\(\int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx\) [57]

Optimal result

Integrand size = 29, antiderivative size = 762 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx=\frac {d \left (5 a^2 d f h-a b (11 d f g+6 d e h-7 c f h)+b^2 (12 d e g-c (f g+6 e h))\right ) \sqrt {e+f x}}{4 b (b c-a d)^3 (b e-a f) (c+d x)^2}-\frac {(b g-a h) \sqrt {e+f x}}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}+\frac {\left (3 a^2 d f h-a b (7 d f g+4 d e h-5 c f h)+b^2 (8 d e g-c (f g+4 e h))\right ) \sqrt {e+f x}}{4 b (b c-a d)^2 (b e-a f) (a+b x) (c+d x)^2}+\frac {d \left (a^2 d f (d f g+11 d e h-12 c f h)+b^2 \left (24 d^2 e^2 g-12 c d e (2 f g+e h)+c^2 f (f g+11 e h)\right )-2 a b \left (6 c^2 f^2 h+6 d^2 e (2 f g+e h)-c d f (11 f g+13 e h)\right )\right ) \sqrt {e+f x}}{4 (b c-a d)^4 (b e-a f) (d e-c f) (c+d x)}+\frac {\sqrt {b} \left (15 a^3 d^2 f^2 h-5 a^2 b d f (7 d f g+8 d e h-6 c f h)-b^3 \left (48 d^2 e^2 g-c^2 f (f g-4 e h)-12 c d e (f g+2 e h)\right )+a b^2 \left (3 c^2 f^2 h+12 d^2 e (7 f g+2 e h)-2 c d f (7 f g+26 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{4 (b c-a d)^5 (b e-a f)^{3/2}}-\frac {\sqrt {d} \left (a^2 d^2 f (d f g-4 d e h+3 c f h)-b^2 \left (48 d^3 e^2 g-15 c^3 f^2 h-12 c d^2 e (7 f g+2 e h)+5 c^2 d f (7 f g+8 e h)\right )+2 a b d \left (15 c^2 f^2 h+6 d^2 e (f g+2 e h)-c d f (7 f g+26 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 (b c-a d)^5 (d e-c f)^{3/2}} \] Output:

1/4*d*(5*a^2*d*f*h-a*b*(-7*c*f*h+6*d*e*h+11*d*f*g)+b^2*(12*d*e*g-c*(6*e*h+ 
f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c)^3/(-a*f+b*e)/(d*x+c)^2-1/2*(-a*h+b*g)*(f 
*x+e)^(1/2)/b/(-a*d+b*c)/(b*x+a)^2/(d*x+c)^2+1/4*(3*a^2*d*f*h-a*b*(-5*c*f* 
h+4*d*e*h+7*d*f*g)+b^2*(8*d*e*g-c*(4*e*h+f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c) 
^2/(-a*f+b*e)/(b*x+a)/(d*x+c)^2+1/4*d*(a^2*d*f*(-12*c*f*h+11*d*e*h+d*f*g)+ 
b^2*(24*d^2*e^2*g-12*c*d*e*(e*h+2*f*g)+c^2*f*(11*e*h+f*g))-2*a*b*(6*c^2*f^ 
2*h+6*d^2*e*(e*h+2*f*g)-c*d*f*(13*e*h+11*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c)^4 
/(-a*f+b*e)/(-c*f+d*e)/(d*x+c)+1/4*b^(1/2)*(15*a^3*d^2*f^2*h-5*a^2*b*d*f*( 
-6*c*f*h+8*d*e*h+7*d*f*g)-b^3*(48*d^2*e^2*g-c^2*f*(-4*e*h+f*g)-12*c*d*e*(2 
*e*h+f*g))+a*b^2*(3*c^2*f^2*h+12*d^2*e*(2*e*h+7*f*g)-2*c*d*f*(26*e*h+7*f*g 
)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/(-a*d+b*c)^5/(-a*f+b*e 
)^(3/2)-1/4*d^(1/2)*(a^2*d^2*f*(3*c*f*h-4*d*e*h+d*f*g)-b^2*(48*d^3*e^2*g-1 
5*c^3*f^2*h-12*c*d^2*e*(2*e*h+7*f*g)+5*c^2*d*f*(8*e*h+7*f*g))+2*a*b*d*(15* 
c^2*f^2*h+6*d^2*e*(2*e*h+f*g)-c*d*f*(26*e*h+7*f*g)))*arctanh(d^(1/2)*(f*x+ 
e)^(1/2)/(-c*f+d*e)^(1/2))/(-a*d+b*c)^5/(-c*f+d*e)^(3/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(6193\) vs. \(2(762)=1524\).

Time = 16.29 (sec) , antiderivative size = 6193, normalized size of antiderivative = 8.13 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 817, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {166, 27, 168, 27, 168, 27, 168, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-5 c f h+4 d e h+7 d f g)+b^2 (-4 c e h-c f g+8 d e g)\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int -\frac {5 d f (4 d e-c f) h a^2-b \left (4 e (11 f g+6 e h) d^2-c f (9 f g+32 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-f (f g-4 e h) c^2-12 d e (f g+2 e h) c+48 d^2 e^2 g\right )+5 d f \left (3 d f h a^2-b (7 d f g+4 d e h-5 c f h) a+b^2 (8 d e g-c (f g+4 e h))\right ) x}{2 (a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {5 d f (4 d e-c f) h a^2-b \left (4 e (11 f g+6 e h) d^2-c f (9 f g+32 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-f (f g-4 e h) c^2-12 d e (f g+2 e h) c+48 d^2 e^2 g\right )+5 d f \left (3 d f h a^2-b (7 d f g+4 d e h-5 c f h) a+b^2 (8 d e g-c (f g+4 e h))\right ) x}{(a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-5 c f h+4 d e h+7 d f g)+b^2 (-4 c e h-c f g+8 d e g)\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {\int \frac {2 b (d e-c f) \left (d f (2 d f g+22 d e h-9 c f h) a^2-b \left (24 e (2 f g+e h) d^2-c f (11 f g+34 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-f (f g-4 e h) c^2-12 d e (f g+2 e h) c+48 d^2 e^2 g\right )+3 d f \left (5 d f h a^2-b (11 d f g+6 d e h-7 c f h) a+b^2 (12 d e g-c (f g+6 e h))\right ) x\right )}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (d e-c f)}+\frac {2 d \sqrt {e+f x} \left (5 a^2 d f h-a b (-7 c f h+6 d e h+11 d f g)+b^2 (-6 c e h-c f g+12 d e g)\right )}{(c+d x)^2 (b c-a d)}}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-5 c f h+4 d e h+7 d f g)+b^2 (-4 c e h-c f g+8 d e g)\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {b \int \frac {d f (2 d f g+22 d e h-9 c f h) a^2-b \left (24 e (2 f g+e h) d^2-c f (11 f g+34 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-f (f g-4 e h) c^2-12 d e (f g+2 e h) c+48 d^2 e^2 g\right )+3 d f \left (5 d f h a^2-b (11 d f g+6 d e h-7 c f h) a+b^2 (12 d e g-c (f g+6 e h))\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{b c-a d}+\frac {2 d \sqrt {e+f x} \left (5 a^2 d f h-a b (-7 c f h+6 d e h+11 d f g)+b^2 (-6 c e h-c f g+12 d e g)\right )}{(c+d x)^2 (b c-a d)}}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-5 c f h+4 d e h+7 d f g)+b^2 (-4 c e h-c f g+8 d e g)\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {b \left (\frac {\int \frac {d^2 f^2 (d f g-4 d e h+3 c f h) a^3+b d f \left (e (11 f g+28 e h) d^2-c f (13 f g+44 e h) d+18 c^2 f^2 h\right ) a^2+b^2 \left (-12 e^2 (5 f g+2 e h) d^3+2 c e f (37 f g+32 e h) d^2-c^2 f^2 (13 f g+44 e h) d+3 c^3 f^3 h\right ) a+b^3 (d e-c f) \left (-f (f g-4 e h) c^2-12 d e (f g+2 e h) c+48 d^2 e^2 g\right )+b d f \left (d f (d f g+11 d e h-12 c f h) a^2-2 b \left (6 e (2 f g+e h) d^2-c f (11 f g+13 e h) d+6 c^2 f^2 h\right ) a+b^2 \left (f (f g+11 e h) c^2-12 d e (2 f g+e h) c+24 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}+\frac {2 d \sqrt {e+f x} \left (a^2 d f (-12 c f h+11 d e h+d f g)-2 a b \left (6 c^2 f^2 h-c d f (13 e h+11 f g)+6 d^2 e (e h+2 f g)\right )+b^2 \left (c^2 f (11 e h+f g)-12 c d e (e h+2 f g)+24 d^2 e^2 g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}\right )}{b c-a d}+\frac {2 d \sqrt {e+f x} \left (5 a^2 d f h-a b (-7 c f h+6 d e h+11 d f g)+b^2 (-6 c e h-c f g+12 d e g)\right )}{(c+d x)^2 (b c-a d)}}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-5 c f h+4 d e h+7 d f g)+b^2 (-4 c e h-c f g+8 d e g)\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {b \left (\frac {\frac {d (b e-a f) \left (a^2 d^2 f (3 c f h-4 d e h+d f g)+2 a b d \left (15 c^2 f^2 h-c d f (26 e h+7 f g)+6 d^2 e (2 e h+f g)\right )-\left (b^2 \left (-15 c^3 f^2 h+5 c^2 d f (8 e h+7 f g)-12 c d^2 e (2 e h+7 f g)+48 d^3 e^2 g\right )\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}-\frac {b (d e-c f) \left (15 a^3 d^2 f^2 h-5 a^2 b d f (-6 c f h+8 d e h+7 d f g)+a b^2 \left (3 c^2 f^2 h-2 c d f (26 e h+7 f g)+12 d^2 e (2 e h+7 f g)\right )-b^3 \left (c^2 (-f) (f g-4 e h)-12 c d e (2 e h+f g)+48 d^2 e^2 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}}{(b c-a d) (d e-c f)}+\frac {2 d \sqrt {e+f x} \left (a^2 d f (-12 c f h+11 d e h+d f g)-2 a b \left (6 c^2 f^2 h-c d f (13 e h+11 f g)+6 d^2 e (e h+2 f g)\right )+b^2 \left (c^2 f (11 e h+f g)-12 c d e (e h+2 f g)+24 d^2 e^2 g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}\right )}{b c-a d}+\frac {2 d \sqrt {e+f x} \left (5 a^2 d f h-a b (-7 c f h+6 d e h+11 d f g)+b^2 (-6 c e h-c f g+12 d e g)\right )}{(c+d x)^2 (b c-a d)}}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-5 c f h+4 d e h+7 d f g)+b^2 (-4 c e h-c f g+8 d e g)\right )}{(a+b x) (c+d x)^2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{2 b (a+b x)^2 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 d f h a^2-b (7 d f g+4 d e h-5 c f h) a+b^2 (8 d e g-c f g-4 c e h)\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (5 d f h a^2-b (11 d f g+6 d e h-7 c f h) a+b^2 (12 d e g-c f g-6 c e h)\right )}{(b c-a d) (c+d x)^2}+\frac {b \left (\frac {2 d \sqrt {e+f x} \left (d f (d f g+11 d e h-12 c f h) a^2-2 b \left (6 e (2 f g+e h) d^2-c f (11 f g+13 e h) d+6 c^2 f^2 h\right ) a+b^2 \left (f (f g+11 e h) c^2-12 d e (2 f g+e h) c+24 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {2 d (b e-a f) \left (-\left (\left (-15 f^2 h c^3+5 d f (7 f g+8 e h) c^2-12 d^2 e (7 f g+2 e h) c+48 d^3 e^2 g\right ) b^2\right )+2 a d \left (6 e (f g+2 e h) d^2-c f (7 f g+26 e h) d+15 c^2 f^2 h\right ) b+a^2 d^2 f (d f g-4 d e h+3 c f h)\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{(b c-a d) f}-\frac {2 b (d e-c f) \left (15 d^2 f^2 h a^3-5 b d f (7 d f g+8 d e h-6 c f h) a^2+b^2 \left (12 e (7 f g+2 e h) d^2-2 c f (7 f g+26 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (f g-4 e h) c^2-12 d e (f g+2 e h) c+48 d^2 e^2 g\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{(b c-a d) f}}{(b c-a d) (d e-c f)}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 d f h a^2-b (7 d f g+4 d e h-5 c f h) a+b^2 (8 d e g-c f g-4 c e h)\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\frac {2 d \sqrt {e+f x} \left (5 d f h a^2-b (11 d f g+6 d e h-7 c f h) a+b^2 (12 d e g-c f g-6 c e h)\right )}{(b c-a d) (c+d x)^2}+\frac {b \left (\frac {2 d \sqrt {e+f x} \left (d f (d f g+11 d e h-12 c f h) a^2-2 b \left (6 e (2 f g+e h) d^2-c f (11 f g+13 e h) d+6 c^2 f^2 h\right ) a+b^2 \left (f (f g+11 e h) c^2-12 d e (2 f g+e h) c+24 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {2 \sqrt {b} (d e-c f) \left (15 d^2 f^2 h a^3-5 b d f (7 d f g+8 d e h-6 c f h) a^2+b^2 \left (12 e (7 f g+2 e h) d^2-2 c f (7 f g+26 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (f g-4 e h) c^2-12 d e (f g+2 e h) c+48 d^2 e^2 g\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) \sqrt {b e-a f}}-\frac {2 \sqrt {d} (b e-a f) \left (-\left (\left (-15 f^2 h c^3+5 d f (7 f g+8 e h) c^2-12 d^2 e (7 f g+2 e h) c+48 d^3 e^2 g\right ) b^2\right )+2 a d \left (6 e (f g+2 e h) d^2-c f (7 f g+26 e h) d+15 c^2 f^2 h\right ) b+a^2 d^2 f (d f g-4 d e h+3 c f h)\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d) \sqrt {d e-c f}}}{(b c-a d) (d e-c f)}\right )}{b c-a d}}{2 (b c-a d) (b e-a f)}}{4 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{2 b (b c-a d) (a+b x)^2 (c+d x)^2}\)

Input:

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

Output:

-1/2*((b*g - a*h)*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)^2*(c + d*x)^2) + 
 (((3*a^2*d*f*h + b^2*(8*d*e*g - c*f*g - 4*c*e*h) - a*b*(7*d*f*g + 4*d*e*h 
 - 5*c*f*h))*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)^2 
) + ((2*d*(5*a^2*d*f*h + b^2*(12*d*e*g - c*f*g - 6*c*e*h) - a*b*(11*d*f*g 
+ 6*d*e*h - 7*c*f*h))*Sqrt[e + f*x])/((b*c - a*d)*(c + d*x)^2) + (b*((2*d* 
(a^2*d*f*(d*f*g + 11*d*e*h - 12*c*f*h) + b^2*(24*d^2*e^2*g - 12*c*d*e*(2*f 
*g + e*h) + c^2*f*(f*g + 11*e*h)) - 2*a*b*(6*c^2*f^2*h + 6*d^2*e*(2*f*g + 
e*h) - c*d*f*(11*f*g + 13*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(d*e - c*f)*( 
c + d*x)) + ((2*Sqrt[b]*(d*e - c*f)*(15*a^3*d^2*f^2*h - 5*a^2*b*d*f*(7*d*f 
*g + 8*d*e*h - 6*c*f*h) - b^3*(48*d^2*e^2*g - c^2*f*(f*g - 4*e*h) - 12*c*d 
*e*(f*g + 2*e*h)) + a*b^2*(3*c^2*f^2*h + 12*d^2*e*(7*f*g + 2*e*h) - 2*c*d* 
f*(7*f*g + 26*e*h)))*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/((b 
*c - a*d)*Sqrt[b*e - a*f]) - (2*Sqrt[d]*(b*e - a*f)*(a^2*d^2*f*(d*f*g - 4* 
d*e*h + 3*c*f*h) - b^2*(48*d^3*e^2*g - 15*c^3*f^2*h - 12*c*d^2*e*(7*f*g + 
2*e*h) + 5*c^2*d*f*(7*f*g + 8*e*h)) + 2*a*b*d*(15*c^2*f^2*h + 6*d^2*e*(f*g 
 + 2*e*h) - c*d*f*(7*f*g + 26*e*h)))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[ 
d*e - c*f]])/((b*c - a*d)*Sqrt[d*e - c*f]))/((b*c - a*d)*(d*e - c*f))))/(b 
*c - a*d))/(2*(b*c - a*d)*(b*e - a*f)))/(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 = 13.44 (sec) , antiderivative size = 1019, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1019\)
default	\(\text {Expression too large to display}\)	\(1019\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(1099\)

Input:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

2*f^4*(-d/f^4/(a*d-b*c)^5*((1/8*d*f*(5*a^2*c*d^2*f*h-4*a^2*d^3*e*h-a^2*d^3 
*f*g+2*a*b*c^2*d*f*h-4*a*b*c*d^2*e*h-10*a*b*c*d^2*f*g+12*a*b*d^3*e*g-7*b^2 
*c^3*f*h+8*b^2*c^2*d*e*h+11*b^2*c^2*d*f*g-12*b^2*c*d^2*e*g)/(c*f-d*e)*(f*x 
+e)^(3/2)+1/8*f*(3*a^2*c*d^2*f*h-4*a^2*d^3*e*h+a^2*d^3*f*g+6*a*b*c^2*d*f*h 
-4*a*b*c*d^2*e*h-14*a*b*c*d^2*f*g+12*a*b*d^3*e*g-9*b^2*c^3*f*h+8*b^2*c^2*d 
*e*h+13*b^2*c^2*d*f*g-12*b^2*c*d^2*e*g)*(f*x+e)^(1/2))/((f*x+e)*d+c*f-d*e) 
^2-1/8*(3*a^2*c*d^2*f^2*h-4*a^2*d^3*e*f*h+a^2*d^3*f^2*g+30*a*b*c^2*d*f^2*h 
-52*a*b*c*d^2*e*f*h-14*a*b*c*d^2*f^2*g+24*a*b*d^3*e^2*h+12*a*b*d^3*e*f*g+1 
5*b^2*c^3*f^2*h-40*b^2*c^2*d*e*f*h-35*b^2*c^2*d*f^2*g+24*b^2*c*d^2*e^2*h+8 
4*b^2*c*d^2*e*f*g-48*b^2*d^3*e^2*g)/(c*f-d*e)/((c*f-d*e)*d)^(1/2)*arctan(d 
*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))-b/f^4/(a*d-b*c)^5*((1/8*b*f*(7*a^3*d^ 
2*f*h-2*a^2*b*c*d*f*h-8*a^2*b*d^2*e*h-11*a^2*b*d^2*f*g-5*a*b^2*c^2*f*h+4*a 
*b^2*c*d*e*h+10*a*b^2*c*d*f*g+12*a*b^2*d^2*e*g+4*b^3*c^2*e*h+b^3*c^2*f*g-1 
2*b^3*c*d*e*g)/(a*f-b*e)*(f*x+e)^(3/2)+1/8*f*(9*a^3*d^2*f*h-6*a^2*b*c*d*f* 
h-8*a^2*b*d^2*e*h-13*a^2*b*d^2*f*g-3*a*b^2*c^2*f*h+4*a*b^2*c*d*e*h+14*a*b^ 
2*c*d*f*g+12*a*b^2*d^2*e*g+4*b^3*c^2*e*h-b^3*c^2*f*g-12*b^3*c*d*e*g)*(f*x+ 
e)^(1/2))/((f*x+e)*b+a*f-b*e)^2+1/8*(15*a^3*d^2*f^2*h+30*a^2*b*c*d*f^2*h-4 
0*a^2*b*d^2*e*f*h-35*a^2*b*d^2*f^2*g+3*a*b^2*c^2*f^2*h-52*a*b^2*c*d*e*f*h- 
14*a*b^2*c*d*f^2*g+24*a*b^2*d^2*e^2*h+84*a*b^2*d^2*e*f*g-4*b^3*c^2*e*f*h+b 
^3*c^2*f^2*g+24*b^3*c*d*e^2*h+12*b^3*c*d*e*f*g-48*b^3*d^2*e^2*g)/(a*f-b...
 

Fricas [F(-1)]

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

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((f*x+e)**(1/2)*(h*x+g)/(b*x+a)**3/(d*x+c)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(1/2)*(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. 3442 vs. \(2 (722) = 1444\).

Time = 0.40 (sec) , antiderivative size = 3442, normalized size of antiderivative = 4.52 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")
 

Output:

1/4*(48*b^4*d^2*e^2*g - 12*b^4*c*d*e*f*g - 84*a*b^3*d^2*e*f*g - b^4*c^2*f^ 
2*g + 14*a*b^3*c*d*f^2*g + 35*a^2*b^2*d^2*f^2*g - 24*b^4*c*d*e^2*h - 24*a* 
b^3*d^2*e^2*h + 4*b^4*c^2*e*f*h + 52*a*b^3*c*d*e*f*h + 40*a^2*b^2*d^2*e*f* 
h - 3*a*b^3*c^2*f^2*h - 30*a^2*b^2*c*d*f^2*h - 15*a^3*b*d^2*f^2*h)*arctan( 
sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^6*c^5*e - 5*a*b^5*c^4*d*e + 10*a 
^2*b^4*c^3*d^2*e - 10*a^3*b^3*c^2*d^3*e + 5*a^4*b^2*c*d^4*e - a^5*b*d^5*e 
- a*b^5*c^5*f + 5*a^2*b^4*c^4*d*f - 10*a^3*b^3*c^3*d^2*f + 10*a^4*b^2*c^2* 
d^3*f - 5*a^5*b*c*d^4*f + a^6*d^5*f)*sqrt(-b^2*e + a*b*f)) - 1/4*(48*b^2*d 
^4*e^2*g - 84*b^2*c*d^3*e*f*g - 12*a*b*d^4*e*f*g + 35*b^2*c^2*d^2*f^2*g + 
14*a*b*c*d^3*f^2*g - a^2*d^4*f^2*g - 24*b^2*c*d^3*e^2*h - 24*a*b*d^4*e^2*h 
 + 40*b^2*c^2*d^2*e*f*h + 52*a*b*c*d^3*e*f*h + 4*a^2*d^4*e*f*h - 15*b^2*c^ 
3*d*f^2*h - 30*a*b*c^2*d^2*f^2*h - 3*a^2*c*d^3*f^2*h)*arctan(sqrt(f*x + e) 
*d/sqrt(-d^2*e + c*d*f))/((b^5*c^5*d*e - 5*a*b^4*c^4*d^2*e + 10*a^2*b^3*c^ 
3*d^3*e - 10*a^3*b^2*c^2*d^4*e + 5*a^4*b*c*d^5*e - a^5*d^6*e - b^5*c^6*f + 
 5*a*b^4*c^5*d*f - 10*a^2*b^3*c^4*d^2*f + 10*a^3*b^2*c^3*d^3*f - 5*a^4*b*c 
^2*d^4*f + a^5*c*d^5*f)*sqrt(-d^2*e + c*d*f)) + 1/4*(24*(f*x + e)^(7/2)*b^ 
4*d^4*e^2*f*g - 72*(f*x + e)^(5/2)*b^4*d^4*e^3*f*g + 72*(f*x + e)^(3/2)*b^ 
4*d^4*e^4*f*g - 24*sqrt(f*x + e)*b^4*d^4*e^5*f*g - 24*(f*x + e)^(7/2)*b^4* 
c*d^3*e*f^2*g - 24*(f*x + e)^(7/2)*a*b^3*d^4*e*f^2*g + 108*(f*x + e)^(5/2) 
*b^4*c*d^3*e^2*f^2*g + 108*(f*x + e)^(5/2)*a*b^3*d^4*e^2*f^2*g - 144*(f...
 

Mupad [B] (verification not implemented)

Time = 38.21 (sec) , antiderivative size = 794225, normalized size of antiderivative = 1042.29 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

int(((e + f*x)^(1/2)*(g + h*x))/((a + b*x)^3*(c + d*x)^3),x)
 

Output:

(((e + f*x)^(3/2)*(a^4*d^4*f^5*g + b^4*c^4*f^5*g - 5*a*b^3*c^4*f^5*h - 5*a 
^4*c*d^3*f^5*h + 4*a^4*d^4*e*f^4*h + 4*b^4*c^4*e*f^4*h + 72*b^4*d^4*e^4*f* 
g - 31*a^2*b^2*c^3*d*f^5*h - 31*a^3*b*c^2*d^2*f^5*h - 144*a*b^3*d^4*e^3*f^ 
2*g - 38*a^3*b*d^4*e^2*f^3*h - 144*b^4*c*d^3*e^3*f^2*g - 38*b^4*c^3*d*e^2* 
f^3*h + 52*a^2*b^2*c^2*d^2*f^5*g + 85*a^2*b^2*d^4*e^2*f^3*g + 69*a^2*b^2*d 
^4*e^3*f^2*h + 85*b^4*c^2*d^2*e^2*f^3*g + 69*b^4*c^2*d^2*e^3*f^2*h + 9*a*b 
^3*c^3*d*f^5*g + 9*a^3*b*c*d^3*f^5*g - 13*a^3*b*d^4*e*f^4*g - 36*a*b^3*d^4 
*e^4*f*h - 13*b^4*c^3*d*e*f^4*g - 36*b^4*c*d^3*e^4*f*h + 73*a*b^3*c^3*d*e* 
f^4*h + 73*a^3*b*c*d^3*e*f^4*h + 262*a*b^3*c*d^3*e^2*f^3*g - 131*a*b^3*c^2 
*d^2*e*f^4*g - 131*a^2*b^2*c*d^3*e*f^4*g + 150*a*b^3*c*d^3*e^3*f^2*h - 178 
*a*b^3*c^2*d^2*e^2*f^3*h - 178*a^2*b^2*c*d^3*e^2*f^3*h + 134*a^2*b^2*c^2*d 
^2*e*f^4*h))/(4*(a*c*f^2 + b*d*e^2 - a*d*e*f - b*c*e*f)*(a^4*d^4 + b^4*c^4 
 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - ((e + f*x)^(1/2)* 
(a^3*d^3*f^4*g + b^3*c^3*f^4*g + 3*a*b^2*c^3*f^4*h + 3*a^3*c*d^2*f^4*h - 4 
*a^3*d^3*e*f^3*h - 4*b^3*c^3*e*f^3*h + 24*b^3*d^3*e^3*f*g - 36*a*b^2*d^3*e 
^2*f^2*g + 17*a^2*b*d^3*e^2*f^2*h - 36*b^3*c*d^2*e^2*f^2*g + 17*b^3*c^2*d* 
e^2*f^2*h - 13*a*b^2*c^2*d*f^4*g - 13*a^2*b*c*d^2*f^4*g + 18*a^2*b*c^2*d*f 
^4*h + 10*a^2*b*d^3*e*f^3*g - 12*a*b^2*d^3*e^3*f*h + 10*b^3*c^2*d*e*f^3*g 
- 12*b^3*c*d^2*e^3*f*h + 52*a*b^2*c*d^2*e*f^3*g - 32*a*b^2*c^2*d*e*f^3*h - 
 32*a^2*b*c*d^2*e*f^3*h + 38*a*b^2*c*d^2*e^2*f^2*h))/(4*(a^4*d^4 + b^4*...
 

Reduce [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 26875, normalized size of antiderivative = 35.27 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3/(d*x+c)^3,x)
 

Output:

( - 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
b*e)))*a**5*c**4*d**2*f**4*h + 30*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*e*f**3*h - 30*sqrt(b)*sqr 
t(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*c**3*d 
**3*f**4*h*x - 15*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**2*f**2*h + 60*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**3*h* 
x - 15*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**4*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**5*c*d**5*e**2*f**2*h*x + 30*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**3*h*x**2 - 15*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**2*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**5*d*f 
**4*h + 100*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*e*f**3*h + 35*sqrt(b)*sqrt(a*f - b*e)*atan((s 
qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*c**4*d**2*f**4*g - 90*sq 
rt(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**4*h*x - 110*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**2*f**2*h - 70*sqrt(b)...