\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx\) [95]

Optimal result

Integrand size = 29, antiderivative size = 1359 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx =\text {Too large to display} \] Output:

-1/64*d*(5*a^4*d^3*f^3*h+5*a^3*b*d^2*f^2*(-19*c*f*h+8*d*e*h+7*d*f*g)-b^4*( 
320*d^3*e^3*g+c^3*f^2*(-8*e*h+3*f*g)+16*c^2*d*e*f*(14*e*h+f*g)-16*c*d^2*e^ 
2*(16*e*h+19*f*g))-a*b^3*(5*c^3*f^3*h-16*d^3*e^2*(4*e*h+41*f*g)+48*c*d^2*e 
*f*(13*e*h+12*f*g)-5*c^2*d*f^2*(88*e*h+5*f*g))-a^2*b^2*d*f*(225*c^2*f^2*h+ 
16*d^2*e*(7*e*h+23*f*g)-c*d*f*(472*e*h+263*f*g)))*(f*x+e)^(1/2)/b^2/(-a*d+ 
b*c)^5/(-a*f+b*e)^2/(d*x+c)-1/24*(a^2*d*f*h-b^2*(-8*c*e*h-3*c*f*g+10*d*e*g 
)+a*b*(-11*c*f*h+2*d*e*h+7*d*f*g))*(f*x+e)^(1/2)/b^2/(-a*d+b*c)^2/(b*x+a)^ 
3/(d*x+c)-1/96*(a^3*d^2*f^2*h+a^2*b*d*f*(-22*c*f*h+12*d*e*h+7*d*f*g)-a*b^2 
*(59*c^2*f^2*h+4*d^2*e*(4*e*h+21*f*g)-2*c*d*f*(46*e*h+35*f*g))+b^3*(80*d^2 
*e^2*g-4*c*d*e*(16*e*h+19*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)-1/192*(5*a^4*d^3*f^3*h+5*a^3*b*d^2 
*f^2*(-21*c*f*h+10*d*e*h+7*d*f*g)-5*a^2*b^2*d*f*(73*c^2*f^2*h-11*c*d*f*(12 
*e*h+7*f*g)+2*d^2*e*(16*e*h+49*f*g))-b^4*(480*d^3*e^3*g+3*c^3*f^2*(-8*e*h+ 
3*f*g)-16*c*d^2*e^2*(24*e*h+31*f*g)+2*c^2*d*e*f*(184*e*h+21*f*g))-a*b^3*(1 
5*c^3*f^3*h-16*d^3*e^2*(6*e*h+59*f*g)+4*c*d^2*e*f*(228*e*h+227*f*g)-c^2*d* 
f^2*(706*e*h+69*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)-1/4*(-a*h+b*g)*(f*x+e)^(3/2)/b/(-a*d+b*c)/(b*x+a)^4/(d*x+c)-1/64* 
(5*a^5*d^4*f^4*h+5*a^4*b*d^3*f^3*(-20*c*f*h+8*d*e*h+7*d*f*g)+b^5*(640*d^4* 
e^4*g+c^4*f^3*(-8*e*h+3*f*g)+16*c^3*d*e*f^2*(-6*e*h+f*g)-256*c*d^3*e^3*(2* 
e*h+3*f*g)+144*c^2*d^2*e^2*f*(4*e*h+f*g))-10*a^3*b^2*d^2*f^2*(45*c^2*f^...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(16895\) vs. \(2(1359)=2718\).

Time = 16.65 (sec) , antiderivative size = 16895, normalized size of antiderivative = 12.43 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 3.45 (sec) , antiderivative size = 1443, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {166, 27, 25, 166, 27, 168, 27, 168, 27, 168, 27, 174, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\int -\frac {\sqrt {e+f x} (10 b d e g-3 b c f g-8 b c e h-2 a d e h+3 a c f h+f (7 b d g-8 b c h+a d h) x)}{2 (a+b x)^4 (c+d x)^2}dx}{4 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -\frac {\sqrt {e+f x} (a (2 d e-3 c f) h-b (10 d e g-3 c f g-8 c e h)-f (7 b d g-8 b c h+a d h) x)}{(a+b x)^4 (c+d x)^2}dx}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\sqrt {e+f x} (a (2 d e-3 c f) h-b (10 d e g-3 c f g-8 c e h)-f (7 b d g-8 b c h+a d h) x)}{(a+b x)^4 (c+d x)^2}dx}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {\frac {\int \frac {12 b e (d e-c f) (5 b d g-4 b c h-a d h)-(2 d e-c f) \left (d f h a^2+b (7 d f g+2 d e h-11 c f h) a-b^2 (10 d e g-3 c f g-8 c e h)\right )-f \left (-\left (\left (48 f h c^2-7 d (9 f g+8 e h) c+70 d^2 e g\right ) b^2\right )-a d (23 c f h-7 d (f g+2 e h)) b+a^2 d^2 f h\right ) x}{2 (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}dx}{3 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-11 c f h+2 d e h+7 d f g)-b^2 (-8 c e h-3 c f g+10 d e g)\right )}{3 b (a+b x)^3 (c+d x) (b c-a d)}}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {12 b e (d e-c f) (5 b d g-4 b c h-a d h)-(2 d e-c f) \left (d f h a^2+b (7 d f g+2 d e h-11 c f h) a-b^2 (10 d e g-3 c f g-8 c e h)\right )-f \left (-\left (\left (48 f h c^2-7 d (9 f g+8 e h) c+70 d^2 e g\right ) b^2\right )-a d (23 c f h-7 d (f g+2 e h)) b+a^2 d^2 f h\right ) x}{(a+b x)^3 (c+d x)^2 \sqrt {e+f x}}dx}{6 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-11 c f h+2 d e h+7 d f g)-b^2 (-8 c e h-3 c f g+10 d e g)\right )}{3 b (a+b x)^3 (c+d x) (b c-a d)}}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {-\frac {\int \frac {5 d^2 f^2 (2 d e-c f) h a^3+5 b d f \left (2 e (7 f g+8 e h) d^2-c f (7 f g+40 e h) d+14 c^2 f^2 h\right ) a^2+b^2 \left (-32 e^2 (17 f g+3 e h) d^3+16 c e f (33 f g+37 e h) d^2-6 c^2 f^2 (9 f g+71 e h) d+15 c^3 f^3 h\right ) a+b^3 \left (3 f^2 (3 f g-8 e h) c^3+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 g\right )+5 d f \left (d^2 f^2 h a^3+b d f (7 d f g+12 d e h-22 c f h) a^2-b^2 \left (4 e (21 f g+4 e h) d^2-2 c f (35 f g+46 e h) d+59 c^2 f^2 h\right ) a+b^3 \left (f (3 f g+56 e h) c^2-4 d e (19 f g+16 e h) c+80 d^2 e^2 g\right )\right ) x}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (a^3 d^2 f^2 h+a^2 b d f (-22 c f h+12 d e h+7 d f g)-a b^2 \left (59 c^2 f^2 h-2 c d f (46 e h+35 f g)+4 d^2 e (4 e h+21 f g)\right )+b^3 \left (c^2 f (56 e h+3 f g)-4 c d e (16 e h+19 f g)+80 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-11 c f h+2 d e h+7 d f g)-b^2 (-8 c e h-3 c f g+10 d e g)\right )}{3 b (a+b x)^3 (c+d x) (b c-a d)}}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\int \frac {5 d^2 f^2 (2 d e-c f) h a^3+5 b d f \left (2 e (7 f g+8 e h) d^2-c f (7 f g+40 e h) d+14 c^2 f^2 h\right ) a^2+b^2 \left (-32 e^2 (17 f g+3 e h) d^3+16 c e f (33 f g+37 e h) d^2-6 c^2 f^2 (9 f g+71 e h) d+15 c^3 f^3 h\right ) a+b^3 \left (3 f^2 (3 f g-8 e h) c^3+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 g\right )+5 d f \left (d^2 f^2 h a^3+b d f (7 d f g+12 d e h-22 c f h) a^2-b^2 \left (4 e (21 f g+4 e h) d^2-2 c f (35 f g+46 e h) d+59 c^2 f^2 h\right ) a+b^3 \left (f (3 f g+56 e h) c^2-4 d e (19 f g+16 e h) c+80 d^2 e^2 g\right )\right ) x}{(a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (a^3 d^2 f^2 h+a^2 b d f (-22 c f h+12 d e h+7 d f g)-a b^2 \left (59 c^2 f^2 h-2 c d f (46 e h+35 f g)+4 d^2 e (4 e h+21 f g)\right )+b^3 \left (c^2 f (56 e h+3 f g)-4 c d e (16 e h+19 f g)+80 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\sqrt {e+f x} \left (a^2 d f h+a b (-11 c f h+2 d e h+7 d f g)-b^2 (-8 c e h-3 c f g+10 d e g)\right )}{3 b (a+b x)^3 (c+d x) (b c-a d)}}{8 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (7 d f g+12 d e h-22 c f h) a^2-b^2 \left (4 e (21 f g+4 e h) d^2-2 c f (35 f g+46 e h) d+59 c^2 f^2 h\right ) a+b^3 \left (f (3 f g+56 e h) c^2-4 d e (19 f g+16 e h) c+80 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x)}-\frac {\frac {\left (5 d^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+10 d e h-21 c f h) a^3-5 b^2 d f \left (2 e (49 f g+16 e h) d^2-11 c f (7 f g+12 e h) d+73 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (59 f g+6 e h) d^3+4 c e f (227 f g+228 e h) d^2-c^2 f^2 (69 f g+706 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+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 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)}-\frac {\int -\frac {3 \left (5 d^3 f^3 (2 d e-c f) h a^4+5 b d^2 f^2 \left (2 e (7 f g+8 e h) d^2-c f (7 f g+44 e h) d+17 c^2 f^2 h\right ) a^3+b^2 d f \left (-32 e^2 (23 f g+7 e h) d^3+4 c e f (193 f g+252 e h) d^2-c^2 f^2 (141 f g+734 e h) d+85 c^3 f^3 h\right ) a^2-b^3 \left (-32 e^3 (41 f g+4 e h) d^4+80 c e^2 f (19 f g+16 e h) d^3-2 c^2 e f^2 (147 f g+608 e h) d^2-c^3 f^3 (19 f g-184 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (f^3 (3 f g-8 e h) c^4+16 d e f^2 (f g-6 e h) c^3+144 d^2 e^2 f (f g+4 e h) c^2-256 d^3 e^3 (3 f g+2 e h) c+640 d^4 e^4 g\right )+d f \left (5 d^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+10 d e h-21 c f h) a^3-5 b^2 d f \left (2 e (49 f g+16 e h) d^2-11 c f (7 f g+12 e h) d+73 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (59 f g+6 e h) d^3+4 c e f (227 f g+228 e h) d^2-c^2 f^2 (69 f g+706 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+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 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)}}{6 b (b c-a d)}-\frac {\left (d f h a^2+b (7 d f g+2 d e h-11 c f h) a-b^2 (10 d e g-3 c f g-8 c e h)\right ) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}}{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)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (7 d f g+12 d e h-22 c f h) a^2-b^2 \left (4 e (21 f g+4 e h) d^2-2 c f (35 f g+46 e h) d+59 c^2 f^2 h\right ) a+b^3 \left (f (3 f g+56 e h) c^2-4 d e (19 f g+16 e h) c+80 d^2 e^2 g\right )\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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+10 d e h-21 c f h) a^3-5 b^2 d f \left (2 e (49 f g+16 e h) d^2-11 c f (7 f g+12 e h) d+73 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (59 f g+6 e h) d^3+4 c e f (227 f g+228 e h) d^2-c^2 f^2 (69 f g+706 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+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x) (c+d x)}+\frac {3 \int \frac {5 d^3 f^3 (2 d e-c f) h a^4+5 b d^2 f^2 \left (2 e (7 f g+8 e h) d^2-c f (7 f g+44 e h) d+17 c^2 f^2 h\right ) a^3+b^2 d f \left (-32 e^2 (23 f g+7 e h) d^3+4 c e f (193 f g+252 e h) d^2-c^2 f^2 (141 f g+734 e h) d+85 c^3 f^3 h\right ) a^2-b^3 \left (-32 e^3 (41 f g+4 e h) d^4+80 c e^2 f (19 f g+16 e h) d^3-2 c^2 e f^2 (147 f g+608 e h) d^2-c^3 f^3 (19 f g-184 e h) d+5 c^4 f^4 h\right ) a-b^4 \left (f^3 (3 f g-8 e h) c^4+16 d e f^2 (f g-6 e h) c^3+144 d^2 e^2 f (f g+4 e h) c^2-256 d^3 e^3 (3 f g+2 e h) c+640 d^4 e^4 g\right )+d f \left (5 d^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+10 d e h-21 c f h) a^3-5 b^2 d f \left (2 e (49 f g+16 e h) d^2-11 c f (7 f g+12 e h) d+73 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (59 f g+6 e h) d^3+4 c e f (227 f g+228 e h) d^2-c^2 f^2 (69 f g+706 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+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 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)}}{4 (b c-a d) (b e-a f)}}{6 b (b c-a d)}-\frac {\left (d f h a^2+b (7 d f g+2 d e h-11 c f h) a-b^2 (10 d e g-3 c f g-8 c e h)\right ) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}}{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)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (7 d f g+12 d e h-22 c f h) a^2-b^2 \left (4 e (21 f g+4 e h) d^2-2 c f (35 f g+46 e h) d+59 c^2 f^2 h\right ) a+b^3 \left (f (3 f g+56 e h) c^2-4 d e (19 f g+16 e h) c+80 d^2 e^2 g\right )\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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+10 d e h-21 c f h) a^3-5 b^2 d f \left (2 e (49 f g+16 e h) d^2-11 c f (7 f g+12 e h) d+73 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (59 f g+6 e h) d^3+4 c e f (227 f g+228 e h) d^2-c^2 f^2 (69 f g+706 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+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+8 d e h-19 c f h) a^3-b^2 d f \left (16 e (23 f g+7 e h) d^2-c f (263 f g+472 e h) d+225 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (41 f g+4 e h) d^3+48 c e f (12 f g+13 e h) d^2-5 c^2 f^2 (5 f g+88 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (3 f g-8 e h) c^3+16 d e f (f g+14 e h) c^2-16 d^2 e^2 (19 f g+16 e h) c+320 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)}+\frac {\int \frac {b (d e-c f) \left (-\left (\left (f^3 (3 f g-8 e h) c^4+16 d e f^2 (f g-6 e h) c^3+144 d^2 e^2 f (f g+4 e h) c^2-256 d^3 e^3 (3 f g+2 e h) c+640 d^4 e^4 g\right ) b^4\right )-a \left (-64 e^3 (23 f g+2 e h) d^4+16 c e^2 f (107 f g+88 e h) d^3-80 c^2 e f^2 (4 f g+17 e h) d^2-25 c^3 f^3 (f g-8 e h) d+5 c^4 f^4 h\right ) b^3+a^2 d f \left (-256 e^2 (4 f g+e h) d^3+48 c e f (23 f g+27 e h) d^2-5 c^2 f^2 (37 f g+200 e h) d+95 c^3 f^3 h\right ) b^2+a^3 d^2 f^2 \left (64 e (3 f g+2 e h) d^2-c f (157 f g+408 e h) d+225 c^2 f^2 h\right ) b+5 a^4 c d^3 f^4 h+d f \left (5 d^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+8 d e h-19 c f h) a^3-b^2 d f \left (16 e (23 f g+7 e h) d^2-c f (263 f g+472 e h) d+225 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (41 f g+4 e h) d^3+48 c e f (12 f g+13 e h) d^2-5 c^2 f^2 (5 f g+88 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (3 f g-8 e h) c^3+16 d e f (f g+14 e h) c^2-16 d^2 e^2 (19 f g+16 e h) c+320 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 )}{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 {\left (d f h a^2+b (7 d f g+2 d e h-11 c f h) a-b^2 (10 d e g-3 c f g-8 c e h)\right ) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}}{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)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (7 d f g+12 d e h-22 c f h) a^2-b^2 \left (4 e (21 f g+4 e h) d^2-2 c f (35 f g+46 e h) d+59 c^2 f^2 h\right ) a+b^3 \left (f (3 f g+56 e h) c^2-4 d e (19 f g+16 e h) c+80 d^2 e^2 g\right )\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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+10 d e h-21 c f h) a^3-5 b^2 d f \left (2 e (49 f g+16 e h) d^2-11 c f (7 f g+12 e h) d+73 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (59 f g+6 e h) d^3+4 c e f (227 f g+228 e h) d^2-c^2 f^2 (69 f g+706 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+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+8 d e h-19 c f h) a^3-b^2 d f \left (16 e (23 f g+7 e h) d^2-c f (263 f g+472 e h) d+225 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (41 f g+4 e h) d^3+48 c e f (12 f g+13 e h) d^2-5 c^2 f^2 (5 f g+88 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (3 f g-8 e h) c^3+16 d e f (f g+14 e h) c^2-16 d^2 e^2 (19 f g+16 e h) c+320 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)}+\frac {b \int \frac {-\left (\left (f^3 (3 f g-8 e h) c^4+16 d e f^2 (f g-6 e h) c^3+144 d^2 e^2 f (f g+4 e h) c^2-256 d^3 e^3 (3 f g+2 e h) c+640 d^4 e^4 g\right ) b^4\right )-a \left (-64 e^3 (23 f g+2 e h) d^4+16 c e^2 f (107 f g+88 e h) d^3-80 c^2 e f^2 (4 f g+17 e h) d^2-25 c^3 f^3 (f g-8 e h) d+5 c^4 f^4 h\right ) b^3+a^2 d f \left (-256 e^2 (4 f g+e h) d^3+48 c e f (23 f g+27 e h) d^2-5 c^2 f^2 (37 f g+200 e h) d+95 c^3 f^3 h\right ) b^2+a^3 d^2 f^2 \left (64 e (3 f g+2 e h) d^2-c f (157 f g+408 e h) d+225 c^2 f^2 h\right ) b+5 a^4 c d^3 f^4 h+d f \left (5 d^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+8 d e h-19 c f h) a^3-b^2 d f \left (16 e (23 f g+7 e h) d^2-c f (263 f g+472 e h) d+225 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (41 f g+4 e h) d^3+48 c e f (12 f g+13 e h) d^2-5 c^2 f^2 (5 f g+88 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (3 f g-8 e h) c^3+16 d e f (f g+14 e h) c^2-16 d^2 e^2 (19 f g+16 e h) c+320 d^3 e^3 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 {\left (d f h a^2+b (7 d f g+2 d e h-11 c f h) a-b^2 (10 d e g-3 c f g-8 c e h)\right ) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}}{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)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (7 d f g+12 d e h-22 c f h) a^2-b^2 \left (4 e (21 f g+4 e h) d^2-2 c f (35 f g+46 e h) d+59 c^2 f^2 h\right ) a+b^3 \left (f (3 f g+56 e h) c^2-4 d e (19 f g+16 e h) c+80 d^2 e^2 g\right )\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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+10 d e h-21 c f h) a^3-5 b^2 d f \left (2 e (49 f g+16 e h) d^2-11 c f (7 f g+12 e h) d+73 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (59 f g+6 e h) d^3+4 c e f (227 f g+228 e h) d^2-c^2 f^2 (69 f g+706 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+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+8 d e h-19 c f h) a^3-b^2 d f \left (16 e (23 f g+7 e h) d^2-c f (263 f g+472 e h) d+225 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (41 f g+4 e h) d^3+48 c e f (12 f g+13 e h) d^2-5 c^2 f^2 (5 f g+88 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (3 f g-8 e h) c^3+16 d e f (f g+14 e h) c^2-16 d^2 e^2 (19 f g+16 e h) c+320 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)}+\frac {b \left (-\frac {64 b d^2 (d e-c f) \left (a d (3 d f g+2 d e h-5 c f h)-b \left (5 f h c^2-d (7 f g+8 e h) c+10 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^4 f^4 h a^5+5 b d^3 f^3 (7 d f g+8 d e h-20 c f h) a^4-10 b^2 d^2 f^2 \left (8 e (7 f g+3 e h) d^2-2 c f (21 f g+44 e h) d+45 c^2 f^2 h\right ) a^3-10 b^3 d f \left (-8 e^2 (21 f g+4 e h) d^3+24 c e f (7 f g+8 e h) d^2-3 c^2 f^2 (7 f g+48 e h) d+10 c^3 f^3 h\right ) a^2+b^4 \left (-128 e^3 (14 f g+e h) d^4+32 c e^2 f (63 f g+52 e h) d^3-48 c^2 e f^2 (7 f g+33 e h) d^2-4 c^3 f^3 (7 f g-52 e h) d+5 c^4 f^4 h\right ) a+b^5 \left (f^3 (3 f g-8 e h) c^4+16 d e f^2 (f g-6 e h) c^3+144 d^2 e^2 f (f g+4 e h) c^2-256 d^3 e^3 (3 f g+2 e h) c+640 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 )}{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 {\left (d f h a^2+b (7 d f g+2 d e h-11 c f h) a-b^2 (10 d e g-3 c f g-8 c e h)\right ) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}}{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)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (7 d f g+12 d e h-22 c f h) a^2-b^2 \left (4 e (21 f g+4 e h) d^2-2 c f (35 f g+46 e h) d+59 c^2 f^2 h\right ) a+b^3 \left (f (3 f g+56 e h) c^2-4 d e (19 f g+16 e h) c+80 d^2 e^2 g\right )\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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+10 d e h-21 c f h) a^3-5 b^2 d f \left (2 e (49 f g+16 e h) d^2-11 c f (7 f g+12 e h) d+73 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (59 f g+6 e h) d^3+4 c e f (227 f g+228 e h) d^2-c^2 f^2 (69 f g+706 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+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+8 d e h-19 c f h) a^3-b^2 d f \left (16 e (23 f g+7 e h) d^2-c f (263 f g+472 e h) d+225 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (41 f g+4 e h) d^3+48 c e f (12 f g+13 e h) d^2-5 c^2 f^2 (5 f g+88 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (3 f g-8 e h) c^3+16 d e f (f g+14 e h) c^2-16 d^2 e^2 (19 f g+16 e h) c+320 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)}+\frac {b \left (-\frac {128 b d^2 (d e-c f) \left (a d (3 d f g+2 d e h-5 c f h)-b \left (5 f h c^2-d (7 f g+8 e h) c+10 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^4 f^4 h a^5+5 b d^3 f^3 (7 d f g+8 d e h-20 c f h) a^4-10 b^2 d^2 f^2 \left (8 e (7 f g+3 e h) d^2-2 c f (21 f g+44 e h) d+45 c^2 f^2 h\right ) a^3-10 b^3 d f \left (-8 e^2 (21 f g+4 e h) d^3+24 c e f (7 f g+8 e h) d^2-3 c^2 f^2 (7 f g+48 e h) d+10 c^3 f^3 h\right ) a^2+b^4 \left (-128 e^3 (14 f g+e h) d^4+32 c e^2 f (63 f g+52 e h) d^3-48 c^2 e f^2 (7 f g+33 e h) d^2-4 c^3 f^3 (7 f g-52 e h) d+5 c^4 f^4 h\right ) a+b^5 \left (f^3 (3 f g-8 e h) c^4+16 d e f^2 (f g-6 e h) c^3+144 d^2 e^2 f (f g+4 e h) c^2-256 d^3 e^3 (3 f g+2 e h) c+640 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 )}{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 {\left (d f h a^2+b (7 d f g+2 d e h-11 c f h) a-b^2 (10 d e g-3 c f g-8 c e h)\right ) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}}{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)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {-\frac {\sqrt {e+f x} \left (d^2 f^2 h a^3+b d f (7 d f g+12 d e h-22 c f h) a^2-b^2 \left (4 e (21 f g+4 e h) d^2-2 c f (35 f g+46 e h) d+59 c^2 f^2 h\right ) a+b^3 \left (f (3 f g+56 e h) c^2-4 d e (19 f g+16 e h) c+80 d^2 e^2 g\right )\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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+10 d e h-21 c f h) a^3-5 b^2 d f \left (2 e (49 f g+16 e h) d^2-11 c f (7 f g+12 e h) d+73 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (59 f g+6 e h) d^3+4 c e f (227 f g+228 e h) d^2-c^2 f^2 (69 f g+706 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+2 d e f (21 f g+184 e h) c^2-16 d^2 e^2 (31 f g+24 e h) c+480 d^3 e^3 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^3 f^3 h a^4+5 b d^2 f^2 (7 d f g+8 d e h-19 c f h) a^3-b^2 d f \left (16 e (23 f g+7 e h) d^2-c f (263 f g+472 e h) d+225 c^2 f^2 h\right ) a^2-b^3 \left (-16 e^2 (41 f g+4 e h) d^3+48 c e f (12 f g+13 e h) d^2-5 c^2 f^2 (5 f g+88 e h) d+5 c^3 f^3 h\right ) a-b^4 \left (f^2 (3 f g-8 e h) c^3+16 d e f (f g+14 e h) c^2-16 d^2 e^2 (19 f g+16 e h) c+320 d^3 e^3 g\right )\right )}{(b c-a d) (c+d x)}+\frac {b \left (\frac {128 b d^{3/2} \sqrt {d e-c f} \left (a d (3 d f g+2 d e h-5 c f h)-b \left (5 f h c^2-d (7 f g+8 e h) c+10 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}+\frac {2 \left (5 d^4 f^4 h a^5+5 b d^3 f^3 (7 d f g+8 d e h-20 c f h) a^4-10 b^2 d^2 f^2 \left (8 e (7 f g+3 e h) d^2-2 c f (21 f g+44 e h) d+45 c^2 f^2 h\right ) a^3-10 b^3 d f \left (-8 e^2 (21 f g+4 e h) d^3+24 c e f (7 f g+8 e h) d^2-3 c^2 f^2 (7 f g+48 e h) d+10 c^3 f^3 h\right ) a^2+b^4 \left (-128 e^3 (14 f g+e h) d^4+32 c e^2 f (63 f g+52 e h) d^3-48 c^2 e f^2 (7 f g+33 e h) d^2-4 c^3 f^3 (7 f g-52 e h) d+5 c^4 f^4 h\right ) a+b^5 \left (f^3 (3 f g-8 e h) c^4+16 d e f^2 (f g-6 e h) c^3+144 d^2 e^2 f (f g+4 e h) c^2-256 d^3 e^3 (3 f g+2 e h) c+640 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 )}{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 {\left (d f h a^2+b (7 d f g+2 d e h-11 c f h) a-b^2 (10 d e g-3 c f g-8 c e h)\right ) \sqrt {e+f x}}{3 b (b c-a d) (a+b x)^3 (c+d x)}}{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)}\)

Input:

Int[((e + f*x)^(3/2)*(g + h*x))/((a + b*x)^5*(c + d*x)^2),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)) + 
 (-1/3*((a^2*d*f*h - b^2*(10*d*e*g - 3*c*f*g - 8*c*e*h) + a*b*(7*d*f*g + 2 
*d*e*h - 11*c*f*h))*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)^3*(c + d*x)) + 
 (-1/2*((a^3*d^2*f^2*h + a^2*b*d*f*(7*d*f*g + 12*d*e*h - 22*c*f*h) - a*b^2 
*(59*c^2*f^2*h + 4*d^2*e*(21*f*g + 4*e*h) - 2*c*d*f*(35*f*g + 46*e*h)) + b 
^3*(80*d^2*e^2*g - 4*c*d*e*(19*f*g + 16*e*h) + c^2*f*(3*f*g + 56*e*h)))*Sq 
rt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)) - (((5*a^4*d^ 
3*f^3*h + 5*a^3*b*d^2*f^2*(7*d*f*g + 10*d*e*h - 21*c*f*h) - 5*a^2*b^2*d*f* 
(73*c^2*f^2*h - 11*c*d*f*(7*f*g + 12*e*h) + 2*d^2*e*(49*f*g + 16*e*h)) - b 
^4*(480*d^3*e^3*g + 3*c^3*f^2*(3*f*g - 8*e*h) - 16*c*d^2*e^2*(31*f*g + 24* 
e*h) + 2*c^2*d*e*f*(21*f*g + 184*e*h)) - a*b^3*(15*c^3*f^3*h - 16*d^3*e^2* 
(59*f*g + 6*e*h) + 4*c*d^2*e*f*(227*f*g + 228*e*h) - c^2*d*f^2*(69*f*g + 7 
06*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^4*d^3*f^3*h + 5*a^3*b*d^2*f^2*(7*d*f*g + 8*d*e*h - 19*c*f*h) 
- b^4*(320*d^3*e^3*g + c^3*f^2*(3*f*g - 8*e*h) + 16*c^2*d*e*f*(f*g + 14*e* 
h) - 16*c*d^2*e^2*(19*f*g + 16*e*h)) - a*b^3*(5*c^3*f^3*h - 16*d^3*e^2*(41 
*f*g + 4*e*h) + 48*c*d^2*e*f*(12*f*g + 13*e*h) - 5*c^2*d*f^2*(5*f*g + 88*e 
*h)) - a^2*b^2*d*f*(225*c^2*f^2*h + 16*d^2*e*(23*f*g + 7*e*h) - c*d*f*(263 
*f*g + 472*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(c + d*x)) + (b*((2*(5*a^5*d 
^4*f^4*h + 5*a^4*b*d^3*f^3*(7*d*f*g + 8*d*e*h - 20*c*f*h) + b^5*(640*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 = 103.17 (sec) , antiderivative size = 1909, normalized size of antiderivative = 1.40

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1909\)
derivativedivides	\(\text {Expression too large to display}\)	\(2949\)
default	\(\text {Expression too large to display}\)	\(2949\)

Input:

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

Output:

5/64*(((c*f-d*e)*d)^(1/2)*(b*x+a)^4*(d*x+c)*((128*d^4*e^4*g-512/5*c*(e*h+3 
/2*f*g)*e^3*d^3+576/5*c^2*(e*h+1/4*f*g)*f*e^2*d^2-96/5*c^3*(e*h-1/6*f*g)*f 
^2*e*d-8/5*c^4*(e*h-3/8*f*g)*f^3)*b^5+a*(-128/5*d^4*e^3*(e*h+14*f*g)+1664/ 
5*c*(e*h+63/52*f*g)*f*e^2*d^3-1584/5*c^2*f^2*e*(e*h+7/33*f*g)*d^2+208/5*(e 
*h-7/52*f*g)*c^3*f^3*d+c^4*f^4*h)*b^4-20*a^2*d*((-16/5*e^3*h-84/5*g*f*e^2) 
*d^3+96/5*c*(e*h+7/8*f*g)*f*e*d^2-72/5*(e*h+7/48*f*g)*c^2*f^2*d+c^3*f^3*h) 
*f*b^3-90*a^3*d^2*((8/15*e^2*h+56/45*e*f*g)*d^2-88/45*(e*h+21/44*f*g)*c*f* 
d+c^2*f^2*h)*f^2*b^2-20*a^4*((-2/5*e*h-7/20*f*g)*d+c*f*h)*d^3*f^3*b+a^5*d^ 
4*f^4*h)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))-((a*f-b*e)*b)^(1/2)*( 
-64*(c*f-d*e)*d^2*((2*d^2*e*g-8/5*c*(e*h+7/8*f*g)*d+c^2*f*h)*b+((-2/5*e*h- 
3/5*f*g)*d+c*f*h)*a*d)*(d*x+c)*(b*x+a)^4*b*(a*f-b*e)^2*arctan(d*(f*x+e)^(1 
/2)/((c*f-d*e)*d)^(1/2))+(a*d-b*c)*((64*d^4*e^3*g*x^4+32*x^3*c*(-19/10*f*g 
*x+e*(-8/5*h*x+g))*e^2*d^3-32/3*x^2*c^2*(-3/10*x^2*g*f^2+31/10*x*(-42/31*h 
*x+g)*e*f+e^2*(12/5*h*x+g))*e*d^2+16/3*x*c^3*(9/80*f^3*g*x^3+21/40*x^2*e*( 
-4/7*h*x+g)*f^2+19/10*x*(46/19*h*x+g)*e^2*f+e^3*(8/5*h*x+g))*d-16/5*c^4*(- 
3/16*f^3*g*x^3+1/8*e*x^2*(4*h*x+g)*f^2+3/2*(14/9*h*x+g)*x*e^2*f+e^3*(4/3*h 
*x+g)))*b^7-16/15*a*((123*e^2*f*g*x^4-210*x^3*(-2/35*h*x+g)*e^3)*d^4-110*x 
^2*c*(54/55*x^2*g*f^2-128/55*x*(-117/256*h*x+g)*e*f+e^2*(-87/55*h*x+g))*e* 
d^3+35*x*c^2*(15/112*f^3*g*x^3-529/280*x^2*e*(-660/529*h*x+g)*f^2+183/70*x 
*e^2*(-404/183*h*x+g)*f+e^3*(86/35*h*x+g))*d^2-17*c^3*(-21/136*x^3*(-5/...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5/(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. 4781 vs. \(2 (1317) = 2634\).

Time = 0.39 (sec) , antiderivative size = 4781, normalized size of antiderivative = 3.52 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/64*(640*b^5*d^4*e^4*g - 768*b^5*c*d^3*e^3*f*g - 1792*a*b^4*d^4*e^3*f*g + 
 144*b^5*c^2*d^2*e^2*f^2*g + 2016*a*b^4*c*d^3*e^2*f^2*g + 1680*a^2*b^3*d^4 
*e^2*f^2*g + 16*b^5*c^3*d*e*f^3*g - 336*a*b^4*c^2*d^2*e*f^3*g - 1680*a^2*b 
^3*c*d^3*e*f^3*g - 560*a^3*b^2*d^4*e*f^3*g + 3*b^5*c^4*f^4*g - 28*a*b^4*c^ 
3*d*f^4*g + 210*a^2*b^3*c^2*d^2*f^4*g + 420*a^3*b^2*c*d^3*f^4*g + 35*a^4*b 
*d^4*f^4*g - 512*b^5*c*d^3*e^4*h - 128*a*b^4*d^4*e^4*h + 576*b^5*c^2*d^2*e 
^3*f*h + 1664*a*b^4*c*d^3*e^3*f*h + 320*a^2*b^3*d^4*e^3*f*h - 96*b^5*c^3*d 
*e^2*f^2*h - 1584*a*b^4*c^2*d^2*e^2*f^2*h - 1920*a^2*b^3*c*d^3*e^2*f^2*h - 
 240*a^3*b^2*d^4*e^2*f^2*h - 8*b^5*c^4*e*f^3*h + 208*a*b^4*c^3*d*e*f^3*h + 
 1440*a^2*b^3*c^2*d^2*e*f^3*h + 880*a^3*b^2*c*d^3*e*f^3*h + 40*a^4*b*d^4*e 
*f^3*h + 5*a*b^4*c^4*f^4*h - 100*a^2*b^3*c^3*d*f^4*h - 450*a^3*b^2*c^2*d^2 
*f^4*h - 100*a^4*b*c*d^3*f^4*h + 5*a^5*d^4*f^4*h)*arctan(sqrt(f*x + e)*b/s 
qrt(-b^2*e + a*b*f))/((b^9*c^6*e^2 - 6*a*b^8*c^5*d*e^2 + 15*a^2*b^7*c^4*d^ 
2*e^2 - 20*a^3*b^6*c^3*d^3*e^2 + 15*a^4*b^5*c^2*d^4*e^2 - 6*a^5*b^4*c*d^5* 
e^2 + a^6*b^3*d^6*e^2 - 2*a*b^8*c^6*e*f + 12*a^2*b^7*c^5*d*e*f - 30*a^3*b^ 
6*c^4*d^2*e*f + 40*a^4*b^5*c^3*d^3*e*f - 30*a^5*b^4*c^2*d^4*e*f + 12*a^6*b 
^3*c*d^5*e*f - 2*a^7*b^2*d^6*e*f + a^2*b^7*c^6*f^2 - 6*a^3*b^6*c^5*d*f^2 + 
 15*a^4*b^5*c^4*d^2*f^2 - 20*a^5*b^4*c^3*d^3*f^2 + 15*a^6*b^3*c^2*d^4*f^2 
- 6*a^7*b^2*c*d^5*f^2 + a^8*b*d^6*f^2)*sqrt(-b^2*e + a*b*f)) - (10*b*d^5*e 
^2*g - 17*b*c*d^4*e*f*g - 3*a*d^5*e*f*g + 7*b*c^2*d^3*f^2*g + 3*a*c*d^4...
 

Mupad [B] (verification not implemented)

Time = 42.89 (sec) , antiderivative size = 1058660, normalized size of antiderivative = 779.00 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(((e + f*x)^(3/2)*(33*b^5*c^4*f^5*g - 15*a^5*d^4*f^5*h + 55*a*b^4*c^4*f^5* 
h + 279*a^4*b*d^4*f^5*g - 88*b^5*c^4*e*f^4*h + 3840*b^5*d^4*e^4*f*g + 1834 
*a^3*b^2*c*d^3*f^5*g - 1030*a^2*b^3*c^3*d*f^5*h - 10272*a*b^4*d^4*e^3*f^2* 
g - 2950*a^3*b^2*d^4*e*f^4*g - 5088*b^5*c*d^3*e^3*f^2*g - 944*b^5*c^3*d*e^ 
2*f^3*h + 1960*a^2*b^3*c^2*d^2*f^5*g - 2440*a^3*b^2*c^2*d^2*f^5*h + 9136*a 
^2*b^3*d^4*e^2*f^3*g + 1824*a^2*b^3*d^4*e^3*f^2*h - 1280*a^3*b^2*d^4*e^2*f 
^3*h + 1360*b^5*c^2*d^2*e^2*f^3*g + 3840*b^5*c^2*d^2*e^3*f^2*h - 266*a*b^4 
*c^3*d*f^5*g - 410*a^4*b*c*d^3*f^5*h - 768*a*b^4*d^4*e^4*f*h + 206*a^4*b*d 
^4*e*f^4*h + 134*b^5*c^3*d*e*f^4*g - 3072*b^5*c*d^3*e^4*f*h + 2106*a*b^4*c 
^3*d*e*f^4*h + 12544*a*b^4*c*d^3*e^2*f^3*g - 3122*a*b^4*c^2*d^2*e*f^4*g - 
9422*a^2*b^3*c*d^3*e*f^4*g + 9696*a*b^4*c*d^3*e^3*f^2*h + 4686*a^3*b^2*c*d 
^3*e*f^4*h - 10048*a*b^4*c^2*d^2*e^2*f^3*h - 10768*a^2*b^3*c*d^3*e^2*f^3*h 
 + 8450*a^2*b^3*c^2*d^2*e*f^4*h))/(192*b*(a^5*d^5 - b^5*c^5 - 10*a^2*b^3*c 
^3*d^2 + 10*a^3*b^2*c^2*d^3 + 5*a*b^4*c^4*d - 5*a^4*b*c*d^4)) + ((e + f*x) 
^(1/2)*(3*a*b^4*c^4*f^6*g - 5*a^5*c*d^3*f^6*h - 3*b^5*c^4*e*f^5*g + 5*a^5* 
d^4*e*f^5*h - 320*b^5*d^4*e^5*f*g + 5*a^2*b^3*c^4*f^6*h + 8*b^5*c^4*e^2*f^ 
4*h - 25*a^2*b^3*c^3*d*f^6*g - 95*a^3*b^2*c^3*d*f^6*h - 225*a^4*b*c^2*d^2* 
f^6*h + 1136*a*b^4*d^4*e^4*f^2*g - 93*a^4*b*d^4*e^2*f^4*h + 464*b^5*c*d^3* 
e^4*f^2*g - 13*b^5*c^3*d*e^2*f^4*g + 88*b^5*c^3*d*e^3*f^3*h + 185*a^3*b^2* 
c^2*d^2*f^6*g - 1472*a^2*b^3*d^4*e^3*f^3*g + 813*a^3*b^2*d^4*e^2*f^4*g ...
 

Reduce [F]

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

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

Output:

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