 c e h)}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (d f (2 d f g+11 d e h-13 c f h) a^2+b \left (-e (19 f g+6 e h) d^2+3 c f (5 f g+3 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (5 f g+6 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (e (2 f g+21 e h) d^2-c f (17 f g+32 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (12 e^2 (3 f g+e h) d^3-2 c e f (34 f g+15 e h) d^2+c^2 f^2 (17 f g+32 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-4 e h) c^3+d e f (2 f g+21 e h) c^2-12 d^2 e^2 (3 f g+e h) c+24 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 (d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (-e (3 f g-4 e h) d^2-c f (17 f g-11 e h) d+5 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (5 f g+9 e h) d^3+c e f (19 f g+20 e h) d^2+2 c^2 f^2 (8 f g-23 e h) d+5 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (4 f g+e h) d^4+2 c e^2 f (19 f g+7 e h) d^3-c^2 e f^2 (19 f g+20 e h) d^2+c^3 f^3 (17 f g-11 e h) d+c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-4 e h) c^4+d e f^2 (3 f g-4 e h) c^3+d^2 e^2 f (5 f g+9 e h) c^2-4 d^3 e^3 (4 f g+e h) c+8 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 \int \frac {d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (-e (3 f g-4 e h) d^2-c f (17 f g-11 e h) d+5 c^2 f^2 h\right ) a^4-b^2 d^2 f^2 \left (e^2 (5 f g-12 e h) d^3-c e f (19 f g-43 e h) d^2-c^2 f^2 (16 f g+17 e h) d+16 c^3 f^3 h\right ) a^3+b^3 d f \left (-e^3 (17 f g+20 e h) d^4+c e^2 f (61 f g+52 e h) d^3-2 c^2 e f^2 (40 f g+17 e h) d^2+c^3 f^3 (16 f g+17 e h) d+5 c^4 f^4 h\right ) a^2-b^4 \left (-4 e^4 (9 f g+2 e h) d^5+2 c e^3 f (47 f g+16 e h) d^4-c^2 e^2 f^2 (61 f g+52 e h) d^3-c^3 e f^3 (19 f g-43 e h) d^2+c^4 f^4 (17 f g-11 e h) d+c^5 f^5 h\right ) a-b^5 (d e-c f)^3 \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 g\right )+b d f \left (d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (-e (3 f g-4 e h) d^2-c f (17 f g-11 e h) d+5 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (5 f g+9 e h) d^3+c e f (19 f g+20 e h) d^2+2 c^2 f^2 (8 f g-23 e h) d+5 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (4 f g+e h) d^4+2 c e^2 f (19 f g+7 e h) d^3-c^2 e f^2 (19 f g+20 e h) d^2+c^3 f^3 (17 f g-11 e h) d+c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-4 e h) c^4+d e f^2 (3 f g-4 e h) c^3+d^2 e^2 f (5 f g+9 e h) c^2-4 d^3 e^3 (4 f g+e h) c+8 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)}}{(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)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (13 d f g+4 d e h+c f h) a+b^2 (8 d e g+5 c f g-4 c e h)}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (d f (2 d f g+11 d e h-13 c f h) a^2+b \left (-e (19 f g+6 e h) d^2+3 c f (5 f g+3 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (5 f g+6 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (e (2 f g+21 e h) d^2-c f (17 f g+32 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (12 e^2 (3 f g+e h) d^3-2 c e f (34 f g+15 e h) d^2+c^2 f^2 (17 f g+32 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-4 e h) c^3+d e f (2 f g+21 e h) c^2-12 d^2 e^2 (3 f g+e h) c+24 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 (d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (-e (3 f g-4 e h) d^2-c f (17 f g-11 e h) d+5 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (5 f g+9 e h) d^3+c e f (19 f g+20 e h) d^2+2 c^2 f^2 (8 f g-23 e h) d+5 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (4 f g+e h) d^4+2 c e^2 f (19 f g+7 e h) d^3-c^2 e f^2 (19 f g+20 e h) d^2+c^3 f^3 (17 f g-11 e h) d+c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-4 e h) c^4+d e f^2 (3 f g-4 e h) c^3+d^2 e^2 f (5 f g+9 e h) c^2-4 d^3 e^3 (4 f g+e h) c+8 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\int \frac {d^4 f^4 (5 d f g-4 d e h-c f h) a^5+b d^3 f^3 \left (-e (3 f g-4 e h) d^2-c f (17 f g-11 e h) d+5 c^2 f^2 h\right ) a^4-b^2 d^2 f^2 \left (e^2 (5 f g-12 e h) d^3-c e f (19 f g-43 e h) d^2-c^2 f^2 (16 f g+17 e h) d+16 c^3 f^3 h\right ) a^3+b^3 d f \left (-e^3 (17 f g+20 e h) d^4+c e^2 f (61 f g+52 e h) d^3-2 c^2 e f^2 (40 f g+17 e h) d^2+c^3 f^3 (16 f g+17 e h) d+5 c^4 f^4 h\right ) a^2-b^4 \left (-4 e^4 (9 f g+2 e h) d^5+2 c e^3 f (47 f g+16 e h) d^4-c^2 e^2 f^2 (61 f g+52 e h) d^3-c^3 e f^3 (19 f g-43 e h) d^2+c^4 f^4 (17 f g-11 e h) d+c^5 f^5 h\right ) a-b^5 (d e-c f)^3 \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 g\right )+b d f \left (d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (-e (3 f g-4 e h) d^2-c f (17 f g-11 e h) d+5 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (5 f g+9 e h) d^3+c e f (19 f g+20 e h) d^2+2 c^2 f^2 (8 f g-23 e h) d+5 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (4 f g+e h) d^4+2 c e^2 f (19 f g+7 e h) d^3-c^2 e f^2 (19 f g+20 e h) d^2+c^3 f^3 (17 f g-11 e h) d+c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-4 e h) c^4+d e f^2 (3 f g-4 e h) c^3+d^2 e^2 f (5 f g+9 e h) c^2-4 d^3 e^3 (4 f g+e h) c+8 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)}}{(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)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (13 d f g+4 d e h+c f h) a+b^2 (8 d e g+5 c f g-4 c e h)}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (d f (2 d f g+11 d e h-13 c f h) a^2+b \left (-e (19 f g+6 e h) d^2+3 c f (5 f g+3 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (5 f g+6 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (e (2 f g+21 e h) d^2-c f (17 f g+32 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (12 e^2 (3 f g+e h) d^3-2 c e f (34 f g+15 e h) d^2+c^2 f^2 (17 f g+32 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-4 e h) c^3+d e f (2 f g+21 e h) c^2-12 d^2 e^2 (3 f g+e h) c+24 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 (d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (-e (3 f g-4 e h) d^2-c f (17 f g-11 e h) d+5 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (5 f g+9 e h) d^3+c e f (19 f g+20 e h) d^2+2 c^2 f^2 (8 f g-23 e h) d+5 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (4 f g+e h) d^4+2 c e^2 f (19 f g+7 e h) d^3-c^2 e f^2 (19 f g+20 e h) d^2+c^3 f^3 (17 f g-11 e h) d+c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-4 e h) c^4+d e f^2 (3 f g-4 e h) c^3+d^2 e^2 f (5 f g+9 e h) c^2-4 d^3 e^3 (4 f g+e h) c+8 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {d^3 \left (\left (-21 f^2 h c^3+3 d f (11 f g+8 e h) c^2-4 d^2 e (11 f g+2 e h) c+16 d^3 e^2 g\right ) b^2+2 a d \left (\left (6 e f g-4 e^2 h\right ) d^2-c f (11 f g-6 e h) d+3 c^2 f^2 h\right ) b+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)^3}{b c-a d}+\frac {b^3 (d e-c f)^3 \left (21 d^2 f^2 h a^3-3 b d f (11 d f g+8 d e h+2 c f h) a^2+b^2 \left (4 e (11 f g+2 e h) d^2+2 c f (11 f g-6 e h) d+c^2 f^2 h\right ) a-b^3 \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 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)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (13 d f g+4 d e h+c f h) a+b^2 (8 d e g+5 c f g-4 c e h)}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (d f (2 d f g+11 d e h-13 c f h) a^2+b \left (-e (19 f g+6 e h) d^2+3 c f (5 f g+3 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (5 f g+6 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (e (2 f g+21 e h) d^2-c f (17 f g+32 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (12 e^2 (3 f g+e h) d^3-2 c e f (34 f g+15 e h) d^2+c^2 f^2 (17 f g+32 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-4 e h) c^3+d e f (2 f g+21 e h) c^2-12 d^2 e^2 (3 f g+e h) c+24 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 (d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (-e (3 f g-4 e h) d^2-c f (17 f g-11 e h) d+5 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (5 f g+9 e h) d^3+c e f (19 f g+20 e h) d^2+2 c^2 f^2 (8 f g-23 e h) d+5 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (4 f g+e h) d^4+2 c e^2 f (19 f g+7 e h) d^3-c^2 e f^2 (19 f g+20 e h) d^2+c^3 f^3 (17 f g-11 e h) d+c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-4 e h) c^4+d e f^2 (3 f g-4 e h) c^3+d^2 e^2 f (5 f g+9 e h) c^2-4 d^3 e^3 (4 f g+e h) c+8 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {2 d^3 \left (\left (-21 f^2 h c^3+3 d f (11 f g+8 e h) c^2-4 d^2 e (11 f g+2 e h) c+16 d^3 e^2 g\right ) b^2+2 a d \left (\left (6 e f g-4 e^2 h\right ) d^2-c f (11 f g-6 e h) d+3 c^2 f^2 h\right ) b+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)^3}{(b c-a d) f}+\frac {2 b^3 (d e-c f)^3 \left (21 d^2 f^2 h a^3-3 b d f (11 d f g+8 d e h+2 c f h) a^2+b^2 \left (4 e (11 f g+2 e h) d^2+2 c f (11 f g-6 e h) d+c^2 f^2 h\right ) a-b^3 \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 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)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}-\frac {-\frac {9 d f h a^2-b (13 d f g+4 d e h+c f h) a+b^2 (8 d e g+5 c f g-4 c e h)}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {2 d \left (d f (2 d f g+11 d e h-13 c f h) a^2+b \left (-e (19 f g+6 e h) d^2+3 c f (5 f g+3 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (5 f g+6 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {2 d \left (d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (e (2 f g+21 e h) d^2-c f (17 f g+32 e h) d+26 c^2 f^2 h\right ) a^2-b^2 \left (12 e^2 (3 f g+e h) d^3-2 c e f (34 f g+15 e h) d^2+c^2 f^2 (17 f g+32 e h) d+c^3 f^3 h\right ) a+b^3 \left (f^2 (5 f g-4 e h) c^3+d e f (2 f g+21 e h) c^2-12 d^2 e^2 (3 f g+e h) c+24 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 (d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (-e (3 f g-4 e h) d^2-c f (17 f g-11 e h) d+5 c^2 f^2 h\right ) a^3+b^2 d f \left (-e^2 (5 f g+9 e h) d^3+c e f (19 f g+20 e h) d^2+2 c^2 f^2 (8 f g-23 e h) d+5 c^3 f^3 h\right ) a^2-b^3 \left (-4 e^3 (4 f g+e h) d^4+2 c e^2 f (19 f g+7 e h) d^3-c^2 e f^2 (19 f g+20 e h) d^2+c^3 f^3 (17 f g-11 e h) d+c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-4 e h) c^4+d e f^2 (3 f g-4 e h) c^3+d^2 e^2 f (5 f g+9 e h) c^2-4 d^3 e^3 (4 f g+e h) c+8 d^4 e^4 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {-\frac {2 d^{5/2} \left (\left (-21 f^2 h c^3+3 d f (11 f g+8 e h) c^2-4 d^2 e (11 f g+2 e h) c+16 d^3 e^2 g\right ) b^2+2 a d \left (\left (6 e f g-4 e^2 h\right ) d^2-c f (11 f g-6 e h) d+3 c^2 f^2 h\right ) b+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)^3}{(b c-a d) \sqrt {d e-c f}}-\frac {2 b^{5/2} (d e-c f)^3 \left (21 d^2 f^2 h a^3-3 b d f (11 d f g+8 d e h+2 c f h) a^2+b^2 \left (4 e (11 f g+2 e h) d^2+2 c f (11 f g-6 e h) d+c^2 f^2 h\right ) a-b^3 \left (f (5 f g-4 e h) c^2+4 d e (3 f g-2 e h) c+16 d^2 e^2 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)}\)

Input:

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

Output:

-1/2*(b*g - a*h)/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)^2*Sqrt[e + 
 f*x]) - (-((9*a^2*d*f*h + b^2*(8*d*e*g + 5*c*f*g - 4*c*e*h) - a*b*(13*d*f 
*g + 4*d*e*h + c*f*h))/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)^2*Sqrt 
[e + f*x])) - ((2*d*(a^2*d*f*(2*d*f*g + 11*d*e*h - 13*c*f*h) + b^2*(12*d^2 
*e^2*g - c^2*f*(5*f*g - 4*e*h) - c*d*e*(5*f*g + 6*e*h)) + a*b*(c^2*f^2*h + 
 3*c*d*f*(5*f*g + 3*e*h) - d^2*e*(19*f*g + 6*e*h))))/((b*c - a*d)*(d*e - c 
*f)*(c + d*x)^2*Sqrt[e + f*x]) + ((2*d*(a^3*d^2*f^2*(5*d*f*g - 4*d*e*h - c 
*f*h) + b^3*(24*d^3*e^3*g + c^3*f^2*(5*f*g - 4*e*h) - 12*c*d^2*e^2*(3*f*g 
+ e*h) + c^2*d*e*f*(2*f*g + 21*e*h)) + a^2*b*d*f*(26*c^2*f^2*h + d^2*e*(2* 
f*g + 21*e*h) - c*d*f*(17*f*g + 32*e*h)) - a*b^2*(c^3*f^3*h + 12*d^3*e^2*( 
3*f*g + e*h) - 2*c*d^2*e*f*(34*f*g + 15*e*h) + c^2*d*f^2*(17*f*g + 32*e*h) 
)))/((b*c - a*d)*(d*e - c*f)*(c + d*x)*Sqrt[e + f*x]) + (3*((-2*f*(a^4*d^3 
*f^3*(5*d*f*g - 4*d*e*h - c*f*h) + a^3*b*d^2*f^2*(5*c^2*f^2*h - c*d*f*(17* 
f*g - 11*e*h) - d^2*e*(3*f*g - 4*e*h)) - b^4*(8*d^4*e^4*g + c^3*d*e*f^2*(3 
*f*g - 4*e*h) - c^4*f^3*(5*f*g - 4*e*h) - 4*c*d^3*e^3*(4*f*g + e*h) + c^2* 
d^2*e^2*f*(5*f*g + 9*e*h)) + a^2*b^2*d*f*(5*c^3*f^3*h + 2*c^2*d*f^2*(8*f*g 
 - 23*e*h) - d^3*e^2*(5*f*g + 9*e*h) + c*d^2*e*f*(19*f*g + 20*e*h)) - a*b^ 
3*(c^4*f^4*h + c^3*d*f^3*(17*f*g - 11*e*h) - 4*d^4*e^3*(4*f*g + e*h) + 2*c 
*d^3*e^2*f*(19*f*g + 7*e*h) - c^2*d^2*e*f^2*(19*f*g + 20*e*h))))/((b*e - a 
*f)*(d*e - c*f)*Sqrt[e + f*x]) - ((-2*b^(5/2)*(d*e - c*f)^3*(21*a^3*d^2...
 

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 = 539.70 (sec) , antiderivative size = 1287, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1287\)
default	\(\text {Expression too large to display}\)	\(1287\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(2473\)

Input:

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

Output:

2*f^4*(-(-e*h+f*g)/(c*f-d*e)^3/(a*f-b*e)^3/(f*x+e)^(1/2)-b^3/(a*d-b*c)^5/f 
^4/(a*f-b*e)^3*(((15/8*a^3*b*d^2*f^2*h-9/4*a^2*b^2*c*d*f^2*h-a^2*b^2*d^2*e 
*f*h-19/8*a^2*b^2*d^2*f^2*g+3/8*a*b^3*c^2*f^2*h+1/2*a*b^3*c*d*e*f*h+13/4*a 
*b^3*c*d*f^2*g+3/2*a*b^3*d^2*e*f*g+1/2*b^4*c^2*e*f*h-7/8*b^4*c^2*f^2*g-3/2 
*b^4*c*d*e*f*g)*(f*x+e)^(3/2)+1/8*f*(17*a^4*d^2*f^2*h-22*a^3*b*c*d*f^2*h-2 
5*a^3*b*d^2*e*f*h-21*a^3*b*d^2*f^2*g+5*a^2*b^2*c^2*f^2*h+26*a^2*b^2*c*d*e* 
f*h+30*a^2*b^2*c*d*f^2*g+8*a^2*b^2*d^2*e^2*h+33*a^2*b^2*d^2*e*f*g-a*b^3*c^ 
2*e*f*h-9*a*b^3*c^2*f^2*g-4*a*b^3*c*d*e^2*h-42*a*b^3*c*d*e*f*g-12*a*b^3*d^ 
2*e^2*g-4*b^4*c^2*e^2*h+9*b^4*c^2*e*f*g+12*b^4*c*d*e^2*g)*(f*x+e)^(1/2))/( 
(f*x+e)*b+a*f-b*e)^2+3/8*(21*a^3*d^2*f^2*h-6*a^2*b*c*d*f^2*h-24*a^2*b*d^2* 
e*f*h-33*a^2*b*d^2*f^2*g+a*b^2*c^2*f^2*h-12*a*b^2*c*d*e*f*h+22*a*b^2*c*d*f 
^2*g+8*a*b^2*d^2*e^2*h+44*a*b^2*d^2*e*f*g+4*b^3*c^2*e*f*h-5*b^3*c^2*f^2*g+ 
8*b^3*c*d*e^2*h-12*b^3*c*d*e*f*g-16*b^3*d^2*e^2*g)/((a*f-b*e)*b)^(1/2)*arc 
tan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2)))+d^3/(a*d-b*c)^5/f^4/(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-9/4*a*b*c^2*d^2 
*f^2*h+1/2*a*b*c*d^3*e*f*h+13/4*a*b*c*d^3*f^2*g-3/2*a*b*d^4*e*f*g+15/8*b^2 
*c^3*d*f^2*h-b^2*c^2*d^2*e*f*h-19/8*b^2*c^2*d^2*f^2*g+3/2*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-22*a*b*c^3*d*f^2*h+26*a*b*c^2*d^2*e*f*h+ 
30*a*b*c^2*d^2*f^2*g-4*a*b*c*d^3*e^2*h-42*a*b*c*d^3*e*f*g+12*a*b*d^4*e^...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((h*x+g)/(b*x+a)^3/(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. 7132 vs. \(2 (1249) = 2498\).

Time = 6.87 (sec) , antiderivative size = 7132, normalized size of antiderivative = 5.52 \[ \int \frac {g+h x}{(a+b x)^3 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

3/4*(16*b^6*d^2*e^2*g + 12*b^6*c*d*e*f*g - 44*a*b^5*d^2*e*f*g + 5*b^6*c^2* 
f^2*g - 22*a*b^5*c*d*f^2*g + 33*a^2*b^4*d^2*f^2*g - 8*b^6*c*d*e^2*h - 8*a* 
b^5*d^2*e^2*h - 4*b^6*c^2*e*f*h + 12*a*b^5*c*d*e*f*h + 24*a^2*b^4*d^2*e*f* 
h - a*b^5*c^2*f^2*h + 6*a^2*b^4*c*d*f^2*h - 21*a^3*b^3*d^2*f^2*h)*arctan(s 
qrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^8*c^5*e^3 - 5*a*b^7*c^4*d*e^3 + 1 
0*a^2*b^6*c^3*d^2*e^3 - 10*a^3*b^5*c^2*d^3*e^3 + 5*a^4*b^4*c*d^4*e^3 - a^5 
*b^3*d^5*e^3 - 3*a*b^7*c^5*e^2*f + 15*a^2*b^6*c^4*d*e^2*f - 30*a^3*b^5*c^3 
*d^2*e^2*f + 30*a^4*b^4*c^2*d^3*e^2*f - 15*a^5*b^3*c*d^4*e^2*f + 3*a^6*b^2 
*d^5*e^2*f + 3*a^2*b^6*c^5*e*f^2 - 15*a^3*b^5*c^4*d*e*f^2 + 30*a^4*b^4*c^3 
*d^2*e*f^2 - 30*a^5*b^3*c^2*d^3*e*f^2 + 15*a^6*b^2*c*d^4*e*f^2 - 3*a^7*b*d 
^5*e*f^2 - a^3*b^5*c^5*f^3 + 5*a^4*b^4*c^4*d*f^3 - 10*a^5*b^3*c^3*d^2*f^3 
+ 10*a^6*b^2*c^2*d^3*f^3 - 5*a^7*b*c*d^4*f^3 + a^8*d^5*f^3)*sqrt(-b^2*e + 
a*b*f)) - 3/4*(16*b^2*d^6*e^2*g - 44*b^2*c*d^5*e*f*g + 12*a*b*d^6*e*f*g + 
33*b^2*c^2*d^4*f^2*g - 22*a*b*c*d^5*f^2*g + 5*a^2*d^6*f^2*g - 8*b^2*c*d^5* 
e^2*h - 8*a*b*d^6*e^2*h + 24*b^2*c^2*d^4*e*f*h + 12*a*b*c*d^5*e*f*h - 4*a^ 
2*d^6*e*f*h - 21*b^2*c^3*d^3*f^2*h + 6*a*b*c^2*d^4*f^2*h - a^2*c*d^5*f^2*h 
)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^5*c^5*d^3*e^3 - 5*a*b^4 
*c^4*d^4*e^3 + 10*a^2*b^3*c^3*d^5*e^3 - 10*a^3*b^2*c^2*d^6*e^3 + 5*a^4*b*c 
*d^7*e^3 - a^5*d^8*e^3 - 3*b^5*c^6*d^2*e^2*f + 15*a*b^4*c^5*d^3*e^2*f - 30 
*a^2*b^3*c^4*d^4*e^2*f + 30*a^3*b^2*c^3*d^5*e^2*f - 15*a^4*b*c^2*d^6*e^...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
                                                                                    
                                                                                    
 

Reduce [F]

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

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

Output:

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