\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^2} \, dx\) [93]

Optimal result

Integrand size = 29, antiderivative size = 520 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^2} \, dx=-\frac {\left (a^2 d^2 f h+a b d (3 d f g+4 d e h-9 c f h)-b^2 \left (12 d^2 e g+4 c^2 f h-c d (9 f g+8 e h)\right )\right ) \sqrt {e+f x}}{4 b^2 (b c-a d)^3 (c+d x)}-\frac {\left (a^2 d f h-b^2 (6 d e g-3 c f g-4 c e h)+a b (3 d f g+2 d e h-7 c f h)\right ) \sqrt {e+f x}}{4 b^2 (b c-a d)^2 (a+b x) (c+d x)}-\frac {(b g-a h) (e+f x)^{3/2}}{2 b (b c-a d) (a+b x)^2 (c+d x)}-\frac {\left (a^3 d^2 f^2 h+a^2 b d f (3 d f g+4 d e h-10 c f h)+b^3 \left (24 d^2 e^2 g-8 c d e (3 f g+2 e h)+3 c^2 f (f g+4 e h)\right )-a b^2 \left (15 c^2 f^2 h+8 d^2 e (3 f g+e h)-2 c d f (9 f g+16 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{4 b^{3/2} (b c-a d)^4 \sqrt {b e-a f}}-\frac {\sqrt {d e-c f} \left (a d (3 d f g+2 d e h-5 c f h)-b \left (6 d^2 e g+c^2 f h-c d (3 f g+4 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (b c-a d)^4} \] Output:

-1/4*(a^2*d^2*f*h+a*b*d*(-9*c*f*h+4*d*e*h+3*d*f*g)-b^2*(12*d^2*e*g+4*c^2*f 
*h-c*d*(8*e*h+9*f*g)))*(f*x+e)^(1/2)/b^2/(-a*d+b*c)^3/(d*x+c)-1/4*(a^2*d*f 
*h-b^2*(-4*c*e*h-3*c*f*g+6*d*e*g)+a*b*(-7*c*f*h+2*d*e*h+3*d*f*g))*(f*x+e)^ 
(1/2)/b^2/(-a*d+b*c)^2/(b*x+a)/(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)-1/4*(a^3*d^2*f^2*h+a^2*b*d*f*(-10*c*f*h+4*d*e*h+3 
*d*f*g)+b^3*(24*d^2*e^2*g-8*c*d*e*(2*e*h+3*f*g)+3*c^2*f*(4*e*h+f*g))-a*b^2 
*(15*c^2*f^2*h+8*d^2*e*(e*h+3*f*g)-2*c*d*f*(16*e*h+9*f*g)))*arctanh(b^(1/2 
)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(3/2)/(-a*d+b*c)^4/(-a*f+b*e)^(1/2)-(- 
c*f+d*e)^(1/2)*(a*d*(-5*c*f*h+2*d*e*h+3*d*f*g)-b*(6*d^2*e*g+c^2*f*h-c*d*(4 
*e*h+3*f*g)))*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(1/2)/(-a* 
d+b*c)^4
 

Mathematica [A] (verified)

Time = 6.08 (sec) , antiderivative size = 510, normalized size of antiderivative = 0.98 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^2} \, dx=\frac {-\frac {(b c-a d) \sqrt {e+f x} \left (-a^3 d f h (c+d x)+a b^2 \left (d^2 x (-18 e g+3 f g x+4 e h x)+c^2 (3 f g+2 e h-17 f h x)+c d \left (-10 e g+14 f g x+14 e h x-9 f h x^2\right )\right )+a^2 b \left (-11 c^2 f h+c d (9 f g+10 e h-6 f h x)+d^2 \left (-4 e g+5 f g x+6 e h x+f h x^2\right )\right )+b^3 \left (-12 d^2 e g x^2+c d x (-6 e g+9 f g x+8 e h x)+c^2 (f x (5 g-4 h x)+2 e (g+2 h x))\right )\right )}{b (a+b x)^2 (c+d x)}+\frac {\left (a^3 d^2 f^2 h+a^2 b d f (3 d f g+4 d e h-10 c f h)+b^3 \left (24 d^2 e^2 g-8 c d e (3 f g+2 e h)+3 c^2 f (f g+4 e h)\right )+a b^2 \left (-15 c^2 f^2 h-8 d^2 e (3 f g+e h)+2 c d f (9 f g+16 e h)\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{b^{3/2} \sqrt {-b e+a f}}+\frac {4 \sqrt {-d e+c f} \left (a d (-3 d f g-2 d e h+5 c f h)+b \left (6 d^2 e g+c^2 f h-c d (3 f g+4 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{\sqrt {d}}}{4 (b c-a d)^4} \] Input:

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

Output:

(-(((b*c - a*d)*Sqrt[e + f*x]*(-(a^3*d*f*h*(c + d*x)) + a*b^2*(d^2*x*(-18* 
e*g + 3*f*g*x + 4*e*h*x) + c^2*(3*f*g + 2*e*h - 17*f*h*x) + c*d*(-10*e*g + 
 14*f*g*x + 14*e*h*x - 9*f*h*x^2)) + a^2*b*(-11*c^2*f*h + c*d*(9*f*g + 10* 
e*h - 6*f*h*x) + d^2*(-4*e*g + 5*f*g*x + 6*e*h*x + f*h*x^2)) + b^3*(-12*d^ 
2*e*g*x^2 + c*d*x*(-6*e*g + 9*f*g*x + 8*e*h*x) + c^2*(f*x*(5*g - 4*h*x) + 
2*e*(g + 2*h*x)))))/(b*(a + b*x)^2*(c + d*x))) + ((a^3*d^2*f^2*h + a^2*b*d 
*f*(3*d*f*g + 4*d*e*h - 10*c*f*h) + b^3*(24*d^2*e^2*g - 8*c*d*e*(3*f*g + 2 
*e*h) + 3*c^2*f*(f*g + 4*e*h)) + a*b^2*(-15*c^2*f^2*h - 8*d^2*e*(3*f*g + e 
*h) + 2*c*d*f*(9*f*g + 16*e*h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e 
) + a*f]])/(b^(3/2)*Sqrt[-(b*e) + a*f]) + (4*Sqrt[-(d*e) + c*f]*(a*d*(-3*d 
*f*g - 2*d*e*h + 5*c*f*h) + b*(6*d^2*e*g + c^2*f*h - c*d*(3*f*g + 4*e*h))) 
*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/Sqrt[d])/(4*(b*c - a* 
d)^4)
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {166, 27, 25, 166, 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.

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\int -\frac {\sqrt {e+f x} (6 b d e g-3 b c f g-4 b c e h-2 a d e h+3 a c f h+f (3 b d g-4 b c h+a d h) x)}{2 (a+b x)^2 (c+d x)^2}dx}{2 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -\frac {\sqrt {e+f x} (a (2 d e-3 c f) h-b (6 d e g-3 c f g-4 c e h)-f (3 b d g-4 b c h+a d h) x)}{(a+b x)^2 (c+d x)^2}dx}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\sqrt {e+f x} (a (2 d e-3 c f) h-b (6 d e g-3 c f g-4 c e h)-f (3 b d g-4 b c h+a d h) x)}{(a+b x)^2 (c+d x)^2}dx}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {-\frac {b \left (-\frac {\left (a^3 d^2 f^2 h+a^2 b d f (-10 c f h+4 d e h+3 d f g)-a b^2 \left (15 c^2 f^2 h-2 c d f (16 e h+9 f g)+8 d^2 e (e h+3 f g)\right )+b^3 \left (3 c^2 f (4 e h+f g)-8 c d e (2 e h+3 f g)+24 d^2 e^2 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {4 b (d e-c f) \left (a d (-5 c f h+2 d e h+3 d f g)-b \left (c^2 f h-c d (4 e h+3 f g)+6 d^2 e g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}\right )}{b c-a d}-\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h+a b d (-9 c f h+4 d e h+3 d f g)-\left (b^2 \left (4 c^2 f h-c d (8 e h+9 f g)+12 d^2 e g\right )\right )\right )}{(c+d x) (b c-a d)}}{2 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-7 c f h+2 d e h+3 d f g)-b^2 (-4 c e h-3 c f g+6 d e g)\right )}{b (a+b x) (c+d x) (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {-\frac {b \left (-\frac {2 \left (a^3 d^2 f^2 h+a^2 b d f (-10 c f h+4 d e h+3 d f g)-a b^2 \left (15 c^2 f^2 h-2 c d f (16 e h+9 f g)+8 d^2 e (e h+3 f g)\right )+b^3 \left (3 c^2 f (4 e h+f g)-8 c d e (2 e h+3 f g)+24 d^2 e^2 g\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}-\frac {8 b (d e-c f) \left (a d (-5 c f h+2 d e h+3 d f g)-b \left (c^2 f h-c d (4 e h+3 f g)+6 d^2 e g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}\right )}{b c-a d}-\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h+a b d (-9 c f h+4 d e h+3 d f g)-\left (b^2 \left (4 c^2 f h-c d (8 e h+9 f g)+12 d^2 e g\right )\right )\right )}{(c+d x) (b c-a d)}}{2 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-7 c f h+2 d e h+3 d f g)-b^2 (-4 c e h-3 c f g+6 d e g)\right )}{b (a+b x) (c+d x) (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {-\frac {2 \sqrt {e+f x} \left (a^2 d^2 f h+a b d (-9 c f h+4 d e h+3 d f g)-\left (b^2 \left (4 c^2 f h-c d (8 e h+9 f g)+12 d^2 e g\right )\right )\right )}{(c+d x) (b c-a d)}-\frac {b \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right ) \left (a^3 d^2 f^2 h+a^2 b d f (-10 c f h+4 d e h+3 d f g)-a b^2 \left (15 c^2 f^2 h-2 c d f (16 e h+9 f g)+8 d^2 e (e h+3 f g)\right )+b^3 \left (3 c^2 f (4 e h+f g)-8 c d e (2 e h+3 f g)+24 d^2 e^2 g\right )\right )}{\sqrt {b} (b c-a d) \sqrt {b e-a f}}+\frac {8 b \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (a d (-5 c f h+2 d e h+3 d f g)-b \left (c^2 f h-c d (4 e h+3 f g)+6 d^2 e g\right )\right )}{\sqrt {d} (b c-a d)}\right )}{b c-a d}}{2 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-7 c f h+2 d e h+3 d f g)-b^2 (-4 c e h-3 c f g+6 d e g)\right )}{b (a+b x) (c+d x) (b c-a d)}}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{2 b (a+b x)^2 (c+d x) (b c-a d)}\)

Input:

Int[((e + f*x)^(3/2)*(g + h*x))/((a + b*x)^3*(c + d*x)^2),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)) + 
 (-(((a^2*d*f*h - b^2*(6*d*e*g - 3*c*f*g - 4*c*e*h) + a*b*(3*d*f*g + 2*d*e 
*h - 7*c*f*h))*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)*(c + d*x))) + ((-2* 
(a^2*d^2*f*h + a*b*d*(3*d*f*g + 4*d*e*h - 9*c*f*h) - b^2*(12*d^2*e*g + 4*c 
^2*f*h - c*d*(9*f*g + 8*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*(3*d*f*g + 4*d*e*h - 10*c*f*h) + b^3*(24*d 
^2*e^2*g - 8*c*d*e*(3*f*g + 2*e*h) + 3*c^2*f*(f*g + 4*e*h)) - a*b^2*(15*c^ 
2*f^2*h + 8*d^2*e*(3*f*g + e*h) - 2*c*d*f*(9*f*g + 16*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]) 
 + (8*b*Sqrt[d*e - c*f]*(a*d*(3*d*f*g + 2*d*e*h - 5*c*f*h) - b*(6*d^2*e*g 
+ c^2*f*h - c*d*(3*f*g + 4*e*h)))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e 
 - c*f]])/(Sqrt[d]*(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 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 = 1.14 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(\frac {\sqrt {\left (c f -d e \right ) d}\, \left (\left (24 d^{2} e^{2} g -16 c \left (e h +\frac {3 f g}{2}\right ) e d +3 \left (4 e f h +f^{2} g \right ) c^{2}\right ) b^{3}-15 a \left (\frac {8 d^{2} e \left (e h +3 f g \right )}{15}-\frac {32 c \left (e h +\frac {9 f g}{16}\right ) f d}{15}+c^{2} f^{2} h \right ) b^{2}-10 a^{2} d \left (\frac {\left (-2 e h -\frac {3 f g}{2}\right ) d}{5}+c f h \right ) f b +a^{3} d^{2} f^{2} h \right ) \left (b x +a \right )^{2} \left (x d +c \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )-\sqrt {\left (a f -b e \right ) b}\, \left (-20 \left (\frac {\left (6 d^{2} e g -4 c \left (e h +\frac {3 f g}{4}\right ) d +c^{2} f h \right ) b}{5}+a d \left (\frac {\left (-2 e h -3 f g \right ) d}{5}+c f h \right )\right ) \left (c f -d e \right ) \left (b x +a \right )^{2} \left (x d +c \right ) b \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}\, \left (2 \left (6 d^{2} e g \,x^{2}+3 x c \left (\left (-\frac {4 e h}{3}-\frac {3 f g}{2}\right ) x +g e \right ) d -\left (-2 h f \,x^{2}+\left (\frac {5 f g}{2}+2 e h \right ) x +g e \right ) c^{2}\right ) b^{3}-2 a \left (\left (\left (2 e h +\frac {3 f g}{2}\right ) x^{2}-9 e g x \right ) d^{2}-5 c \left (\frac {9 h f \,x^{2}}{10}+\frac {7 \left (-e h -f g \right ) x}{5}+g e \right ) d +c^{2} \left (-\frac {17}{2} f h x +\frac {3}{2} f g +e h \right )\right ) b^{2}+11 a^{2} \left (\frac {\left (-h f \,x^{2}+\left (-6 e h -5 f g \right ) x +4 g e \right ) d^{2}}{11}-\frac {10 \left (-\frac {3}{5} f h x +\frac {9}{10} f g +e h \right ) c d}{11}+c^{2} f h \right ) b +a^{3} d f h \left (x d +c \right )\right ) \sqrt {f x +e}\right )}{4 \sqrt {\left (a f -b e \right ) b}\, \sqrt {\left (c f -d e \right ) d}\, \left (x d +c \right ) \left (a d -b c \right )^{4} \left (b x +a \right )^{2} b}\)	\(570\)
derivativedivides	\(2 f^{3} \left (-\frac {\frac {\left (-\frac {1}{8} a^{3} d^{2} f^{2} h +\frac {5}{4} a^{2} b c d \,f^{2} h -\frac {1}{2} a^{2} b \,d^{2} e f h -\frac {3}{8} a^{2} b \,d^{2} f^{2} g -\frac {9}{8} a \,b^{2} c^{2} f^{2} h -\frac {1}{4} a \,b^{2} c d \,f^{2} g +a \,b^{2} d^{2} e f g +\frac {1}{2} b^{3} c^{2} e f h +\frac {5}{8} b^{3} c^{2} f^{2} g -b^{3} c d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\frac {f \left (a^{4} d^{2} f^{2} h +6 a^{3} b c d \,f^{2} h -5 a^{3} b \,d^{2} e f h -5 a^{3} b \,d^{2} f^{2} g -7 a^{2} b^{2} c^{2} f^{2} h -6 a^{2} b^{2} c d e f h +2 a^{2} b^{2} c d \,f^{2} g +4 a^{2} b^{2} d^{2} e^{2} h +13 a^{2} b^{2} d^{2} e f g +11 a \,b^{3} c^{2} e f h +3 a \,b^{3} c^{2} f^{2} g -10 a \,b^{3} c d e f g -8 a \,b^{3} d^{2} e^{2} g -4 b^{4} c^{2} e^{2} h -3 b^{4} c^{2} e f g +8 b^{4} c d \,e^{2} g \right ) \sqrt {f x +e}}{8 b}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}-\frac {\left (a^{3} d^{2} f^{2} h -10 a^{2} b c d \,f^{2} h +4 a^{2} b \,d^{2} e f h +3 a^{2} b \,d^{2} f^{2} g -15 a \,b^{2} c^{2} f^{2} h +32 a \,b^{2} c d e f h +18 a \,b^{2} c d \,f^{2} g -8 a \,b^{2} d^{2} e^{2} h -24 a \,b^{2} d^{2} e f g +12 b^{3} c^{2} e f h +3 b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -24 b^{3} c d e f g +24 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 b \sqrt {\left (a f -b e \right ) b}}}{f^{3} \left (a d -b c \right )^{4}}+\frac {\left (c f -d e \right ) \left (\frac {\left (-\frac {1}{2} a c d f h +\frac {1}{2} a \,d^{2} f g +\frac {1}{2} b \,c^{2} f h -\frac {1}{2} b c d f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (5 a c d f h -2 a \,d^{2} e h -3 a \,d^{2} f g +b \,c^{2} f h -4 b c d e h -3 b c d f g +6 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{f^{3} \left (a d -b c \right )^{4}}\right )\)	\(758\)
default	\(2 f^{3} \left (-\frac {\frac {\left (-\frac {1}{8} a^{3} d^{2} f^{2} h +\frac {5}{4} a^{2} b c d \,f^{2} h -\frac {1}{2} a^{2} b \,d^{2} e f h -\frac {3}{8} a^{2} b \,d^{2} f^{2} g -\frac {9}{8} a \,b^{2} c^{2} f^{2} h -\frac {1}{4} a \,b^{2} c d \,f^{2} g +a \,b^{2} d^{2} e f g +\frac {1}{2} b^{3} c^{2} e f h +\frac {5}{8} b^{3} c^{2} f^{2} g -b^{3} c d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\frac {f \left (a^{4} d^{2} f^{2} h +6 a^{3} b c d \,f^{2} h -5 a^{3} b \,d^{2} e f h -5 a^{3} b \,d^{2} f^{2} g -7 a^{2} b^{2} c^{2} f^{2} h -6 a^{2} b^{2} c d e f h +2 a^{2} b^{2} c d \,f^{2} g +4 a^{2} b^{2} d^{2} e^{2} h +13 a^{2} b^{2} d^{2} e f g +11 a \,b^{3} c^{2} e f h +3 a \,b^{3} c^{2} f^{2} g -10 a \,b^{3} c d e f g -8 a \,b^{3} d^{2} e^{2} g -4 b^{4} c^{2} e^{2} h -3 b^{4} c^{2} e f g +8 b^{4} c d \,e^{2} g \right ) \sqrt {f x +e}}{8 b}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}-\frac {\left (a^{3} d^{2} f^{2} h -10 a^{2} b c d \,f^{2} h +4 a^{2} b \,d^{2} e f h +3 a^{2} b \,d^{2} f^{2} g -15 a \,b^{2} c^{2} f^{2} h +32 a \,b^{2} c d e f h +18 a \,b^{2} c d \,f^{2} g -8 a \,b^{2} d^{2} e^{2} h -24 a \,b^{2} d^{2} e f g +12 b^{3} c^{2} e f h +3 b^{3} c^{2} f^{2} g -16 b^{3} c d \,e^{2} h -24 b^{3} c d e f g +24 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 b \sqrt {\left (a f -b e \right ) b}}}{f^{3} \left (a d -b c \right )^{4}}+\frac {\left (c f -d e \right ) \left (\frac {\left (-\frac {1}{2} a c d f h +\frac {1}{2} a \,d^{2} f g +\frac {1}{2} b \,c^{2} f h -\frac {1}{2} b c d f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (5 a c d f h -2 a \,d^{2} e h -3 a \,d^{2} f g +b \,c^{2} f h -4 b c d e h -3 b c d f g +6 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{f^{3} \left (a d -b c \right )^{4}}\right )\)	\(758\)

Input:

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

Output:

1/4*(((c*f-d*e)*d)^(1/2)*((24*d^2*e^2*g-16*c*(e*h+3/2*f*g)*e*d+3*(4*e*f*h+ 
f^2*g)*c^2)*b^3-15*a*(8/15*d^2*e*(e*h+3*f*g)-32/15*c*(e*h+9/16*f*g)*f*d+c^ 
2*f^2*h)*b^2-10*a^2*d*(1/5*(-2*e*h-3/2*f*g)*d+c*f*h)*f*b+a^3*d^2*f^2*h)*(b 
*x+a)^2*(d*x+c)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))-((a*f-b*e)*b)^ 
(1/2)*(-20*(1/5*(6*d^2*e*g-4*c*(e*h+3/4*f*g)*d+c^2*f*h)*b+a*d*(1/5*(-2*e*h 
-3*f*g)*d+c*f*h))*(c*f-d*e)*(b*x+a)^2*(d*x+c)*b*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)*(2*(6*d^2*e*g*x^2+3*x*c*(( 
-4/3*e*h-3/2*f*g)*x+g*e)*d-(-2*h*f*x^2+(5/2*f*g+2*e*h)*x+g*e)*c^2)*b^3-2*a 
*(((2*e*h+3/2*f*g)*x^2-9*e*g*x)*d^2-5*c*(9/10*h*f*x^2+7/5*(-e*h-f*g)*x+g*e 
)*d+c^2*(-17/2*f*h*x+3/2*f*g+e*h))*b^2+11*a^2*(1/11*(-h*f*x^2+(-6*e*h-5*f* 
g)*x+4*g*e)*d^2-10/11*(-3/5*f*h*x+9/10*f*g+e*h)*c*d+c^2*f*h)*b+a^3*d*f*h*( 
d*x+c))*(f*x+e)^(1/2)))/((a*f-b*e)*b)^(1/2)/((c*f-d*e)*d)^(1/2)/(d*x+c)/(a 
*d-b*c)^4/(b*x+a)^2/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3104 vs. \(2 (486) = 972\).

Time = 23.60 (sec) , antiderivative size = 12461, normalized size of antiderivative = 23.96 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^3/(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 [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 958, normalized size of antiderivative = 1.84 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(24*b^3*d^2*e^2*g - 24*b^3*c*d*e*f*g - 24*a*b^2*d^2*e*f*g + 3*b^3*c^2* 
f^2*g + 18*a*b^2*c*d*f^2*g + 3*a^2*b*d^2*f^2*g - 16*b^3*c*d*e^2*h - 8*a*b^ 
2*d^2*e^2*h + 12*b^3*c^2*e*f*h + 32*a*b^2*c*d*e*f*h + 4*a^2*b*d^2*e*f*h - 
15*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^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 
- 4*a^3*b^2*c*d^3 + a^4*b*d^4)*sqrt(-b^2*e + a*b*f)) - (6*b*d^3*e^2*g - 9* 
b*c*d^2*e*f*g - 3*a*d^3*e*f*g + 3*b*c^2*d*f^2*g + 3*a*c*d^2*f^2*g - 4*b*c* 
d^2*e^2*h - 2*a*d^3*e^2*h + 5*b*c^2*d*e*f*h + 7*a*c*d^2*e*f*h - b*c^3*f^2* 
h - 5*a*c^2*d*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((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)*sqrt(-d^2 
*e + c*d*f)) + (sqrt(f*x + e)*d^2*e*f*g - sqrt(f*x + e)*c*d*f^2*g - sqrt(f 
*x + e)*c*d*e*f*h + sqrt(f*x + e)*c^2*f^2*h)/((b^3*c^3 - 3*a*b^2*c^2*d + 3 
*a^2*b*c*d^2 - a^3*d^3)*((f*x + e)*d - d*e + c*f)) + 1/4*(8*(f*x + e)^(3/2 
)*b^3*d*e*f*g - 8*sqrt(f*x + e)*b^3*d*e^2*f*g - 5*(f*x + e)^(3/2)*b^3*c*f^ 
2*g - 3*(f*x + e)^(3/2)*a*b^2*d*f^2*g + 3*sqrt(f*x + e)*b^3*c*e*f^2*g + 13 
*sqrt(f*x + e)*a*b^2*d*e*f^2*g - 3*sqrt(f*x + e)*a*b^2*c*f^3*g - 5*sqrt(f* 
x + e)*a^2*b*d*f^3*g - 4*(f*x + e)^(3/2)*b^3*c*e*f*h - 4*(f*x + e)^(3/2)*a 
*b^2*d*e*f*h + 4*sqrt(f*x + e)*b^3*c*e^2*f*h + 4*sqrt(f*x + e)*a*b^2*d*e^2 
*f*h + 9*(f*x + e)^(3/2)*a*b^2*c*f^2*h - (f*x + e)^(3/2)*a^2*b*d*f^2*h - 1 
1*sqrt(f*x + e)*a*b^2*c*e*f^2*h - 5*sqrt(f*x + e)*a^2*b*d*e*f^2*h + 7*s...
 

Mupad [B] (verification not implemented)

Time = 114.84 (sec) , antiderivative size = 97983, normalized size of antiderivative = 188.43 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(log(- (((((b*d^2*f^3*(c*f - d*e)*(8*b^2*c*e*h + 3*b^2*c*f*g - 12*b^2*d*e* 
g - a^2*d*f*h - 11*a*b*c*f*h + 4*a*b*d*e*h + 9*a*b*d*f*g))/(a*d - b*c) + ( 
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)*(-(2*( 
f^6*(a*d - b*c)^24*(16*b^2*c*e*h^2 + a^2*d*f*h^2 + 9*b^2*d*f*g^2 - 16*a*b* 
c*f*h^2 + 8*a*b*d*e*h^2 - 24*b^2*d*e*g*h + 6*a*b*d*f*g*h)^2)^(1/2) + 2*a^1 
4*d^13*f^4*h^2 + 2304*a^8*b^6*d^13*e^4*g^2 + 256*a^10*b^4*d^13*e^4*h^2 + 1 
8*a^12*b^2*d^13*f^4*g^2 + 2304*b^14*c^8*d^5*e^4*g^2 + 1024*b^14*c^10*d^3*e 
^4*h^2 + 32*a*b^13*c^13*f^4*h^2 + 18*b^14*c^12*d*f^4*g^2 - 32*b^14*c^13*e* 
f^3*h^2 - 56*a^13*b*c*d^12*f^4*h^2 + 16*a^13*b*d^13*e*f^3*h^2 - 1536*a^9*b 
^5*d^13*e^4*g*h - 3072*b^14*c^9*d^4*e^4*g*h - 18432*a*b^13*c^7*d^6*e^4*g^2 
 - 18432*a^7*b^7*c*d^12*e^4*g^2 - 7168*a*b^13*c^9*d^4*e^4*h^2 + 360*a*b^13 
*c^11*d^2*f^4*g^2 - 1024*a^9*b^5*c*d^12*e^4*h^2 + 360*a^11*b^3*c*d^12*f^4* 
g^2 + 514*a^2*b^12*c^12*d*f^4*h^2 - 4608*a^9*b^5*d^13*e^3*f*g^2 - 576*a^11 
*b^3*d^13*e*f^3*g^2 - 256*a^11*b^3*d^13*e^3*f*h^2 - 4608*b^14*c^9*d^4*e^3* 
f*g^2 - 576*b^14*c^11*d^2*e*f^3*g^2 - 1536*b^14*c^11*d^2*e^3*f*h^2 + 576*b 
^14*c^12*d*e^2*f^2*h^2 + 12*a^13*b*d^13*f^4*g*h + 64512*a^2*b^12*c^6*d^7*e 
^4*g^2 - 129024*a^3*b^11*c^5*d^8*e^4*g^2 + 161280*a^4*b^10*c^4*d^9*e^4*g^2 
 - 129024*a^5*b^9*c^3*d^10*e^4*g^2 + 64512*a^6*b^8*c^2*d^11*e^4*g^2 + 2073 
6*a^2*b^12*c^8*d^5*e^4*h^2 - 2268*a^2*b^12*c^10*d^3*f^4*g^2 - 30720*a^3*b^ 
11*c^7*d^6*e^4*h^2 + 3528*a^3*b^11*c^9*d^4*f^4*g^2 + 21504*a^4*b^10*c^6...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 8595, normalized size of antiderivative = 16.53 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^3 (c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e))) 
*a**5*c*d**3*f**2*h + sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt 
(b)*sqrt(a*f - b*e)))*a**5*d**4*f**2*h*x - 10*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**2*f**2*h + 4* 
sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))* 
a**4*b*c*d**3*e*f*h + 3*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*d**3*f**2*g - 8*sqrt(b)*sqrt(a*f - b*e)*a 
tan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*c*d**3*f**2*h*x + 
4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)) 
)*a**4*b*d**4*e*f*h*x + 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/( 
sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d**4*f**2*g*x + 2*sqrt(b)*sqrt(a*f - b*e) 
*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d**4*f**2*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**3*b**2*c**3*d*f**2*h + 32*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c**2*d**2*e*f*h + 18*sqrt(b)* 
sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b** 
2*c**2*d**2*f**2*g - 35*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq 
rt(b)*sqrt(a*f - b*e)))*a**3*b**2*c**2*d**2*f**2*h*x - 8*sqrt(b)*sqrt(a*f 
- b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c*d**3* 
e**2*h - 24*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqr...