\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx\) [104]

Optimal result

Integrand size = 29, antiderivative size = 1700 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx =\text {Too large to display} \] Output:

1/192*d*(35*a^4*d^3*f^3*h-35*a^3*b*d^2*f^2*(-19*c*f*h+14*d*e*h+9*d*f*g)+b^ 
4*(1440*d^3*e^3*g+3*c^3*f^2*(-8*e*h+3*f*g)-240*c*d^2*e^2*(4*e*h+5*f*g)+22* 
c^2*d*e*f*(32*e*h+3*f*g))+a*b^3*(15*c^3*f^3*h-240*d^3*e^2*(2*e*h+13*f*g)+4 
*c*d^2*e*f*(668*e*h+567*f*g)-c^2*d*f^2*(1402*e*h+93*f*g))+a^2*b^2*d*f*(725 
*c^2*f^2*h+2*d^2*e*(472*e*h+993*f*g)-c*d*f*(2404*e*h+1041*f*g)))*(f*x+e)^( 
1/2)/b^2/(-a*d+b*c)^5/(-a*f+b*e)^2/(d*x+c)^2+1/24*(a^2*d*f*h+b^2*(-8*c*e*h 
-3*c*f*g+12*d*e*g)-a*b*(-11*c*f*h+4*d*e*h+9*d*f*g))*(f*x+e)^(1/2)/b^2/(-a* 
d+b*c)^2/(b*x+a)^3/(d*x+c)^2+1/96*(3*a^3*d^2*f^2*h-a^2*b*d*f*(-58*c*f*h+40 
*d*e*h+27*d*f*g)+a*b^2*(59*c^2*f^2*h+8*d^2*e*(5*e*h+18*f*g)-18*c*d*f*(8*e* 
h+5*f*g))-b^3*(120*d^2*e^2*g-16*c*d*e*(5*e*h+6*f*g)+c^2*f*(56*e*h+3*f*g))) 
*(f*x+e)^(1/2)/b^2/(-a*d+b*c)^3/(-a*f+b*e)/(b*x+a)^2/(d*x+c)^2+1/192*(21*a 
^4*d^3*f^3*h-7*a^3*b*d^2*f^2*(-59*c*f*h+44*d*e*h+27*d*f*g)+b^4*(960*d^3*e^ 
3*g+3*c^3*f^2*(-8*e*h+3*f*g)-40*c*d^2*e^2*(16*e*h+21*f*g)+4*c^2*d*e*f*(124 
*e*h+15*f*g))+7*a^2*b^2*d*f*(73*c^2*f^2*h+4*d^2*e*(22*e*h+45*f*g)-c*d*f*(2 
24*e*h+99*f*g))+a*b^3*(15*c^3*f^3*h-40*d^3*e^2*(8*e*h+51*f*g)+104*c*d^2*e* 
f*(17*e*h+15*f*g)-c^2*d*f^2*(980*e*h+87*f*g)))*(f*x+e)^(1/2)/b^2/(-a*d+b*c 
)^4/(-a*f+b*e)^2/(b*x+a)/(d*x+c)^2+1/64*d*(35*a^4*d^3*f^3*h-a^3*b*d^2*f^2* 
(-495*c*f*h+368*d*e*h+267*d*f*g)+b^4*(960*d^3*e^3*g+c^3*f^2*(-8*e*h+3*f*g) 
-80*c*d^2*e^2*(8*e*h+9*f*g)+8*c^2*d*e*f*(52*e*h+3*f*g))+a*b^3*(5*c^3*f^3*h 
-80*d^3*e^2*(4*e*h+27*f*g)+16*c*d^2*e*f*(113*e*h+87*f*g)-c^2*d*f^2*(832...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(29680\) vs. \(2(1700)=3400\).

Time = 17.18 (sec) , antiderivative size = 29680, normalized size of antiderivative = 17.46 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx=\text {Result too large to show} \] Input:

Integrate[((e + f*x)^(3/2)*(g + h*x))/((a + b*x)^5*(c + d*x)^3),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 4.75 (sec) , antiderivative size = 1781, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {166, 27, 25, 166, 27, 168, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\frac {\int \frac {24 b e (d e-c f) (3 b d g-2 b c h-a d h)+(4 d e-c f) \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )+3 f \left (\left (16 f h c^2-3 d (9 f g+8 e h) c+36 d^2 e g\right ) b^2+a d (19 c f h-3 d (3 f g+4 e h)) b+a^2 d^2 f h\right ) x}{2 (a+b x)^3 (c+d x)^3 \sqrt {e+f x}}dx}{3 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-11 c f h+4 d e h+9 d f g)+b^2 (-8 c e h-3 c f g+12 d e g)\right )}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {24 b e (d e-c f) (3 b d g-2 b c h-a d h)+(4 d e-c f) \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )+3 f \left (\left (16 f h c^2-3 d (9 f g+8 e h) c+36 d^2 e g\right ) b^2+a d (19 c f h-3 d (3 f g+4 e h)) b+a^2 d^2 f h\right ) x}{(a+b x)^3 (c+d x)^3 \sqrt {e+f x}}dx}{6 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-11 c f h+4 d e h+9 d f g)+b^2 (-8 c e h-3 c f g+12 d e g)\right )}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {\sqrt {e+f x} \left (3 a^3 d^2 f^2 h-a^2 b d f (-58 c f h+40 d e h+27 d f g)+a b^2 \left (59 c^2 f^2 h-18 c d f (8 e h+5 f g)+8 d^2 e (5 e h+18 f g)\right )-b^3 \left (c^2 f (56 e h+3 f g)-16 c d e (5 e h+6 f g)+120 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int -\frac {7 d^2 f^2 (4 d e-c f) h a^3-7 b d f \left (12 e (3 f g+4 e h) d^2-c f (9 f g+80 e h) d+14 c^2 f^2 h\right ) a^2-b^2 \left (-80 e^2 (15 f g+4 e h) d^3+8 c e f (111 f g+151 e h) d^2-6 c^2 f^2 (11 f g+98 e h) d+15 c^3 f^3 h\right ) a-b^3 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )+7 d f \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right ) x}{2 (a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{6 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-11 c f h+4 d e h+9 d f g)+b^2 (-8 c e h-3 c f g+12 d e g)\right )}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {7 d^2 f^2 (4 d e-c f) h a^3-7 b d f \left (12 e (3 f g+4 e h) d^2-c f (9 f g+80 e h) d+14 c^2 f^2 h\right ) a^2-b^2 \left (-80 e^2 (15 f g+4 e h) d^3+8 c e f (111 f g+151 e h) d^2-6 c^2 f^2 (11 f g+98 e h) d+15 c^3 f^3 h\right ) a-b^3 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )+7 d f \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right ) x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^3 d^2 f^2 h-a^2 b d f (-58 c f h+40 d e h+27 d f g)+a b^2 \left (59 c^2 f^2 h-18 c d f (8 e h+5 f g)+8 d^2 e (5 e h+18 f g)\right )-b^3 \left (c^2 f (56 e h+3 f g)-16 c d e (5 e h+6 f g)+120 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}}{6 b (b c-a d)}+\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-11 c f h+4 d e h+9 d f g)+b^2 (-8 c e h-3 c f g+12 d e g)\right )}{3 b (a+b x)^3 (c+d x)^2 (b c-a d)}}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}-\frac {\int -\frac {35 d^3 f^3 (4 d e-c f) h a^4-35 b d^2 f^2 \left (4 e (9 f g+14 e h) d^2-c f (9 f g+88 e h) d+17 c^2 f^2 h\right ) a^3-b^2 d f \left (-8 e^2 (993 f g+472 e h) d^3+8 c e f (726 f g+1289 e h) d^2-c^2 f^2 (699 f g+4676 e h) d+345 c^3 f^3 h\right ) a^2+b^3 \left (-960 e^3 (13 f g+2 e h) d^4+344 c e^2 f (33 f g+32 e h) d^3-4 c^2 e f^2 (411 f g+1864 e h) d^2-3 c^3 f^3 (21 f g-256 e h) d+15 c^4 f^4 h\right ) a+3 b^4 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 g\right )+5 d f \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\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 c-a d) (b e-a f)}}{6 b (b c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2}+\frac {\int \frac {35 d^3 f^3 (4 d e-c f) h a^4-35 b d^2 f^2 \left (4 e (9 f g+14 e h) d^2-c f (9 f g+88 e h) d+17 c^2 f^2 h\right ) a^3-b^2 d f \left (-8 e^2 (993 f g+472 e h) d^3+8 c e f (726 f g+1289 e h) d^2-c^2 f^2 (699 f g+4676 e h) d+345 c^3 f^3 h\right ) a^2+b^3 \left (-960 e^3 (13 f g+2 e h) d^4+344 c e^2 f (33 f g+32 e h) d^3-4 c^2 e f^2 (411 f g+1864 e h) d^2-3 c^3 f^3 (21 f g-256 e h) d+15 c^4 f^4 h\right ) a+3 b^4 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 g\right )+5 d f \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {\int \frac {6 b (d e-c f) \left (35 d^3 f^3 (2 d e-c f) h a^4-b d^2 f^2 \left (2 e (267 f g+368 e h) d^2-3 c f (73 f g+412 e h) d+325 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (183 f g+82 e h) d^3+4 c e f (567 f g+928 e h) d^2-5 c^2 f^2 (57 f g+358 e h) d+125 c^3 f^3 h\right ) a^2+b^3 \left (-160 e^3 (27 f g+4 e h) d^4+16 c e^2 f (249 f g+236 e h) d^3-2 c^2 e f^2 (291 f g+1304 e h) d^2-c^3 f^3 (27 f g-272 e h) d+5 c^4 f^4 h\right ) a+b^4 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 g\right )+d f \left (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right ) x\right )}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \int \frac {35 d^3 f^3 (2 d e-c f) h a^4-b d^2 f^2 \left (2 e (267 f g+368 e h) d^2-3 c f (73 f g+412 e h) d+325 c^2 f^2 h\right ) a^3-b^2 d f \left (-16 e^2 (183 f g+82 e h) d^3+4 c e f (567 f g+928 e h) d^2-5 c^2 f^2 (57 f g+358 e h) d+125 c^3 f^3 h\right ) a^2+b^3 \left (-160 e^3 (27 f g+4 e h) d^4+16 c e^2 f (249 f g+236 e h) d^3-2 c^2 e f^2 (291 f g+1304 e h) d^2-c^3 f^3 (27 f g-272 e h) d+5 c^4 f^4 h\right ) a+b^4 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 g\right )+d f \left (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{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 c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \sqrt {e+f x} \left (35 d^3 f^3 h a^4-b d^2 f^2 (267 d f g+368 d e h-495 c f h) a^3+b^2 d f \left (8 e (183 f g+82 e h) d^2-c f (663 f g+1672 e h) d+425 c^2 f^2 h\right ) a^2+b^3 \left (-80 e^2 (27 f g+4 e h) d^3+16 c e f (87 f g+113 e h) d^2-c^2 f^2 (33 f g+832 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (3 f g-8 e h) c^3+8 d e f (3 f g+52 e h) c^2-80 d^2 e^2 (9 f g+8 e h) c+960 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)}+\frac {\int -\frac {b (d e-c f) \left (d^3 f^3 (205 c f h-48 d (f g+4 e h)) a^4+b d^2 f^2 \left (32 e (33 f g+32 e h) d^2-3 c f (199 f g+656 e h) d+625 c^2 f^2 h\right ) a^3+b^2 d f \left (-16 e^2 (243 f g+92 e h) d^3+8 c e f (393 f g+572 e h) d^2-5 c^2 f^2 (69 f g+472 e h) d+135 c^3 f^3 h\right ) a^2-b^3 \left (-320 e^3 (15 f g+2 e h) d^4+16 c e^2 f (279 f g+256 e h) d^3-16 c^2 e f^2 (39 f g+181 e h) d^2-3 c^3 f^3 (11 f g-96 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 g\right )-d f \left (35 d^3 f^3 h a^4-b d^2 f^2 (267 d f g+368 d e h-495 c f h) a^3+b^2 d f \left (8 e (183 f g+82 e h) d^2-c f (663 f g+1672 e h) d+425 c^2 f^2 h\right ) a^2+b^3 \left (-80 e^2 (27 f g+4 e h) d^3+16 c e f (87 f g+113 e h) d^2-c^2 f^2 (33 f g+832 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (3 f g-8 e h) c^3+8 d e f (3 f g+52 e h) c^2-80 d^2 e^2 (9 f g+8 e h) c+960 d^3 e^3 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 )}{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 c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \left (35 d^3 f^3 h a^4-b d^2 f^2 (267 d f g+368 d e h-495 c f h) a^3+b^2 d f \left (8 e (183 f g+82 e h) d^2-c f (663 f g+1672 e h) d+425 c^2 f^2 h\right ) a^2+b^3 \left (-80 e^2 (27 f g+4 e h) d^3+16 c e f (87 f g+113 e h) d^2-c^2 f^2 (33 f g+832 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (3 f g-8 e h) c^3+8 d e f (3 f g+52 e h) c^2-80 d^2 e^2 (9 f g+8 e h) c+960 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {\int \frac {b (d e-c f) \left (d^3 f^3 (205 c f h-48 d (f g+4 e h)) a^4+b d^2 f^2 \left (32 e (33 f g+32 e h) d^2-3 c f (199 f g+656 e h) d+625 c^2 f^2 h\right ) a^3+b^2 d f \left (-16 e^2 (243 f g+92 e h) d^3+8 c e f (393 f g+572 e h) d^2-5 c^2 f^2 (69 f g+472 e h) d+135 c^3 f^3 h\right ) a^2-b^3 \left (-320 e^3 (15 f g+2 e h) d^4+16 c e^2 f (279 f g+256 e h) d^3-16 c^2 e f^2 (39 f g+181 e h) d^2-3 c^3 f^3 (11 f g-96 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 g\right )-d f \left (35 d^3 f^3 h a^4-b d^2 f^2 (267 d f g+368 d e h-495 c f h) a^3+b^2 d f \left (8 e (183 f g+82 e h) d^2-c f (663 f g+1672 e h) d+425 c^2 f^2 h\right ) a^2+b^3 \left (-80 e^2 (27 f g+4 e h) d^3+16 c e f (87 f g+113 e h) d^2-c^2 f^2 (33 f g+832 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (3 f g-8 e h) c^3+8 d e f (3 f g+52 e h) c^2-80 d^2 e^2 (9 f g+8 e h) c+960 d^3 e^3 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 )}{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 c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \left (35 d^3 f^3 h a^4-b d^2 f^2 (267 d f g+368 d e h-495 c f h) a^3+b^2 d f \left (8 e (183 f g+82 e h) d^2-c f (663 f g+1672 e h) d+425 c^2 f^2 h\right ) a^2+b^3 \left (-80 e^2 (27 f g+4 e h) d^3+16 c e f (87 f g+113 e h) d^2-c^2 f^2 (33 f g+832 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (3 f g-8 e h) c^3+8 d e f (3 f g+52 e h) c^2-80 d^2 e^2 (9 f g+8 e h) c+960 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \int \frac {d^3 f^3 (205 c f h-48 d (f g+4 e h)) a^4+b d^2 f^2 \left (32 e (33 f g+32 e h) d^2-3 c f (199 f g+656 e h) d+625 c^2 f^2 h\right ) a^3+b^2 d f \left (-16 e^2 (243 f g+92 e h) d^3+8 c e f (393 f g+572 e h) d^2-5 c^2 f^2 (69 f g+472 e h) d+135 c^3 f^3 h\right ) a^2-b^3 \left (-320 e^3 (15 f g+2 e h) d^4+16 c e^2 f (279 f g+256 e h) d^3-16 c^2 e f^2 (39 f g+181 e h) d^2-3 c^3 f^3 (11 f g-96 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 g\right )-d f \left (35 d^3 f^3 h a^4-b d^2 f^2 (267 d f g+368 d e h-495 c f h) a^3+b^2 d f \left (8 e (183 f g+82 e h) d^2-c f (663 f g+1672 e h) d+425 c^2 f^2 h\right ) a^2+b^3 \left (-80 e^2 (27 f g+4 e h) d^3+16 c e f (87 f g+113 e h) d^2-c^2 f^2 (33 f g+832 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (3 f g-8 e h) c^3+8 d e f (3 f g+52 e h) c^2-80 d^2 e^2 (9 f g+8 e h) c+960 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{b c-a d}\right )}{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 c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \left (35 d^3 f^3 h a^4-b d^2 f^2 (267 d f g+368 d e h-495 c f h) a^3+b^2 d f \left (8 e (183 f g+82 e h) d^2-c f (663 f g+1672 e h) d+425 c^2 f^2 h\right ) a^2+b^3 \left (-80 e^2 (27 f g+4 e h) d^3+16 c e f (87 f g+113 e h) d^2-c^2 f^2 (33 f g+832 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (3 f g-8 e h) c^3+8 d e f (3 f g+52 e h) c^2-80 d^2 e^2 (9 f g+8 e h) c+960 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {16 d^2 \left (\left (-35 f^2 h c^3+7 d f (9 f g+16 e h) c^2-20 d^2 e (9 f g+4 e h) c+120 d^3 e^2 g\right ) b^2-2 a d \left (10 e (3 f g+2 e h) d^2-c f (27 f g+58 e h) d+35 c^2 f^2 h\right ) b+3 a^2 d^2 f (d f g+4 d e h-5 c f h)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx (b e-a f)^2}{b c-a d}+\frac {\left (35 d^4 f^4 h a^5-35 b d^3 f^3 (9 d f g+16 d e h-20 c f h) a^4+70 b^2 d^2 f^2 \left (12 e (3 f g+2 e h) d^2-2 c f (9 f g+26 e h) d+15 c^2 f^2 h\right ) a^3+14 b^3 d f \left (-16 e^2 (27 f g+8 e h) d^3+12 c e f (27 f g+38 e h) d^2-3 c^2 f^2 (9 f g+76 e h) d+10 c^3 f^3 h\right ) a^2-b^4 \left (-640 e^3 (9 f g+e h) d^4+64 c e^2 f (81 f g+74 e h) d^3-72 c^2 e f^2 (9 f g+46 e h) d^2-4 c^3 f^3 (9 f g-74 e h) d+5 c^4 f^4 h\right ) a-b^5 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}\right )}{b c-a d}\right )}{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 c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \left (35 d^3 f^3 h a^4-b d^2 f^2 (267 d f g+368 d e h-495 c f h) a^3+b^2 d f \left (8 e (183 f g+82 e h) d^2-c f (663 f g+1672 e h) d+425 c^2 f^2 h\right ) a^2+b^3 \left (-80 e^2 (27 f g+4 e h) d^3+16 c e f (87 f g+113 e h) d^2-c^2 f^2 (33 f g+832 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (3 f g-8 e h) c^3+8 d e f (3 f g+52 e h) c^2-80 d^2 e^2 (9 f g+8 e h) c+960 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {32 d^2 \left (\left (-35 f^2 h c^3+7 d f (9 f g+16 e h) c^2-20 d^2 e (9 f g+4 e h) c+120 d^3 e^2 g\right ) b^2-2 a d \left (10 e (3 f g+2 e h) d^2-c f (27 f g+58 e h) d+35 c^2 f^2 h\right ) b+3 a^2 d^2 f (d f g+4 d e h-5 c f h)\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x} (b e-a f)^2}{(b c-a d) f}+\frac {2 \left (35 d^4 f^4 h a^5-35 b d^3 f^3 (9 d f g+16 d e h-20 c f h) a^4+70 b^2 d^2 f^2 \left (12 e (3 f g+2 e h) d^2-2 c f (9 f g+26 e h) d+15 c^2 f^2 h\right ) a^3+14 b^3 d f \left (-16 e^2 (27 f g+8 e h) d^3+12 c e f (27 f g+38 e h) d^2-3 c^2 f^2 (9 f g+76 e h) d+10 c^3 f^3 h\right ) a^2-b^4 \left (-640 e^3 (9 f g+e h) d^4+64 c e^2 f (81 f g+74 e h) d^3-72 c^2 e f^2 (9 f g+46 e h) d^2-4 c^3 f^3 (9 f g-74 e h) d+5 c^4 f^4 h\right ) a-b^5 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 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 )}{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 c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+4 d e h-11 c f h) a+b^2 (12 d e g-3 c f g-8 c e h)\right )}{3 b (b c-a d) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (3 d^2 f^2 h a^3-b d f (27 d f g+40 d e h-58 c f h) a^2+b^2 \left (8 e (18 f g+5 e h) d^2-18 c f (5 f g+8 e h) d+59 c^2 f^2 h\right ) a-b^3 \left (f (3 f g+56 e h) c^2-16 d e (6 f g+5 e h) c+120 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (21 d^3 f^3 h a^4-7 b d^2 f^2 (27 d f g+44 d e h-59 c f h) a^3+7 b^2 d f \left (4 e (45 f g+22 e h) d^2-c f (99 f g+224 e h) d+73 c^2 f^2 h\right ) a^2+b^3 \left (-40 e^2 (51 f g+8 e h) d^3+104 c e f (15 f g+17 e h) d^2-c^2 f^2 (87 f g+980 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+4 d e f (15 f g+124 e h) c^2-40 d^2 e^2 (21 f g+16 e h) c+960 d^3 e^3 g\right )\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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+14 d e h-19 c f h) a^3+b^2 d f \left (2 e (993 f g+472 e h) d^2-c f (1041 f g+2404 e h) d+725 c^2 f^2 h\right ) a^2+b^3 \left (-240 e^2 (13 f g+2 e h) d^3+4 c e f (567 f g+668 e h) d^2-c^2 f^2 (93 f g+1402 e h) d+15 c^3 f^3 h\right ) a+b^4 \left (3 f^2 (3 f g-8 e h) c^3+22 d e f (3 f g+32 e h) c^2-240 d^2 e^2 (5 f g+4 e h) c+1440 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)^2}+\frac {3 b \left (\frac {2 d \left (35 d^3 f^3 h a^4-b d^2 f^2 (267 d f g+368 d e h-495 c f h) a^3+b^2 d f \left (8 e (183 f g+82 e h) d^2-c f (663 f g+1672 e h) d+425 c^2 f^2 h\right ) a^2+b^3 \left (-80 e^2 (27 f g+4 e h) d^3+16 c e f (87 f g+113 e h) d^2-c^2 f^2 (33 f g+832 e h) d+5 c^3 f^3 h\right ) a+b^4 \left (f^2 (3 f g-8 e h) c^3+8 d e f (3 f g+52 e h) c^2-80 d^2 e^2 (9 f g+8 e h) c+960 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (-\frac {32 d^{3/2} \left (\left (-35 f^2 h c^3+7 d f (9 f g+16 e h) c^2-20 d^2 e (9 f g+4 e h) c+120 d^3 e^2 g\right ) b^2-2 a d \left (10 e (3 f g+2 e h) d^2-c f (27 f g+58 e h) d+35 c^2 f^2 h\right ) b+3 a^2 d^2 f (d f g+4 d e h-5 c f h)\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 (35 d^4 f^4 h a^5-35 b d^3 f^3 (9 d f g+16 d e h-20 c f h) a^4+70 b^2 d^2 f^2 \left (12 e (3 f g+2 e h) d^2-2 c f (9 f g+26 e h) d+15 c^2 f^2 h\right ) a^3+14 b^3 d f \left (-16 e^2 (27 f g+8 e h) d^3+12 c e f (27 f g+38 e h) d^2-3 c^2 f^2 (9 f g+76 e h) d+10 c^3 f^3 h\right ) a^2-b^4 \left (-640 e^3 (9 f g+e h) d^4+64 c e^2 f (81 f g+74 e h) d^3-72 c^2 e f^2 (9 f g+46 e h) d^2-4 c^3 f^3 (9 f g-74 e h) d+5 c^4 f^4 h\right ) a-b^5 \left (f^3 (3 f g-8 e h) c^4+24 d e f^2 (f g-6 e h) c^3+288 d^2 e^2 f (f g+4 e h) c^2-640 d^3 e^3 (3 f g+2 e h) c+1920 d^4 e^4 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 )}{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 c-a d)}}{8 b (b c-a d)}-\frac {(b g-a h) (e+f x)^{3/2}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

Input:

Int[((e + f*x)^(3/2)*(g + h*x))/((a + b*x)^5*(c + d*x)^3),x]
 

Output:

-1/4*((b*g - a*h)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(a + b*x)^4*(c + d*x)^2) 
 + (((a^2*d*f*h + b^2*(12*d*e*g - 3*c*f*g - 8*c*e*h) - a*b*(9*d*f*g + 4*d* 
e*h - 11*c*f*h))*Sqrt[e + f*x])/(3*b*(b*c - a*d)*(a + b*x)^3*(c + d*x)^2) 
+ (((3*a^3*d^2*f^2*h - a^2*b*d*f*(27*d*f*g + 40*d*e*h - 58*c*f*h) + a*b^2* 
(59*c^2*f^2*h + 8*d^2*e*(18*f*g + 5*e*h) - 18*c*d*f*(5*f*g + 8*e*h)) - b^3 
*(120*d^2*e^2*g - 16*c*d*e*(6*f*g + 5*e*h) + c^2*f*(3*f*g + 56*e*h)))*Sqrt 
[e + f*x])/(2*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)^2) + (((21*a^4 
*d^3*f^3*h - 7*a^3*b*d^2*f^2*(27*d*f*g + 44*d*e*h - 59*c*f*h) + b^4*(960*d 
^3*e^3*g + 3*c^3*f^2*(3*f*g - 8*e*h) - 40*c*d^2*e^2*(21*f*g + 16*e*h) + 4* 
c^2*d*e*f*(15*f*g + 124*e*h)) + 7*a^2*b^2*d*f*(73*c^2*f^2*h + 4*d^2*e*(45* 
f*g + 22*e*h) - c*d*f*(99*f*g + 224*e*h)) + a*b^3*(15*c^3*f^3*h - 40*d^3*e 
^2*(51*f*g + 8*e*h) + 104*c*d^2*e*f*(15*f*g + 17*e*h) - c^2*d*f^2*(87*f*g 
+ 980*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)^2 
) + ((2*d*(35*a^4*d^3*f^3*h - 35*a^3*b*d^2*f^2*(9*d*f*g + 14*d*e*h - 19*c* 
f*h) + b^4*(1440*d^3*e^3*g + 3*c^3*f^2*(3*f*g - 8*e*h) - 240*c*d^2*e^2*(5* 
f*g + 4*e*h) + 22*c^2*d*e*f*(3*f*g + 32*e*h)) + a*b^3*(15*c^3*f^3*h - 240* 
d^3*e^2*(13*f*g + 2*e*h) + 4*c*d^2*e*f*(567*f*g + 668*e*h) - c^2*d*f^2*(93 
*f*g + 1402*e*h)) + a^2*b^2*d*f*(725*c^2*f^2*h + 2*d^2*e*(993*f*g + 472*e* 
h) - c*d*f*(1041*f*g + 2404*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(c + d*x)^2 
) + (3*b*((2*d*(35*a^4*d^3*f^3*h - a^3*b*d^2*f^2*(267*d*f*g + 368*d*e*h...
 

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 = 252.77 (sec) , antiderivative size = 2761, normalized size of antiderivative = 1.62

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(2761\)
derivativedivides	\(\text {Expression too large to display}\)	\(3386\)
default	\(\text {Expression too large to display}\)	\(3386\)

Input:

int((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

35/64*(((-384/7*d^4*e^4*g+256/7*c*(e*h+3/2*f*g)*e^3*d^3-1152/35*c^2*(e*h+1 
/4*f*g)*f*e^2*d^2+144/35*c^3*(e*h-1/6*f*g)*f^2*e*d+8/35*c^4*(e*h-3/8*f*g)* 
f^3)*b^5-1/7*a*((-128*e^4*h-1152*e^3*f*g)*d^4+4736/5*c*(e*h+81/74*f*g)*f*e 
^2*d^3-3312/5*c^2*(e*h+9/46*f*g)*f^2*e*d^2+296/5*(e*h-9/74*f*g)*c^3*f^3*d+ 
c^4*f^4*h)*b^4+4*a^2*d*f*((-64/5*e^3*h-216/5*g*f*e^2)*d^3+228/5*(e*h+27/38 
*f*g)*c*f*e*d^2-114/5*c^2*(e*h+9/76*f*g)*f^2*d+c^3*f^3*h)*b^3+30*a^3*d^2*f 
^2*((8/5*e^2*h+12/5*e*f*g)*d^2-52/15*(e*h+9/26*f*g)*c*f*d+c^2*f^2*h)*b^2+2 
0*a^4*((-4/5*e*h-9/20*f*g)*d+c*f*h)*d^3*f^3*b+a^5*d^4*f^4*h)*((c*f-d*e)*d) 
^(1/2)*(d*x+c)^2*(b*x+a)^4*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))+41/ 
7*(-48/41*((-8*d^3*e^2*g+16/3*c*(e*h+9/4*f*g)*e*d^2-112/15*c^2*(e*h+9/16*f 
*g)*f*d+7/3*c^3*f^2*h)*b^2+14/3*a*((4/7*e^2*h+6/7*e*f*g)*d^2-58/35*c*(e*h+ 
27/58*f*g)*f*d+c^2*f^2*h)*d*b+a^2*d^2*((-4/5*e*h-1/5*f*g)*d+c*f*h)*f)*d^2* 
(d*x+c)^2*(b*x+a)^4*(a*f-b*e)^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)*((192/41*d^5*e^3*g*x^5+288/41*x^4*c*((-4/9 
*e*h-1/2*f*g)*x+g*e)*e^2*d^4+64/41*x^3*c^2*((13/10*e*f*h+3/40*f^2*g)*x^2+( 
-3*e^2*h-7/2*e*f*g)*x+e^2*g)*e*d^3-16/41*x^2*c^3*((1/10*h*f^2*e-3/80*g*f^3 
)*x^3+(-122/15*h*e^2*f-23/40*e*f^2*g)*x^2+(8/3*e^3*h+7/2*g*f*e^2)*x+e^3*g) 
*d^2+32/205*x*c^4*((-1/2*h*f^2*e+3/16*g*f^3)*x^3+(31/6*h*e^2*f+5/8*e*f^2*g 
)*x^2+(5/3*e^3*h+2*g*f*e^2)*x+e^3*g)*d-16/205*c^5*((1/2*h*f^2*e-3/16*g*f^3 
)*x^3+(7/3*h*e^2*f+1/8*e*f^2*g)*x^2+(4/3*e^3*h+3/2*g*f*e^2)*x+e^3*g))*b...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)**(3/2)*(h*x+g)/(b*x+a)**5/(d*x+c)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5/(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. 5220 vs. \(2 (1652) = 3304\).

Time = 0.48 (sec) , antiderivative size = 5220, normalized size of antiderivative = 3.07 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x, algorithm="giac")
 

Output:

1/64*(1920*b^5*d^4*e^4*g - 1920*b^5*c*d^3*e^3*f*g - 5760*a*b^4*d^4*e^3*f*g 
 + 288*b^5*c^2*d^2*e^2*f^2*g + 5184*a*b^4*c*d^3*e^2*f^2*g + 6048*a^2*b^3*d 
^4*e^2*f^2*g + 24*b^5*c^3*d*e*f^3*g - 648*a*b^4*c^2*d^2*e*f^3*g - 4536*a^2 
*b^3*c*d^3*e*f^3*g - 2520*a^3*b^2*d^4*e*f^3*g + 3*b^5*c^4*f^4*g - 36*a*b^4 
*c^3*d*f^4*g + 378*a^2*b^3*c^2*d^2*f^4*g + 1260*a^3*b^2*c*d^3*f^4*g + 315* 
a^4*b*d^4*f^4*g - 1280*b^5*c*d^3*e^4*h - 640*a*b^4*d^4*e^4*h + 1152*b^5*c^ 
2*d^2*e^3*f*h + 4736*a*b^4*c*d^3*e^3*f*h + 1792*a^2*b^3*d^4*e^3*f*h - 144* 
b^5*c^3*d*e^2*f^2*h - 3312*a*b^4*c^2*d^2*e^2*f^2*h - 6384*a^2*b^3*c*d^3*e^ 
2*f^2*h - 1680*a^3*b^2*d^4*e^2*f^2*h - 8*b^5*c^4*e*f^3*h + 296*a*b^4*c^3*d 
*e*f^3*h + 3192*a^2*b^3*c^2*d^2*e*f^3*h + 3640*a^3*b^2*c*d^3*e*f^3*h + 560 
*a^4*b*d^4*e*f^3*h + 5*a*b^4*c^4*f^4*h - 140*a^2*b^3*c^3*d*f^4*h - 1050*a^ 
3*b^2*c^2*d^2*f^4*h - 700*a^4*b*c*d^3*f^4*h - 35*a^5*d^4*f^4*h)*arctan(sqr 
t(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^9*c^7*e^2 - 7*a*b^8*c^6*d*e^2 + 21* 
a^2*b^7*c^5*d^2*e^2 - 35*a^3*b^6*c^4*d^3*e^2 + 35*a^4*b^5*c^3*d^4*e^2 - 21 
*a^5*b^4*c^2*d^5*e^2 + 7*a^6*b^3*c*d^6*e^2 - a^7*b^2*d^7*e^2 - 2*a*b^8*c^7 
*e*f + 14*a^2*b^7*c^6*d*e*f - 42*a^3*b^6*c^5*d^2*e*f + 70*a^4*b^5*c^4*d^3* 
e*f - 70*a^5*b^4*c^3*d^4*e*f + 42*a^6*b^3*c^2*d^5*e*f - 14*a^7*b^2*c*d^6*e 
*f + 2*a^8*b*d^7*e*f + a^2*b^7*c^7*f^2 - 7*a^3*b^6*c^6*d*f^2 + 21*a^4*b^5* 
c^5*d^2*f^2 - 35*a^5*b^4*c^4*d^3*f^2 + 35*a^6*b^3*c^3*d^4*f^2 - 21*a^7*b^2 
*c^2*d^5*f^2 + 7*a^8*b*c*d^6*f^2 - a^9*d^7*f^2)*sqrt(-b^2*e + a*b*f)) -...
 

Mupad [B] (verification not implemented)

Time = 51.16 (sec) , antiderivative size = 1321612, normalized size of antiderivative = 777.42 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

int(((e + f*x)^(3/2)*(g + h*x))/((a + b*x)^5*(c + d*x)^3),x)
 

Output:

atan(((((49152*a^20*b^2*d^20*f^9*g - 76800*a^19*b^3*c*d^19*f^9*g - 209920* 
a^20*b^2*c*d^19*f^9*h - 906240*a^19*b^3*d^20*e*f^8*g + 160768*a^20*b^2*d^2 
0*e*f^8*h + 3072*a^2*b^20*c^18*d^2*f^9*g - 76800*a^3*b^19*c^17*d^3*f^9*g + 
 1105920*a^4*b^18*c^16*d^4*f^9*g - 8527872*a^5*b^17*c^15*d^5*f^9*g + 39014 
400*a^6*b^16*c^14*d^6*f^9*g - 113627136*a^7*b^15*c^13*d^7*f^9*g + 21245952 
0*a^8*b^14*c^12*d^8*f^9*g - 225239040*a^9*b^13*c^11*d^9*f^9*g + 11421696*a 
^10*b^12*c^10*d^10*f^9*g + 417331200*a^11*b^11*c^9*d^11*f^9*g - 818847744* 
a^12*b^10*c^8*d^12*f^9*g + 924917760*a^13*b^9*c^7*d^13*f^9*g - 710062080*a 
^14*b^8*c^6*d^14*f^9*g + 381825024*a^15*b^7*c^5*d^15*f^9*g - 140697600*a^1 
6*b^6*c^4*d^16*f^9*g + 32759808*a^17*b^5*c^3*d^17*f^9*g - 3732480*a^18*b^4 
*c^2*d^18*f^9*g + 5120*a^3*b^19*c^18*d^2*f^9*h - 209920*a^4*b^18*c^17*d^3* 
f^9*h + 1761280*a^5*b^17*c^16*d^4*f^9*h - 5693440*a^6*b^16*c^15*d^5*f^9*h 
+ 143360*a^7*b^15*c^14*d^6*f^9*h + 65228800*a^8*b^14*c^13*d^7*f^9*h - 2720 
97280*a^9*b^13*c^12*d^8*f^9*h + 638443520*a^10*b^12*c^11*d^9*f^9*h - 10118 
45120*a^11*b^11*c^10*d^10*f^9*h + 1140705280*a^12*b^10*c^9*d^11*f^9*h - 91 
9592960*a^13*b^9*c^8*d^12*f^9*h + 510648320*a^14*b^8*c^7*d^13*f^9*h - 1695 
94880*a^15*b^7*c^6*d^14*f^9*h + 10178560*a^16*b^6*c^5*d^15*f^9*h + 2011136 
0*a^17*b^5*c^4*d^16*f^9*h - 10280960*a^18*b^4*c^3*d^17*f^9*h + 2298880*a^1 
9*b^3*c^2*d^18*f^9*h + 983040*a^14*b^8*d^20*e^6*f^3*g - 4669440*a^15*b^7*d 
^20*e^5*f^4*g + 8871936*a^16*b^6*d^20*e^4*f^5*g - 8475648*a^17*b^5*d^20...
 

Reduce [F]

\[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx=\int \frac {\left (f x +e \right )^{\frac {3}{2}} \left (h x +g \right )}{\left (b x +a \right )^{5} \left (d x +c \right )^{3}}d x \] Input:

int((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x)
 

Output:

int((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x)