\(\int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx\) [59]

Optimal result

Integrand size = 29, antiderivative size = 1821 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx =\text {Too large to display} \] Output:

1/192*d*(315*a^4*d^3*f^3*h-a^3*b*d^2*f^2*(-1041*c*f*h+1178*d*e*h+1123*d*f* 
g)+b^4*(1440*d^3*e^3*g-2*c^2*d*e*f*(-64*e*h+31*f*g)-3*c^3*f^2*(-8*e*h+5*f* 
g)-240*c*d^2*e^2*(4*e*h+f*g))-a*b^3*(9*c^3*f^3*h-c^2*d*f^2*(-266*e*h+107*f 
*g)+240*d^3*e^2*(2*e*h+17*f*g)-4*c*d^2*e*f*(716*e*h+151*f*g))+a^2*b^2*d*f* 
(93*c^2*f^2*h+2*d^2*e*(664*e*h+1889*f*g)-c*d*f*(2900*e*h+409*f*g)))*(f*x+e 
)^(1/2)/b/(-a*d+b*c)^5/(-a*f+b*e)^3/(d*x+c)^2-1/4*(-a*h+b*g)*(f*x+e)^(1/2) 
/b/(-a*d+b*c)/(b*x+a)^4/(d*x+c)^2+1/24*(3*a^2*d*f*h-a*b*(-9*c*f*h+4*d*e*h+ 
11*d*f*g)+b^2*(12*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)^2+1/96*(27*a^3*d^2*f^2*h-9*a^2*b*d*f*(-10*c*f*h+8* 
d*e*h+11*d*f*g)-b^3*(120*d^2*e^2*g-c^2*f*(-8*e*h+5*f*g)-16*c*d*e*(5*e*h+f* 
g))+a*b^2*(3*c^2*f^2*h+8*d^2*e*(5*e*h+28*f*g)-2*c*d*f*(80*e*h+13*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)^2+1/192*(189*a^4 
*d^3*f^3*h-63*a^3*b*d^2*f^2*(-11*c*f*h+12*d*e*h+11*d*f*g)+b^4*(960*d^3*e^3 
*g-4*c^2*d*e*f*(-28*e*h+13*f*g)-3*c^3*f^2*(-8*e*h+5*f*g)-40*c*d^2*e^2*(16* 
e*h+5*f*g))-a*b^3*(9*c^3*f^3*h-c^2*d*f^2*(-244*e*h+97*f*g)+40*d^3*e^2*(8*e 
*h+67*f*g)-24*c*d^2*e*f*(79*e*h+21*f*g))+a^2*b^2*d*f*(87*c^2*f^2*h+4*d^2*e 
*(218*e*h+607*f*g)-c*d*f*(1904*e*h+349*f*g)))*(f*x+e)^(1/2)/b/(-a*d+b*c)^4 
/(-a*f+b*e)^3/(b*x+a)/(d*x+c)^2+1/64*d*(a^4*d^3*f^3*(-267*c*f*h+251*d*e*h+ 
16*d*f*g)+b^4*(960*d^4*e^4*g+c^3*d*e*f^2*(-40*e*h+19*f*g)+c^4*f^3*(-8*e*h+ 
5*f*g)-80*c*d^3*e^3*(8*e*h+13*f*g)+24*c^2*d^2*e^2*f*(28*e*h+3*f*g))+a*b...
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(24162\) vs. \(2(1821)=3642\).

Time = 16.94 (sec) , antiderivative size = 24162, normalized size of antiderivative = 13.27 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sqrt[e + f*x]*(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 = 1924, 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, 27, 168, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-9 c f h+4 d e h+11 d f g)+b^2 (-8 c e h-c f g+12 d e g)\right )}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int -\frac {9 d f (4 d e-c f) h a^2-b \left (4 e (29 f g+10 e h) d^2-c f (17 f g+88 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-8 e h) c^2-16 d e (f g+5 e h) c+120 d^2 e^2 g\right )+9 d f \left (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c (f g+8 e h))\right ) x}{2 (a+b x)^3 (c+d x)^3 \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {9 d f (4 d e-c f) h a^2-b \left (4 e (29 f g+10 e h) d^2-c f (17 f g+88 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-8 e h) c^2-16 d e (f g+5 e h) c+120 d^2 e^2 g\right )+9 d f \left (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c (f g+8 e h))\right ) x}{(a+b x)^3 (c+d x)^3 \sqrt {e+f x}}dx}{6 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-9 c f h+4 d e h+11 d f g)+b^2 (-8 c e h-c f g+12 d e g)\right )}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {\sqrt {e+f x} \left (27 a^3 d^2 f^2 h-9 a^2 b d f (-10 c f h+8 d e h+11 d f g)+a b^2 \left (3 c^2 f^2 h-2 c d f (80 e h+13 f g)+8 d^2 e (5 e h+28 f g)\right )-b^3 \left (c^2 (-f) (5 f g-8 e h)-16 c d e (5 e h+f g)+120 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}-\frac {\int -\frac {63 d^2 f^2 (4 d e-c f) h a^3-b d f \left (4 e (215 f g+148 e h) d^2-c f (167 f g+784 e h) d+66 c^2 f^2 h\right ) a^2+b^2 \left (80 e^2 (23 f g+4 e h) d^3-8 c e f (49 f g+167 e h) d^2-2 c^2 f^2 (31 f g-94 e h) d+9 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 f g+16 e h) c+960 d^3 e^3 g\right )+7 d f \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (f g+5 e h) c+120 d^2 e^2 g\right )\right ) x}{2 (a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-9 c f h+4 d e h+11 d f g)+b^2 (-8 c e h-c f g+12 d e g)\right )}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {63 d^2 f^2 (4 d e-c f) h a^3-b d f \left (4 e (215 f g+148 e h) d^2-c f (167 f g+784 e h) d+66 c^2 f^2 h\right ) a^2+b^2 \left (80 e^2 (23 f g+4 e h) d^3-8 c e f (49 f g+167 e h) d^2-2 c^2 f^2 (31 f g-94 e h) d+9 c^3 f^3 h\right ) a-b^3 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 f g+16 e h) c+960 d^3 e^3 g\right )+7 d f \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (f g+5 e h) c+120 d^2 e^2 g\right )\right ) x}{(a+b x)^2 (c+d x)^3 \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (27 a^3 d^2 f^2 h-9 a^2 b d f (-10 c f h+8 d e h+11 d f g)+a b^2 \left (3 c^2 f^2 h-2 c d f (80 e h+13 f g)+8 d^2 e (5 e h+28 f g)\right )-b^3 \left (c^2 (-f) (5 f g-8 e h)-16 c d e (5 e h+f g)+120 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (c+d x)^2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (3 a^2 d f h-a b (-9 c f h+4 d e h+11 d f g)+b^2 (-8 c e h-c f g+12 d e g)\right )}{3 (a+b x)^3 (c+d x)^2 (b c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {\sqrt {e+f x} (b g-a h)}{4 b (a+b x)^4 (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (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 (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 {315 d^3 f^3 (4 d e-c f) h a^4-b d^2 f^2 \left (4 e (1123 f g+1178 e h) d^2-13 c f (79 f g+392 e h) d+699 c^2 f^2 h\right ) a^3+b^2 d f \left (8 e^2 (1889 f g+664 e h) d^3-24 c e f (192 f g+523 e h) d^2-c^2 f^2 (109 f g-2452 e h) d+63 c^3 f^3 h\right ) a^2-b^3 \left (960 e^3 (17 f g+2 e h) d^4-184 c e^2 f (29 f g+64 e h) d^3-76 c^2 e f^2 (7 f g-40 e h) d^2-3 c^3 f^3 (19 f g-64 e h) d+9 c^4 f^4 h\right ) a+3 b^4 \left (-f^3 (5 f g-8 e h) c^4-24 d e f^2 (f g-2 e h) c^3-96 d^2 e^2 f (f g-4 e h) c^2-640 d^3 e^3 (f g+2 e h) c+1920 d^4 e^4 g\right )+5 d f \left (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{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 (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (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 (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 {315 d^3 f^3 (4 d e-c f) h a^4-b d^2 f^2 \left (4 e (1123 f g+1178 e h) d^2-13 c f (79 f g+392 e h) d+699 c^2 f^2 h\right ) a^3+b^2 d f \left (8 e^2 (1889 f g+664 e h) d^3-24 c e f (192 f g+523 e h) d^2-c^2 f^2 (109 f g-2452 e h) d+63 c^3 f^3 h\right ) a^2-b^3 \left (960 e^3 (17 f g+2 e h) d^4-184 c e^2 f (29 f g+64 e h) d^3-76 c^2 e f^2 (7 f g-40 e h) d^2-3 c^3 f^3 (19 f g-64 e h) d+9 c^4 f^4 h\right ) a+3 b^4 \left (-f^3 (5 f g-8 e h) c^4-24 d e f^2 (f g-2 e h) c^3-96 d^2 e^2 f (f g-4 e h) c^2-640 d^3 e^3 (f g+2 e h) c+1920 d^4 e^4 g\right )+5 d f \left (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{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 (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (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 (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 (315 d^3 f^3 h a^4-b d^2 f^2 (1123 d f g+1178 d e h-1041 c f h) a^3+b^2 d f \left (2 e (1889 f g+664 e h) d^2-c f (409 f g+2900 e h) d+93 c^2 f^2 h\right ) a^2-b^3 \left (240 e^2 (17 f g+2 e h) d^3-4 c e f (151 f g+716 e h) d^2-c^2 f^2 (107 f g-266 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 (31 f g-64 e h) c^2-240 d^2 e^2 (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 (d^3 f^3 (32 d f g+502 d e h-219 c f h) a^4-b d^2 f^2 \left (2 e (883 f g+848 e h) d^2-c f (515 f g+1972 e h) d+285 c^2 f^2 h\right ) a^3+b^2 d f \left (48 e^2 (113 f g+38 e h) d^3-4 c e f (443 f g+1120 e h) d^2-c^2 f^2 (91 f g-958 e h) d+27 c^3 f^3 h\right ) a^2-b^3 \left (160 e^3 (35 f g+4 e h) d^4-144 c e^2 f (13 f g+28 e h) d^3-2 c^2 e f^2 (101 f g-536 e h) d^2-c^3 f^3 (29 f g-80 e h) d+3 c^4 f^4 h\right ) a+b^4 \left (-f^3 (5 f g-8 e h) c^4-24 d e f^2 (f g-2 e h) c^3-96 d^2 e^2 f (f g-4 e h) c^2-640 d^3 e^3 (f g+2 e h) c+1920 d^4 e^4 g\right )+d f \left (315 d^3 f^3 h a^4-b d^2 f^2 (1123 d f g+1178 d e h-1041 c f h) a^3+b^2 d f \left (2 e (1889 f g+664 e h) d^2-c f (409 f g+2900 e h) d+93 c^2 f^2 h\right ) a^2-b^3 \left (240 e^2 (17 f g+2 e h) d^3-4 c e f (151 f g+716 e h) d^2-c^2 f^2 (107 f g-266 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 (31 f g-64 e h) c^2-240 d^2 e^2 (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 c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{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 (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (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 (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 (315 d^3 f^3 h a^4-b d^2 f^2 (1123 d f g+1178 d e h-1041 c f h) a^3+b^2 d f \left (2 e (1889 f g+664 e h) d^2-c f (409 f g+2900 e h) d+93 c^2 f^2 h\right ) a^2-b^3 \left (240 e^2 (17 f g+2 e h) d^3-4 c e f (151 f g+716 e h) d^2-c^2 f^2 (107 f g-266 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 (31 f g-64 e h) c^2-240 d^2 e^2 (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 {d^3 f^3 (32 d f g+502 d e h-219 c f h) a^4-b d^2 f^2 \left (2 e (883 f g+848 e h) d^2-c f (515 f g+1972 e h) d+285 c^2 f^2 h\right ) a^3+b^2 d f \left (48 e^2 (113 f g+38 e h) d^3-4 c e f (443 f g+1120 e h) d^2-c^2 f^2 (91 f g-958 e h) d+27 c^3 f^3 h\right ) a^2-b^3 \left (160 e^3 (35 f g+4 e h) d^4-144 c e^2 f (13 f g+28 e h) d^3-2 c^2 e f^2 (101 f g-536 e h) d^2-c^3 f^3 (29 f g-80 e h) d+3 c^4 f^4 h\right ) a+b^4 \left (-f^3 (5 f g-8 e h) c^4-24 d e f^2 (f g-2 e h) c^3-96 d^2 e^2 f (f g-4 e h) c^2-640 d^3 e^3 (f g+2 e h) c+1920 d^4 e^4 g\right )+d f \left (315 d^3 f^3 h a^4-b d^2 f^2 (1123 d f g+1178 d e h-1041 c f h) a^3+b^2 d f \left (2 e (1889 f g+664 e h) d^2-c f (409 f g+2900 e h) d+93 c^2 f^2 h\right ) a^2-b^3 \left (240 e^2 (17 f g+2 e h) d^3-4 c e f (151 f g+716 e h) d^2-c^2 f^2 (107 f g-266 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 (31 f g-64 e h) c^2-240 d^2 e^2 (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 c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{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 (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (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 (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 (315 d^3 f^3 h a^4-b d^2 f^2 (1123 d f g+1178 d e h-1041 c f h) a^3+b^2 d f \left (2 e (1889 f g+664 e h) d^2-c f (409 f g+2900 e h) d+93 c^2 f^2 h\right ) a^2-b^3 \left (240 e^2 (17 f g+2 e h) d^3-4 c e f (151 f g+716 e h) d^2-c^2 f^2 (107 f g-266 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 (31 f g-64 e h) c^2-240 d^2 e^2 (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 (d^3 f^3 (16 d f g+251 d e h-267 c f h) a^4-b d^2 f^2 \left (e (883 f g+848 e h) d^2-63 c f (13 f g+25 e h) d+663 c^2 f^2 h\right ) a^3-3 b^2 d f \left (-8 e^2 (113 f g+38 e h) d^3+c e f (925 f g+968 e h) d^2-c^2 f^2 (53 f g+643 e h) d+11 c^3 f^3 h\right ) a^2+b^3 \left (-80 e^3 (35 f g+4 e h) d^4+48 c e^2 f (62 f g+47 e h) d^3-3 c^2 e f^2 (67 f g+656 e h) d^2-3 c^3 f^3 (13 f g-31 e h) d+3 c^4 f^4 h\right ) a+b^4 \left (f^3 (5 f g-8 e h) c^4+d e f^2 (19 f g-40 e h) c^3+24 d^2 e^2 f (3 f g+28 e h) c^2-80 d^3 e^3 (13 f g+8 e h) c+960 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\int \frac {16 d^4 f^4 (d f g-4 d e h+3 c f h) a^5+b d^3 f^3 \left (16 e (17 f g+52 e h) d^2-21 c f (16 f g+65 e h) d+597 c^2 f^2 h\right ) a^4+3 b^2 d^2 f^2 \left (-16 e^2 (59 f g+44 e h) d^3+c e f (1231 f g+1664 e h) d^2-3 c^2 f^2 (85 f g+369 e h) d+115 c^3 f^3 h\right ) a^3-b^3 d f \left (-16 e^3 (419 f g+124 e h) d^4+168 c e^2 f (53 f g+44 e h) d^3-3 c^2 e f^2 (659 f g+2200 e h) d^2-3 c^3 f^3 (53 f g-365 e h) d+33 c^4 f^4 h\right ) a^2+b^4 \left (-320 e^4 (19 f g+2 e h) d^5+624 c e^3 f (13 f g+8 e h) d^4-48 c^2 e^2 f^2 (37 f g+115 e h) d^3-3 c^3 e f^3 (67 f g-352 e h) d^2-3 c^4 f^4 (13 f g-31 e h) d+3 c^5 f^5 h\right ) a+b^5 (d e-c f) \left (-f^3 (5 f g-8 e h) c^4-24 d e f^2 (f g-2 e h) c^3-96 d^2 e^2 f (f g-4 e h) c^2-640 d^3 e^3 (f g+2 e h) c+1920 d^4 e^4 g\right )+b d f \left (d^3 f^3 (16 d f g+251 d e h-267 c f h) a^4-b d^2 f^2 \left (e (883 f g+848 e h) d^2-63 c f (13 f g+25 e h) d+663 c^2 f^2 h\right ) a^3-3 b^2 d f \left (-8 e^2 (113 f g+38 e h) d^3+c e f (925 f g+968 e h) d^2-c^2 f^2 (53 f g+643 e h) d+11 c^3 f^3 h\right ) a^2+b^3 \left (-80 e^3 (35 f g+4 e h) d^4+48 c e^2 f (62 f g+47 e h) d^3-3 c^2 e f^2 (67 f g+656 e h) d^2-3 c^3 f^3 (13 f g-31 e h) d+3 c^4 f^4 h\right ) a+b^4 \left (f^3 (5 f g-8 e h) c^4+d e f^2 (19 f g-40 e h) c^3+24 d^2 e^2 f (3 f g+28 e h) c^2-80 d^3 e^3 (13 f g+8 e h) c+960 d^4 e^4 g\right )\right ) x}{(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 c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{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 (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (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 (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 (315 d^3 f^3 h a^4-b d^2 f^2 (1123 d f g+1178 d e h-1041 c f h) a^3+b^2 d f \left (2 e (1889 f g+664 e h) d^2-c f (409 f g+2900 e h) d+93 c^2 f^2 h\right ) a^2-b^3 \left (240 e^2 (17 f g+2 e h) d^3-4 c e f (151 f g+716 e h) d^2-c^2 f^2 (107 f g-266 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 (31 f g-64 e h) c^2-240 d^2 e^2 (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 (d^3 f^3 (16 d f g+251 d e h-267 c f h) a^4-b d^2 f^2 \left (e (883 f g+848 e h) d^2-63 c f (13 f g+25 e h) d+663 c^2 f^2 h\right ) a^3-3 b^2 d f \left (-8 e^2 (113 f g+38 e h) d^3+c e f (925 f g+968 e h) d^2-c^2 f^2 (53 f g+643 e h) d+11 c^3 f^3 h\right ) a^2+b^3 \left (-80 e^3 (35 f g+4 e h) d^4+48 c e^2 f (62 f g+47 e h) d^3-3 c^2 e f^2 (67 f g+656 e h) d^2-3 c^3 f^3 (13 f g-31 e h) d+3 c^4 f^4 h\right ) a+b^4 \left (f^3 (5 f g-8 e h) c^4+d e f^2 (19 f g-40 e h) c^3+24 d^2 e^2 f (3 f g+28 e h) c^2-80 d^3 e^3 (13 f g+8 e h) c+960 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {16 d^3 (b e-a f)^3 \left (-\left (\left (-63 f^2 h c^3+9 d f (11 f g+16 e h) c^2-20 d^2 e (11 f g+4 e h) c+120 d^3 e^2 g\right ) b^2\right )+2 a d \left (10 e (f g+2 e h) d^2-c f (11 f g+46 e h) d+27 c^2 f^2 h\right ) b+a^2 d^2 f (d f g-4 d e h+3 c f h)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}-\frac {b (d e-c f) \left (315 d^4 f^4 h a^5-105 b d^3 f^3 (11 d f g+16 d e h-12 c f h) a^4+42 b^2 d^2 f^2 \left (12 e (11 f g+6 e h) d^2-2 c f (11 f g+58 e h) d+9 c^2 f^2 h\right ) a^3-18 b^3 d f \left (16 e^2 (33 f g+8 e h) d^3-12 c e f (11 f g+34 e h) d^2-c^2 f^2 (11 f g-68 e h) d+2 c^3 f^3 h\right ) a^2+b^4 \left (640 e^3 (11 f g+e h) d^4-192 c e^2 f (11 f g+26 e h) d^3-24 c^2 e f^2 (11 f g-50 e h) d^2-4 c^3 f^3 (11 f g-26 e h) d+3 c^4 f^4 h\right ) a-b^5 \left (-f^3 (5 f g-8 e h) c^4-24 d e f^2 (f g-2 e h) c^3-96 d^2 e^2 f (f g-4 e h) c^2-640 d^3 e^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}}{(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 c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{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 (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (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 (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 (315 d^3 f^3 h a^4-b d^2 f^2 (1123 d f g+1178 d e h-1041 c f h) a^3+b^2 d f \left (2 e (1889 f g+664 e h) d^2-c f (409 f g+2900 e h) d+93 c^2 f^2 h\right ) a^2-b^3 \left (240 e^2 (17 f g+2 e h) d^3-4 c e f (151 f g+716 e h) d^2-c^2 f^2 (107 f g-266 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 (31 f g-64 e h) c^2-240 d^2 e^2 (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 (d^3 f^3 (16 d f g+251 d e h-267 c f h) a^4-b d^2 f^2 \left (e (883 f g+848 e h) d^2-63 c f (13 f g+25 e h) d+663 c^2 f^2 h\right ) a^3-3 b^2 d f \left (-8 e^2 (113 f g+38 e h) d^3+c e f (925 f g+968 e h) d^2-c^2 f^2 (53 f g+643 e h) d+11 c^3 f^3 h\right ) a^2+b^3 \left (-80 e^3 (35 f g+4 e h) d^4+48 c e^2 f (62 f g+47 e h) d^3-3 c^2 e f^2 (67 f g+656 e h) d^2-3 c^3 f^3 (13 f g-31 e h) d+3 c^4 f^4 h\right ) a+b^4 \left (f^3 (5 f g-8 e h) c^4+d e f^2 (19 f g-40 e h) c^3+24 d^2 e^2 f (3 f g+28 e h) c^2-80 d^3 e^3 (13 f g+8 e h) c+960 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {32 d^3 (b e-a f)^3 \left (-\left (\left (-63 f^2 h c^3+9 d f (11 f g+16 e h) c^2-20 d^2 e (11 f g+4 e h) c+120 d^3 e^2 g\right ) b^2\right )+2 a d \left (10 e (f g+2 e h) d^2-c f (11 f g+46 e h) d+27 c^2 f^2 h\right ) b+a^2 d^2 f (d f g-4 d e h+3 c f h)\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}-\frac {2 b (d e-c f) \left (315 d^4 f^4 h a^5-105 b d^3 f^3 (11 d f g+16 d e h-12 c f h) a^4+42 b^2 d^2 f^2 \left (12 e (11 f g+6 e h) d^2-2 c f (11 f g+58 e h) d+9 c^2 f^2 h\right ) a^3-18 b^3 d f \left (16 e^2 (33 f g+8 e h) d^3-12 c e f (11 f g+34 e h) d^2-c^2 f^2 (11 f g-68 e h) d+2 c^3 f^3 h\right ) a^2+b^4 \left (640 e^3 (11 f g+e h) d^4-192 c e^2 f (11 f g+26 e h) d^3-24 c^2 e f^2 (11 f g-50 e h) d^2-4 c^3 f^3 (11 f g-26 e h) d+3 c^4 f^4 h\right ) a-b^5 \left (-f^3 (5 f g-8 e h) c^4-24 d e f^2 (f g-2 e h) c^3-96 d^2 e^2 f (f g-4 e h) c^2-640 d^3 e^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}}{(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 c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{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 (3 d f h a^2-b (11 d f g+4 d e h-9 c f h) a+b^2 (12 d e g-c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3 (c+d x)^2}+\frac {\frac {\sqrt {e+f x} \left (27 d^2 f^2 h a^3-9 b d f (11 d f g+8 d e h-10 c f h) a^2+b^2 \left (8 e (28 f g+5 e h) d^2-2 c f (13 f g+80 e h) d+3 c^2 f^2 h\right ) a-b^3 \left (-f (5 f g-8 e h) c^2-16 d e (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 (189 d^3 f^3 h a^4-63 b d^2 f^2 (11 d f g+12 d e h-11 c f h) a^3+b^2 d f \left (4 e (607 f g+218 e h) d^2-c f (349 f g+1904 e h) d+87 c^2 f^2 h\right ) a^2-b^3 \left (40 e^2 (67 f g+8 e h) d^3-24 c e f (21 f g+79 e h) d^2-c^2 f^2 (97 f g-244 e h) d+9 c^3 f^3 h\right ) a+b^4 \left (-3 f^2 (5 f g-8 e h) c^3-4 d e f (13 f g-28 e h) c^2-40 d^2 e^2 (5 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 (315 d^3 f^3 h a^4-b d^2 f^2 (1123 d f g+1178 d e h-1041 c f h) a^3+b^2 d f \left (2 e (1889 f g+664 e h) d^2-c f (409 f g+2900 e h) d+93 c^2 f^2 h\right ) a^2-b^3 \left (240 e^2 (17 f g+2 e h) d^3-4 c e f (151 f g+716 e h) d^2-c^2 f^2 (107 f g-266 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 (31 f g-64 e h) c^2-240 d^2 e^2 (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 (d^3 f^3 (16 d f g+251 d e h-267 c f h) a^4-b d^2 f^2 \left (e (883 f g+848 e h) d^2-63 c f (13 f g+25 e h) d+663 c^2 f^2 h\right ) a^3-3 b^2 d f \left (-8 e^2 (113 f g+38 e h) d^3+c e f (925 f g+968 e h) d^2-c^2 f^2 (53 f g+643 e h) d+11 c^3 f^3 h\right ) a^2+b^3 \left (-80 e^3 (35 f g+4 e h) d^4+48 c e^2 f (62 f g+47 e h) d^3-3 c^2 e f^2 (67 f g+656 e h) d^2-3 c^3 f^3 (13 f g-31 e h) d+3 c^4 f^4 h\right ) a+b^4 \left (f^3 (5 f g-8 e h) c^4+d e f^2 (19 f g-40 e h) c^3+24 d^2 e^2 f (3 f g+28 e h) c^2-80 d^3 e^3 (13 f g+8 e h) c+960 d^4 e^4 g\right )\right )}{(b c-a d) (d e-c f) (c+d x)}+\frac {\frac {2 \sqrt {b} (d e-c f) \left (315 d^4 f^4 h a^5-105 b d^3 f^3 (11 d f g+16 d e h-12 c f h) a^4+42 b^2 d^2 f^2 \left (12 e (11 f g+6 e h) d^2-2 c f (11 f g+58 e h) d+9 c^2 f^2 h\right ) a^3-18 b^3 d f \left (16 e^2 (33 f g+8 e h) d^3-12 c e f (11 f g+34 e h) d^2-c^2 f^2 (11 f g-68 e h) d+2 c^3 f^3 h\right ) a^2+b^4 \left (640 e^3 (11 f g+e h) d^4-192 c e^2 f (11 f g+26 e h) d^3-24 c^2 e f^2 (11 f g-50 e h) d^2-4 c^3 f^3 (11 f g-26 e h) d+3 c^4 f^4 h\right ) a-b^5 \left (-f^3 (5 f g-8 e h) c^4-24 d e f^2 (f g-2 e h) c^3-96 d^2 e^2 f (f g-4 e h) c^2-640 d^3 e^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 )}{(b c-a d) \sqrt {b e-a f}}-\frac {32 d^{5/2} (b e-a f)^3 \left (-\left (\left (-63 f^2 h c^3+9 d f (11 f g+16 e h) c^2-20 d^2 e (11 f g+4 e h) c+120 d^3 e^2 g\right ) b^2\right )+2 a d \left (10 e (f g+2 e h) d^2-c f (11 f g+46 e h) d+27 c^2 f^2 h\right ) b+a^2 d^2 f (d f g-4 d e h+3 c f h)\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}}}{(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 c-a d) (b e-a f)}}{8 b (b c-a d)}-\frac {(b g-a h) \sqrt {e+f x}}{4 b (b c-a d) (a+b x)^4 (c+d x)^2}\)

Input:

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

Output:

-1/4*((b*g - a*h)*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)^4*(c + d*x)^2) + 
 (((3*a^2*d*f*h + b^2*(12*d*e*g - c*f*g - 8*c*e*h) - a*b*(11*d*f*g + 4*d*e 
*h - 9*c*f*h))*Sqrt[e + f*x])/(3*(b*c - a*d)*(b*e - a*f)*(a + b*x)^3*(c + 
d*x)^2) + (((27*a^3*d^2*f^2*h - 9*a^2*b*d*f*(11*d*f*g + 8*d*e*h - 10*c*f*h 
) - b^3*(120*d^2*e^2*g - c^2*f*(5*f*g - 8*e*h) - 16*c*d*e*(f*g + 5*e*h)) + 
 a*b^2*(3*c^2*f^2*h + 8*d^2*e*(28*f*g + 5*e*h) - 2*c*d*f*(13*f*g + 80*e*h) 
))*Sqrt[e + f*x])/(2*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)^2) + (( 
(189*a^4*d^3*f^3*h - 63*a^3*b*d^2*f^2*(11*d*f*g + 12*d*e*h - 11*c*f*h) + b 
^4*(960*d^3*e^3*g - 4*c^2*d*e*f*(13*f*g - 28*e*h) - 3*c^3*f^2*(5*f*g - 8*e 
*h) - 40*c*d^2*e^2*(5*f*g + 16*e*h)) - a*b^3*(9*c^3*f^3*h - c^2*d*f^2*(97* 
f*g - 244*e*h) + 40*d^3*e^2*(67*f*g + 8*e*h) - 24*c*d^2*e*f*(21*f*g + 79*e 
*h)) + a^2*b^2*d*f*(87*c^2*f^2*h + 4*d^2*e*(607*f*g + 218*e*h) - c*d*f*(34 
9*f*g + 1904*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + 
 d*x)^2) + ((2*d*(315*a^4*d^3*f^3*h - a^3*b*d^2*f^2*(1123*d*f*g + 1178*d*e 
*h - 1041*c*f*h) + b^4*(1440*d^3*e^3*g - 2*c^2*d*e*f*(31*f*g - 64*e*h) - 3 
*c^3*f^2*(5*f*g - 8*e*h) - 240*c*d^2*e^2*(f*g + 4*e*h)) - a*b^3*(9*c^3*f^3 
*h - c^2*d*f^2*(107*f*g - 266*e*h) + 240*d^3*e^2*(17*f*g + 2*e*h) - 4*c*d^ 
2*e*f*(151*f*g + 716*e*h)) + a^2*b^2*d*f*(93*c^2*f^2*h + 2*d^2*e*(1889*f*g 
 + 664*e*h) - c*d*f*(409*f*g + 2900*e*h)))*Sqrt[e + f*x])/((b*c - a*d)*(c 
+ d*x)^2) + (3*b*((2*d*(a^4*d^3*f^3*(16*d*f*g + 251*d*e*h - 267*c*f*h) ...
 

Defintions of rubi rules used

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 = 362.92 (sec) , antiderivative size = 3167, normalized size of antiderivative = 1.74

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3167\)
default	\(\text {Expression too large to display}\)	\(3167\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(3232\)

Input:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

2*f^6*(-d^3/f^6/(a*d-b*c)^7*((1/8*d*f*(5*a^2*c*d^2*f*h-4*a^2*d^3*e*h-a^2*d 
^3*f*g+10*a*b*c^2*d*f*h-12*a*b*c*d^2*e*h-18*a*b*c*d^2*f*g+20*a*b*d^3*e*g-1 
5*b^2*c^3*f*h+16*b^2*c^2*d*e*h+19*b^2*c^2*d*f*g-20*b^2*c*d^2*e*g)/(c*f-d*e 
)*(f*x+e)^(3/2)+1/8*f*(3*a^2*c*d^2*f*h-4*a^2*d^3*e*h+a^2*d^3*f*g+14*a*b*c^ 
2*d*f*h-12*a*b*c*d^2*e*h-22*a*b*c*d^2*f*g+20*a*b*d^3*e*g-17*b^2*c^3*f*h+16 
*b^2*c^2*d*e*h+21*b^2*c^2*d*f*g-20*b^2*c*d^2*e*g)*(f*x+e)^(1/2))/((f*x+e)* 
d+c*f-d*e)^2-1/8*(3*a^2*c*d^2*f^2*h-4*a^2*d^3*e*f*h+a^2*d^3*f^2*g+54*a*b*c 
^2*d*f^2*h-92*a*b*c*d^2*e*f*h-22*a*b*c*d^2*f^2*g+40*a*b*d^3*e^2*h+20*a*b*d 
^3*e*f*g+63*b^2*c^3*f^2*h-144*b^2*c^2*d*e*f*h-99*b^2*c^2*d*f^2*g+80*b^2*c* 
d^2*e^2*h+220*b^2*c*d^2*e*f*g-120*b^2*d^3*e^2*g)/(c*f-d*e)/((c*f-d*e)*d)^( 
1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))-b/f^6/(a*d-b*c)^7*((1/12 
8*b^3*f*(187*a^5*d^4*f^3*h+236*a^4*b*c*d^3*f^3*h-656*a^4*b*d^4*e*f^2*h-515 
*a^4*b*d^4*f^3*g-390*a^3*b^2*c^2*d^2*f^3*h-520*a^3*b^2*c*d^3*e*f^2*h+356*a 
^3*b^2*c*d^3*f^3*g+720*a^3*b^2*d^4*e^2*f*h+1704*a^3*b^2*d^4*e*f^2*g-36*a^2 
*b^3*c^3*d*f^3*h+1080*a^2*b^3*c^2*d^2*e*f^2*h+198*a^2*b^3*c^2*d^2*f^3*g+43 
2*a^2*b^3*c*d^3*e^2*f*h-1464*a^2*b^3*c*d^3*e*f^2*g-256*a^2*b^3*d^4*e^3*h-1 
824*a^2*b^3*d^4*e^2*f*g+3*a*b^4*c^4*f^3*h+104*a*b^4*c^3*d*e*f^2*h-44*a*b^4 
*c^3*d*f^3*g-1104*a*b^4*c^2*d^2*e^2*f*h-264*a*b^4*c^2*d^2*e*f^2*g-128*a*b^ 
4*c*d^3*e^3*h+1728*a*b^4*c*d^3*e^2*f*g+640*a*b^4*d^4*e^3*g-8*b^5*c^4*e*f^2 
*h+5*b^5*c^4*f^3*g-48*b^5*c^3*d*e^2*f*h+24*b^5*c^3*d*e*f^2*g+384*b^5*c^...
 

Fricas [F(-1)]

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

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((f*x+e)**(1/2)*(h*x+g)/(b*x+a)**5/(d*x+c)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(1/2)*(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. 5728 vs. \(2 (1773) = 3546\).

Time = 0.53 (sec) , antiderivative size = 5728, normalized size of antiderivative = 3.15 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^5 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x, algorithm="giac")
 

Output:

1/64*(1920*b^6*d^4*e^4*g - 640*b^6*c*d^3*e^3*f*g - 7040*a*b^5*d^4*e^3*f*g 
- 96*b^6*c^2*d^2*e^2*f^2*g + 2112*a*b^5*c*d^3*e^2*f^2*g + 9504*a^2*b^4*d^4 
*e^2*f^2*g - 24*b^6*c^3*d*e*f^3*g + 264*a*b^5*c^2*d^2*e*f^3*g - 2376*a^2*b 
^4*c*d^3*e*f^3*g - 5544*a^3*b^3*d^4*e*f^3*g - 5*b^6*c^4*f^4*g + 44*a*b^5*c 
^3*d*f^4*g - 198*a^2*b^4*c^2*d^2*f^4*g + 924*a^3*b^3*c*d^3*f^4*g + 1155*a^ 
4*b^2*d^4*f^4*g - 1280*b^6*c*d^3*e^4*h - 640*a*b^5*d^4*e^4*h + 384*b^6*c^2 
*d^2*e^3*f*h + 4992*a*b^5*c*d^3*e^3*f*h + 2304*a^2*b^4*d^4*e^3*f*h + 48*b^ 
6*c^3*d*e^2*f^2*h - 1200*a*b^5*c^2*d^2*e^2*f^2*h - 7344*a^2*b^4*c*d^3*e^2* 
f^2*h - 3024*a^3*b^3*d^4*e^2*f^2*h + 8*b^6*c^4*e*f^3*h - 104*a*b^5*c^3*d*e 
*f^3*h + 1224*a^2*b^4*c^2*d^2*e*f^3*h + 4872*a^3*b^3*c*d^3*e*f^3*h + 1680* 
a^4*b^2*d^4*e*f^3*h - 3*a*b^5*c^4*f^4*h + 36*a^2*b^4*c^3*d*f^4*h - 378*a^3 
*b^3*c^2*d^2*f^4*h - 1260*a^4*b^2*c*d^3*f^4*h - 315*a^5*b*d^4*f^4*h)*arcta 
n(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^10*c^7*e^3 - 7*a*b^9*c^6*d*e^3 
 + 21*a^2*b^8*c^5*d^2*e^3 - 35*a^3*b^7*c^4*d^3*e^3 + 35*a^4*b^6*c^3*d^4*e^ 
3 - 21*a^5*b^5*c^2*d^5*e^3 + 7*a^6*b^4*c*d^6*e^3 - a^7*b^3*d^7*e^3 - 3*a*b 
^9*c^7*e^2*f + 21*a^2*b^8*c^6*d*e^2*f - 63*a^3*b^7*c^5*d^2*e^2*f + 105*a^4 
*b^6*c^4*d^3*e^2*f - 105*a^5*b^5*c^3*d^4*e^2*f + 63*a^6*b^4*c^2*d^5*e^2*f 
- 21*a^7*b^3*c*d^6*e^2*f + 3*a^8*b^2*d^7*e^2*f + 3*a^2*b^8*c^7*e*f^2 - 21* 
a^3*b^7*c^6*d*e*f^2 + 63*a^4*b^6*c^5*d^2*e*f^2 - 105*a^5*b^5*c^4*d^3*e*f^2 
 + 105*a^6*b^4*c^3*d^4*e*f^2 - 63*a^7*b^3*c^2*d^5*e*f^2 + 21*a^8*b^2*c*...
 

Mupad [F(-1)]

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

int(((e + f*x)^(1/2)*(g + h*x))/((a + b*x)^5*(c + d*x)^3),x)
 

Output:

\text{Hanged}
                                                                                    
                                                                                    
 

Reduce [F]

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

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x)
 

Output:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^5/(d*x+c)^3,x)