\(\int \frac {g+h x}{(a+b x)^5 (c+d x) \sqrt {e+f x}} \, dx\) [131]

Optimal result

Integrand size = 29, antiderivative size = 946 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x) \sqrt {e+f x}} \, dx=-\frac {(b g-a h) \sqrt {e+f x}}{4 (b c-a d) (b e-a f) (a+b x)^4}+\frac {\left (7 a^2 d f h-a b f (15 d g-c h)+b^2 (8 d e g+7 c f g-8 c e h)\right ) \sqrt {e+f x}}{24 (b c-a d)^2 (b e-a f)^2 (a+b x)^3}+\frac {\left (35 a^3 d^2 f^2 h-3 a^2 b d f^2 (41 d g-6 c h)+a b^2 f \left (136 d^2 e g-5 c^2 f h+2 c d (55 f g-68 e h)\right )-b^3 \left (48 d^2 e^2 g+5 c^2 f (7 f g-8 e h)+8 c d e (5 f g-6 e h)\right )\right ) \sqrt {e+f x}}{96 (b c-a d)^3 (b e-a f)^3 (a+b x)^2}+\frac {\left (35 a^4 d^3 f^3 h-a^3 b d^2 f^3 (187 d g-47 c h)+a^2 b^2 d f^2 \left (328 d^2 e g-23 c^2 f h+c d (233 f g-328 e h)\right )-a b^3 f \left (240 d^3 e^2 g-5 c^3 f^2 h+c^2 d f (145 f g-176 e h)+16 c d^2 e (11 f g-15 e h)\right )+b^4 \left (64 d^3 e^3 g+5 c^3 f^2 (7 f g-8 e h)+8 c^2 d e f (5 f g-6 e h)+16 c d^2 e^2 (3 f g-4 e h)\right )\right ) \sqrt {e+f x}}{64 (b c-a d)^4 (b e-a f)^4 (a+b x)}+\frac {\left (35 a^5 d^4 f^4 h-35 a^4 b d^3 f^4 (9 d g-4 c h)+70 a^3 b^2 d^2 f^3 \left (12 d^2 e g-c^2 f h+6 c d (f g-2 e h)\right )-14 a^2 b^3 d f^2 \left (72 d^3 e^2 g-2 c^3 f^2 h+9 c^2 d f (3 f g-4 e h)+36 c d^2 e (f g-2 e h)\right )+a b^4 f \left (576 d^4 e^3 g-5 c^4 f^3 h+36 c^3 d f^2 (5 f g-6 e h)+72 c^2 d^2 e f (3 f g-4 e h)+288 c d^3 e^2 (f g-2 e h)\right )-b^5 \left (128 d^4 e^4 g+5 c^4 f^3 (7 f g-8 e h)+8 c^3 d e f^2 (5 f g-6 e h)+16 c^2 d^2 e^2 f (3 f g-4 e h)+64 c d^3 e^3 (f g-2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{64 \sqrt {b} (b c-a d)^5 (b e-a f)^{9/2}}+\frac {2 d^{7/2} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d)^5 \sqrt {d e-c f}} \] Output:

-1/4*(-a*h+b*g)*(f*x+e)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^4+1/24*(7*a^2* 
d*f*h-a*b*f*(-c*h+15*d*g)+b^2*(-8*c*e*h+7*c*f*g+8*d*e*g))*(f*x+e)^(1/2)/(- 
a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^3+1/96*(35*a^3*d^2*f^2*h-3*a^2*b*d*f^2*(-6 
*c*h+41*d*g)+a*b^2*f*(136*d^2*e*g-5*c^2*f*h+2*c*d*(-68*e*h+55*f*g))-b^3*(4 
8*d^2*e^2*g+5*c^2*f*(-8*e*h+7*f*g)+8*c*d*e*(-6*e*h+5*f*g)))*(f*x+e)^(1/2)/ 
(-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)^2+1/64*(35*a^4*d^3*f^3*h-a^3*b*d^2*f^3*( 
-47*c*h+187*d*g)+a^2*b^2*d*f^2*(328*d^2*e*g-23*c^2*f*h+c*d*(-328*e*h+233*f 
*g))-a*b^3*f*(240*d^3*e^2*g-5*c^3*f^2*h+c^2*d*f*(-176*e*h+145*f*g)+16*c*d^ 
2*e*(-15*e*h+11*f*g))+b^4*(64*d^3*e^3*g+5*c^3*f^2*(-8*e*h+7*f*g)+8*c^2*d*e 
*f*(-6*e*h+5*f*g)+16*c*d^2*e^2*(-4*e*h+3*f*g)))*(f*x+e)^(1/2)/(-a*d+b*c)^4 
/(-a*f+b*e)^4/(b*x+a)+1/64*(35*a^5*d^4*f^4*h-35*a^4*b*d^3*f^4*(-4*c*h+9*d* 
g)+70*a^3*b^2*d^2*f^3*(12*d^2*e*g-c^2*f*h+6*c*d*(-2*e*h+f*g))-14*a^2*b^3*d 
*f^2*(72*d^3*e^2*g-2*c^3*f^2*h+9*c^2*d*f*(-4*e*h+3*f*g)+36*c*d^2*e*(-2*e*h 
+f*g))+a*b^4*f*(576*d^4*e^3*g-5*c^4*f^3*h+36*c^3*d*f^2*(-6*e*h+5*f*g)+72*c 
^2*d^2*e*f*(-4*e*h+3*f*g)+288*c*d^3*e^2*(-2*e*h+f*g))-b^5*(128*d^4*e^4*g+5 
*c^4*f^3*(-8*e*h+7*f*g)+8*c^3*d*e*f^2*(-6*e*h+5*f*g)+16*c^2*d^2*e^2*f*(-4* 
e*h+3*f*g)+64*c*d^3*e^3*(-2*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f 
+b*e)^(1/2))/b^(1/2)/(-a*d+b*c)^5/(-a*f+b*e)^(9/2)+2*d^(7/2)*(-c*h+d*g)*ar 
ctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/(-a*d+b*c)^5/(-c*f+d*e)^(1/2 
)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4159\) vs. \(2(946)=1892\).

Time = 16.16 (sec) , antiderivative size = 4159, normalized size of antiderivative = 4.40 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x) \sqrt {e+f x}} \, dx=\text {Result too large to show} \] Input:

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

Output:

-1/4*((b*g - a*h)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^4) - ( 
-1/3*(((-7*a*d*f*(b*g - a*h))/2 + (b*(-(a*f*(8*d*g - c*h)) + b*(8*d*e*g + 
7*c*f*g - 8*c*e*h)))/2)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^ 
3) - (-1/2*(((-5*a*d*f*(7*a^2*d*f*h - a*b*f*(15*d*g - c*h) + b^2*(8*d*e*g 
+ 7*c*f*g - 8*c*e*h)))/4 + (b*(a^2*d*f^2*(48*d*g - 13*c*h) - a*b*f*(96*d^2 
*e*g + 75*c*d*f*g - 96*c*d*e*h - 5*c^2*f*h) + b^2*(48*d^2*e^2*g + 5*c^2*f* 
(7*f*g - 8*e*h) + 8*c*d*e*(5*f*g - 6*e*h))))/4)*Sqrt[e + f*x])/((b*c - a*d 
)*(b*e - a*f)*(a + b*x)^2) - (-((((-3*a*d*f*((-5*a*d*f*(7*a^2*d*f*h - a*b* 
f*(15*d*g - c*h) + b^2*(8*d*e*g + 7*c*f*g - 8*c*e*h)))/4 + (b*(a^2*d*f^2*( 
48*d*g - 13*c*h) - a*b*f*(96*d^2*e*g + 75*c*d*f*g - 96*c*d*e*h - 5*c^2*f*h 
) + b^2*(48*d^2*e^2*g + 5*c^2*f*(7*f*g - 8*e*h) + 8*c*d*e*(5*f*g - 6*e*h)) 
))/4))/2 + b*(-1/2*(c*f*((-5*a*d*f*(7*a^2*d*f*h - a*b*f*(15*d*g - c*h) + b 
^2*(8*d*e*g + 7*c*f*g - 8*c*e*h)))/4 + (b*(a^2*d*f^2*(48*d*g - 13*c*h) - a 
*b*f*(96*d^2*e*g + 75*c*d*f*g - 96*c*d*e*h - 5*c^2*f*h) + b^2*(48*d^2*e^2* 
g + 5*c^2*f*(7*f*g - 8*e*h) + 8*c*d*e*(5*f*g - 6*e*h))))/4)) - 2*((5*b*c*d 
*e*f*(7*a^2*d*f*h - a*b*f*(15*d*g - c*h) + b^2*(8*d*e*g + 7*c*f*g - 8*c*e* 
h)))/4 + (a*d*f*(a^2*d*f^2*(48*d*g - 13*c*h) - a*b*f*(96*d^2*e*g + 75*c*d* 
f*g - 96*c*d*e*h - 5*c^2*f*h) + b^2*(48*d^2*e^2*g + 5*c^2*f*(7*f*g - 8*e*h 
) + 8*c*d*e*(5*f*g - 6*e*h))))/4 - (b*(d*e + c*f)*(a^2*d*f^2*(48*d*g - 13* 
c*h) - a*b*f*(96*d^2*e*g + 75*c*d*f*g - 96*c*d*e*h - 5*c^2*f*h) + b^2*(...
 

Rubi [A] (verified)

Time = 2.33 (sec) , antiderivative size = 1051, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {168, 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.

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (7 a^2 d f h-a b f (15 d g-c h)+b^2 (-8 c e h+7 c f g+8 d e g)\right )}{3 (a+b x)^3 (b c-a d) (b e-a f)}-\frac {\int -\frac {\left (5 f (7 f g-8 e h) c^2+8 d e (5 f g-6 e h) c+48 d^2 e^2 g\right ) b^2-a f \left (-5 f h c^2+75 d f g c-96 d e h c+96 d^2 e g\right ) b+a^2 d f^2 (48 d g-13 c h)+5 d f \left (7 d f h a^2-b f (15 d g-c h) a+b^2 (8 d e g+7 c f g-8 c e h)\right ) x}{2 (a+b x)^3 (c+d x) \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (5 f (7 f g-8 e h) c^2+8 d e (5 f g-6 e h) c+48 d^2 e^2 g\right ) b^2-a f \left (-5 f h c^2+75 d f g c-96 d e h c+96 d^2 e g\right ) b+a^2 d f^2 (48 d g-13 c h)+5 d f \left (7 d f h a^2-b f (15 d g-c h) a+b^2 (8 d e g+7 c f g-8 c e h)\right ) x}{(a+b x)^3 (c+d x) \sqrt {e+f x}}dx}{6 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (7 a^2 d f h-a b f (15 d g-c h)+b^2 (-8 c e h+7 c f g+8 d e g)\right )}{3 (a+b x)^3 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {\sqrt {e+f x} \left (35 a^3 d^2 f^2 h-3 a^2 b d f^2 (41 d g-6 c h)+a b^2 f \left (-5 c^2 f h+2 c d (55 f g-68 e h)+136 d^2 e g\right )-b^3 \left (5 c^2 f (7 f g-8 e h)+8 c d e (5 f g-6 e h)+48 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (b c-a d) (b e-a f)}-\frac {\int -\frac {3 \left (-\left (\left (5 f^2 (7 f g-8 e h) c^3+8 d e f (5 f g-6 e h) c^2+16 d^2 e^2 (3 f g-4 e h) c+64 d^3 e^3 g\right ) b^3\right )+a f \left (-5 f^2 h c^3+2 d f (55 f g-68 e h) c^2+8 d^2 e (17 f g-24 e h) c+192 d^3 e^2 g\right ) b^2-3 a^2 d f^2 \left (-6 f h c^2+d (41 f g-64 e h) c+64 d^2 e g\right ) b+a^3 d^2 f^3 (64 d g-29 c h)+d f \left (35 d^2 f^2 h a^3-3 b d f^2 (41 d g-6 c h) a^2+b^2 f \left (-5 f h c^2+2 d (55 f g-68 e h) c+136 d^2 e g\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+8 d e (5 f g-6 e h) c+48 d^2 e^2 g\right )\right ) x\right )}{2 (a+b x)^2 (c+d x) \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (7 a^2 d f h-a b f (15 d g-c h)+b^2 (-8 c e h+7 c f g+8 d e g)\right )}{3 (a+b x)^3 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {-\left (\left (5 f^2 (7 f g-8 e h) c^3+8 d e f (5 f g-6 e h) c^2+16 d^2 e^2 (3 f g-4 e h) c+64 d^3 e^3 g\right ) b^3\right )+a f \left (-5 f^2 h c^3+2 d f (55 f g-68 e h) c^2+8 d^2 e (17 f g-24 e h) c+192 d^3 e^2 g\right ) b^2-3 a^2 d f^2 \left (-6 f h c^2+d (41 f g-64 e h) c+64 d^2 e g\right ) b+a^3 d^2 f^3 (64 d g-29 c h)+d f \left (35 d^2 f^2 h a^3-3 b d f^2 (41 d g-6 c h) a^2+b^2 f \left (-5 f h c^2+2 d (55 f g-68 e h) c+136 d^2 e g\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+8 d e (5 f g-6 e h) c+48 d^2 e^2 g\right )\right ) x}{(a+b x)^2 (c+d x) \sqrt {e+f x}}dx}{4 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (35 a^3 d^2 f^2 h-3 a^2 b d f^2 (41 d g-6 c h)+a b^2 f \left (-5 c^2 f h+2 c d (55 f g-68 e h)+136 d^2 e g\right )-b^3 \left (5 c^2 f (7 f g-8 e h)+8 c d e (5 f g-6 e h)+48 d^2 e^2 g\right )\right )}{2 (a+b x)^2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}+\frac {\sqrt {e+f x} \left (7 a^2 d f h-a b f (15 d g-c h)+b^2 (-8 c e h+7 c f g+8 d e g)\right )}{3 (a+b x)^3 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {\sqrt {e+f x} (b g-a h)}{4 (a+b x)^4 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (7 d f h a^2-b f (15 d g-c h) a+b^2 (8 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3}+\frac {\frac {\sqrt {e+f x} \left (35 d^2 f^2 h a^3-3 b d f^2 (41 d g-6 c h) a^2+b^2 f \left (-5 f h c^2+2 d (55 f g-68 e h) c+136 d^2 e g\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+8 d e (5 f g-6 e h) c+48 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2}+\frac {3 \left (\frac {\left (35 d^3 f^3 h a^4-b d^2 f^3 (187 d g-47 c h) a^3+b^2 d f^2 \left (-23 f h c^2+d (233 f g-328 e h) c+328 d^2 e g\right ) a^2-b^3 f \left (-5 f^2 h c^3+d f (145 f g-176 e h) c^2+16 d^2 e (11 f g-15 e h) c+240 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-8 e h) c^3+8 d e f (5 f g-6 e h) c^2+16 d^2 e^2 (3 f g-4 e h) c+64 d^3 e^3 g\right )\right ) \sqrt {e+f x}}{(b c-a d) (b e-a f) (a+b x)}-\frac {\int -\frac {\left (5 f^3 (7 f g-8 e h) c^4+8 d e f^2 (5 f g-6 e h) c^3+16 d^2 e^2 f (3 f g-4 e h) c^2+64 d^3 e^3 (f g-2 e h) c+128 d^4 e^4 g\right ) b^4-a f \left (-5 f^3 h c^4+d f^2 (145 f g-176 e h) c^3+16 d^2 e f (11 f g-15 e h) c^2+16 d^3 e^2 (15 f g-32 e h) c+512 d^4 e^3 g\right ) b^3+a^2 d f^2 \left (-23 f^2 h c^3+d f (233 f g-328 e h) c^2+8 d^2 e (41 f g-96 e h) c+768 d^3 e^2 g\right ) b^2-a^3 d^2 f^3 \left (-47 f h c^2+d (187 f g-512 e h) c+512 d^2 e g\right ) b+a^4 d^3 f^4 (128 d g-93 c h)+d f \left (35 d^3 f^3 h a^4-b d^2 f^3 (187 d g-47 c h) a^3+b^2 d f^2 \left (-23 f h c^2+d (233 f g-328 e h) c+328 d^2 e g\right ) a^2-b^3 f \left (-5 f^2 h c^3+d f (145 f g-176 e h) c^2+16 d^2 e (11 f g-15 e h) c+240 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-8 e h) c^3+8 d e f (5 f g-6 e h) c^2+16 d^2 e^2 (3 f g-4 e h) c+64 d^3 e^3 g\right )\right ) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}\right )}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {(b g-a h) \sqrt {e+f x}}{4 (b c-a d) (b e-a f) (a+b x)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (7 d f h a^2-b f (15 d g-c h) a+b^2 (8 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3}+\frac {\frac {\sqrt {e+f x} \left (35 d^2 f^2 h a^3-3 b d f^2 (41 d g-6 c h) a^2+b^2 f \left (-5 f h c^2+2 d (55 f g-68 e h) c+136 d^2 e g\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+8 d e (5 f g-6 e h) c+48 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2}+\frac {3 \left (\frac {\sqrt {e+f x} \left (35 d^3 f^3 h a^4-b d^2 f^3 (187 d g-47 c h) a^3+b^2 d f^2 \left (-23 f h c^2+d (233 f g-328 e h) c+328 d^2 e g\right ) a^2-b^3 f \left (-5 f^2 h c^3+d f (145 f g-176 e h) c^2+16 d^2 e (11 f g-15 e h) c+240 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-8 e h) c^3+8 d e f (5 f g-6 e h) c^2+16 d^2 e^2 (3 f g-4 e h) c+64 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x)}+\frac {\int \frac {\left (5 f^3 (7 f g-8 e h) c^4+8 d e f^2 (5 f g-6 e h) c^3+16 d^2 e^2 f (3 f g-4 e h) c^2+64 d^3 e^3 (f g-2 e h) c+128 d^4 e^4 g\right ) b^4-a f \left (-5 f^3 h c^4+d f^2 (145 f g-176 e h) c^3+16 d^2 e f (11 f g-15 e h) c^2+16 d^3 e^2 (15 f g-32 e h) c+512 d^4 e^3 g\right ) b^3+a^2 d f^2 \left (-23 f^2 h c^3+d f (233 f g-328 e h) c^2+8 d^2 e (41 f g-96 e h) c+768 d^3 e^2 g\right ) b^2-a^3 d^2 f^3 \left (-47 f h c^2+187 d f g c-512 d e h c+512 d^2 e g\right ) b+a^4 d^3 f^4 (128 d g-93 c h)+d f \left (35 d^3 f^3 h a^4-b d^2 f^3 (187 d g-47 c h) a^3+b^2 d f^2 \left (-23 f h c^2+d (233 f g-328 e h) c+328 d^2 e g\right ) a^2-b^3 f \left (-5 f^2 h c^3+d f (145 f g-176 e h) c^2+16 d^2 e (11 f g-15 e h) c+240 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-8 e h) c^3+8 d e f (5 f g-6 e h) c^2+16 d^2 e^2 (3 f g-4 e h) c+64 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}\right )}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {(b g-a h) \sqrt {e+f x}}{4 (b c-a d) (b e-a f) (a+b x)^4}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (7 d f h a^2-b f (15 d g-c h) a+b^2 (8 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3}+\frac {\frac {\sqrt {e+f x} \left (35 d^2 f^2 h a^3-3 b d f^2 (41 d g-6 c h) a^2+b^2 f \left (-5 f h c^2+2 d (55 f g-68 e h) c+136 d^2 e g\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+8 d e (5 f g-6 e h) c+48 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2}+\frac {3 \left (\frac {\sqrt {e+f x} \left (35 d^3 f^3 h a^4-b d^2 f^3 (187 d g-47 c h) a^3+b^2 d f^2 \left (-23 f h c^2+d (233 f g-328 e h) c+328 d^2 e g\right ) a^2-b^3 f \left (-5 f^2 h c^3+d f (145 f g-176 e h) c^2+16 d^2 e (11 f g-15 e h) c+240 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-8 e h) c^3+8 d e f (5 f g-6 e h) c^2+16 d^2 e^2 (3 f g-4 e h) c+64 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x)}+\frac {-\frac {128 d^4 (d g-c h) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx (b e-a f)^4}{b c-a d}-\frac {\left (35 d^4 f^4 h a^5-35 b d^3 f^4 (9 d g-4 c h) a^4+70 b^2 d^2 f^3 \left (-f h c^2+6 d (f g-2 e h) c+12 d^2 e g\right ) a^3-14 b^3 d f^2 \left (-2 f^2 h c^3+9 d f (3 f g-4 e h) c^2+36 d^2 e (f g-2 e h) c+72 d^3 e^2 g\right ) a^2+b^4 f \left (-5 f^3 h c^4+36 d f^2 (5 f g-6 e h) c^3+72 d^2 e f (3 f g-4 e h) c^2+288 d^3 e^2 (f g-2 e h) c+576 d^4 e^3 g\right ) a-b^5 \left (5 f^3 (7 f g-8 e h) c^4+8 d e f^2 (5 f g-6 e h) c^3+16 d^2 e^2 f (3 f g-4 e h) c^2+64 d^3 e^3 (f g-2 e h) c+128 d^4 e^4 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}}{2 (b c-a d) (b e-a f)}\right )}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {(b g-a h) \sqrt {e+f x}}{4 (b c-a d) (b e-a f) (a+b x)^4}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (7 d f h a^2-b f (15 d g-c h) a+b^2 (8 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3}+\frac {\frac {\sqrt {e+f x} \left (35 d^2 f^2 h a^3-3 b d f^2 (41 d g-6 c h) a^2+b^2 f \left (-5 f h c^2+2 d (55 f g-68 e h) c+136 d^2 e g\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+8 d e (5 f g-6 e h) c+48 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2}+\frac {3 \left (\frac {\sqrt {e+f x} \left (35 d^3 f^3 h a^4-b d^2 f^3 (187 d g-47 c h) a^3+b^2 d f^2 \left (-23 f h c^2+d (233 f g-328 e h) c+328 d^2 e g\right ) a^2-b^3 f \left (-5 f^2 h c^3+d f (145 f g-176 e h) c^2+16 d^2 e (11 f g-15 e h) c+240 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-8 e h) c^3+8 d e f (5 f g-6 e h) c^2+16 d^2 e^2 (3 f g-4 e h) c+64 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x)}+\frac {-\frac {256 d^4 (d g-c h) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x} (b e-a f)^4}{(b c-a d) f}-\frac {2 \left (35 d^4 f^4 h a^5-35 b d^3 f^4 (9 d g-4 c h) a^4+70 b^2 d^2 f^3 \left (-f h c^2+6 d (f g-2 e h) c+12 d^2 e g\right ) a^3-14 b^3 d f^2 \left (-2 f^2 h c^3+9 d f (3 f g-4 e h) c^2+36 d^2 e (f g-2 e h) c+72 d^3 e^2 g\right ) a^2+b^4 f \left (-5 f^3 h c^4+36 d f^2 (5 f g-6 e h) c^3+72 d^2 e f (3 f g-4 e h) c^2+288 d^3 e^2 (f g-2 e h) c+576 d^4 e^3 g\right ) a-b^5 \left (5 f^3 (7 f g-8 e h) c^4+8 d e f^2 (5 f g-6 e h) c^3+16 d^2 e^2 f (3 f g-4 e h) c^2+64 d^3 e^3 (f g-2 e h) c+128 d^4 e^4 g\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{(b c-a d) f}}{2 (b c-a d) (b e-a f)}\right )}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {(b g-a h) \sqrt {e+f x}}{4 (b c-a d) (b e-a f) (a+b x)^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {e+f x} \left (7 d f h a^2-b f (15 d g-c h) a+b^2 (8 d e g+7 c f g-8 c e h)\right )}{3 (b c-a d) (b e-a f) (a+b x)^3}+\frac {\frac {\sqrt {e+f x} \left (35 d^2 f^2 h a^3-3 b d f^2 (41 d g-6 c h) a^2+b^2 f \left (-5 f h c^2+2 d (55 f g-68 e h) c+136 d^2 e g\right ) a-b^3 \left (5 f (7 f g-8 e h) c^2+8 d e (5 f g-6 e h) c+48 d^2 e^2 g\right )\right )}{2 (b c-a d) (b e-a f) (a+b x)^2}+\frac {3 \left (\frac {\sqrt {e+f x} \left (35 d^3 f^3 h a^4-b d^2 f^3 (187 d g-47 c h) a^3+b^2 d f^2 \left (-23 f h c^2+d (233 f g-328 e h) c+328 d^2 e g\right ) a^2-b^3 f \left (-5 f^2 h c^3+d f (145 f g-176 e h) c^2+16 d^2 e (11 f g-15 e h) c+240 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-8 e h) c^3+8 d e f (5 f g-6 e h) c^2+16 d^2 e^2 (3 f g-4 e h) c+64 d^3 e^3 g\right )\right )}{(b c-a d) (b e-a f) (a+b x)}+\frac {\frac {256 d^{7/2} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) (b e-a f)^4}{(b c-a d) \sqrt {d e-c f}}+\frac {2 \left (35 d^4 f^4 h a^5-35 b d^3 f^4 (9 d g-4 c h) a^4+70 b^2 d^2 f^3 \left (-f h c^2+6 d (f g-2 e h) c+12 d^2 e g\right ) a^3-14 b^3 d f^2 \left (-2 f^2 h c^3+9 d f (3 f g-4 e h) c^2+36 d^2 e (f g-2 e h) c+72 d^3 e^2 g\right ) a^2+b^4 f \left (-5 f^3 h c^4+36 d f^2 (5 f g-6 e h) c^3+72 d^2 e f (3 f g-4 e h) c^2+288 d^3 e^2 (f g-2 e h) c+576 d^4 e^3 g\right ) a-b^5 \left (5 f^3 (7 f g-8 e h) c^4+8 d e f^2 (5 f g-6 e h) c^3+16 d^2 e^2 f (3 f g-4 e h) c^2+64 d^3 e^3 (f g-2 e h) c+128 d^4 e^4 g\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b c-a d) \sqrt {b e-a f}}}{2 (b c-a d) (b e-a f)}\right )}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}}{8 (b c-a d) (b e-a f)}-\frac {(b g-a h) \sqrt {e+f x}}{4 (b c-a d) (b e-a f) (a+b x)^4}\)

Input:

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

Output:

-1/4*((b*g - a*h)*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^4) + ( 
((7*a^2*d*f*h - a*b*f*(15*d*g - c*h) + b^2*(8*d*e*g + 7*c*f*g - 8*c*e*h))* 
Sqrt[e + f*x])/(3*(b*c - a*d)*(b*e - a*f)*(a + b*x)^3) + (((35*a^3*d^2*f^2 
*h - 3*a^2*b*d*f^2*(41*d*g - 6*c*h) + a*b^2*f*(136*d^2*e*g - 5*c^2*f*h + 2 
*c*d*(55*f*g - 68*e*h)) - b^3*(48*d^2*e^2*g + 5*c^2*f*(7*f*g - 8*e*h) + 8* 
c*d*e*(5*f*g - 6*e*h)))*Sqrt[e + f*x])/(2*(b*c - a*d)*(b*e - a*f)*(a + b*x 
)^2) + (3*(((35*a^4*d^3*f^3*h - a^3*b*d^2*f^3*(187*d*g - 47*c*h) + a^2*b^2 
*d*f^2*(328*d^2*e*g - 23*c^2*f*h + c*d*(233*f*g - 328*e*h)) - a*b^3*f*(240 
*d^3*e^2*g - 5*c^3*f^2*h + c^2*d*f*(145*f*g - 176*e*h) + 16*c*d^2*e*(11*f* 
g - 15*e*h)) + b^4*(64*d^3*e^3*g + 5*c^3*f^2*(7*f*g - 8*e*h) + 8*c^2*d*e*f 
*(5*f*g - 6*e*h) + 16*c*d^2*e^2*(3*f*g - 4*e*h)))*Sqrt[e + f*x])/((b*c - a 
*d)*(b*e - a*f)*(a + b*x)) + ((2*(35*a^5*d^4*f^4*h - 35*a^4*b*d^3*f^4*(9*d 
*g - 4*c*h) + 70*a^3*b^2*d^2*f^3*(12*d^2*e*g - c^2*f*h + 6*c*d*(f*g - 2*e* 
h)) - 14*a^2*b^3*d*f^2*(72*d^3*e^2*g - 2*c^3*f^2*h + 9*c^2*d*f*(3*f*g - 4* 
e*h) + 36*c*d^2*e*(f*g - 2*e*h)) + a*b^4*f*(576*d^4*e^3*g - 5*c^4*f^3*h + 
36*c^3*d*f^2*(5*f*g - 6*e*h) + 72*c^2*d^2*e*f*(3*f*g - 4*e*h) + 288*c*d^3* 
e^2*(f*g - 2*e*h)) - b^5*(128*d^4*e^4*g + 5*c^4*f^3*(7*f*g - 8*e*h) + 8*c^ 
3*d*e*f^2*(5*f*g - 6*e*h) + 16*c^2*d^2*e^2*f*(3*f*g - 4*e*h) + 64*c*d^3*e^ 
3*(f*g - 2*e*h)))*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[ 
b]*(b*c - a*d)*Sqrt[b*e - a*f]) + (256*d^(7/2)*(b*e - a*f)^4*(d*g - c*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 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 = 15.54 (sec) , antiderivative size = 1405, normalized size of antiderivative = 1.49

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1405\)
derivativedivides	\(\text {Expression too large to display}\)	\(2523\)
default	\(\text {Expression too large to display}\)	\(2523\)

Input:

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

Output:

93/64/((a*f-b*e)*b)^(1/2)*(35/93*((c*f-d*e)*d)^(1/2)*(b*x+a)^4*(((a^5*h-9* 
a^4*b*g)*f^4+24*a^3*b^2*e*f^3*g-144/5*a^2*b^3*e^2*f^2*g+576/35*a*b^4*e^3*f 
*g-128/35*b^5*e^4*g)*d^4+4*c*((a^4*h+3*a^3*b*g)*f^4-6*a^2*(a*h+3/5*b*g)*b* 
e*f^3+36/5*a*(a*h+2/7*b*g)*b^2*e^2*f^2-144/35*(a*h+1/9*b*g)*b^3*e^3*f+32/3 
5*b^4*e^4*h)*b*d^3-2*(a^2*(a*h+27/5*b*g)*f^3-36/5*a*(a*h+3/7*b*g)*b*e*f^2+ 
144/35*(a*h+1/6*b*g)*b^2*e^2*f-32/35*b^3*e^3*h)*c^2*b^2*f*d^2+4/5*c^3*(a*( 
a*h+45/7*b*g)*f^2-54/7*b*(a*h+5/27*b*g)*e*f+12/7*b^2*e^2*h)*b^3*f^2*d-1/7* 
c^4*((a*h+7*b*g)*f-8*e*h*b)*b^4*f^3)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^ 
(1/2))+(-128/93*d^4*(b*x+a)^4*(a*f-b*e)^4*(c*h-d*g)*arctan(d*(f*x+e)^(1/2) 
/((c*f-d*e)*d)^(1/2))+(a*d-b*c)*((a^3*(-187/93*b^4*g*x^3-643/93*a*x^2*(-35 
/643*h*x+g)*b^3-255/31*a^2*(-77/459*h*x+g)*x*b^2-325/93*a^3*(-511/975*h*x+ 
g)*b+a^4*h)*f^3-326/279*a^2*b*(-492/163*b^4*g*x^3-1735/163*a*b^3*g*x^2-214 
6/163*a^2*x*(-35/2146*h*x+g)*b^2-975/163*a^3*(-42/325*h*x+g)*b+a^4*h)*e*f^ 
2+200/279*a*(-18/5*b^4*g*x^3-316/25*a*b^3*g*x^2-393/25*a^2*b^2*g*x-37/5*a^ 
3*(-7/185*h*x+g)*b+a^4*h)*b^2*e^2*f-16/93*(-4*b^4*g*x^3-14*a*b^3*g*x^2-52/ 
3*a^2*b^2*g*x+a^4*h-25/3*a^3*b*g)*b^3*e^3)*d^3-47/93*c*(a^2*(-233/47*b^4*g 
*x^3+(-2563/141*g*x^2-h*x^3)*a*b^3-3325/141*a^2*x*(389/3325*h*x+g)*b^2-535 
/47*a^3*(251/1605*h*x+g)*b+a^4*h)*f^3+646/141*a*(264/323*b^4*g*x^3+1201/32 
3*a*x^2*(492/1201*h*x+g)*b^3+1970/323*a^2*x*(319/394*h*x+g)*b^2+1249/323*a 
^3*(1642/1249*h*x+g)*b+a^4*h)*b*e*f^2-680/141*(18/85*b^4*g*x^3+22/17*a*...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((h*x+g)/(b*x+a)^5/(d*x+c)/(f*x+e)^(1/2),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. 4455 vs. \(2 (908) = 1816\).

Time = 0.33 (sec) , antiderivative size = 4455, normalized size of antiderivative = 4.71 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x) \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/64*(128*b^5*d^4*e^4*g + 64*b^5*c*d^3*e^3*f*g - 576*a*b^4*d^4*e^3*f*g + 4 
8*b^5*c^2*d^2*e^2*f^2*g - 288*a*b^4*c*d^3*e^2*f^2*g + 1008*a^2*b^3*d^4*e^2 
*f^2*g + 40*b^5*c^3*d*e*f^3*g - 216*a*b^4*c^2*d^2*e*f^3*g + 504*a^2*b^3*c* 
d^3*e*f^3*g - 840*a^3*b^2*d^4*e*f^3*g + 35*b^5*c^4*f^4*g - 180*a*b^4*c^3*d 
*f^4*g + 378*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 - 128*b^5*c*d^3*e^4*h - 64*b^5*c^2*d^2*e^3*f*h + 576*a*b^4*c*d^3* 
e^3*f*h - 48*b^5*c^3*d*e^2*f^2*h + 288*a*b^4*c^2*d^2*e^2*f^2*h - 1008*a^2* 
b^3*c*d^3*e^2*f^2*h - 40*b^5*c^4*e*f^3*h + 216*a*b^4*c^3*d*e*f^3*h - 504*a 
^2*b^3*c^2*d^2*e*f^3*h + 840*a^3*b^2*c*d^3*e*f^3*h + 5*a*b^4*c^4*f^4*h - 2 
8*a^2*b^3*c^3*d*f^4*h + 70*a^3*b^2*c^2*d^2*f^4*h - 140*a^4*b*c*d^3*f^4*h - 
 35*a^5*d^4*f^4*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^9*c^5* 
e^4 - 5*a*b^8*c^4*d*e^4 + 10*a^2*b^7*c^3*d^2*e^4 - 10*a^3*b^6*c^2*d^3*e^4 
+ 5*a^4*b^5*c*d^4*e^4 - a^5*b^4*d^5*e^4 - 4*a*b^8*c^5*e^3*f + 20*a^2*b^7*c 
^4*d*e^3*f - 40*a^3*b^6*c^3*d^2*e^3*f + 40*a^4*b^5*c^2*d^3*e^3*f - 20*a^5* 
b^4*c*d^4*e^3*f + 4*a^6*b^3*d^5*e^3*f + 6*a^2*b^7*c^5*e^2*f^2 - 30*a^3*b^6 
*c^4*d*e^2*f^2 + 60*a^4*b^5*c^3*d^2*e^2*f^2 - 60*a^5*b^4*c^2*d^3*e^2*f^2 + 
 30*a^6*b^3*c*d^4*e^2*f^2 - 6*a^7*b^2*d^5*e^2*f^2 - 4*a^3*b^6*c^5*e*f^3 + 
20*a^4*b^5*c^4*d*e*f^3 - 40*a^5*b^4*c^3*d^2*e*f^3 + 40*a^6*b^3*c^2*d^3*e*f 
^3 - 20*a^7*b^2*c*d^4*e*f^3 + 4*a^8*b*d^5*e*f^3 + a^4*b^5*c^5*f^4 - 5*a^5* 
b^4*c^4*d*f^4 + 10*a^6*b^3*c^3*d^2*f^4 - 10*a^7*b^2*c^2*d^3*f^4 + 5*a^8...
 

Mupad [B] (verification not implemented)

Time = 43.57 (sec) , antiderivative size = 998307, normalized size of antiderivative = 1055.29 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x) \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((16384*a^18*b^2*d^16*f^11*g - 187776*a^17*b^3*c*d^15*f^11*g - 1190 
4*a^18*b^2*c*d^15*f^11*h - 107136*a^17*b^3*d^16*e*f^10*g - 4480*a^18*b^2*d 
^16*e*f^10*h + 4480*a^4*b^16*c^14*d^2*f^11*g - 63360*a^5*b^15*c^13*d^3*f^1 
1*g + 417024*a^6*b^14*c^12*d^4*f^11*g - 1694976*a^7*b^13*c^11*d^5*f^11*g + 
 4765824*a^8*b^12*c^10*d^6*f^11*g - 9846400*a^9*b^11*c^9*d^7*f^11*g + 1549 
0560*a^10*b^10*c^8*d^8*f^11*g - 18943488*a^11*b^9*c^7*d^9*f^11*g + 1816435 
2*a^12*b^8*c^6*d^10*f^11*g - 13614208*a^13*b^7*c^5*d^11*f^11*g + 7845120*a 
^14*b^6*c^4*d^12*f^11*g - 3360000*a^15*b^5*c^3*d^13*f^11*g + 1006464*a^16* 
b^4*c^2*d^14*f^11*g + 640*a^5*b^15*c^14*d^2*f^11*h - 9344*a^6*b^14*c^13*d^ 
3*f^11*h + 64256*a^7*b^13*c^12*d^4*f^11*h - 281344*a^8*b^12*c^11*d^5*f^11* 
h + 877440*a^9*b^11*c^10*d^6*f^11*h - 2037120*a^10*b^10*c^9*d^7*f^11*h + 3 
568128*a^11*b^9*c^8*d^8*f^11*h - 4710912*a^12*b^8*c^7*d^9*f^11*h + 4645248 
*a^13*b^7*c^6*d^10*f^11*h - 3360640*a^14*b^6*c^5*d^11*f^11*h + 1729280*a^1 
5*b^5*c^4*d^12*f^11*h - 598784*a^16*b^4*c^3*d^13*f^11*h + 125056*a^17*b^3* 
c^2*d^14*f^11*h + 8192*a^10*b^10*d^16*e^8*f^3*g - 67584*a^11*b^9*d^16*e^7* 
f^4*g + 244736*a^12*b^8*d^16*e^6*f^5*g - 508544*a^13*b^7*d^16*e^5*f^6*g + 
668160*a^14*b^6*d^16*e^4*f^7*g - 575232*a^15*b^5*d^16*e^3*f^8*g + 321024*a 
^16*b^4*d^16*e^2*f^9*g - 4480*a^14*b^6*d^16*e^5*f^6*h + 17920*a^15*b^5*d^1 
6*e^4*f^7*h - 26880*a^16*b^4*d^16*e^3*f^8*h + 17920*a^17*b^3*d^16*e^2*f^9* 
h + 8192*b^20*c^10*d^6*e^8*f^3*g + 2048*b^20*c^11*d^5*e^7*f^4*g + 1024*...
 

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 24455, normalized size of antiderivative = 25.85 \[ \int \frac {g+h x}{(a+b x)^5 (c+d x) \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

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

Output:

(105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* 
e)))*a**9*c*d**4*f**5*h - 105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)* 
b)/(sqrt(b)*sqrt(a*f - b*e)))*a**9*d**5*e*f**4*h + 420*sqrt(b)*sqrt(a*f - 
b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**8*b*c**2*d**3*f* 
*5*h - 420*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a* 
f - b*e)))*a**8*b*c*d**4*e*f**4*h - 945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt 
(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**8*b*c*d**4*f**5*g + 420*sqrt(b) 
*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**8*b* 
c*d**4*f**5*h*x + 945*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt 
(b)*sqrt(a*f - b*e)))*a**8*b*d**5*e*f**4*g - 420*sqrt(b)*sqrt(a*f - b*e)*a 
tan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**8*b*d**5*e*f**4*h*x - 
210*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e 
)))*a**7*b**2*c**3*d**2*f**5*h - 2310*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e 
 + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b**2*c**2*d**3*e*f**4*h + 1260* 
sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))* 
a**7*b**2*c**2*d**3*f**5*g + 1680*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f 
*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b**2*c**2*d**3*f**5*h*x + 2520*sqrt 
(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7 
*b**2*c*d**4*e**2*f**3*h + 1260*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x 
)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*b**2*c*d**4*e*f**4*g - 1680*sqrt(b...