\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx\) [102]

Optimal result

Integrand size = 29, antiderivative size = 686 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx=\frac {\left (a^2 d^2 f h-a b d (5 d f g+6 d e h-9 c f h)+b^2 \left (12 d^2 e g+2 c^2 f h-c d (7 f g+6 e h)\right )\right ) \sqrt {e+f x}}{4 b^2 (b c-a d)^3 (c+d x)^2}+\frac {\left (a^2 d f h+b^2 (8 d e g-3 c f g-4 c e h)-a b (5 d f g+4 d e h-7 c f h)\right ) \sqrt {e+f x}}{4 b^2 (b c-a d)^2 (a+b x) (c+d x)^2}+\frac {3 \left (a^2 d^2 f h+2 a b d (3 c f h-2 d (f g+e h))+b^2 \left (8 d^2 e g+c^2 f h-4 c d (f g+e h)\right )\right ) \sqrt {e+f x}}{4 b (b c-a d)^4 (c+d x)}-\frac {(b g-a h) (e+f x)^{3/2}}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}+\frac {3 \left (a^3 d^2 f^2 h-a^2 b d f (5 d f g+8 d e h-10 c f h)+a b^2 \left (5 c^2 f^2 h-10 c d f (f g+2 e h)+4 d^2 e (5 f g+2 e h)\right )-b^3 \left (16 d^2 e^2 g-4 c d e (3 f g+2 e h)+c^2 f (f g+4 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{4 \sqrt {b} (b c-a d)^5 \sqrt {b e-a f}}+\frac {3 \left (a^2 d^2 f (d f g+4 d e h-5 c f h)-2 a b d \left (5 c^2 f^2 h-5 c d f (f g+2 e h)+2 d^2 e (3 f g+2 e h)\right )+b^2 \left (16 d^3 e^2 g-c^3 f^2 h-4 c d^2 e (5 f g+2 e h)+c^2 d f (5 f g+8 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 \sqrt {d} (b c-a d)^5 \sqrt {d e-c f}} \] Output:

1/4*(a^2*d^2*f*h-a*b*d*(-9*c*f*h+6*d*e*h+5*d*f*g)+b^2*(12*d^2*e*g+2*c^2*f* 
h-c*d*(6*e*h+7*f*g)))*(f*x+e)^(1/2)/b^2/(-a*d+b*c)^3/(d*x+c)^2+1/4*(a^2*d* 
f*h+b^2*(-4*c*e*h-3*c*f*g+8*d*e*g)-a*b*(-7*c*f*h+4*d*e*h+5*d*f*g))*(f*x+e) 
^(1/2)/b^2/(-a*d+b*c)^2/(b*x+a)/(d*x+c)^2+3/4*(a^2*d^2*f*h+2*a*b*d*(3*c*f* 
h-2*d*(e*h+f*g))+b^2*(8*d^2*e*g+c^2*f*h-4*c*d*(e*h+f*g)))*(f*x+e)^(1/2)/b/ 
(-a*d+b*c)^4/(d*x+c)-1/2*(-a*h+b*g)*(f*x+e)^(3/2)/b/(-a*d+b*c)/(b*x+a)^2/( 
d*x+c)^2+3/4*(a^3*d^2*f^2*h-a^2*b*d*f*(-10*c*f*h+8*d*e*h+5*d*f*g)+a*b^2*(5 
*c^2*f^2*h-10*c*d*f*(2*e*h+f*g)+4*d^2*e*(2*e*h+5*f*g))-b^3*(16*d^2*e^2*g-4 
*c*d*e*(2*e*h+3*f*g)+c^2*f*(4*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a 
*f+b*e)^(1/2))/b^(1/2)/(-a*d+b*c)^5/(-a*f+b*e)^(1/2)+3/4*(a^2*d^2*f*(-5*c* 
f*h+4*d*e*h+d*f*g)-2*a*b*d*(5*c^2*f^2*h-5*c*d*f*(2*e*h+f*g)+2*d^2*e*(2*e*h 
+3*f*g))+b^2*(16*d^3*e^2*g-c^3*f^2*h-4*c*d^2*e*(2*e*h+5*f*g)+c^2*d*f*(8*e* 
h+5*f*g)))*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(1/2)/(-a*d+b 
*c)^5/(-c*f+d*e)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(6339\) vs. \(2(686)=1372\).

Time = 16.36 (sec) , antiderivative size = 6339, normalized size of antiderivative = 9.24 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {166, 27, 166, 27, 168, 27, 168, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\frac {\int \frac {8 b e (d e-c f) (2 b d g-b c h-a d h)+(4 d e-c f) \left (d f h a^2-b (5 d f g+4 d e h-7 c f h) a+b^2 (8 d e g-3 c f g-4 c e h)\right )+f \left (\left (8 f h c^2-5 d (5 f g+4 e h) c+40 d^2 e g\right ) b^2+a d (29 c f h-5 d (3 f g+4 e h)) b+3 a^2 d^2 f h\right ) x}{2 (a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+4 d e h+5 d f g)+b^2 (-4 c e h-3 c f g+8 d e g)\right )}{b (a+b x) (c+d x)^2 (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {8 b e (d e-c f) (2 b d g-b c h-a d h)+(4 d e-c f) \left (d f h a^2-b (5 d f g+4 d e h-7 c f h) a+b^2 (8 d e g-3 c f g-4 c e h)\right )+f \left (\left (8 f h c^2-5 d (5 f g+4 e h) c+40 d^2 e g\right ) b^2+a d (29 c f h-5 d (3 f g+4 e h)) b+3 a^2 d^2 f h\right ) x}{(a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{2 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+4 d e h+5 d f g)+b^2 (-4 c e h-3 c f g+8 d e g)\right )}{b (a+b x) (c+d x)^2 (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {\int \frac {6 b (d e-c f) \left (d f (2 d e-c f) h a^2-b \left (8 e (f g+e h) d^2-c f (3 f g+14 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (f (f g+4 e h) c^2-4 d e (3 f g+2 e h) c+16 d^2 e^2 g\right )+f \left (\left (2 f h c^2-d (7 f g+6 e h) c+12 d^2 e g\right ) b^2-a d (5 d f g+6 d e h-9 c f h) b+a^2 d^2 f h\right ) x\right )}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (d e-c f)}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h-a b d (-9 c f h+6 d e h+5 d f g)+b^2 \left (2 c^2 f h-c d (6 e h+7 f g)+12 d^2 e g\right )\right )}{(c+d x)^2 (b c-a d)}}{2 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+4 d e h+5 d f g)+b^2 (-4 c e h-3 c f g+8 d e g)\right )}{b (a+b x) (c+d x)^2 (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 b \int \frac {d f (2 d e-c f) h a^2-b \left (8 e (f g+e h) d^2-c f (3 f g+14 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (f (f g+4 e h) c^2-4 d e (3 f g+2 e h) c+16 d^2 e^2 g\right )+f \left (\left (2 f h c^2-d (7 f g+6 e h) c+12 d^2 e g\right ) b^2-a d (5 d f g+6 d e h-9 c f h) b+a^2 d^2 f h\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{b c-a d}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h-a b d (-9 c f h+6 d e h+5 d f g)+b^2 \left (2 c^2 f h-c d (6 e h+7 f g)+12 d^2 e g\right )\right )}{(c+d x)^2 (b c-a d)}}{2 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+4 d e h+5 d f g)+b^2 (-4 c e h-3 c f g+8 d e g)\right )}{b (a+b x) (c+d x)^2 (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {3 b \left (\frac {\int \frac {b (d e-c f) \left (d f (d f g+4 d e h-4 c f h) a^2-2 b \left (2 e (3 f g+2 e h) d^2-c f (3 f g+8 e h) d+2 c^2 f^2 h\right ) a+b^2 \left (f (f g+4 e h) c^2-4 d e (3 f g+2 e h) c+16 d^2 e^2 g\right )+f \left (\left (f h c^2-4 d (f g+e h) c+8 d^2 e g\right ) b^2+2 a d (3 c f h-2 d (f g+e h)) b+a^2 d^2 f h\right ) x\right )}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h+2 a b d (3 c f h-2 d (e h+f g))+b^2 \left (c^2 f h-4 c d (e h+f g)+8 d^2 e g\right )\right )}{(c+d x) (b c-a d)}\right )}{b c-a d}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h-a b d (-9 c f h+6 d e h+5 d f g)+b^2 \left (2 c^2 f h-c d (6 e h+7 f g)+12 d^2 e g\right )\right )}{(c+d x)^2 (b c-a d)}}{2 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+4 d e h+5 d f g)+b^2 (-4 c e h-3 c f g+8 d e g)\right )}{b (a+b x) (c+d x)^2 (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 b \left (\frac {b \int \frac {d f (d f g+4 d e h-4 c f h) a^2-2 b \left (2 e (3 f g+2 e h) d^2-c f (3 f g+8 e h) d+2 c^2 f^2 h\right ) a+b^2 \left (f (f g+4 e h) c^2-4 d e (3 f g+2 e h) c+16 d^2 e^2 g\right )+f \left (\left (f h c^2-4 d (f g+e h) c+8 d^2 e g\right ) b^2+2 a d (3 c f h-2 d (f g+e h)) b+a^2 d^2 f h\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{b c-a d}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h+2 a b d (3 c f h-2 d (e h+f g))+b^2 \left (c^2 f h-4 c d (e h+f g)+8 d^2 e g\right )\right )}{(c+d x) (b c-a d)}\right )}{b c-a d}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h-a b d (-9 c f h+6 d e h+5 d f g)+b^2 \left (2 c^2 f h-c d (6 e h+7 f g)+12 d^2 e g\right )\right )}{(c+d x)^2 (b c-a d)}}{2 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+4 d e h+5 d f g)+b^2 (-4 c e h-3 c f g+8 d e g)\right )}{b (a+b x) (c+d x)^2 (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {3 b \left (\frac {b \left (-\frac {\left (a^2 d^2 f (-5 c f h+4 d e h+d f g)-2 a b d \left (5 c^2 f^2 h-5 c d f (2 e h+f g)+2 d^2 e (2 e h+3 f g)\right )+b^2 \left (c^3 \left (-f^2\right ) h+c^2 d f (8 e h+5 f g)-4 c d^2 e (2 e h+5 f g)+16 d^3 e^2 g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}-\frac {\left (a^3 d^2 f^2 h-a^2 b d f (-10 c f h+8 d e h+5 d f g)+a b^2 \left (5 c^2 f^2 h-10 c d f (2 e h+f g)+4 d^2 e (2 e h+5 f g)\right )-b^3 \left (c^2 f (4 e h+f g)-4 c d e (2 e h+3 f g)+16 d^2 e^2 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}\right )}{b c-a d}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h+2 a b d (3 c f h-2 d (e h+f g))+b^2 \left (c^2 f h-4 c d (e h+f g)+8 d^2 e g\right )\right )}{(c+d x) (b c-a d)}\right )}{b c-a d}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h-a b d (-9 c f h+6 d e h+5 d f g)+b^2 \left (2 c^2 f h-c d (6 e h+7 f g)+12 d^2 e g\right )\right )}{(c+d x)^2 (b c-a d)}}{2 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+4 d e h+5 d f g)+b^2 (-4 c e h-3 c f g+8 d e g)\right )}{b (a+b x) (c+d x)^2 (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {3 b \left (\frac {b \left (-\frac {2 \left (a^2 d^2 f (-5 c f h+4 d e h+d f g)-2 a b d \left (5 c^2 f^2 h-5 c d f (2 e h+f g)+2 d^2 e (2 e h+3 f g)\right )+b^2 \left (c^3 \left (-f^2\right ) h+c^2 d f (8 e h+5 f g)-4 c d^2 e (2 e h+5 f g)+16 d^3 e^2 g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}-\frac {2 \left (a^3 d^2 f^2 h-a^2 b d f (-10 c f h+8 d e h+5 d f g)+a b^2 \left (5 c^2 f^2 h-10 c d f (2 e h+f g)+4 d^2 e (2 e h+5 f g)\right )-b^3 \left (c^2 f (4 e h+f g)-4 c d e (2 e h+3 f g)+16 d^2 e^2 g\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}\right )}{b c-a d}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h+2 a b d (3 c f h-2 d (e h+f g))+b^2 \left (c^2 f h-4 c d (e h+f g)+8 d^2 e g\right )\right )}{(c+d x) (b c-a d)}\right )}{b c-a d}+\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h-a b d (-9 c f h+6 d e h+5 d f g)+b^2 \left (2 c^2 f h-c d (6 e h+7 f g)+12 d^2 e g\right )\right )}{(c+d x)^2 (b c-a d)}}{2 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+4 d e h+5 d f g)+b^2 (-4 c e h-3 c f g+8 d e g)\right )}{b (a+b x) (c+d x)^2 (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+4 d e h+5 d f g)+b^2 (-4 c e h-3 c f g+8 d e g)\right )}{b (a+b x) (c+d x)^2 (b c-a d)}+\frac {\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h-a b d (-9 c f h+6 d e h+5 d f g)+b^2 \left (2 c^2 f h-c d (6 e h+7 f g)+12 d^2 e g\right )\right )}{(c+d x)^2 (b c-a d)}+\frac {3 b \left (\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h+2 a b d (3 c f h-2 d (e h+f g))+b^2 \left (c^2 f h-4 c d (e h+f g)+8 d^2 e g\right )\right )}{(c+d x) (b c-a d)}+\frac {b \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (a^2 d^2 f (-5 c f h+4 d e h+d f g)-2 a b d \left (5 c^2 f^2 h-5 c d f (2 e h+f g)+2 d^2 e (2 e h+3 f g)\right )+b^2 \left (c^3 \left (-f^2\right ) h+c^2 d f (8 e h+5 f g)-4 c d^2 e (2 e h+5 f g)+16 d^3 e^2 g\right )\right )}{\sqrt {d} (b c-a d) \sqrt {d e-c f}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right ) \left (a^3 d^2 f^2 h-a^2 b d f (-10 c f h+8 d e h+5 d f g)+a b^2 \left (5 c^2 f^2 h-10 c d f (2 e h+f g)+4 d^2 e (2 e h+5 f g)\right )-b^3 \left (c^2 f (4 e h+f g)-4 c d e (2 e h+3 f g)+16 d^2 e^2 g\right )\right )}{\sqrt {b} (b c-a d) \sqrt {b e-a f}}\right )}{b c-a d}\right )}{b c-a d}}{2 b (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

Input:

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

Output:

-1/2*((b*g - a*h)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(a + b*x)^2*(c + d*x)^2) 
 + (((a^2*d*f*h + b^2*(8*d*e*g - 3*c*f*g - 4*c*e*h) - a*b*(5*d*f*g + 4*d*e 
*h - 7*c*f*h))*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)*(c + d*x)^2) + ((2* 
(a^2*d^2*f*h - a*b*d*(5*d*f*g + 6*d*e*h - 9*c*f*h) + b^2*(12*d^2*e*g + 2*c 
^2*f*h - c*d*(7*f*g + 6*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(c + d*x)^2) + 
(3*b*((2*(a^2*d^2*f*h + 2*a*b*d*(3*c*f*h - 2*d*(f*g + e*h)) + b^2*(8*d^2*e 
*g + c^2*f*h - 4*c*d*(f*g + e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(c + d*x)) 
+ (b*((2*(a^3*d^2*f^2*h - a^2*b*d*f*(5*d*f*g + 8*d*e*h - 10*c*f*h) + a*b^2 
*(5*c^2*f^2*h - 10*c*d*f*(f*g + 2*e*h) + 4*d^2*e*(5*f*g + 2*e*h)) - b^3*(1 
6*d^2*e^2*g - 4*c*d*e*(3*f*g + 2*e*h) + c^2*f*(f*g + 4*e*h)))*ArcTanh[(Sqr 
t[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b*c - a*d)*Sqrt[b*e - a*f] 
) + (2*(a^2*d^2*f*(d*f*g + 4*d*e*h - 5*c*f*h) - 2*a*b*d*(5*c^2*f^2*h - 5*c 
*d*f*(f*g + 2*e*h) + 2*d^2*e*(3*f*g + 2*e*h)) + b^2*(16*d^3*e^2*g - c^3*f^ 
2*h - 4*c*d^2*e*(5*f*g + 2*e*h) + c^2*d*f*(5*f*g + 8*e*h)))*ArcTanh[(Sqrt[ 
d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(Sqrt[d]*(b*c - a*d)*Sqrt[d*e - c*f])) 
)/(b*c - a*d)))/(b*c - a*d))/(2*b*(b*c - a*d)))/(4*b*(b*c - a*d))
 

Defintions of rubi rules used

rule 27

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

rule 73

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

rule 166

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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

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 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 = 2.94 (sec) , antiderivative size = 775, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {\frac {3 \left (\left (-16 d^{2} e^{2} g +8 c \left (e h +\frac {3 f g}{2}\right ) e d -4 c^{2} e f h -g \,f^{2} c^{2}\right ) b^{3}+5 a \left (4 \left (\frac {2}{5} e^{2} h +e f g \right ) d^{2}+2 c \left (-2 e f h -f^{2} g \right ) d +c^{2} f^{2} h \right ) b^{2}+10 a^{2} d \left (\left (-\frac {4 e h}{5}-\frac {f g}{2}\right ) d +c f h \right ) f b +a^{3} d^{2} f^{2} h \right ) \sqrt {\left (c f -d e \right ) d}\, \left (x d +c \right )^{2} \left (b x +a \right )^{2} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4}+3 \sqrt {\left (a f -b e \right ) b}\, \left (-\frac {5 \left (\left (-\frac {16 d^{3} e^{2} g}{5}+4 c \left (\frac {2}{5} e^{2} h +e f g \right ) d^{2}+c^{2} \left (-\frac {8}{5} e f h -f^{2} g \right ) d +\frac {c^{3} f^{2} h}{5}\right ) b^{2}+2 a d \left (\frac {2 \left (2 e^{2} h +3 e f g \right ) d^{2}}{5}+c \left (-2 e f h -f^{2} g \right ) d +c^{2} f^{2} h \right ) b +a^{2} d^{2} \left (\frac {\left (-4 e h -f g \right ) d}{5}+c f h \right ) f \right ) \left (x d +c \right )^{2} \left (b x +a \right )^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{4}+\left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}\, \left (\left (2 d^{3} e g \,x^{3}+3 x^{2} c \left (-\frac {f g x}{3}+e \left (-\frac {h x}{3}+g \right )\right ) d^{2}+\frac {2 x \,c^{2} \left (\frac {\left (3 h \,x^{2}-19 g x \right ) f}{8}+e \left (-\frac {9 h x}{4}+g \right )\right ) d}{3}-\frac {c^{3} \left (\frac {5 x \left (-h x +g \right ) f}{2}+e \left (2 h x +g \right )\right )}{6}\right ) b^{3}-\frac {a \left (6 \left (f g \,x^{3}-3 x^{2} \left (-\frac {h x}{3}+g \right ) e \right ) d^{3}-28 \left (-\frac {17 x \left (-\frac {9 h x}{17}+g \right ) f}{28}+e \left (-\frac {9 h x}{14}+g \right )\right ) x c \,d^{2}-7 c^{2} \left (-\frac {31 x \left (-h x +g \right ) f}{14}+e \left (-\frac {16 h x}{7}+g \right )\right ) d +c^{3} \left (\frac {\left (-19 h x +3 g \right ) f}{2}+e h \right )\right ) b^{2}}{6}+a^{2} \left (\frac {2 x \left (\frac {\left (3 h \,x^{2}-19 g x \right ) f}{8}+e \left (-\frac {9 h x}{4}+g \right )\right ) d^{3}}{3}+\frac {7 c \left (-\frac {31 x \left (-h x +g \right ) f}{14}+e \left (-\frac {16 h x}{7}+g \right )\right ) d^{2}}{6}-\frac {5 c^{2} \left (\frac {\left (-17 h x +9 g \right ) f}{10}+e h \right ) d}{3}+c^{3} f h \right ) b +a^{3} d \left (\frac {\left (-\frac {5 x \left (-h x +g \right ) f}{2}-e \left (2 h x +g \right )\right ) d^{2}}{6}-\frac {c \left (\frac {\left (-19 h x +3 g \right ) f}{2}+e h \right ) d}{6}+c^{2} f h \right )\right )\right )}{\sqrt {\left (c f -d e \right ) d}\, \sqrt {\left (a f -b e \right ) b}\, \left (a d -b c \right )^{5} \left (b x +a \right )^{2} \left (x d +c \right )^{2}}\)	\(775\)
derivativedivides	\(\text {Expression too large to display}\)	\(1224\)
default	\(\text {Expression too large to display}\)	\(1224\)

Input:

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

Output:

3/((c*f-d*e)*d)^(1/2)*(1/4*((-16*d^2*e^2*g+8*c*(e*h+3/2*f*g)*e*d-4*c^2*e*f 
*h-g*f^2*c^2)*b^3+5*a*(4*(2/5*e^2*h+e*f*g)*d^2+2*c*(-2*e*f*h-f^2*g)*d+c^2* 
f^2*h)*b^2+10*a^2*d*((-4/5*e*h-1/2*f*g)*d+c*f*h)*f*b+a^3*d^2*f^2*h)*((c*f- 
d*e)*d)^(1/2)*(d*x+c)^2*(b*x+a)^2*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/ 
2))+((a*f-b*e)*b)^(1/2)*(-5/4*((-16/5*d^3*e^2*g+4*c*(2/5*e^2*h+e*f*g)*d^2+ 
c^2*(-8/5*e*f*h-f^2*g)*d+1/5*c^3*f^2*h)*b^2+2*a*d*(2/5*(2*e^2*h+3*e*f*g)*d 
^2+c*(-2*e*f*h-f^2*g)*d+c^2*f^2*h)*b+a^2*d^2*(1/5*(-4*e*h-f*g)*d+c*f*h)*f) 
*(d*x+c)^2*(b*x+a)^2*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+(a*d-b*c) 
*((c*f-d*e)*d)^(1/2)*(f*x+e)^(1/2)*((2*d^3*e*g*x^3+3*x^2*c*(-1/3*f*g*x+e*( 
-1/3*h*x+g))*d^2+2/3*x*c^2*(1/8*(3*h*x^2-19*g*x)*f+e*(-9/4*h*x+g))*d-1/6*c 
^3*(5/2*x*(-h*x+g)*f+e*(2*h*x+g)))*b^3-1/6*a*(6*(f*g*x^3-3*x^2*(-1/3*h*x+g 
)*e)*d^3-28*(-17/28*x*(-9/17*h*x+g)*f+e*(-9/14*h*x+g))*x*c*d^2-7*c^2*(-31/ 
14*x*(-h*x+g)*f+e*(-16/7*h*x+g))*d+c^3*(1/2*(-19*h*x+3*g)*f+e*h))*b^2+a^2* 
(2/3*x*(1/8*(3*h*x^2-19*g*x)*f+e*(-9/4*h*x+g))*d^3+7/6*c*(-31/14*x*(-h*x+g 
)*f+e*(-16/7*h*x+g))*d^2-5/3*c^2*(1/10*(-17*h*x+9*g)*f+e*h)*d+c^3*f*h)*b+a 
^3*d*(1/6*(-5/2*x*(-h*x+g)*f-e*(2*h*x+g))*d^2-1/6*c*(1/2*(-19*h*x+3*g)*f+e 
*h)*d+c^2*f*h))))/((a*f-b*e)*b)^(1/2)/(a*d-b*c)^5/(b*x+a)^2/(d*x+c)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6917 vs. \(2 (646) = 1292\).

Time = 54.03 (sec) , antiderivative size = 27720, normalized size of antiderivative = 40.41 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^3/(d*x+c)^3,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. 2182 vs. \(2 (646) = 1292\).

Time = 0.41 (sec) , antiderivative size = 2182, normalized size of antiderivative = 3.18 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

3/4*(16*b^3*d^2*e^2*g - 12*b^3*c*d*e*f*g - 20*a*b^2*d^2*e*f*g + b^3*c^2*f^ 
2*g + 10*a*b^2*c*d*f^2*g + 5*a^2*b*d^2*f^2*g - 8*b^3*c*d*e^2*h - 8*a*b^2*d 
^2*e^2*h + 4*b^3*c^2*e*f*h + 20*a*b^2*c*d*e*f*h + 8*a^2*b*d^2*e*f*h - 5*a* 
b^2*c^2*f^2*h - 10*a^2*b*c*d*f^2*h - a^3*d^2*f^2*h)*arctan(sqrt(f*x + e)*b 
/sqrt(-b^2*e + a*b*f))/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10 
*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*sqrt(-b^2*e + a*b*f)) - 3/4*(1 
6*b^2*d^3*e^2*g - 20*b^2*c*d^2*e*f*g - 12*a*b*d^3*e*f*g + 5*b^2*c^2*d*f^2* 
g + 10*a*b*c*d^2*f^2*g + a^2*d^3*f^2*g - 8*b^2*c*d^2*e^2*h - 8*a*b*d^3*e^2 
*h + 8*b^2*c^2*d*e*f*h + 20*a*b*c*d^2*e*f*h + 4*a^2*d^3*e*f*h - b^2*c^3*f^ 
2*h - 10*a*b*c^2*d*f^2*h - 5*a^2*c*d^2*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt( 
-d^2*e + c*d*f))/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b 
^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*sqrt(-d^2*e + c*d*f)) + 1/4*(24*(f*x 
 + e)^(7/2)*b^3*d^3*e*f*g - 72*(f*x + e)^(5/2)*b^3*d^3*e^2*f*g + 72*(f*x + 
 e)^(3/2)*b^3*d^3*e^3*f*g - 24*sqrt(f*x + e)*b^3*d^3*e^4*f*g - 12*(f*x + e 
)^(7/2)*b^3*c*d^2*f^2*g - 12*(f*x + e)^(7/2)*a*b^2*d^3*f^2*g + 72*(f*x + e 
)^(5/2)*b^3*c*d^2*e*f^2*g + 72*(f*x + e)^(5/2)*a*b^2*d^3*e*f^2*g - 108*(f* 
x + e)^(3/2)*b^3*c*d^2*e^2*f^2*g - 108*(f*x + e)^(3/2)*a*b^2*d^3*e^2*f^2*g 
 + 48*sqrt(f*x + e)*b^3*c*d^2*e^3*f^2*g + 48*sqrt(f*x + e)*a*b^2*d^3*e^3*f 
^2*g - 19*(f*x + e)^(5/2)*b^3*c^2*d*f^3*g - 34*(f*x + e)^(5/2)*a*b^2*c*d^2 
*f^3*g - 19*(f*x + e)^(5/2)*a^2*b*d^3*f^3*g + 46*(f*x + e)^(3/2)*b^3*c^...
 

Mupad [B] (verification not implemented)

Time = 153.83 (sec) , antiderivative size = 133067, normalized size of antiderivative = 193.98 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(log((((((3*b^2*d^2*f^3*(a^2*d^2*f^2*g + b^2*c^2*f^2*g + 8*b^2*d^2*e^2*g - 
 4*a*b*c^2*f^2*h - 4*a*b*d^2*e^2*h - 4*a^2*c*d*f^2*h - 4*b^2*c*d*e^2*h + 3 
*a^2*d^2*e*f*h + 3*b^2*c^2*e*f*h + 6*a*b*c*d*f^2*g - 8*a*b*d^2*e*f*g - 8*b 
^2*c*d*e*f*g + 10*a*b*c*d*e*f*h))/(a*d - b*c)^2 - (b^2*d^2*f^2*(e + f*x)^( 
1/2)*(a*d - b*c)^2*(a*d*f + b*c*f - 2*b*d*e)*(-(288*(f^8*(a*d - b*c)^30*(a 
*d*e*h^2 - a*c*f*h^2 + b*c*e*h^2 + b*d*f*g^2 - 2*b*d*e*g*h)^2)^(1/2) - 147 
456*a^10*b^6*d^16*e^5*g^2 - 36864*a^12*b^4*d^16*e^5*h^2 - 147456*b^16*c^10 
*d^6*e^5*g^2 - 36864*b^16*c^12*d^4*e^5*h^2 + 288*a*b^15*c^16*f^5*h^2 + 288 
*a^15*b*d^16*f^5*g^2 + 288*a^16*c*d^15*f^5*h^2 + 288*b^16*c^15*d*f^5*g^2 - 
 288*a^16*d^16*e*f^4*h^2 - 288*b^16*c^16*e*f^4*h^2 + 147456*a^11*b^5*d^16* 
e^5*g*h + 147456*b^16*c^11*d^5*e^5*g*h + 1474560*a*b^15*c^9*d^7*e^5*g^2 + 
1474560*a^9*b^7*c*d^15*e^5*g^2 + 294912*a*b^15*c^11*d^5*e^5*h^2 + 10080*a* 
b^15*c^14*d^2*f^5*g^2 + 294912*a^11*b^5*c*d^15*e^5*h^2 + 10080*a^14*b^2*c* 
d^15*f^5*g^2 + 10080*a^2*b^14*c^15*d*f^5*h^2 + 10080*a^15*b*c^2*d^14*f^5*h 
^2 + 368640*a^11*b^5*d^16*e^4*f*g^2 - 14400*a^14*b^2*d^16*e*f^4*g^2 + 7372 
8*a^13*b^3*d^16*e^4*f*h^2 + 9216*a^15*b*d^16*e^2*f^3*h^2 + 368640*b^16*c^1 
1*d^5*e^4*f*g^2 - 14400*b^16*c^14*d^2*e*f^4*g^2 + 73728*b^16*c^13*d^3*e^4* 
f*h^2 + 9216*b^16*c^15*d*e^2*f^3*h^2 - 6635520*a^2*b^14*c^8*d^8*e^5*g^2 + 
17694720*a^3*b^13*c^7*d^9*e^5*g^2 - 30965760*a^4*b^12*c^6*d^10*e^5*g^2 + 3 
7158912*a^5*b^11*c^5*d^11*e^5*g^2 - 30965760*a^6*b^10*c^4*d^12*e^5*g^2 ...
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 19308, normalized size of antiderivative = 28.15 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) 
))*a**5*c**3*d**3*f**3*h - 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b 
)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*c**2*d**4*e*f**2*h + 6*sqrt(b)*sqrt(a*f 
- b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*c**2*d**4*f* 
*3*h*x - 6*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a* 
f - b*e)))*a**5*c*d**5*e*f**2*h*x + 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e 
 + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*c*d**5*f**3*h*x**2 - 3*sqrt(b)* 
sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*d** 
6*e*f**2*h*x**2 + 30*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt( 
b)*sqrt(a*f - b*e)))*a**4*b*c**4*d**2*f**3*h - 54*sqrt(b)*sqrt(a*f - b*e)* 
atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*c**3*d**3*e*f**2* 
h - 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
b*e)))*a**4*b*c**3*d**3*f**3*g + 66*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
 f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*c**3*d**3*f**3*h*x + 24*sqrt(b) 
*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b* 
c**2*d**4*e**2*f*h + 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq 
rt(b)*sqrt(a*f - b*e)))*a**4*b*c**2*d**4*e*f**2*g - 114*sqrt(b)*sqrt(a*f - 
 b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*c**2*d**4*e 
*f**2*h*x - 30*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqr 
t(a*f - b*e)))*a**4*b*c**2*d**4*f**3*g*x + 42*sqrt(b)*sqrt(a*f - b*e)*a...