\(\int \frac {g+h x}{(a+b x)^4 (c+d x) (e+f x)^{3/2}} \, dx\) [175]

Optimal result

Integrand size = 29, antiderivative size = 831 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x) (e+f x)^{3/2}} \, dx=-\frac {f \left (a^3 d^2 f^2 (16 d f g-35 d e h+19 c f h)+a^2 b d f^2 \left (41 d^2 e g-16 c^2 f h-c d (89 f g-64 e h)\right )-a b^2 f \left (30 d^3 e^2 g-5 c^3 f^2 h-c^2 d f (100 f g-83 e h)+2 c d^2 e (11 f g-15 e h)\right )+b^3 \left (8 d^3 e^3 g+c^2 d e f (5 f g-6 e h)-5 c^3 f^2 (7 f g-6 e h)+2 c d^2 e^2 (3 f g-4 e h)\right )\right )}{8 (b c-a d)^3 (b e-a f)^4 (d e-c f) \sqrt {e+f x}}-\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 \sqrt {e+f x}}+\frac {7 a^2 d f h-a b f (13 d g+c h)+b^2 (6 d e g+7 c f g-6 c e h)}{12 (b c-a d)^2 (b e-a f)^2 (a+b x)^2 \sqrt {e+f x}}+\frac {35 a^3 d^2 f^2 h-a^2 b d f^2 (89 d g+16 c h)+a b^2 f \left (78 d^2 e g+5 c^2 f h+2 c d (50 f g-39 e h)\right )-b^3 \left (24 d^2 e^2 g+5 c^2 f (7 f g-6 e h)+6 c d e (5 f g-4 e h)\right )}{24 (b c-a d)^3 (b e-a f)^3 (a+b x) \sqrt {e+f x}}+\frac {\sqrt {b} \left (35 a^4 d^3 f^3 h-35 a^3 b d^2 f^3 (3 d g+c h)+21 a^2 b^2 d f^2 \left (6 d^2 e g+c^2 f h+c d (9 f g-6 e h)\right )-a b^3 f \left (72 d^3 e^2 g+5 c^3 f^2 h+27 c^2 d f (5 f g-4 e h)+36 c d^2 e (3 f g-2 e h)\right )+b^4 \left (16 d^3 e^3 g+5 c^3 f^2 (7 f g-6 e h)+6 c^2 d e f (5 f g-4 e h)+8 c d^2 e^2 (3 f g-2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 (b c-a d)^4 (b e-a f)^{9/2}}-\frac {2 d^{7/2} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d)^4 (d e-c f)^{3/2}} \] Output:

-1/8*f*(a^3*d^2*f^2*(19*c*f*h-35*d*e*h+16*d*f*g)+a^2*b*d*f^2*(41*d^2*e*g-1 
6*c^2*f*h-c*d*(-64*e*h+89*f*g))-a*b^2*f*(30*d^3*e^2*g-5*c^3*f^2*h-c^2*d*f* 
(-83*e*h+100*f*g)+2*c*d^2*e*(-15*e*h+11*f*g))+b^3*(8*d^3*e^3*g+c^2*d*e*f*( 
-6*e*h+5*f*g)-5*c^3*f^2*(-6*e*h+7*f*g)+2*c*d^2*e^2*(-4*e*h+3*f*g)))/(-a*d+ 
b*c)^3/(-a*f+b*e)^4/(-c*f+d*e)/(f*x+e)^(1/2)-1/3*(-a*h+b*g)/(-a*d+b*c)/(-a 
*f+b*e)/(b*x+a)^3/(f*x+e)^(1/2)+1/12*(7*a^2*d*f*h-a*b*f*(c*h+13*d*g)+b^2*( 
-6*c*e*h+7*c*f*g+6*d*e*g))/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^2/(f*x+e)^(1/ 
2)+1/24*(35*a^3*d^2*f^2*h-a^2*b*d*f^2*(16*c*h+89*d*g)+a*b^2*f*(78*d^2*e*g+ 
5*c^2*f*h+2*c*d*(-39*e*h+50*f*g))-b^3*(24*d^2*e^2*g+5*c^2*f*(-6*e*h+7*f*g) 
+6*c*d*e*(-4*e*h+5*f*g)))/(-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)/(f*x+e)^(1/2)+ 
1/8*b^(1/2)*(35*a^4*d^3*f^3*h-35*a^3*b*d^2*f^3*(c*h+3*d*g)+21*a^2*b^2*d*f^ 
2*(6*d^2*e*g+c^2*f*h+c*d*(-6*e*h+9*f*g))-a*b^3*f*(72*d^3*e^2*g+5*c^3*f^2*h 
+27*c^2*d*f*(-4*e*h+5*f*g)+36*c*d^2*e*(-2*e*h+3*f*g))+b^4*(16*d^3*e^3*g+5* 
c^3*f^2*(-6*e*h+7*f*g)+6*c^2*d*e*f*(-4*e*h+5*f*g)+8*c*d^2*e^2*(-2*e*h+3*f* 
g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/(-a*d+b*c)^4/(-a*f+b* 
e)^(9/2)-2*d^(7/2)*(-c*h+d*g)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/ 
2))/(-a*d+b*c)^4/(-c*f+d*e)^(3/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.66 (sec) , antiderivative size = 565, normalized size of antiderivative = 0.68 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x) (e+f x)^{3/2}} \, dx=\frac {-\frac {8 (b c-a d) (b e-a f) (b g-a h)}{(a+b x)^3}+\frac {2 \left (7 a^2 d f h-a b f (13 d g+c h)+b^2 (6 d e g+7 c f g-6 c e h)\right )}{(a+b x)^2}+\frac {35 a^3 d^2 f^2 h-a^2 b d f^2 (89 d g+16 c h)+a b^2 f \left (78 d^2 e g+5 c^2 f h+2 c d (50 f g-39 e h)\right )+b^3 \left (-24 d^2 e^2 g+6 c d e (-5 f g+4 e h)+5 c^2 f (-7 f g+6 e h)\right )}{(b c-a d) (b e-a f) (a+b x)}+\frac {3 (d e-c f) \left (35 a^4 d^3 f^3 h-35 a^3 b d^2 f^3 (3 d g+c h)+21 a^2 b^2 d f^2 \left (6 d^2 e g+9 c d f g-6 c d e h+c^2 f h\right )-a b^3 f \left (72 d^3 e^2 g+5 c^3 f^2 h+27 c^2 d f (5 f g-4 e h)-36 c d^2 e (-3 f g+2 e h)\right )+b^4 \left (16 d^3 e^3 g+5 c^3 f^2 (7 f g-6 e h)-8 c d^2 e^2 (-3 f g+2 e h)-6 c^2 d e f (-5 f g+4 e h)\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {b (e+f x)}{b e-a f}\right )-48 d^3 (b e-a f)^4 (d g-c h) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {d (e+f x)}{d e-c f}\right )}{(b c-a d)^2 (b e-a f)^2 (-d e+c f)}}{24 (b c-a d)^2 (b e-a f)^2 \sqrt {e+f x}} \] Input:

Integrate[(g + h*x)/((a + b*x)^4*(c + d*x)*(e + f*x)^(3/2)),x]
 

Output:

((-8*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))/(a + b*x)^3 + (2*(7*a^2*d*f*h - 
a*b*f*(13*d*g + c*h) + b^2*(6*d*e*g + 7*c*f*g - 6*c*e*h)))/(a + b*x)^2 + ( 
35*a^3*d^2*f^2*h - a^2*b*d*f^2*(89*d*g + 16*c*h) + a*b^2*f*(78*d^2*e*g + 5 
*c^2*f*h + 2*c*d*(50*f*g - 39*e*h)) + b^3*(-24*d^2*e^2*g + 6*c*d*e*(-5*f*g 
 + 4*e*h) + 5*c^2*f*(-7*f*g + 6*e*h)))/((b*c - a*d)*(b*e - a*f)*(a + b*x)) 
 + (3*(d*e - c*f)*(35*a^4*d^3*f^3*h - 35*a^3*b*d^2*f^3*(3*d*g + c*h) + 21* 
a^2*b^2*d*f^2*(6*d^2*e*g + 9*c*d*f*g - 6*c*d*e*h + c^2*f*h) - a*b^3*f*(72* 
d^3*e^2*g + 5*c^3*f^2*h + 27*c^2*d*f*(5*f*g - 4*e*h) - 36*c*d^2*e*(-3*f*g 
+ 2*e*h)) + b^4*(16*d^3*e^3*g + 5*c^3*f^2*(7*f*g - 6*e*h) - 8*c*d^2*e^2*(- 
3*f*g + 2*e*h) - 6*c^2*d*e*f*(-5*f*g + 4*e*h)))*Hypergeometric2F1[-1/2, 1, 
 1/2, (b*(e + f*x))/(b*e - a*f)] - 48*d^3*(b*e - a*f)^4*(d*g - c*h)*Hyperg 
eometric2F1[-1/2, 1, 1/2, (d*(e + f*x))/(d*e - c*f)])/((b*c - a*d)^2*(b*e 
- a*f)^2*(-(d*e) + c*f)))/(24*(b*c - a*d)^2*(b*e - a*f)^2*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 2.13 (sec) , antiderivative size = 929, normalized size of antiderivative = 1.12, 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, 169, 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.

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\int -\frac {a f (6 d g+c h)-b (6 d e g+7 c f g-6 c e h)-7 d f (b g-a h) x}{2 (a+b x)^3 (c+d x) (e+f x)^{3/2}}dx}{3 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a f (6 d g+c h)-b (6 d e g+7 c f g-6 c e h)-7 d f (b g-a h) x}{(a+b x)^3 (c+d x) (e+f x)^{3/2}}dx}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {7 a^2 d f h-a b f (c h+13 d g)+b^2 (-6 c e h+7 c f g+6 d e g)}{2 (a+b x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}-\frac {\int -\frac {\left (5 f (7 f g-6 e h) c^2+6 d e (5 f g-4 e h) c+24 d^2 e^2 g\right ) b^2-a f \left (5 f h c^2+d (65 f g-48 e h) c+48 d^2 e g\right ) b+a^2 d f^2 (24 d g+11 c h)+5 d f \left (7 d f h a^2-b f (13 d g+c h) a+b^2 (6 d e g+7 c f g-6 c e h)\right ) x}{2 (a+b x)^2 (c+d x) (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (5 f (7 f g-6 e h) c^2+6 d e (5 f g-4 e h) c+24 d^2 e^2 g\right ) b^2-a f \left (5 f h c^2+d (65 f g-48 e h) c+48 d^2 e g\right ) b+a^2 d f^2 (24 d g+11 c h)+5 d f \left (7 d f h a^2-b f (13 d g+c h) a+b^2 (6 d e g+7 c f g-6 c e h)\right ) x}{(a+b x)^2 (c+d x) (e+f x)^{3/2}}dx}{4 (b c-a d) (b e-a f)}+\frac {7 a^2 d f h-a b f (c h+13 d g)+b^2 (-6 c e h+7 c f g+6 d e g)}{2 (a+b x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\frac {35 a^3 d^2 f^2 h-a^2 b d f^2 (16 c h+89 d g)+a b^2 f \left (5 c^2 f h+2 c d (50 f g-39 e h)+78 d^2 e g\right )-b^3 \left (5 c^2 f (7 f g-6 e h)+6 c d e (5 f g-4 e h)+24 d^2 e^2 g\right )}{(a+b x) \sqrt {e+f x} (b c-a d) (b e-a f)}-\frac {\int -\frac {3 \left (-\left (\left (5 f^2 (7 f g-6 e h) c^3+6 d e f (5 f g-4 e h) c^2+8 d^2 e^2 (3 f g-2 e h) c+16 d^3 e^3 g\right ) b^3\right )+a f \left (5 f^2 h c^3+2 d f (50 f g-39 e h) c^2+6 d^2 e (13 f g-8 e h) c+48 d^3 e^2 g\right ) b^2-a^2 d f^2 \left (16 f h c^2+d (89 f g-48 e h) c+48 d^2 e g\right ) b+a^3 d^2 f^3 (16 d g+19 c h)+d f \left (35 d^2 f^2 h a^3-b d f^2 (89 d g+16 c h) a^2+b^2 f \left (5 f h c^2+2 d (50 f g-39 e h) c+78 d^2 e g\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+6 d e (5 f g-4 e h) c+24 d^2 e^2 g\right )\right ) x\right )}{2 (a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}+\frac {7 a^2 d f h-a b f (c h+13 d g)+b^2 (-6 c e h+7 c f g+6 d e g)}{2 (a+b x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {-\left (\left (5 f^2 (7 f g-6 e h) c^3+6 d e f (5 f g-4 e h) c^2+8 d^2 e^2 (3 f g-2 e h) c+16 d^3 e^3 g\right ) b^3\right )+a f \left (5 f^2 h c^3+2 d f (50 f g-39 e h) c^2+6 d^2 e (13 f g-8 e h) c+48 d^3 e^2 g\right ) b^2-a^2 d f^2 \left (16 f h c^2+d (89 f g-48 e h) c+48 d^2 e g\right ) b+a^3 d^2 f^3 (16 d g+19 c h)+d f \left (35 d^2 f^2 h a^3-b d f^2 (89 d g+16 c h) a^2+b^2 f \left (5 f h c^2+2 d (50 f g-39 e h) c+78 d^2 e g\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+6 d e (5 f g-4 e h) c+24 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}+\frac {35 a^3 d^2 f^2 h-a^2 b d f^2 (16 c h+89 d g)+a b^2 f \left (5 c^2 f h+2 c d (50 f g-39 e h)+78 d^2 e g\right )-b^3 \left (5 c^2 f (7 f g-6 e h)+6 c d e (5 f g-4 e h)+24 d^2 e^2 g\right )}{(a+b x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}+\frac {7 a^2 d f h-a b f (c h+13 d g)+b^2 (-6 c e h+7 c f g+6 d e g)}{2 (a+b x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (a+b x)^3 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {7 d f h a^2-b f (13 d g+c h) a+b^2 (6 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 \sqrt {e+f x}}+\frac {\frac {35 d^2 f^2 h a^3-b d f^2 (89 d g+16 c h) a^2+b^2 f \left (5 f h c^2+2 d (50 f g-39 e h) c+78 d^2 e g\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+6 d e (5 f g-4 e h) c+24 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (d^2 f^2 (16 d f g-35 d e h+19 c f h) a^3+b d f^2 \left (-16 f h c^2-89 d f g c+64 d e h c+41 d^2 e g\right ) a^2-b^2 f \left (-5 f^2 h c^3-d f (100 f g-83 e h) c^2+2 d^2 e (11 f g-15 e h) c+30 d^3 e^2 g\right ) a+b^3 \left (-5 f^2 (7 f g-6 e h) c^3+d e f (5 f g-6 e h) c^2+2 d^2 e^2 (3 f g-4 e h) c+8 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 \int \frac {(d e-c f) \left (5 f^2 (7 f g-6 e h) c^3+6 d e f (5 f g-4 e h) c^2+8 d^2 e^2 (3 f g-2 e h) c+16 d^3 e^3 g\right ) b^4-a f \left (-5 f^3 h c^4-d f^2 (100 f g-83 e h) c^3+2 d^2 e f (11 f g-15 e h) c^2+2 d^3 e^2 (15 f g-32 e h) c+64 d^4 e^3 g\right ) b^3+a^2 d f^2 \left (-16 f^2 h c^3-d f (89 f g-64 e h) c^2+d^2 e (41 f g-96 e h) c+96 d^3 e^2 g\right ) b^2-a^3 d^2 f^3 \left (-19 f h c^2-d (16 f g+29 e h) c+64 d^2 e g\right ) b+d f \left (d^2 f^2 (16 d f g-35 d e h+19 c f h) a^3+b d f^2 \left (-16 f h c^2-89 d f g c+64 d e h c+41 d^2 e g\right ) a^2-b^2 f \left (-5 f^2 h c^3-d f (100 f g-83 e h) c^2+2 d^2 e (11 f g-15 e h) c+30 d^3 e^2 g\right ) a+b^3 \left (-5 f^2 (7 f g-6 e h) c^3+d e f (5 f g-6 e h) c^2+2 d^2 e^2 (3 f g-4 e h) c+8 d^3 e^3 g\right )\right ) x b+16 a^4 d^3 f^4 (d g-c h)}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 \sqrt {e+f x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {7 d f h a^2-b f (13 d g+c h) a+b^2 (6 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 \sqrt {e+f x}}+\frac {\frac {35 d^2 f^2 h a^3-b d f^2 (89 d g+16 c h) a^2+b^2 f \left (5 f h c^2+2 d (50 f g-39 e h) c+78 d^2 e g\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+6 d e (5 f g-4 e h) c+24 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (d^2 f^2 (16 d f g-35 d e h+19 c f h) a^3+b d f^2 \left (-16 f h c^2-89 d f g c+64 d e h c+41 d^2 e g\right ) a^2-b^2 f \left (-5 f^2 h c^3-d f (100 f g-83 e h) c^2+2 d^2 e (11 f g-15 e h) c+30 d^3 e^2 g\right ) a+b^3 \left (-5 f^2 (7 f g-6 e h) c^3+d e f (5 f g-6 e h) c^2+2 d^2 e^2 (3 f g-4 e h) c+8 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\int \frac {(d e-c f) \left (5 f^2 (7 f g-6 e h) c^3+6 d e f (5 f g-4 e h) c^2+8 d^2 e^2 (3 f g-2 e h) c+16 d^3 e^3 g\right ) b^4-a f \left (-5 f^3 h c^4-d f^2 (100 f g-83 e h) c^3+2 d^2 e f (11 f g-15 e h) c^2+2 d^3 e^2 (15 f g-32 e h) c+64 d^4 e^3 g\right ) b^3+a^2 d f^2 \left (-16 f^2 h c^3-d f (89 f g-64 e h) c^2+d^2 e (41 f g-96 e h) c+96 d^3 e^2 g\right ) b^2-a^3 d^2 f^3 \left (-19 f h c^2-d (16 f g+29 e h) c+64 d^2 e g\right ) b+d f \left (d^2 f^2 (16 d f g-35 d e h+19 c f h) a^3+b d f^2 \left (-16 f h c^2-89 d f g c+64 d e h c+41 d^2 e g\right ) a^2-b^2 f \left (-5 f^2 h c^3-d f (100 f g-83 e h) c^2+2 d^2 e (11 f g-15 e h) c+30 d^3 e^2 g\right ) a+b^3 \left (-5 f^2 (7 f g-6 e h) c^3+d e f (5 f g-6 e h) c^2+2 d^2 e^2 (3 f g-4 e h) c+8 d^3 e^3 g\right )\right ) x b+16 a^4 d^3 f^4 (d g-c h)}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 \sqrt {e+f x}}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {7 d f h a^2-b f (13 d g+c h) a+b^2 (6 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 \sqrt {e+f x}}+\frac {\frac {35 d^2 f^2 h a^3-b d f^2 (89 d g+16 c h) a^2+b^2 f \left (5 f h c^2+2 d (50 f g-39 e h) c+78 d^2 e g\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+6 d e (5 f g-4 e h) c+24 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (d^2 f^2 (16 d f g-35 d e h+19 c f h) a^3+b d f^2 \left (-16 f h c^2-89 d f g c+64 d e h c+41 d^2 e g\right ) a^2-b^2 f \left (-5 f^2 h c^3-d f (100 f g-83 e h) c^2+2 d^2 e (11 f g-15 e h) c+30 d^3 e^2 g\right ) a+b^3 \left (-5 f^2 (7 f g-6 e h) c^3+d e f (5 f g-6 e h) c^2+2 d^2 e^2 (3 f g-4 e h) c+8 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {b (d e-c f) \left (35 d^3 f^3 h a^4-35 b d^2 f^3 (3 d g+c h) a^3+21 b^2 d f^2 \left (f h c^2+9 d f g c-6 d e h c+6 d^2 e g\right ) a^2-b^3 f \left (5 f^2 h c^3+27 d f (5 f g-4 e h) c^2+36 d^2 e (3 f g-2 e h) c+72 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-6 e h) c^3+6 d e f (5 f g-4 e h) c^2+8 d^2 e^2 (3 f g-2 e h) c+16 d^3 e^3 g\right )\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {16 d^4 (b e-a f)^4 (d g-c h) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}}{(b e-a f) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 \sqrt {e+f x}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {7 d f h a^2-b f (13 d g+c h) a+b^2 (6 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 \sqrt {e+f x}}+\frac {\frac {35 d^2 f^2 h a^3-b d f^2 (89 d g+16 c h) a^2+b^2 f \left (5 f h c^2+2 d (50 f g-39 e h) c+78 d^2 e g\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+6 d e (5 f g-4 e h) c+24 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (d^2 f^2 (16 d f g-35 d e h+19 c f h) a^3+b d f^2 \left (-16 f h c^2-89 d f g c+64 d e h c+41 d^2 e g\right ) a^2-b^2 f \left (-5 f^2 h c^3-d f (100 f g-83 e h) c^2+2 d^2 e (11 f g-15 e h) c+30 d^3 e^2 g\right ) a+b^3 \left (-5 f^2 (7 f g-6 e h) c^3+d e f (5 f g-6 e h) c^2+2 d^2 e^2 (3 f g-4 e h) c+8 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {2 b (d e-c f) \left (35 d^3 f^3 h a^4-35 b d^2 f^3 (3 d g+c h) a^3+21 b^2 d f^2 \left (f h c^2+9 d f g c-6 d e h c+6 d^2 e g\right ) a^2-b^3 f \left (5 f^2 h c^3+27 d f (5 f g-4 e h) c^2+36 d^2 e (3 f g-2 e h) c+72 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-6 e h) c^3+6 d e f (5 f g-4 e h) c^2+8 d^2 e^2 (3 f g-2 e h) c+16 d^3 e^3 g\right )\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{(b c-a d) f}-\frac {32 d^4 (b e-a f)^4 (d g-c h) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{(b c-a d) f}}{(b e-a f) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 \sqrt {e+f x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {7 d f h a^2-b f (13 d g+c h) a+b^2 (6 d e g+7 c f g-6 c e h)}{2 (b c-a d) (b e-a f) (a+b x)^2 \sqrt {e+f x}}+\frac {\frac {35 d^2 f^2 h a^3-b d f^2 (89 d g+16 c h) a^2+b^2 f \left (5 f h c^2+2 d (50 f g-39 e h) c+78 d^2 e g\right ) a-b^3 \left (5 f (7 f g-6 e h) c^2+6 d e (5 f g-4 e h) c+24 d^2 e^2 g\right )}{(b c-a d) (b e-a f) (a+b x) \sqrt {e+f x}}+\frac {3 \left (-\frac {2 f \left (d^2 f^2 (16 d f g-35 d e h+19 c f h) a^3+b d f^2 \left (-16 f h c^2-89 d f g c+64 d e h c+41 d^2 e g\right ) a^2-b^2 f \left (-5 f^2 h c^3-d f (100 f g-83 e h) c^2+2 d^2 e (11 f g-15 e h) c+30 d^3 e^2 g\right ) a+b^3 \left (-5 f^2 (7 f g-6 e h) c^3+d e f (5 f g-6 e h) c^2+2 d^2 e^2 (3 f g-4 e h) c+8 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\frac {32 d^{7/2} (b e-a f)^4 (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d) \sqrt {d e-c f}}-\frac {2 \sqrt {b} (d e-c f) \left (35 d^3 f^3 h a^4-35 b d^2 f^3 (3 d g+c h) a^3+21 b^2 d f^2 \left (f h c^2+9 d f g c-6 d e h c+6 d^2 e g\right ) a^2-b^3 f \left (5 f^2 h c^3+27 d f (5 f g-4 e h) c^2+36 d^2 e (3 f g-2 e h) c+72 d^3 e^2 g\right ) a+b^4 \left (5 f^2 (7 f g-6 e h) c^3+6 d e f (5 f g-4 e h) c^2+8 d^2 e^2 (3 f g-2 e h) c+16 d^3 e^3 g\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) \sqrt {b e-a f}}}{(b e-a f) (d e-c f)}\right )}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}}{6 (b c-a d) (b e-a f)}-\frac {b g-a h}{3 (b c-a d) (b e-a f) (a+b x)^3 \sqrt {e+f x}}\)

Input:

Int[(g + h*x)/((a + b*x)^4*(c + d*x)*(e + f*x)^(3/2)),x]
 

Output:

-1/3*(b*g - a*h)/((b*c - a*d)*(b*e - a*f)*(a + b*x)^3*Sqrt[e + f*x]) + ((7 
*a^2*d*f*h - a*b*f*(13*d*g + c*h) + b^2*(6*d*e*g + 7*c*f*g - 6*c*e*h))/(2* 
(b*c - a*d)*(b*e - a*f)*(a + b*x)^2*Sqrt[e + f*x]) + ((35*a^3*d^2*f^2*h - 
a^2*b*d*f^2*(89*d*g + 16*c*h) + a*b^2*f*(78*d^2*e*g + 5*c^2*f*h + 2*c*d*(5 
0*f*g - 39*e*h)) - b^3*(24*d^2*e^2*g + 5*c^2*f*(7*f*g - 6*e*h) + 6*c*d*e*( 
5*f*g - 4*e*h)))/((b*c - a*d)*(b*e - a*f)*(a + b*x)*Sqrt[e + f*x]) + (3*(( 
-2*f*(a^3*d^2*f^2*(16*d*f*g - 35*d*e*h + 19*c*f*h) + a^2*b*d*f^2*(41*d^2*e 
*g - 89*c*d*f*g + 64*c*d*e*h - 16*c^2*f*h) - a*b^2*f*(30*d^3*e^2*g - 5*c^3 
*f^2*h - c^2*d*f*(100*f*g - 83*e*h) + 2*c*d^2*e*(11*f*g - 15*e*h)) + b^3*( 
8*d^3*e^3*g + c^2*d*e*f*(5*f*g - 6*e*h) - 5*c^3*f^2*(7*f*g - 6*e*h) + 2*c* 
d^2*e^2*(3*f*g - 4*e*h))))/((b*e - a*f)*(d*e - c*f)*Sqrt[e + f*x]) - ((-2* 
Sqrt[b]*(d*e - c*f)*(35*a^4*d^3*f^3*h - 35*a^3*b*d^2*f^3*(3*d*g + c*h) + 2 
1*a^2*b^2*d*f^2*(6*d^2*e*g + 9*c*d*f*g - 6*c*d*e*h + c^2*f*h) - a*b^3*f*(7 
2*d^3*e^2*g + 5*c^3*f^2*h + 27*c^2*d*f*(5*f*g - 4*e*h) + 36*c*d^2*e*(3*f*g 
 - 2*e*h)) + b^4*(16*d^3*e^3*g + 5*c^3*f^2*(7*f*g - 6*e*h) + 6*c^2*d*e*f*( 
5*f*g - 4*e*h) + 8*c*d^2*e^2*(3*f*g - 2*e*h)))*ArcTanh[(Sqrt[b]*Sqrt[e + f 
*x])/Sqrt[b*e - a*f]])/((b*c - a*d)*Sqrt[b*e - a*f]) + (32*d^(7/2)*(b*e - 
a*f)^4*(d*g - c*h)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/((b*c 
 - a*d)*Sqrt[d*e - c*f]))/((b*e - a*f)*(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))
 

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 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 = 30.70 (sec) , antiderivative size = 1318, normalized size of antiderivative = 1.59

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1318\)
derivativedivides	\(\text {Expression too large to display}\)	\(1698\)
default	\(\text {Expression too large to display}\)	\(1698\)

Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)/(f*x+e)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2*(-35/16*((c*f-d*e)*d)^(1/2)*(b*x+a)^3*(c*f-d*e)*(f*x+e)^(1/2)*b*(((a^4*h 
-3*a^3*b*g)*d^3-a^2*(a*h-27/5*b*g)*c*b*d^2+3/5*a*c^2*(a*h-45/7*b*g)*b^2*d- 
1/7*b^3*c^3*(a*h-7*b*g))*f^3-18/5*(a^2*d^2-6/7*a*b*c*d+5/21*b^2*c^2)*(c*h- 
d*g)*b^2*e*f^2+72/35*d*(c*h-d*g)*(a*d-1/3*b*c)*b^3*e^2*f-16/35*b^4*d^2*e^3 
*(c*h-d*g))*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))+((f*x+e)^(1/2)*d^4 
*(b*x+a)^3*(a*f-b*e)^4*(c*h-d*g)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2 
))+((-a^3*g*(b*x+a)^3*d^3+3*a^2*c*(89/48*b^3*g*x^3+89/18*a*x^2*(-57/712*h* 
x+g)*b^2+199/48*a^2*x*(-136/597*h*x+g)*b+a^3*(-29/48*h*x+g))*b*d^2-3*a*c^2 
*(25/12*b^3*g*x^3+50/9*a*x^2*(-3/50*h*x+g)*b^2+55/12*(-32/165*h*x+g)*a^2*x 
*b+a^3*(-2/3*h*x+g))*b^2*d+c^3*(35/16*b^3*g*x^3+35/6*a*x^2*(-3/56*h*x+g)*b 
^2+77/16*a^2*x*(-40/231*h*x+g)*b+a^3*(g-11/16*h*x))*b^3)*f^4+(a^2*(-41/16* 
b^4*g*x^3-35/6*a*x^2*(-3/8*h*x+g)*b^3-55/16*a^2*x*(-56/33*h*x+g)*b^2+77/16 
*a^3*b*h*x+a^4*h)*d^3-77/16*a*(-2/7*b^4*g*x^3-265/231*a*x^2*(-192/265*h*x+ 
g)*b^3-356/231*a^2*x*(-595/356*h*x+g)*b^2-5/7*a^3*(-626/165*h*x+g)*b+a^4*h 
)*c*b*d^2+5*(-1/16*b^4*g*x^3-7/12*a*x^2*(-249/140*h*x+g)*b^3-313/240*a^2*x 
*(-680/313*h*x+g)*b^2-19/20*a^3*(-631/228*h*x+g)*b+a^4*h)*c^2*b^2*d-27/16* 
c^3*(-35/81*x^2*(-18/7*h*x+g)*b^3-98/81*a*x*(-5/2*h*x+g)*b^2-29/27*(-212/8 
7*h*x+g)*a^2*b+h*a^3)*b^3)*e*f^3+29/16*(a*(30/29*b^4*g*x^3+103/87*a*b^3*g* 
x^2-128/87*(-35/128*h*x+g)*a^2*x*b^2-55/29*a^3*(-98/165*h*x+g)*b+a^4*h)*d^ 
3-2/3*c*(9/29*b^4*g*x^3+13/29*a*x^2*(45/13*h*x+g)*b^3-22/29*a^2*x*(-52/...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x) (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)/(f*x+e)^(3/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x) (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)**4/(d*x+c)/(f*x+e)**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x) (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)/(f*x+e)^(3/2),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. 2492 vs. \(2 (794) = 1588\).

Time = 0.44 (sec) , antiderivative size = 2492, normalized size of antiderivative = 3.00 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x) (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)^4/(d*x+c)/(f*x+e)^(3/2),x, algorithm="giac")
 

Output:

-1/8*(16*b^5*d^3*e^3*g + 24*b^5*c*d^2*e^2*f*g - 72*a*b^4*d^3*e^2*f*g + 30* 
b^5*c^2*d*e*f^2*g - 108*a*b^4*c*d^2*e*f^2*g + 126*a^2*b^3*d^3*e*f^2*g + 35 
*b^5*c^3*f^3*g - 135*a*b^4*c^2*d*f^3*g + 189*a^2*b^3*c*d^2*f^3*g - 105*a^3 
*b^2*d^3*f^3*g - 16*b^5*c*d^2*e^3*h - 24*b^5*c^2*d*e^2*f*h + 72*a*b^4*c*d^ 
2*e^2*f*h - 30*b^5*c^3*e*f^2*h + 108*a*b^4*c^2*d*e*f^2*h - 126*a^2*b^3*c*d 
^2*e*f^2*h - 5*a*b^4*c^3*f^3*h + 21*a^2*b^3*c^2*d*f^3*h - 35*a^3*b^2*c*d^2 
*f^3*h + 35*a^4*b*d^3*f^3*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/ 
((b^8*c^4*e^4 - 4*a*b^7*c^3*d*e^4 + 6*a^2*b^6*c^2*d^2*e^4 - 4*a^3*b^5*c*d^ 
3*e^4 + a^4*b^4*d^4*e^4 - 4*a*b^7*c^4*e^3*f + 16*a^2*b^6*c^3*d*e^3*f - 24* 
a^3*b^5*c^2*d^2*e^3*f + 16*a^4*b^4*c*d^3*e^3*f - 4*a^5*b^3*d^4*e^3*f + 6*a 
^2*b^6*c^4*e^2*f^2 - 24*a^3*b^5*c^3*d*e^2*f^2 + 36*a^4*b^4*c^2*d^2*e^2*f^2 
 - 24*a^5*b^3*c*d^3*e^2*f^2 + 6*a^6*b^2*d^4*e^2*f^2 - 4*a^3*b^5*c^4*e*f^3 
+ 16*a^4*b^4*c^3*d*e*f^3 - 24*a^5*b^3*c^2*d^2*e*f^3 + 16*a^6*b^2*c*d^3*e*f 
^3 - 4*a^7*b*d^4*e*f^3 + a^4*b^4*c^4*f^4 - 4*a^5*b^3*c^3*d*f^4 + 6*a^6*b^2 
*c^2*d^2*f^4 - 4*a^7*b*c*d^3*f^4 + a^8*d^4*f^4)*sqrt(-b^2*e + a*b*f)) + 2* 
(d^5*g - c*d^4*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^4*c^4*d 
*e - 4*a*b^3*c^3*d^2*e + 6*a^2*b^2*c^2*d^3*e - 4*a^3*b*c*d^4*e + a^4*d^5*e 
 - b^4*c^5*f + 4*a*b^3*c^4*d*f - 6*a^2*b^2*c^3*d^2*f + 4*a^3*b*c^2*d^3*f - 
 a^4*c*d^4*f)*sqrt(-d^2*e + c*d*f)) + 2*(f^4*g - e*f^3*h)/((b^4*d*e^5 - b^ 
4*c*e^4*f - 4*a*b^3*d*e^4*f + 4*a*b^3*c*e^3*f^2 + 6*a^2*b^2*d*e^3*f^2 -...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x)^4 (c+d x) (e+f x)^{3/2}} \, dx=\text {Hanged} \] Input:

int((g + h*x)/((e + f*x)^(3/2)*(a + b*x)^4*(c + d*x)),x)
 

Output:

\text{Hanged}
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 20501, normalized size of antiderivative = 24.67 \[ \int \frac {g+h x}{(a+b x)^4 (c+d x) (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

int((h*x+g)/(b*x+a)^4/(d*x+c)/(f*x+e)^(3/2),x)
 

Output:

( - 105*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt 
(b)*sqrt(a*f - b*e)))*a**7*c**2*d**3*f**5*h + 210*sqrt(b)*sqrt(e + f*x)*sq 
rt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*c*d** 
4*e*f**4*h - 105*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x) 
*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**7*d**5*e**2*f**3*h + 105*sqrt(b)*sqrt(e 
+ f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a 
**6*b*c**3*d**2*f**5*h - 210*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((s 
qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**2*d**3*e*f**4*h + 315 
*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqr 
t(a*f - b*e)))*a**6*b*c**2*d**3*f**5*g - 315*sqrt(b)*sqrt(e + f*x)*sqrt(a* 
f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c**2*d** 
3*f**5*h*x + 105*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x) 
*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c*d**4*e**2*f**3*h - 630*sqrt(b)*sqr 
t(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) 
))*a**6*b*c*d**4*e*f**4*g + 630*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan 
((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*c*d**4*e*f**4*h*x + 3 
15*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*s 
qrt(a*f - b*e)))*a**6*b*d**5*e**2*f**3*g - 315*sqrt(b)*sqrt(e + f*x)*sqrt( 
a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**6*b*d**5*e 
**2*f**3*h*x - 63*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + ...