\(\int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx\) [49]

Optimal result

Integrand size = 29, antiderivative size = 872 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\frac {d \left (5 a^3 d^2 f^2 h-a^2 b d f (27 d f g+14 d e h-26 c f h)-b^3 \left (32 d^2 e^2 g-c^2 f (f g-2 e h)-4 c d e (f g+6 e h)\right )+a b^2 \left (c^2 f^2 h+4 d^2 e (15 f g+2 e h)-6 c d f (f g+8 e h)\right )\right ) \sqrt {e+f x}}{8 b (b c-a d)^4 (b e-a f)^2 (c+d x)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}+\frac {\left (a^2 d f h-a b (7 d f g+2 d e h-7 c f h)+b^2 (8 d e g-c (f g+6 e h))\right ) \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f) (a+b x)^2 (c+d x)}+\frac {\left (5 a^3 d^2 f^2 h-5 a^2 b d f (7 d f g+4 d e h-8 c f h)-b^3 \left (48 d^2 e^2 g-3 c^2 f (f g-2 e h)-2 c d e (5 f g+18 e h)\right )+a b^2 \left (3 c^2 f^2 h+2 d^2 e (43 f g+6 e h)-2 c d f (8 f g+35 e h)\right )\right ) \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^2 (a+b x) (c+d x)}+\frac {\left (5 a^4 d^3 f^3 h-5 a^3 b d^2 f^2 (7 d f g+6 d e h-9 c f h)+b^4 \left (64 d^3 e^3 g-4 c^2 d e f (f g-4 e h)-c^3 f^2 (f g-2 e h)-24 c d^2 e^2 (f g+2 e h)\right )-a b^3 \left (c^3 f^3 h-c^2 d f^2 (7 f g-34 e h)+8 d^3 e^2 (21 f g+2 e h)-8 c d^2 e f (7 f g+17 e h)\right )+5 a^2 b^2 d f \left (3 c^2 f^2 h+4 d^2 e (7 f g+2 e h)-c d f (7 f g+26 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 \sqrt {b} (b c-a d)^5 (b e-a f)^{5/2}}+\frac {d^{3/2} \left (a d (d f g+2 d e h-3 c f h)-b \left (8 d^2 e g+5 c^2 f h-c d (7 f g+6 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d)^5 \sqrt {d e-c f}} \] Output:

1/8*d*(5*a^3*d^2*f^2*h-a^2*b*d*f*(-26*c*f*h+14*d*e*h+27*d*f*g)-b^3*(32*d^2 
*e^2*g-c^2*f*(-2*e*h+f*g)-4*c*d*e*(6*e*h+f*g))+a*b^2*(c^2*f^2*h+4*d^2*e*(2 
*e*h+15*f*g)-6*c*d*f*(8*e*h+f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c)^4/(-a*f+b*e) 
^2/(d*x+c)-1/3*(-a*h+b*g)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(b*x+a)^3/(d*x+c)+1/1 
2*(a^2*d*f*h-a*b*(-7*c*f*h+2*d*e*h+7*d*f*g)+b^2*(8*d*e*g-c*(6*e*h+f*g)))*( 
f*x+e)^(1/2)/b/(-a*d+b*c)^2/(-a*f+b*e)/(b*x+a)^2/(d*x+c)+1/24*(5*a^3*d^2*f 
^2*h-5*a^2*b*d*f*(-8*c*f*h+4*d*e*h+7*d*f*g)-b^3*(48*d^2*e^2*g-3*c^2*f*(-2* 
e*h+f*g)-2*c*d*e*(18*e*h+5*f*g))+a*b^2*(3*c^2*f^2*h+2*d^2*e*(6*e*h+43*f*g) 
-2*c*d*f*(35*e*h+8*f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c)^3/(-a*f+b*e)^2/(b*x+a 
)/(d*x+c)+1/8*(5*a^4*d^3*f^3*h-5*a^3*b*d^2*f^2*(-9*c*f*h+6*d*e*h+7*d*f*g)+ 
b^4*(64*d^3*e^3*g-4*c^2*d*e*f*(-4*e*h+f*g)-c^3*f^2*(-2*e*h+f*g)-24*c*d^2*e 
^2*(2*e*h+f*g))-a*b^3*(c^3*f^3*h-c^2*d*f^2*(-34*e*h+7*f*g)+8*d^3*e^2*(2*e* 
h+21*f*g)-8*c*d^2*e*f*(17*e*h+7*f*g))+5*a^2*b^2*d*f*(3*c^2*f^2*h+4*d^2*e*( 
2*e*h+7*f*g)-c*d*f*(26*e*h+7*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)^(5/2)+d^(3/2)*(a*d*(-3*c*f*h+2*d 
*e*h+d*f*g)-b*(8*d^2*e*g+5*c^2*f*h-c*d*(6*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)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(6117\) vs. \(2(872)=1744\).

Time = 16.30 (sec) , antiderivative size = 6117, normalized size of antiderivative = 7.01 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.24 (sec) , antiderivative size = 938, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {166, 27, 168, 27, 168, 27, 168, 25, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+2 d e h+7 d f g)+b^2 (-6 c e h-c f g+8 d e g)\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}-\frac {\int -\frac {5 d f (2 d e-c f) h a^2-b \left (2 e (23 f g+6 e h) d^2-c f (11 f g+40 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )+5 d f \left (d f h a^2-b (7 d f g+2 d e h-7 c f h) a+b^2 (8 d e g-c (f g+6 e h))\right ) x}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {5 d f (2 d e-c f) h a^2-b \left (2 e (23 f g+6 e h) d^2-c f (11 f g+40 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )+5 d f \left (d f h a^2-b (7 d f g+2 d e h-7 c f h) a+b^2 (8 d e g-c (f g+6 e h))\right ) x}{(a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+2 d e h+7 d f g)+b^2 (-6 c e h-c f g+8 d e g)\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {\sqrt {e+f x} \left (5 a^3 d^2 f^2 h-5 a^2 b d f (-8 c f h+4 d e h+7 d f g)+a b^2 \left (3 c^2 f^2 h-2 c d f (35 e h+8 f g)+2 d^2 e (6 e h+43 f g)\right )-b^3 \left (-3 c^2 f (f g-2 e h)-2 c d e (18 e h+5 f g)+48 d^2 e^2 g\right )\right )}{(a+b x) (c+d x) (b c-a d) (b e-a f)}-\frac {\int -\frac {3 \left (5 d^2 f^2 (2 d e-c f) h a^3-b d f \left (2 e (27 f g+14 e h) d^2-c f (19 f g+60 e h) d+12 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (15 f g+2 e h) d^3-2 c e f (23 f g+50 e h) d^2-4 c^2 f^2 (f g-7 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (f g-2 e h) c^3-4 d e f (f g-4 e h) c^2-24 d^2 e^2 (f g+2 e h) c+64 d^3 e^3 g\right )+d f \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+4 d e h-8 c f h) a^2+b^2 \left (2 e (43 f g+6 e h) d^2-2 c f (8 f g+35 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )\right ) x\right )}{2 (a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+2 d e h+7 d f g)+b^2 (-6 c e h-c f g+8 d e g)\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {5 d^2 f^2 (2 d e-c f) h a^3-b d f \left (2 e (27 f g+14 e h) d^2-c f (19 f g+60 e h) d+12 c^2 f^2 h\right ) a^2+b^2 \left (8 e^2 (15 f g+2 e h) d^3-2 c e f (23 f g+50 e h) d^2-4 c^2 f^2 (f g-7 e h) d+c^3 f^3 h\right ) a-b^3 \left (-f^2 (f g-2 e h) c^3-4 d e f (f g-4 e h) c^2-24 d^2 e^2 (f g+2 e h) c+64 d^3 e^3 g\right )+d f \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+4 d e h-8 c f h) a^2+b^2 \left (2 e (43 f g+6 e h) d^2-2 c f (8 f g+35 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (5 a^3 d^2 f^2 h-5 a^2 b d f (-8 c f h+4 d e h+7 d f g)+a b^2 \left (3 c^2 f^2 h-2 c d f (35 e h+8 f g)+2 d^2 e (6 e h+43 f g)\right )-b^3 \left (-3 c^2 f (f g-2 e h)-2 c d e (18 e h+5 f g)+48 d^2 e^2 g\right )\right )}{(a+b x) (c+d x) (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-7 c f h+2 d e h+7 d f g)+b^2 (-6 c e h-c f g+8 d e g)\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{3 b (a+b x)^3 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+2 d e h-7 c f h) a+b^2 (8 d e g-c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}+\frac {\frac {\sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+4 d e h-8 c f h) a^2+b^2 \left (2 e (43 f g+6 e h) d^2-2 c f (8 f g+35 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}+\frac {3 \left (\frac {2 d \sqrt {e+f x} \left (5 d^2 f^2 h a^3-b d f (27 d f g+14 d e h-26 c f h) a^2+b^2 \left (4 e (15 f g+2 e h) d^2-6 c f (f g+8 e h) d+c^2 f^2 h\right ) a-b^3 \left (-f (f g-2 e h) c^2-4 d e (f g+6 e h) c+32 d^2 e^2 g\right )\right )}{(b c-a d) (c+d x)}+\frac {\int -\frac {b (d e-c f) \left (d^2 f^2 (19 c f h-8 d (f g+2 e h)) a^3+b d f \left (16 e (5 f g+2 e h) d^2-c f (29 f g+82 e h) d+14 c^2 f^2 h\right ) a^2-b^2 \left (8 e^2 (17 f g+2 e h) d^3-4 c e f (13 f g+28 e h) d^2-2 c^2 f^2 (3 f g-16 e h) d+c^3 f^3 h\right ) a+b^3 \left (-f^2 (f g-2 e h) c^3-4 d e f (f g-4 e h) c^2-24 d^2 e^2 (f g+2 e h) c+64 d^3 e^3 g\right )-d f \left (5 d^2 f^2 h a^3-b d f (27 d f g+14 d e h-26 c f h) a^2+b^2 \left (4 e (15 f g+2 e h) d^2-6 c f (f g+8 e h) d+c^2 f^2 h\right ) a-b^3 \left (-f (f g-2 e h) c^2-4 d e (f g+6 e h) c+32 d^2 e^2 g\right )\right ) x\right )}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+2 d e h-7 c f h) a+b^2 (8 d e g-c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}+\frac {\frac {\sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+4 d e h-8 c f h) a^2+b^2 \left (2 e (43 f g+6 e h) d^2-2 c f (8 f g+35 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}+\frac {3 \left (\frac {2 d \left (5 d^2 f^2 h a^3-b d f (27 d f g+14 d e h-26 c f h) a^2+b^2 \left (4 e (15 f g+2 e h) d^2-6 c f (f g+8 e h) d+c^2 f^2 h\right ) a-b^3 \left (-f (f g-2 e h) c^2-4 d e (f g+6 e h) c+32 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {\int \frac {b (d e-c f) \left (d^2 f^2 (19 c f h-8 d (f g+2 e h)) a^3+b d f \left (16 e (5 f g+2 e h) d^2-c f (29 f g+82 e h) d+14 c^2 f^2 h\right ) a^2-b^2 \left (8 e^2 (17 f g+2 e h) d^3-4 c e f (13 f g+28 e h) d^2-2 c^2 f^2 (3 f g-16 e h) d+c^3 f^3 h\right ) a+b^3 \left (-f^2 (f g-2 e h) c^3-4 d e f (f g-4 e h) c^2-24 d^2 e^2 (f g+2 e h) c+64 d^3 e^3 g\right )-d f \left (5 d^2 f^2 h a^3-b d f (27 d f g+14 d e h-26 c f h) a^2+b^2 \left (4 e (15 f g+2 e h) d^2-6 c f (f g+8 e h) d+c^2 f^2 h\right ) a-b^3 \left (-f (f g-2 e h) c^2-4 d e (f g+6 e h) c+32 d^2 e^2 g\right )\right ) x\right )}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+2 d e h-7 c f h) a+b^2 (8 d e g-c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}+\frac {\frac {\sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+4 d e h-8 c f h) a^2+b^2 \left (2 e (43 f g+6 e h) d^2-2 c f (8 f g+35 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}+\frac {3 \left (\frac {2 d \left (5 d^2 f^2 h a^3-b d f (27 d f g+14 d e h-26 c f h) a^2+b^2 \left (4 e (15 f g+2 e h) d^2-6 c f (f g+8 e h) d+c^2 f^2 h\right ) a-b^3 \left (-f (f g-2 e h) c^2-4 d e (f g+6 e h) c+32 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \int \frac {d^2 f^2 (19 c f h-8 d (f g+2 e h)) a^3+b d f \left (16 e (5 f g+2 e h) d^2-c f (29 f g+82 e h) d+14 c^2 f^2 h\right ) a^2-b^2 \left (8 e^2 (17 f g+2 e h) d^3-4 c e f (13 f g+28 e h) d^2-2 c^2 f^2 (3 f g-16 e h) d+c^3 f^3 h\right ) a+b^3 \left (-f^2 (f g-2 e h) c^3-4 d e f (f g-4 e h) c^2-24 d^2 e^2 (f g+2 e h) c+64 d^3 e^3 g\right )-d f \left (5 d^2 f^2 h a^3-b d f (27 d f g+14 d e h-26 c f h) a^2+b^2 \left (4 e (15 f g+2 e h) d^2-6 c f (f g+8 e h) d+c^2 f^2 h\right ) a-b^3 \left (-f (f g-2 e h) c^2-4 d e (f g+6 e h) c+32 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{b c-a d}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+2 d e h-7 c f h) a+b^2 (8 d e g-c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}+\frac {\frac {\sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+4 d e h-8 c f h) a^2+b^2 \left (2 e (43 f g+6 e h) d^2-2 c f (8 f g+35 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}+\frac {3 \left (\frac {2 d \left (5 d^2 f^2 h a^3-b d f (27 d f g+14 d e h-26 c f h) a^2+b^2 \left (4 e (15 f g+2 e h) d^2-6 c f (f g+8 e h) d+c^2 f^2 h\right ) a-b^3 \left (-f (f g-2 e h) c^2-4 d e (f g+6 e h) c+32 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {8 d^2 \left (a d (d f g+2 d e h-3 c f h)-b \left (5 f h c^2-d (7 f g+6 e h) c+8 d^2 e g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx (b e-a f)^2}{b c-a d}+\frac {\left (5 d^3 f^3 h a^4-5 b d^2 f^2 (7 d f g+6 d e h-9 c f h) a^3+5 b^2 d f \left (4 e (7 f g+2 e h) d^2-c f (7 f g+26 e h) d+3 c^2 f^2 h\right ) a^2-b^3 \left (8 e^2 (21 f g+2 e h) d^3-8 c e f (7 f g+17 e h) d^2-c^2 f^2 (7 f g-34 e h) d+c^3 f^3 h\right ) a+b^4 \left (-f^2 (f g-2 e h) c^3-4 d e f (f g-4 e h) c^2-24 d^2 e^2 (f g+2 e h) c+64 d^3 e^3 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}\right )}{b c-a d}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+2 d e h-7 c f h) a+b^2 (8 d e g-c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}+\frac {\frac {\sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+4 d e h-8 c f h) a^2+b^2 \left (2 e (43 f g+6 e h) d^2-2 c f (8 f g+35 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}+\frac {3 \left (\frac {2 d \left (5 d^2 f^2 h a^3-b d f (27 d f g+14 d e h-26 c f h) a^2+b^2 \left (4 e (15 f g+2 e h) d^2-6 c f (f g+8 e h) d+c^2 f^2 h\right ) a-b^3 \left (-f (f g-2 e h) c^2-4 d e (f g+6 e h) c+32 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {16 d^2 \left (a d (d f g+2 d e h-3 c f h)-b \left (5 f h c^2-d (7 f g+6 e h) c+8 d^2 e g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x} (b e-a f)^2}{(b c-a d) f}+\frac {2 \left (5 d^3 f^3 h a^4-5 b d^2 f^2 (7 d f g+6 d e h-9 c f h) a^3+5 b^2 d f \left (4 e (7 f g+2 e h) d^2-c f (7 f g+26 e h) d+3 c^2 f^2 h\right ) a^2-b^3 \left (8 e^2 (21 f g+2 e h) d^3-8 c e f (7 f g+17 e h) d^2-c^2 f^2 (7 f g-34 e h) d+c^3 f^3 h\right ) a+b^4 \left (-f^2 (f g-2 e h) c^3-4 d e f (f g-4 e h) c^2-24 d^2 e^2 (f g+2 e h) c+64 d^3 e^3 g\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{(b c-a d) f}\right )}{b c-a d}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (7 d f g+2 d e h-7 c f h) a+b^2 (8 d e g-c f g-6 c e h)\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}+\frac {\frac {\sqrt {e+f x} \left (5 d^2 f^2 h a^3-5 b d f (7 d f g+4 d e h-8 c f h) a^2+b^2 \left (2 e (43 f g+6 e h) d^2-2 c f (8 f g+35 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-3 f (f g-2 e h) c^2-2 d e (5 f g+18 e h) c+48 d^2 e^2 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}+\frac {3 \left (\frac {2 d \left (5 d^2 f^2 h a^3-b d f (27 d f g+14 d e h-26 c f h) a^2+b^2 \left (4 e (15 f g+2 e h) d^2-6 c f (f g+8 e h) d+c^2 f^2 h\right ) a-b^3 \left (-f (f g-2 e h) c^2-4 d e (f g+6 e h) c+32 d^2 e^2 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (-\frac {16 d^{3/2} \left (a d (d f g+2 d e h-3 c f h)-b \left (5 f h c^2-d (7 f g+6 e h) c+8 d^2 e g\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) (b e-a f)^2}{(b c-a d) \sqrt {d e-c f}}-\frac {2 \left (5 d^3 f^3 h a^4-5 b d^2 f^2 (7 d f g+6 d e h-9 c f h) a^3+5 b^2 d f \left (4 e (7 f g+2 e h) d^2-c f (7 f g+26 e h) d+3 c^2 f^2 h\right ) a^2-b^3 \left (8 e^2 (21 f g+2 e h) d^3-8 c e f (7 f g+17 e h) d^2-c^2 f^2 (7 f g-34 e h) d+c^3 f^3 h\right ) a+b^4 \left (-f^2 (f g-2 e h) c^3-4 d e f (f g-4 e h) c^2-24 d^2 e^2 (f g+2 e h) c+64 d^3 e^3 g\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b c-a d) \sqrt {b e-a f}}\right )}{b c-a d}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}\)

Input:

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

Output:

-1/3*((b*g - a*h)*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)^3*(c + d*x)) + ( 
((a^2*d*f*h + b^2*(8*d*e*g - c*f*g - 6*c*e*h) - a*b*(7*d*f*g + 2*d*e*h - 7 
*c*f*h))*Sqrt[e + f*x])/(2*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)) 
+ (((5*a^3*d^2*f^2*h - 5*a^2*b*d*f*(7*d*f*g + 4*d*e*h - 8*c*f*h) - b^3*(48 
*d^2*e^2*g - 3*c^2*f*(f*g - 2*e*h) - 2*c*d*e*(5*f*g + 18*e*h)) + a*b^2*(3* 
c^2*f^2*h + 2*d^2*e*(43*f*g + 6*e*h) - 2*c*d*f*(8*f*g + 35*e*h)))*Sqrt[e + 
 f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)) + (3*((2*d*(5*a^3*d^2 
*f^2*h - a^2*b*d*f*(27*d*f*g + 14*d*e*h - 26*c*f*h) - b^3*(32*d^2*e^2*g - 
c^2*f*(f*g - 2*e*h) - 4*c*d*e*(f*g + 6*e*h)) + a*b^2*(c^2*f^2*h + 4*d^2*e* 
(15*f*g + 2*e*h) - 6*c*d*f*(f*g + 8*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(c 
+ d*x)) - (b*((-2*(5*a^4*d^3*f^3*h - 5*a^3*b*d^2*f^2*(7*d*f*g + 6*d*e*h - 
9*c*f*h) + b^4*(64*d^3*e^3*g - 4*c^2*d*e*f*(f*g - 4*e*h) - c^3*f^2*(f*g - 
2*e*h) - 24*c*d^2*e^2*(f*g + 2*e*h)) - a*b^3*(c^3*f^3*h - c^2*d*f^2*(7*f*g 
 - 34*e*h) + 8*d^3*e^2*(21*f*g + 2*e*h) - 8*c*d^2*e*f*(7*f*g + 17*e*h)) + 
5*a^2*b^2*d*f*(3*c^2*f^2*h + 4*d^2*e*(7*f*g + 2*e*h) - c*d*f*(7*f*g + 26*e 
*h)))*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b*c - a* 
d)*Sqrt[b*e - a*f]) - (16*d^(3/2)*(b*e - a*f)^2*(a*d*(d*f*g + 2*d*e*h - 3* 
c*f*h) - b*(8*d^2*e*g + 5*c^2*f*h - c*d*(7*f*g + 6*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)))/(2*(b*c - a*d)*(b*e - a*f)))/(4*(b*c - a*d)*(b*e - a*f)))/(6*b*(b...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 = 10.83 (sec) , antiderivative size = 1143, normalized size of antiderivative = 1.31

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1143\)
derivativedivides	\(\text {Expression too large to display}\)	\(1458\)
default	\(\text {Expression too large to display}\)	\(1458\)

Input:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

5/8/((a*f-b*e)*b)^(1/2)*((d*x+c)*((c*f-d*e)*d)^(1/2)*(b*x+a)^3*((64/5*d^3* 
e^3*g-48/5*c*(e*h+1/2*f*g)*e^2*d^2+16/5*(e*h-1/4*f*g)*c^2*f*e*d+2/5*(e*h-1 
/2*f*g)*c^3*f^2)*b^4-1/5*a*((16*e^3*h+168*e^2*f*g)*d^3+(-136*c*e^2*f*h-56* 
c*e*f^2*g)*d^2+(34*c^2*e*f^2*h-7*c^2*f^3*g)*d+c^3*f^3*h)*b^3+3*a^2*d*f*((8 
/3*e^2*h+28/3*e*f*g)*d^2-26/3*c*(e*h+7/26*f*g)*f*d+c^2*f^2*h)*b^2+9*a^3*d^ 
2*f^2*((-2/3*e*h-7/9*f*g)*d+c*f*h)*b+a^4*d^3*f^3*h)*arctan(b*(f*x+e)^(1/2) 
/((a*f-b*e)*b)^(1/2))+19/5*((a*f-b*e)*b)^(1/2)*(-24/19*d^2*(d*x+c)*(b*x+a) 
^3*(a*f-b*e)^2*((8/3*d^2*e*g-2*c*(e*h+7/6*f*g)*d+5/3*c^2*f*h)*b+a*d*((-2/3 
*e*h-1/3*f*g)*d+c*f*h))*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)*((-32/19*d^3*e^2*g*x^3-16/19*x^2*c*e*((-3/2*e*h-1/ 
4*f*g)*x+g*e)*d^2+16/57*x*c^2*((-3/8*e*f*h+3/16*f^2*g)*x^2+(9/4*e^2*h+5/8* 
e*f*g)*x+e^2*g)*d-8/57*((3/4*e*f*h-3/8*f^2*g)*x^2+(3/2*e^2*h+1/4*e*f*g)*x+ 
e^2*g)*c^3)*b^5-4/57*a*(((-6*e^2*h-45*e*f*g)*x^3+60*e^2*g*x^2)*d^3+32*x*c* 
((9/8*e*f*h+9/64*f^2*g)*x^2+(-3/2*e^2*h-7/8*e*f*g)*x+e^2*g)*d^2-10*(3/40*f 
^2*h*x^3+(-41/20*e*f*h-1/4*f^2*g)*x^2+(23/10*e^2*h-3/20*e*f*g)*x+e^2*g)*c^ 
2*d+c^3*(-3/4*f^2*h*x^2+(-7/2*e*f*h-2*f^2*g)*x+e*(e*h-7/2*f*g)))*b^4+4/57* 
a^2*(((-21/2*e*f*h-81/4*f^2*g)*x^3+(15*e^2*h+227/2*e*f*g)*x^2-44*e^2*g*x)* 
d^3-26*(-3/4*f^2*h*x^3+(191/52*e*f*h+73/104*f^2*g)*x^2-41/26*(e*h+3/2*f*g) 
*e*x+e^2*g)*c*d^2+8*c^2*(23/16*f^2*h*x^2+(-6*e*f*h-19/32*f^2*g)*x+e^2*h-2* 
e*f*g)*d+c^3*f*(-2*f*h*x-3/4*f*g+e*h))*b^3-1/19*a^3*((-5*f^2*h*x^3+(106...
 

Fricas [F(-1)]

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

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((f*x+e)**(1/2)*(h*x+g)/(b*x+a)**4/(d*x+c)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^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. 2352 vs. \(2 (834) = 1668\).

Time = 0.26 (sec) , antiderivative size = 2352, normalized size of antiderivative = 2.70 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^2,x, algorithm="giac")
 

Output:

-1/8*(64*b^4*d^3*e^3*g - 24*b^4*c*d^2*e^2*f*g - 168*a*b^3*d^3*e^2*f*g - 4* 
b^4*c^2*d*e*f^2*g + 56*a*b^3*c*d^2*e*f^2*g + 140*a^2*b^2*d^3*e*f^2*g - b^4 
*c^3*f^3*g + 7*a*b^3*c^2*d*f^3*g - 35*a^2*b^2*c*d^2*f^3*g - 35*a^3*b*d^3*f 
^3*g - 48*b^4*c*d^2*e^3*h - 16*a*b^3*d^3*e^3*h + 16*b^4*c^2*d*e^2*f*h + 13 
6*a*b^3*c*d^2*e^2*f*h + 40*a^2*b^2*d^3*e^2*f*h + 2*b^4*c^3*e*f^2*h - 34*a* 
b^3*c^2*d*e*f^2*h - 130*a^2*b^2*c*d^2*e*f^2*h - 30*a^3*b*d^3*e*f^2*h - a*b 
^3*c^3*f^3*h + 15*a^2*b^2*c^2*d*f^3*h + 45*a^3*b*c*d^2*f^3*h + 5*a^4*d^3*f 
^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^7*c^5*e^2 - 5*a*b^6 
*c^4*d*e^2 + 10*a^2*b^5*c^3*d^2*e^2 - 10*a^3*b^4*c^2*d^3*e^2 + 5*a^4*b^3*c 
*d^4*e^2 - a^5*b^2*d^5*e^2 - 2*a*b^6*c^5*e*f + 10*a^2*b^5*c^4*d*e*f - 20*a 
^3*b^4*c^3*d^2*e*f + 20*a^4*b^3*c^2*d^3*e*f - 10*a^5*b^2*c*d^4*e*f + 2*a^6 
*b*d^5*e*f + a^2*b^5*c^5*f^2 - 5*a^3*b^4*c^4*d*f^2 + 10*a^4*b^3*c^3*d^2*f^ 
2 - 10*a^5*b^2*c^2*d^3*f^2 + 5*a^6*b*c*d^4*f^2 - a^7*d^5*f^2)*sqrt(-b^2*e 
+ a*b*f)) + (8*b*d^4*e*g - 7*b*c*d^3*f*g - a*d^4*f*g - 6*b*c*d^3*e*h - 2*a 
*d^4*e*h + 5*b*c^2*d^2*f*h + 3*a*c*d^3*f*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)) - (sqrt(f*x + e) 
*d^3*f*g - sqrt(f*x + e)*c*d^2*f*h)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2* 
c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*((f*x + e)*d - d*e + c*f)) - 1/24*(72*( 
f*x + e)^(5/2)*b^5*d^2*e^2*f*g - 144*(f*x + e)^(3/2)*b^5*d^2*e^3*f*g + ...
 

Mupad [B] (verification not implemented)

Time = 35.04 (sec) , antiderivative size = 783233, normalized size of antiderivative = 898.20 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

int(((e + f*x)^(1/2)*(g + h*x))/((a + b*x)^4*(c + d*x)^2),x)
 

Output:

atan(((((256*a^15*b^2*d^15*f^8*g - 1632*a^14*b^3*c*d^14*f^8*g - 608*a^15*b 
^2*c*d^14*f^8*h - 2208*a^14*b^3*d^15*e*f^7*g + 352*a^15*b^2*d^15*e*f^7*h + 
 32*a^2*b^15*c^13*d^2*f^8*g - 512*a^3*b^14*c^12*d^3*f^8*g + 4288*a^4*b^13* 
c^11*d^4*f^8*g - 21504*a^5*b^12*c^10*d^5*f^8*g + 68960*a^6*b^11*c^9*d^6*f^ 
8*g - 148224*a^7*b^10*c^8*d^7*f^8*g + 219264*a^8*b^9*c^7*d^8*f^8*g - 22425 
6*a^9*b^8*c^6*d^9*f^8*g + 154848*a^10*b^7*c^5*d^10*f^8*g - 66560*a^11*b^6* 
c^4*d^11*f^8*g + 12992*a^12*b^5*c^3*d^12*f^8*g + 2048*a^13*b^4*c^2*d^13*f^ 
8*g + 32*a^3*b^14*c^13*d^2*f^8*h - 768*a^4*b^13*c^12*d^3*f^8*h + 5312*a^5* 
b^12*c^11*d^4*f^8*h - 17920*a^6*b^11*c^10*d^5*f^8*h + 33120*a^7*b^10*c^9*d 
^6*f^8*h - 29184*a^8*b^9*c^8*d^7*f^8*h - 8064*a^9*b^8*c^7*d^8*f^8*h + 5529 
6*a^10*b^7*c^6*d^9*f^8*h - 72480*a^11*b^6*c^5*d^10*f^8*h + 52480*a^12*b^5* 
c^4*d^11*f^8*h - 22848*a^13*b^4*c^3*d^12*f^8*h + 5632*a^14*b^3*c^2*d^13*f^ 
8*h - 1024*a^10*b^7*d^15*e^5*f^3*g + 4480*a^11*b^6*d^15*e^4*f^4*g - 7584*a 
^12*b^5*d^15*e^3*f^5*g + 6080*a^13*b^4*d^15*e^2*f^6*g + 256*a^11*b^6*d^15* 
e^5*f^3*h - 1088*a^12*b^5*d^15*e^4*f^4*h + 1760*a^13*b^4*d^15*e^3*f^5*h - 
1280*a^14*b^3*d^15*e^2*f^6*h - 1024*b^17*c^10*d^5*e^5*f^3*g + 640*b^17*c^1 
1*d^4*e^4*f^4*g + 96*b^17*c^12*d^3*e^3*f^5*g + 32*b^17*c^13*d^2*e^2*f^6*g 
+ 768*b^17*c^11*d^4*e^5*f^3*h - 448*b^17*c^12*d^3*e^4*f^4*h - 64*b^17*c^13 
*d^2*e^3*f^5*h - 46080*a^2*b^15*c^8*d^7*e^5*f^3*g - 16000*a^2*b^15*c^9*d^6 
*e^4*f^4*g + 24256*a^2*b^15*c^10*d^5*e^3*f^5*g + 9792*a^2*b^15*c^11*d^4...
 

Reduce [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 25836, normalized size of antiderivative = 29.63 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^4 (c+d x)^2} \, dx =\text {Too large to display} \] Input:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^4/(d*x+c)^2,x)
 

Output:

(15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e 
)))*a**7*c**2*d**3*f**4*h - 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x) 
*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c*d**4*e*f**3*h + 15*sqrt(b)*sqrt(a*f 
- b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c*d**4*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**7*d**5*e*f**3*h*x + 135*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
 f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**3*d**2*f**4*h - 225*sqrt(b)* 
sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c 
**2*d**3*e*f**3*h - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq 
rt(b)*sqrt(a*f - b*e)))*a**6*b*c**2*d**3*f**4*g + 180*sqrt(b)*sqrt(a*f - b 
*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**2*d**3*f** 
4*h*x + 90*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a* 
f - b*e)))*a**6*b*c*d**4*e**2*f**2*h + 105*sqrt(b)*sqrt(a*f - b*e)*atan((s 
qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c*d**4*e*f**3*g - 270*sq 
rt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a* 
*6*b*c*d**4*e*f**3*h*x - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b 
)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c*d**4*f**4*g*x + 45*sqrt(b)*sqrt(a*f 
- b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c*d**4*f** 
4*h*x**2 + 90*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt 
(a*f - b*e)))*a**6*b*d**5*e**2*f**2*h*x + 105*sqrt(b)*sqrt(a*f - b*e)*a...