\(\int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx\) [50]

Optimal result

Integrand size = 29, antiderivative size = 1368 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx =\text {Too large to display} \] Output:

1/64*d*(35*a^4*d^3*f^3*h-a^3*b*d^2*f^2*(-263*c*f*h+152*d*e*h+251*d*f*g)+b^ 
4*(320*d^3*e^3*g-c^3*f^2*(-8*e*h+5*f*g)-16*c^2*d*e*f*(-2*e*h+f*g)-16*c*d^2 
*e^2*(16*e*h+3*f*g))-a*b^3*(3*c^3*f^3*h-c^2*d*f^2*(-72*e*h+31*f*g)+16*d^3* 
e^2*(4*e*h+57*f*g)-16*c*d^2*e*f*(47*e*h+8*f*g))+a^2*b^2*d*f*(25*c^2*f^2*h+ 
16*d^2*e*(11*e*h+53*f*g)-c*d*f*(744*e*h+95*f*g)))*(f*x+e)^(1/2)/b/(-a*d+b* 
c)^5/(-a*f+b*e)^3/(d*x+c)-1/4*(-a*h+b*g)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(b*x+a 
)^4/(d*x+c)+1/24*(a^2*d*f*h-a*b*(-9*c*f*h+2*d*e*h+9*d*f*g)+b^2*(10*d*e*g-c 
*(8*e*h+f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c)^2/(-a*f+b*e)/(b*x+a)^3/(d*x+c)+1 
/96*(7*a^3*d^2*f^2*h-7*a^2*b*d*f*(-10*c*f*h+4*d*e*h+9*d*f*g)-b^3*(80*d^2*e 
^2*g-c^2*f*(-8*e*h+5*f*g)-4*c*d*e*(16*e*h+3*f*g))+a*b^2*(3*c^2*f^2*h+4*d^2 
*e*(4*e*h+37*f*g)-2*c*d*f*(62*e*h+11*f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c)^3/( 
-a*f+b*e)^2/(b*x+a)^2/(d*x+c)+1/192*(35*a^4*d^3*f^3*h-35*a^3*b*d^2*f^2*(-1 
1*c*f*h+6*d*e*h+9*d*f*g)+b^4*(480*d^3*e^3*g-2*c^2*d*e*f*(-40*e*h+19*f*g)-3 
*c^3*f^2*(-8*e*h+5*f*g)-16*c*d^2*e^2*(24*e*h+7*f*g))-a*b^3*(9*c^3*f^3*h-c^ 
2*d*f^2*(-194*e*h+83*f*g)+16*d^3*e^2*(6*e*h+83*f*g)-12*c*d^2*e*f*(92*e*h+2 
5*f*g))+a^2*b^2*d*f*(69*c^2*f^2*h+2*d^2*e*(128*e*h+589*f*g)-c*d*f*(1060*e* 
h+233*f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c)^4/(-a*f+b*e)^3/(b*x+a)/(d*x+c)+1/6 
4*(35*a^5*d^4*f^4*h-35*a^4*b*d^3*f^3*(-12*c*f*h+8*d*e*h+9*d*f*g)-b^5*(640* 
d^4*e^4*g-c^4*f^3*(-8*e*h+5*f*g)-48*c^2*d^2*e^2*f*(-4*e*h+f*g)-16*c^3*d*e* 
f^2*(-2*e*h+f*g)-256*c*d^3*e^3*(2*e*h+f*g))+70*a^3*b^2*d^2*f^2*(3*c^2*f...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(12952\) vs. \(2(1368)=2736\).

Time = 16.49 (sec) , antiderivative size = 12952, normalized size of antiderivative = 9.47 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 3.79 (sec) , antiderivative size = 1456, 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, 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.

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\int -\frac {10 b d e g-b c f g-8 b c e h-2 a d e h+a c f h+f (9 b d g-8 b c h-a d h) x}{2 (a+b x)^4 (c+d x)^2 \sqrt {e+f x}}dx}{4 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {10 b d e g-a (2 d e-c f) h-b c (f g+8 e h)+f (9 b d g-8 b c h-a d h) x}{(a+b x)^4 (c+d x)^2 \sqrt {e+f x}}dx}{8 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {7 d f (2 d e-c f) h a^2-b \left (2 e (39 f g+8 e h) d^2-c f (15 f g+68 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-8 e h) c^2-4 d e (3 f g+16 e h) c+80 d^2 e^2 g\right )+7 d f \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right ) x}{2 (a+b x)^3 (c+d x)^2 \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-9 c f h+2 d e h+9 d f g)+b^2 (10 d e g-c (8 e h+f g))\right )}{3 (a+b x)^3 (c+d x) (b c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {7 d f (2 d e-c f) h a^2-b \left (2 e (39 f g+8 e h) d^2-c f (15 f g+68 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-8 e h) c^2-4 d e (3 f g+16 e h) c+80 d^2 e^2 g\right )+7 d f \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right ) x}{(a+b x)^3 (c+d x)^2 \sqrt {e+f x}}dx}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-9 c f h+2 d e h+9 d f g)+b^2 (10 d e g-c (8 e h+f g))\right )}{3 (a+b x)^3 (c+d x) (b c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {\sqrt {e+f x} \left (7 a^3 d^2 f^2 h-7 a^2 b d f (-10 c f h+4 d e h+9 d f g)+a b^2 \left (3 c^2 f^2 h-2 c d f (62 e h+11 f g)+4 d^2 e (4 e h+37 f g)\right )-b^3 \left (c^2 (-f) (5 f g-8 e h)-4 c d e (16 e h+3 f g)+80 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}-\frac {\int -\frac {35 d^2 f^2 (2 d e-c f) h a^3-b d f \left (2 e (219 f g+88 e h) d^2-c f (123 f g+440 e h) d+54 c^2 f^2 h\right ) a^2+b^2 \left (32 e^2 (29 f g+3 e h) d^3-16 c e f (15 f g+49 e h) d^2-2 c^2 f^2 (29 f g-77 e h) d+9 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 f g+24 e h) c+480 d^3 e^3 g\right )+5 d f \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 f g+16 e h) c+80 d^2 e^2 g\right )\right ) x}{2 (a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-9 c f h+2 d e h+9 d f g)+b^2 (10 d e g-c (8 e h+f g))\right )}{3 (a+b x)^3 (c+d x) (b c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\int \frac {35 d^2 f^2 (2 d e-c f) h a^3-b d f \left (2 e (219 f g+88 e h) d^2-c f (123 f g+440 e h) d+54 c^2 f^2 h\right ) a^2+b^2 \left (32 e^2 (29 f g+3 e h) d^3-16 c e f (15 f g+49 e h) d^2-2 c^2 f^2 (29 f g-77 e h) d+9 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 f g+24 e h) c+480 d^3 e^3 g\right )+5 d f \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 f g+16 e h) c+80 d^2 e^2 g\right )\right ) x}{(a+b x)^2 (c+d x)^2 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (7 a^3 d^2 f^2 h-7 a^2 b d f (-10 c f h+4 d e h+9 d f g)+a b^2 \left (3 c^2 f^2 h-2 c d f (62 e h+11 f g)+4 d^2 e (4 e h+37 f g)\right )-b^3 \left (c^2 (-f) (5 f g-8 e h)-4 c d e (16 e h+3 f g)+80 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x) (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} \left (a^2 d f h-a b (-9 c f h+2 d e h+9 d f g)+b^2 (10 d e g-c (8 e h+f g))\right )}{3 (a+b x)^3 (c+d x) (b c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 b (b c-a d) (a+b x)^4 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {\frac {\sqrt {e+f x} \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 (35 d^3 f^3 (2 d e-c f) h a^4-b d^2 f^2 \left (2 e (251 f g+152 e h) d^2-c f (187 f g+620 e h) d+141 c^2 f^2 h\right ) a^3+b^2 d f \left (32 e^2 (53 f g+11 e h) d^3-12 c e f (59 f g+132 e h) d^2-c^2 f^2 (43 f g-478 e h) d+19 c^3 f^3 h\right ) a^2-b^3 \left (32 e^3 (57 f g+4 e h) d^4-16 c e^2 f (47 f g+96 e h) d^3-2 c^2 e f^2 (53 f g-272 e h) d^2-7 c^3 f^3 (3 f g-8 e h) d+3 c^4 f^4 h\right ) a+b^4 \left (-f^3 (5 f g-8 e h) c^4-16 d e f^2 (f g-2 e h) c^3-48 d^2 e^2 f (f g-4 e h) c^2-256 d^3 e^3 (f g+2 e h) c+640 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+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 c-a d) (b e-a f)}}{8 b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 b (b c-a d) (a+b x)^4 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {\frac {\sqrt {e+f x} \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 {35 d^3 f^3 (2 d e-c f) h a^4-b d^2 f^2 \left (2 e (251 f g+152 e h) d^2-c f (187 f g+620 e h) d+141 c^2 f^2 h\right ) a^3+b^2 d f \left (32 e^2 (53 f g+11 e h) d^3-12 c e f (59 f g+132 e h) d^2-c^2 f^2 (43 f g-478 e h) d+19 c^3 f^3 h\right ) a^2-b^3 \left (32 e^3 (57 f g+4 e h) d^4-16 c e^2 f (47 f g+96 e h) d^3-2 c^2 e f^2 (53 f g-272 e h) d^2-7 c^3 f^3 (3 f g-8 e h) d+3 c^4 f^4 h\right ) a+b^4 \left (-f^3 (5 f g-8 e h) c^4-16 d e f^2 (f g-2 e h) c^3-48 d^2 e^2 f (f g-4 e h) c^2-256 d^3 e^3 (f g+2 e h) c+640 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+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 c-a d) (b e-a f)}}{8 b (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 b (b c-a d) (a+b x)^4 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {\frac {\sqrt {e+f x} \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 (35 d^3 f^3 h a^4-b d^2 f^2 (251 d f g+152 d e h-263 c f h) a^3+b^2 d f \left (16 e (53 f g+11 e h) d^2-c f (95 f g+744 e h) d+25 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (57 f g+4 e h) d^3-16 c e f (8 f g+47 e h) d^2-c^2 f^2 (31 f g-72 e h) d+3 c^3 f^3 h\right ) a+b^4 \left (-f^2 (5 f g-8 e h) c^3-16 d e f (f g-2 e h) c^2-16 d^2 e^2 (3 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 (d^3 f^3 (157 c f h-64 d (f g+2 e h)) a^4+b d^2 f^2 \left (64 e (13 f g+6 e h) d^2-13 c f (25 f g+72 e h) d+185 c^2 f^2 h\right ) a^3-b^2 d f \left (192 e^2 (11 f g+2 e h) d^3-176 c e f (5 f g+11 e h) d^2-5 c^2 f^2 (19 f g-120 e h) d+25 c^3 f^3 h\right ) a^2+b^3 \left (64 e^3 (31 f g+2 e h) d^4-16 c e^2 f (51 f g+104 e h) d^3-16 c^2 e f^2 (8 f g-37 e h) d^2-c^3 f^3 (31 f g-72 e h) d+3 c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-8 e h) c^4-16 d e f^2 (f g-2 e h) c^3-48 d^2 e^2 f (f g-4 e h) c^2-256 d^3 e^3 (f g+2 e h) c+640 d^4 e^4 g\right )-d f \left (35 d^3 f^3 h a^4-b d^2 f^2 (251 d f g+152 d e h-263 c f h) a^3+b^2 d f \left (16 e (53 f g+11 e h) d^2-c f (95 f g+744 e h) d+25 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (57 f g+4 e h) d^3-16 c e f (8 f g+47 e h) d^2-c^2 f^2 (31 f g-72 e h) d+3 c^3 f^3 h\right ) a+b^4 \left (-f^2 (5 f g-8 e h) c^3-16 d e f (f g-2 e h) c^2-16 d^2 e^2 (3 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 c-a d) (b e-a f)}}{8 b (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 b (b c-a d) (a+b x)^4 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {\frac {\sqrt {e+f x} \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 \left (35 d^3 f^3 h a^4-b d^2 f^2 (251 d f g+152 d e h-263 c f h) a^3+b^2 d f \left (16 e (53 f g+11 e h) d^2-c f (95 f g+744 e h) d+25 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (57 f g+4 e h) d^3-16 c e f (8 f g+47 e h) d^2-c^2 f^2 (31 f g-72 e h) d+3 c^3 f^3 h\right ) a+b^4 \left (-f^2 (5 f g-8 e h) c^3-16 d e f (f g-2 e h) c^2-16 d^2 e^2 (3 f g+16 e h) c+320 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 (157 c f h-64 d (f g+2 e h)) a^4+b d^2 f^2 \left (64 e (13 f g+6 e h) d^2-13 c f (25 f g+72 e h) d+185 c^2 f^2 h\right ) a^3-b^2 d f \left (192 e^2 (11 f g+2 e h) d^3-176 c e f (5 f g+11 e h) d^2-5 c^2 f^2 (19 f g-120 e h) d+25 c^3 f^3 h\right ) a^2+b^3 \left (64 e^3 (31 f g+2 e h) d^4-16 c e^2 f (51 f g+104 e h) d^3-16 c^2 e f^2 (8 f g-37 e h) d^2-c^3 f^3 (31 f g-72 e h) d+3 c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-8 e h) c^4-16 d e f^2 (f g-2 e h) c^3-48 d^2 e^2 f (f g-4 e h) c^2-256 d^3 e^3 (f g+2 e h) c+640 d^4 e^4 g\right )-d f \left (35 d^3 f^3 h a^4-b d^2 f^2 (251 d f g+152 d e h-263 c f h) a^3+b^2 d f \left (16 e (53 f g+11 e h) d^2-c f (95 f g+744 e h) d+25 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (57 f g+4 e h) d^3-16 c e f (8 f g+47 e h) d^2-c^2 f^2 (31 f g-72 e h) d+3 c^3 f^3 h\right ) a+b^4 \left (-f^2 (5 f g-8 e h) c^3-16 d e f (f g-2 e h) c^2-16 d^2 e^2 (3 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 c-a d) (b e-a f)}}{8 b (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 b (b c-a d) (a+b x)^4 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {\frac {\sqrt {e+f x} \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 \left (35 d^3 f^3 h a^4-b d^2 f^2 (251 d f g+152 d e h-263 c f h) a^3+b^2 d f \left (16 e (53 f g+11 e h) d^2-c f (95 f g+744 e h) d+25 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (57 f g+4 e h) d^3-16 c e f (8 f g+47 e h) d^2-c^2 f^2 (31 f g-72 e h) d+3 c^3 f^3 h\right ) a+b^4 \left (-f^2 (5 f g-8 e h) c^3-16 d e f (f g-2 e h) c^2-16 d^2 e^2 (3 f g+16 e h) c+320 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 (157 c f h-64 d (f g+2 e h)) a^4+b d^2 f^2 \left (64 e (13 f g+6 e h) d^2-13 c f (25 f g+72 e h) d+185 c^2 f^2 h\right ) a^3-b^2 d f \left (192 e^2 (11 f g+2 e h) d^3-176 c e f (5 f g+11 e h) d^2-5 c^2 f^2 (19 f g-120 e h) d+25 c^3 f^3 h\right ) a^2+b^3 \left (64 e^3 (31 f g+2 e h) d^4-16 c e^2 f (51 f g+104 e h) d^3-16 c^2 e f^2 (8 f g-37 e h) d^2-c^3 f^3 (31 f g-72 e h) d+3 c^4 f^4 h\right ) a-b^4 \left (-f^3 (5 f g-8 e h) c^4-16 d e f^2 (f g-2 e h) c^3-48 d^2 e^2 f (f g-4 e h) c^2-256 d^3 e^3 (f g+2 e h) c+640 d^4 e^4 g\right )-d f \left (35 d^3 f^3 h a^4-b d^2 f^2 (251 d f g+152 d e h-263 c f h) a^3+b^2 d f \left (16 e (53 f g+11 e h) d^2-c f (95 f g+744 e h) d+25 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (57 f g+4 e h) d^3-16 c e f (8 f g+47 e h) d^2-c^2 f^2 (31 f g-72 e h) d+3 c^3 f^3 h\right ) a+b^4 \left (-f^2 (5 f g-8 e h) c^3-16 d e f (f g-2 e h) c^2-16 d^2 e^2 (3 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 c-a d) (b e-a f)}}{8 b (b c-a d)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 b (b c-a d) (a+b x)^4 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {\frac {\sqrt {e+f x} \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 \left (35 d^3 f^3 h a^4-b d^2 f^2 (251 d f g+152 d e h-263 c f h) a^3+b^2 d f \left (16 e (53 f g+11 e h) d^2-c f (95 f g+744 e h) d+25 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (57 f g+4 e h) d^3-16 c e f (8 f g+47 e h) d^2-c^2 f^2 (31 f g-72 e h) d+3 c^3 f^3 h\right ) a+b^4 \left (-f^2 (5 f g-8 e h) c^3-16 d e f (f g-2 e h) c^2-16 d^2 e^2 (3 f g+16 e h) c+320 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {\left (35 d^4 f^4 h a^5-35 b d^3 f^3 (9 d f g+8 d e h-12 c f h) a^4+70 b^2 d^2 f^2 \left (8 e (3 f g+e h) d^2-6 c f (f g+4 e h) d+3 c^2 f^2 h\right ) a^3-14 b^3 d f \left (8 e^2 (27 f g+4 e h) d^3-24 c e f (3 f g+8 e h) d^2-3 c^2 f^2 (3 f g-16 e h) d+2 c^3 f^3 h\right ) a^2+b^4 \left (128 e^3 (18 f g+e h) d^4-96 c e^2 f (9 f g+20 e h) d^3-48 c^2 e f^2 (3 f g-13 e h) d^2-4 c^3 f^3 (9 f g-20 e h) d+3 c^4 f^4 h\right ) a-b^5 \left (-f^3 (5 f g-8 e h) c^4-16 d e f^2 (f g-2 e h) c^3-48 d^2 e^2 f (f g-4 e h) c^2-256 d^3 e^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}-\frac {64 d^3 (b e-a f)^3 \left (a d (d f g+2 d e h-3 c f h)-b \left (7 f h c^2-d (9 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 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 c-a d) (b e-a f)}}{8 b (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 b (b c-a d) (a+b x)^4 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {\frac {\sqrt {e+f x} \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 \left (35 d^3 f^3 h a^4-b d^2 f^2 (251 d f g+152 d e h-263 c f h) a^3+b^2 d f \left (16 e (53 f g+11 e h) d^2-c f (95 f g+744 e h) d+25 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (57 f g+4 e h) d^3-16 c e f (8 f g+47 e h) d^2-c^2 f^2 (31 f g-72 e h) d+3 c^3 f^3 h\right ) a+b^4 \left (-f^2 (5 f g-8 e h) c^3-16 d e f (f g-2 e h) c^2-16 d^2 e^2 (3 f g+16 e h) c+320 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {2 \left (35 d^4 f^4 h a^5-35 b d^3 f^3 (9 d f g+8 d e h-12 c f h) a^4+70 b^2 d^2 f^2 \left (8 e (3 f g+e h) d^2-6 c f (f g+4 e h) d+3 c^2 f^2 h\right ) a^3-14 b^3 d f \left (8 e^2 (27 f g+4 e h) d^3-24 c e f (3 f g+8 e h) d^2-3 c^2 f^2 (3 f g-16 e h) d+2 c^3 f^3 h\right ) a^2+b^4 \left (128 e^3 (18 f g+e h) d^4-96 c e^2 f (9 f g+20 e h) d^3-48 c^2 e f^2 (3 f g-13 e h) d^2-4 c^3 f^3 (9 f g-20 e h) d+3 c^4 f^4 h\right ) a-b^5 \left (-f^3 (5 f g-8 e h) c^4-16 d e f^2 (f g-2 e h) c^3-48 d^2 e^2 f (f g-4 e h) c^2-256 d^3 e^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}-\frac {128 d^3 (b e-a f)^3 \left (a d (d f g+2 d e h-3 c f h)-b \left (7 f h c^2-d (9 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 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 c-a d) (b e-a f)}}{8 b (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\sqrt {e+f x} (b g-a h)}{4 b (b c-a d) (a+b x)^4 (c+d x)}-\frac {-\frac {\sqrt {e+f x} \left (d f h a^2-b (9 d f g+2 d e h-9 c f h) a+b^2 (10 d e g-c (f g+8 e h))\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)}-\frac {\frac {\sqrt {e+f x} \left (7 d^2 f^2 h a^3-7 b d f (9 d f g+4 d e h-10 c f h) a^2+b^2 \left (4 e (37 f g+4 e h) d^2-2 c f (11 f g+62 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-4 d e (3 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 (35 d^3 f^3 h a^4-35 b d^2 f^2 (9 d f g+6 d e h-11 c f h) a^3+b^2 d f \left (2 e (589 f g+128 e h) d^2-c f (233 f g+1060 e h) d+69 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (83 f g+6 e h) d^3-12 c e f (25 f g+92 e h) d^2-c^2 f^2 (83 f g-194 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-2 d e f (19 f g-40 e h) c^2-16 d^2 e^2 (7 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 \left (35 d^3 f^3 h a^4-b d^2 f^2 (251 d f g+152 d e h-263 c f h) a^3+b^2 d f \left (16 e (53 f g+11 e h) d^2-c f (95 f g+744 e h) d+25 c^2 f^2 h\right ) a^2-b^3 \left (16 e^2 (57 f g+4 e h) d^3-16 c e f (8 f g+47 e h) d^2-c^2 f^2 (31 f g-72 e h) d+3 c^3 f^3 h\right ) a+b^4 \left (-f^2 (5 f g-8 e h) c^3-16 d e f (f g-2 e h) c^2-16 d^2 e^2 (3 f g+16 e h) c+320 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (c+d x)}-\frac {b \left (\frac {128 d^{5/2} (b e-a f)^3 \left (a d (d f g+2 d e h-3 c f h)-b \left (7 f h c^2-d (9 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 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+8 d e h-12 c f h) a^4+70 b^2 d^2 f^2 \left (8 e (3 f g+e h) d^2-6 c f (f g+4 e h) d+3 c^2 f^2 h\right ) a^3-14 b^3 d f \left (8 e^2 (27 f g+4 e h) d^3-24 c e f (3 f g+8 e h) d^2-3 c^2 f^2 (3 f g-16 e h) d+2 c^3 f^3 h\right ) a^2+b^4 \left (128 e^3 (18 f g+e h) d^4-96 c e^2 f (9 f g+20 e h) d^3-48 c^2 e f^2 (3 f g-13 e h) d^2-4 c^3 f^3 (9 f g-20 e h) d+3 c^4 f^4 h\right ) a-b^5 \left (-f^3 (5 f g-8 e h) c^4-16 d e f^2 (f g-2 e h) c^3-48 d^2 e^2 f (f g-4 e h) c^2-256 d^3 e^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 c-a d) (b e-a f)}}{8 b (b c-a d)}\)

Input:

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

Output:

-1/4*((b*g - a*h)*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)^4*(c + d*x)) - ( 
-1/3*((a^2*d*f*h - a*b*(9*d*f*g + 2*d*e*h - 9*c*f*h) + b^2*(10*d*e*g - c*( 
f*g + 8*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^3*(c + d* 
x)) - (((7*a^3*d^2*f^2*h - 7*a^2*b*d*f*(9*d*f*g + 4*d*e*h - 10*c*f*h) - b^ 
3*(80*d^2*e^2*g - c^2*f*(5*f*g - 8*e*h) - 4*c*d*e*(3*f*g + 16*e*h)) + a*b^ 
2*(3*c^2*f^2*h + 4*d^2*e*(37*f*g + 4*e*h) - 2*c*d*f*(11*f*g + 62*e*h)))*Sq 
rt[e + f*x])/(2*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)) + (((35*a^4 
*d^3*f^3*h - 35*a^3*b*d^2*f^2*(9*d*f*g + 6*d*e*h - 11*c*f*h) + b^4*(480*d^ 
3*e^3*g - 2*c^2*d*e*f*(19*f*g - 40*e*h) - 3*c^3*f^2*(5*f*g - 8*e*h) - 16*c 
*d^2*e^2*(7*f*g + 24*e*h)) - a*b^3*(9*c^3*f^3*h - c^2*d*f^2*(83*f*g - 194* 
e*h) + 16*d^3*e^2*(83*f*g + 6*e*h) - 12*c*d^2*e*f*(25*f*g + 92*e*h)) + a^2 
*b^2*d*f*(69*c^2*f^2*h + 2*d^2*e*(589*f*g + 128*e*h) - c*d*f*(233*f*g + 10 
60*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)) + ( 
3*((2*d*(35*a^4*d^3*f^3*h - a^3*b*d^2*f^2*(251*d*f*g + 152*d*e*h - 263*c*f 
*h) + b^4*(320*d^3*e^3*g - c^3*f^2*(5*f*g - 8*e*h) - 16*c^2*d*e*f*(f*g - 2 
*e*h) - 16*c*d^2*e^2*(3*f*g + 16*e*h)) - a*b^3*(3*c^3*f^3*h - c^2*d*f^2*(3 
1*f*g - 72*e*h) + 16*d^3*e^2*(57*f*g + 4*e*h) - 16*c*d^2*e*f*(8*f*g + 47*e 
*h)) + a^2*b^2*d*f*(25*c^2*f^2*h + 16*d^2*e*(53*f*g + 11*e*h) - c*d*f*(95* 
f*g + 744*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(c + d*x)) - (b*((-2*(35*a^5* 
d^4*f^4*h - 35*a^4*b*d^3*f^3*(9*d*f*g + 8*d*e*h - 12*c*f*h) - b^5*(640*...
 

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 = 69.90 (sec) , antiderivative size = 1940, normalized size of antiderivative = 1.42

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1940\)
derivativedivides	\(\text {Expression too large to display}\)	\(2822\)
default	\(\text {Expression too large to display}\)	\(2822\)

Input:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

35/64/((a*f-b*e)*b)^(1/2)/((c*f-d*e)*d)^(1/2)*((d*x+c)*((c*f-d*e)*d)^(1/2) 
*(8/7*(-16*d^4*e^4*g+64/5*(e*h+1/2*f*g)*c*e^3*d^3-24/5*(e*h-1/4*f*g)*c^2*f 
*e^2*d^2-4/5*(e*h-1/2*f*g)*c^3*f^2*e*d-1/5*c^4*(e*h-5/8*f*g)*f^3)*b^5+3/35 
*a*(128*(1/3*e^4*h+6*e^3*f*g)*d^4-640*c*(e*h+9/20*f*g)*f*e^2*d^3+16*c^2*(1 
3*e^2*f^2*h-3*e*f^3*g)*d^2+4*c^3*(20/3*e*f^3*h-3*f^4*g)*d+c^4*f^4*h)*b^4-4 
/5*a^2*d*(4*(4*e^3*h+27*e^2*f*g)*d^3-96*c*(e*h+3/8*f*g)*f*e*d^2+24*c^2*(e* 
h-3/16*f*g)*f^2*d+c^3*f^3*h)*f*b^3+6*a^3*d^2*(8*(1/3*e^2*h+e*f*g)*d^2+2*c* 
(-4*e*f*h-f^2*g)*d+c^2*f^2*h)*f^2*b^2+12*a^4*d^3*((-2/3*e*h-3/4*f*g)*d+c*f 
*h)*f^3*b+a^5*d^4*f^4*h)*(b*x+a)^4*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1 
/2))+157/35*((a*f-b*e)*b)^(1/2)*(-192/157*d^3*(d*x+c)*((10/3*d^2*e*g+c*(-8 
/3*e*h-3*f*g)*d+7/3*c^2*f*h)*b+a*d*(1/3*(-2*e*h-f*g)*d+c*f*h))*(b*x+a)^4*( 
a*f-b*e)^3*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+(16/157*(20*d^4*e^3 
*g*x^4+10*x^3*c*(-3/10*f*g*x+e*(-8/5*h*x+g))*e^2*d^3-10/3*x^2*c^2*(3/10*x^ 
2*g*f^2+7/10*x*(-6/7*h*x+g)*e*f+e^2*(12/5*h*x+g))*e*d^2+5/3*x*c^3*(-3/16*f 
^3*g*x^3-19/40*x^2*(-12/19*h*x+g)*e*f^2+3/10*x*(10/3*h*x+g)*e^2*f+e^3*(8/5 
*h*x+g))*d-c^4*(5/16*f^3*g*x^3-5/24*x^2*(12/5*h*x+g)*e*f^2+1/6*e^2*x*(2*h* 
x+g)*f+e^3*(4/3*h*x+g)))*b^7-16/471*a*(3*(57*e^2*f*g*x^4-70*x^3*(-2/35*h*x 
+g)*e^3)*d^4-110*x^2*c*(12/55*x^2*g*f^2-56/55*x*(-141/112*h*x+g)*e*f+e^2*( 
-87/55*h*x+g))*e*d^3+35*x*c^2*(-93/560*f^3*g*x^3-73/280*x^2*(-108/73*h*x+g 
)*e*f^2-9/70*x*(176/9*h*x+g)*e^2*f+e^3*(86/35*h*x+g))*d^2-17*c^3*(19/13...
 

Fricas [F(-1)]

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

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((f*x+e)**(1/2)*(h*x+g)/(b*x+a)**5/(d*x+c)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(1/2)*(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. 4916 vs. \(2 (1326) = 2652\).

Time = 0.37 (sec) , antiderivative size = 4916, normalized size of antiderivative = 3.59 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^(1/2)*(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 - 256*b^5*c*d^3*e^3*f*g - 2304*a*b^4*d^4*e^3*f*g - 
 48*b^5*c^2*d^2*e^2*f^2*g + 864*a*b^4*c*d^3*e^2*f^2*g + 3024*a^2*b^3*d^4*e 
^2*f^2*g - 16*b^5*c^3*d*e*f^3*g + 144*a*b^4*c^2*d^2*e*f^3*g - 1008*a^2*b^3 
*c*d^3*e*f^3*g - 1680*a^3*b^2*d^4*e*f^3*g - 5*b^5*c^4*f^4*g + 36*a*b^4*c^3 
*d*f^4*g - 126*a^2*b^3*c^2*d^2*f^4*g + 420*a^3*b^2*c*d^3*f^4*g + 315*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 + 192*b^5*c^2*d^2*e 
^3*f*h + 1920*a*b^4*c*d^3*e^3*f*h + 448*a^2*b^3*d^4*e^3*f*h + 32*b^5*c^3*d 
*e^2*f^2*h - 624*a*b^4*c^2*d^2*e^2*f^2*h - 2688*a^2*b^3*c*d^3*e^2*f^2*h - 
560*a^3*b^2*d^4*e^2*f^2*h + 8*b^5*c^4*e*f^3*h - 80*a*b^4*c^3*d*e*f^3*h + 6 
72*a^2*b^3*c^2*d^2*e*f^3*h + 1680*a^3*b^2*c*d^3*e*f^3*h + 280*a^4*b*d^4*e* 
f^3*h - 3*a*b^4*c^4*f^4*h + 28*a^2*b^3*c^3*d*f^4*h - 210*a^3*b^2*c^2*d^2*f 
^4*h - 420*a^4*b*c*d^3*f^4*h - 35*a^5*d^4*f^4*h)*arctan(sqrt(f*x + e)*b/sq 
rt(-b^2*e + a*b*f))/((b^9*c^6*e^3 - 6*a*b^8*c^5*d*e^3 + 15*a^2*b^7*c^4*d^2 
*e^3 - 20*a^3*b^6*c^3*d^3*e^3 + 15*a^4*b^5*c^2*d^4*e^3 - 6*a^5*b^4*c*d^5*e 
^3 + a^6*b^3*d^6*e^3 - 3*a*b^8*c^6*e^2*f + 18*a^2*b^7*c^5*d*e^2*f - 45*a^3 
*b^6*c^4*d^2*e^2*f + 60*a^4*b^5*c^3*d^3*e^2*f - 45*a^5*b^4*c^2*d^4*e^2*f + 
 18*a^6*b^3*c*d^5*e^2*f - 3*a^7*b^2*d^6*e^2*f + 3*a^2*b^7*c^6*e*f^2 - 18*a 
^3*b^6*c^5*d*e*f^2 + 45*a^4*b^5*c^4*d^2*e*f^2 - 60*a^5*b^4*c^3*d^3*e*f^2 + 
 45*a^6*b^3*c^2*d^4*e*f^2 - 18*a^7*b^2*c*d^5*e*f^2 + 3*a^8*b*d^6*e*f^2 - a 
^3*b^6*c^6*f^3 + 6*a^4*b^5*c^5*d*f^3 - 15*a^5*b^4*c^4*d^2*f^3 + 20*a^6*...
 

Mupad [B] (verification not implemented)

Time = 46.61 (sec) , antiderivative size = 1195213, normalized size of antiderivative = 873.69 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^5 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

int(((e + f*x)^(1/2)*(g + h*x))/((a + b*x)^5*(c + d*x)^2),x)
 

Output:

atan(((((16384*a^19*b^2*d^18*f^10*g - 113408*a^18*b^3*c*d^17*f^10*g - 4019 
2*a^19*b^2*c*d^17*f^10*h - 197888*a^18*b^3*d^18*e*f^9*g + 23808*a^19*b^2*d 
^18*e*f^9*h - 1280*a^3*b^18*c^16*d^2*f^10*g + 23296*a^4*b^17*c^15*d^3*f^10 
*g - 204032*a^5*b^16*c^14*d^4*f^10*g + 1180416*a^6*b^15*c^13*d^5*f^10*g - 
4966656*a^7*b^14*c^12*d^6*f^10*g + 15587072*a^8*b^13*c^11*d^7*f^10*g - 367 
29088*a^9*b^12*c^10*d^8*f^10*g + 65187584*a^10*b^11*c^9*d^9*f^10*g - 87174 
912*a^11*b^10*c^8*d^10*f^10*g + 87372032*a^12*b^9*c^7*d^11*f^10*g - 646243 
84*a^13*b^8*c^6*d^12*f^10*g + 34097408*a^14*b^7*c^5*d^13*f^10*g - 11895552 
*a^15*b^6*c^4*d^14*f^10*g + 2186496*a^16*b^5*c^3*d^15*f^10*g + 58624*a^17* 
b^4*c^2*d^16*f^10*g - 768*a^4*b^17*c^16*d^2*f^10*h + 15616*a^5*b^16*c^15*d 
^3*f^10*h - 174848*a^6*b^15*c^14*d^4*f^10*h + 1119488*a^7*b^14*c^13*d^5*f^ 
10*h - 4431616*a^8*b^13*c^12*d^6*f^10*h + 11542784*a^9*b^12*c^11*d^7*f^10* 
h - 20379392*a^10*b^11*c^10*d^8*f^10*h + 24135936*a^11*b^10*c^9*d^9*f^10*h 
 - 17377536*a^12*b^9*c^8*d^10*f^10*h + 3708672*a^13*b^8*c^7*d^11*f^10*h + 
6930176*a^14*b^7*c^6*d^12*f^10*h - 9044224*a^15*b^6*c^5*d^13*f^10*h + 5638 
912*a^16*b^5*c^4*d^14*f^10*h - 2077952*a^17*b^4*c^3*d^15*f^10*h + 434944*a 
^18*b^3*c^2*d^16*f^10*h - 81920*a^12*b^9*d^18*e^7*f^3*g + 520192*a^13*b^8* 
d^18*e^6*f^4*g - 1392640*a^14*b^7*d^18*e^5*f^5*g + 2024704*a^15*b^6*d^18*e 
^4*f^6*g - 1707776*a^16*b^5*d^18*e^3*f^7*g + 818944*a^17*b^4*d^18*e^2*f^8* 
g + 16384*a^13*b^8*d^18*e^7*f^3*h - 102400*a^14*b^7*d^18*e^6*f^4*h + 26...
 

Reduce [F]

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

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^2,x)
 

Output:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^2,x)