\(\int \frac {g+h x}{(a+b x)^2 (c+d x)^3 (e+f x)^{3/2}} \, dx\) [191]

Optimal result

Integrand size = 29, antiderivative size = 838 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 (e+f x)^{3/2}} \, dx=-\frac {f \left (3 a^3 d^2 f^2 (5 d f g-4 d e h-c f h)+b^3 \left (12 d^3 e^3 g-4 c^3 f^2 (3 f g-2 e h)-c d^2 e^2 (27 f g+4 e h)+c^2 d e f (12 f g+11 e h)\right )+a b^2 \left (4 c^3 f^3 h+2 c^2 d f^2 (12 f g-29 e h)-d^3 e^2 (9 f g+8 e h)+c d^2 e f (30 f g+17 e h)\right )+a^2 b d f \left (11 c^2 f^2 h-13 c d f (3 f g-2 e h)-d^2 \left (6 e f g-8 e^2 h\right )\right )\right )}{4 (b c-a d)^3 (b e-a f)^2 (d e-c f)^3 \sqrt {e+f x}}-\frac {d (3 b d e g-b c (2 f g+e h)-a (d f g+2 d e h-3 c f h))}{2 (b c-a d)^2 (b e-a f) (d e-c f) (c+d x)^2 \sqrt {e+f x}}-\frac {b g-a h}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}+\frac {d \left (a^2 d f (5 d f g-4 d e h-c f h)-b^2 \left (12 d^2 e^2 g-c d e (21 f g+4 e h)+c^2 f (4 f g+9 e h)\right )+a b \left (13 c^2 f^2 h+d^2 e (3 f g+8 e h)-c d f (13 f g+11 e h)\right )\right )}{4 (b c-a d)^3 (b e-a f) (d e-c f)^2 (c+d x) \sqrt {e+f x}}+\frac {b^{5/2} \left (7 a^2 d f h+b^2 (6 d e g+3 c f g-2 c e h)-a b (9 d f g+4 d e h+c f h)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d)^4 (b e-a f)^{5/2}}-\frac {d^{3/2} \left (3 a^2 d^2 f (5 d f g-4 d e h-c f h)+2 a b d \left (7 c^2 f^2 h-c d f (27 f g-16 e h)+4 d^2 e (3 f g-2 e h)\right )+b^2 \left (24 d^3 e^2 g-35 c^3 f^2 h-8 c d^2 e (9 f g+e h)+7 c^2 d f (9 f g+4 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 (b c-a d)^4 (d e-c f)^{7/2}} \] Output:

-1/4*f*(3*a^3*d^2*f^2*(-c*f*h-4*d*e*h+5*d*f*g)+b^3*(12*d^3*e^3*g-4*c^3*f^2 
*(-2*e*h+3*f*g)-c*d^2*e^2*(4*e*h+27*f*g)+c^2*d*e*f*(11*e*h+12*f*g))+a*b^2* 
(4*c^3*f^3*h+2*c^2*d*f^2*(-29*e*h+12*f*g)-d^3*e^2*(8*e*h+9*f*g)+c*d^2*e*f* 
(17*e*h+30*f*g))+a^2*b*d*f*(11*c^2*f^2*h-13*c*d*f*(-2*e*h+3*f*g)-d^2*(-8*e 
^2*h+6*e*f*g)))/(-a*d+b*c)^3/(-a*f+b*e)^2/(-c*f+d*e)^3/(f*x+e)^(1/2)-1/2*d 
*(3*b*d*e*g-b*c*(e*h+2*f*g)-a*(-3*c*f*h+2*d*e*h+d*f*g))/(-a*d+b*c)^2/(-a*f 
+b*e)/(-c*f+d*e)/(d*x+c)^2/(f*x+e)^(1/2)-(-a*h+b*g)/(-a*d+b*c)/(-a*f+b*e)/ 
(b*x+a)/(d*x+c)^2/(f*x+e)^(1/2)+1/4*d*(a^2*d*f*(-c*f*h-4*d*e*h+5*d*f*g)-b^ 
2*(12*d^2*e^2*g-c*d*e*(4*e*h+21*f*g)+c^2*f*(9*e*h+4*f*g))+a*b*(13*c^2*f^2* 
h+d^2*e*(8*e*h+3*f*g)-c*d*f*(11*e*h+13*f*g)))/(-a*d+b*c)^3/(-a*f+b*e)/(-c* 
f+d*e)^2/(d*x+c)/(f*x+e)^(1/2)+b^(5/2)*(7*a^2*d*f*h+b^2*(-2*c*e*h+3*c*f*g+ 
6*d*e*g)-a*b*(c*f*h+4*d*e*h+9*d*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)^(5/2)-1/4*d^(3/2)*(3*a^2*d^2*f*(-c*f*h 
-4*d*e*h+5*d*f*g)+2*a*b*d*(7*c^2*f^2*h-c*d*f*(-16*e*h+27*f*g)+4*d^2*e*(-2* 
e*h+3*f*g))+b^2*(24*d^3*e^2*g-35*c^3*f^2*h-8*c*d^2*e*(e*h+9*f*g)+7*c^2*d*f 
*(4*e*h+9*f*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)^(7/2)
 

Mathematica [C] (verified)

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

Time = 11.12 (sec) , antiderivative size = 544, normalized size of antiderivative = 0.65 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 (e+f x)^{3/2}} \, dx=\frac {\frac {d (3 b d e g-b c (2 f g+e h)-a (d f g+2 d e h-3 c f h))}{(b c-a d) (-d e+c f)}+\frac {2 (-b g+a h)}{a+b x}+\frac {(c+d x) \left (d (b c-a d) (b e-a f) (-d e+c f) \left (a^2 d f (-5 d f g+4 d e h+c f h)+b^2 \left (12 d^2 e^2 g-c d e (21 f g+4 e h)+c^2 f (4 f g+9 e h)\right )+a b \left (-13 c^2 f^2 h-d^2 e (3 f g+8 e h)+c d f (13 f g+11 e h)\right )\right )-(c+d x) \left (4 b^2 (d e-c f)^3 \left (7 a^2 d f h+b^2 (6 d e g+3 c f g-2 c e h)-a b (9 d f g+4 d e h+c f h)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {b (e+f x)}{b e-a f}\right )-d (b e-a f)^2 \left (-3 a^2 d^2 f (-5 d f g+4 d e h+c f h)+b^2 \left (24 d^3 e^2 g-35 c^3 f^2 h-8 c d^2 e (9 f g+e h)+7 c^2 d f (9 f g+4 e h)\right )+2 a b d \left (7 c^2 f^2 h+4 d^2 e (3 f g-2 e h)+c d f (-27 f g+16 e h)\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {d (e+f x)}{d e-c f}\right )\right )\right )}{2 (b c-a d)^3 (b e-a f) (d e-c f)^3}}{2 (b c-a d) (b e-a f) (c+d x)^2 \sqrt {e+f x}} \] Input:

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

Output:

((d*(3*b*d*e*g - b*c*(2*f*g + e*h) - a*(d*f*g + 2*d*e*h - 3*c*f*h)))/((b*c 
 - a*d)*(-(d*e) + c*f)) + (2*(-(b*g) + a*h))/(a + b*x) + ((c + d*x)*(d*(b* 
c - a*d)*(b*e - a*f)*(-(d*e) + c*f)*(a^2*d*f*(-5*d*f*g + 4*d*e*h + c*f*h) 
+ b^2*(12*d^2*e^2*g - c*d*e*(21*f*g + 4*e*h) + c^2*f*(4*f*g + 9*e*h)) + a* 
b*(-13*c^2*f^2*h - d^2*e*(3*f*g + 8*e*h) + c*d*f*(13*f*g + 11*e*h))) - (c 
+ d*x)*(4*b^2*(d*e - c*f)^3*(7*a^2*d*f*h + b^2*(6*d*e*g + 3*c*f*g - 2*c*e* 
h) - a*b*(9*d*f*g + 4*d*e*h + c*f*h))*Hypergeometric2F1[-1/2, 1, 1/2, (b*( 
e + f*x))/(b*e - a*f)] - d*(b*e - a*f)^2*(-3*a^2*d^2*f*(-5*d*f*g + 4*d*e*h 
 + c*f*h) + b^2*(24*d^3*e^2*g - 35*c^3*f^2*h - 8*c*d^2*e*(9*f*g + e*h) + 7 
*c^2*d*f*(9*f*g + 4*e*h)) + 2*a*b*d*(7*c^2*f^2*h + 4*d^2*e*(3*f*g - 2*e*h) 
 + c*d*f*(-27*f*g + 16*e*h)))*Hypergeometric2F1[-1/2, 1, 1/2, (d*(e + f*x) 
)/(d*e - c*f)])))/(2*(b*c - a*d)^3*(b*e - a*f)*(d*e - c*f)^3))/(2*(b*c - a 
*d)*(b*e - a*f)*(c + d*x)^2*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 2.20 (sec) , antiderivative size = 915, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {168, 27, 168, 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.

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\int \frac {b (6 d e g+3 c f g-2 c e h)-a (2 d f g+4 d e h+c f h)+7 d f (b g-a h) x}{2 (a+b x) (c+d x)^3 (e+f x)^{3/2}}dx}{(b c-a d) (b e-a f)}-\frac {b g-a h}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {b (6 d e g+3 c f g-2 c e h)-a (2 d f g+4 d e h+c f h)+7 d f (b g-a h) x}{(a+b x) (c+d x)^3 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}-\frac {b g-a h}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {\frac {\frac {\int \frac {-3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3-b d f \left (e (9 f g-4 e h) d^2-c f (39 f g-23 e h) d+11 c^2 f^2 h\right ) a^2-b^2 \left (16 d^3 h e^3-c d^2 f (9 f g+16 e h) e+4 c^3 f^3 h+c^2 d f^2 (24 f g-19 e h)\right ) a+4 b^3 (d e-c f)^2 (6 d e g+3 c f g-2 c e h)-3 b d f \left (d f (5 d f g-4 d e h-c f h) a^2+b \left (e (3 f g+8 e h) d^2-c f (13 f g+11 e h) d+13 c^2 f^2 h\right ) a-b^2 \left (f (4 f g+9 e h) c^2-d e (21 f g+4 e h) c+12 d^2 e^2 g\right )\right ) x}{2 (a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}-\frac {d \left (a^2 d f (-c f h-4 d e h+5 d f g)+a b \left (13 c^2 f^2 h-c d f (11 e h+13 f g)+d^2 e (8 e h+3 f g)\right )-b^2 \left (c^2 f (9 e h+4 f g)-c d e (4 e h+21 f g)+12 d^2 e^2 g\right )\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {d (b (-c e h-2 c f g+3 d e g)-a (-3 c f h+2 d e h+d f g))}{(c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {b g-a h}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {\int \frac {-3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3-b d f \left (e (9 f g-4 e h) d^2-c f (39 f g-23 e h) d+11 c^2 f^2 h\right ) a^2-b^2 \left (16 d^3 h e^3-c d^2 f (9 f g+16 e h) e+4 c^3 f^3 h+c^2 d f^2 (24 f g-19 e h)\right ) a+4 b^3 (d e-c f)^2 (6 d e g+3 c f g-2 c e h)-3 b d f \left (d f (5 d f g-4 d e h-c f h) a^2+b \left (e (3 f g+8 e h) d^2-c f (13 f g+11 e h) d+13 c^2 f^2 h\right ) a-b^2 \left (f (4 f g+9 e h) c^2-d e (21 f g+4 e h) c+12 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{2 (b c-a d) (d e-c f)}-\frac {d \left (a^2 d f (-c f h-4 d e h+5 d f g)+a b \left (13 c^2 f^2 h-c d f (11 e h+13 f g)+d^2 e (8 e h+3 f g)\right )-b^2 \left (c^2 f (9 e h+4 f g)-c d e (4 e h+21 f g)+12 d^2 e^2 g\right )\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{2 (b c-a d) (d e-c f)}+\frac {d (b (-c e h-2 c f g+3 d e g)-a (-3 c f h+2 d e h+d f g))}{(c+d x)^2 \sqrt {e+f x} (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {b g-a h}{(a+b x) (c+d x)^2 \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {b g-a h}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {d (b (3 d e g-2 c f g-c e h)-a (d f g+2 d e h-3 c f h))}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {\frac {2 f \left (3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (-2 e (3 f g-4 e h) d^2-13 c f (3 f g-2 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (-e^2 (9 f g+8 e h) d^3+c e f (30 f g+17 e h) d^2+2 c^2 f^2 (12 f g-29 e h) d+4 c^3 f^3 h\right ) a+b^3 \left (-4 f^2 (3 f g-2 e h) c^3+d e f (12 f g+11 e h) c^2-d^2 e^2 (27 f g+4 e h) c+12 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 \int -\frac {3 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (-\left (\left (6 e f g-8 e^2 h\right ) d^2\right )-13 c f (3 f g-2 e h) d+11 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (9 f g-20 e h) d^3-c e f (30 f g-67 e h) d^2-2 c^2 f^2 (12 f g+13 e h) d+24 c^3 f^3 h\right ) a^2+b^3 \left (-8 e^3 (3 f g+2 e h) d^4+c e^2 f (81 f g+40 e h) d^3-c^2 e f^2 (96 f g+25 e h) d^2+12 c^3 f^3 (2 f g+e h) d+4 c^4 f^4 h\right ) a+4 b^4 (d e-c f)^3 (6 d e g+3 c f g-2 c e h)+b d f \left (3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (-2 e (3 f g-4 e h) d^2-13 c f (3 f g-2 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (-e^2 (9 f g+8 e h) d^3+c e f (30 f g+17 e h) d^2+2 c^2 f^2 (12 f g-29 e h) d+4 c^3 f^3 h\right ) a+b^3 \left (-4 f^2 (3 f g-2 e h) c^3+d e f (12 f g+11 e h) c^2-d^2 e^2 (27 f g+4 e h) c+12 d^3 e^3 g\right )\right ) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}-\frac {d \left (d f (5 d f g-4 d e h-c f h) a^2+b \left (e (3 f g+8 e h) d^2-c f (13 f g+11 e h) d+13 c^2 f^2 h\right ) a-b^2 \left (f (4 f g+9 e h) c^2-d e (21 f g+4 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}}{2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {d (b (3 d e g-2 c f g-c e h)-a (d f g+2 d e h-3 c f h))}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {\frac {2 f \left (3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (-2 e (3 f g-4 e h) d^2-13 c f (3 f g-2 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (-e^2 (9 f g+8 e h) d^3+c e f (30 f g+17 e h) d^2+2 c^2 f^2 (12 f g-29 e h) d+4 c^3 f^3 h\right ) a+b^3 \left (-4 f^2 (3 f g-2 e h) c^3+d e f (12 f g+11 e h) c^2-d^2 e^2 (27 f g+4 e h) c+12 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}+\frac {\int \frac {3 d^3 f^3 (5 d f g-4 d e h-c f h) a^4+b d^2 f^2 \left (-2 e (3 f g-4 e h) d^2-13 c f (3 f g-2 e h) d+11 c^2 f^2 h\right ) a^3-b^2 d f \left (e^2 (9 f g-20 e h) d^3-c e f (30 f g-67 e h) d^2-2 c^2 f^2 (12 f g+13 e h) d+24 c^3 f^3 h\right ) a^2+b^3 \left (-8 e^3 (3 f g+2 e h) d^4+c e^2 f (81 f g+40 e h) d^3-c^2 e f^2 (96 f g+25 e h) d^2+12 c^3 f^3 (2 f g+e h) d+4 c^4 f^4 h\right ) a+4 b^4 (d e-c f)^3 (6 d e g+3 c f g-2 c e h)+b d f \left (3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (-2 e (3 f g-4 e h) d^2-13 c f (3 f g-2 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (-e^2 (9 f g+8 e h) d^3+c e f (30 f g+17 e h) d^2+2 c^2 f^2 (12 f g-29 e h) d+4 c^3 f^3 h\right ) a+b^3 \left (-4 f^2 (3 f g-2 e h) c^3+d e f (12 f g+11 e h) c^2-d^2 e^2 (27 f g+4 e h) c+12 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}-\frac {d \left (d f (5 d f g-4 d e h-c f h) a^2+b \left (e (3 f g+8 e h) d^2-c f (13 f g+11 e h) d+13 c^2 f^2 h\right ) a-b^2 \left (f (4 f g+9 e h) c^2-d e (21 f g+4 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}}{2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {b g-a h}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {d (b (3 d e g-2 c f g-c e h)-a (d f g+2 d e h-3 c f h))}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {\frac {2 f \left (3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (-2 e (3 f g-4 e h) d^2-13 c f (3 f g-2 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (-e^2 (9 f g+8 e h) d^3+c e f (30 f g+17 e h) d^2+2 c^2 f^2 (12 f g-29 e h) d+4 c^3 f^3 h\right ) a+b^3 \left (-4 f^2 (3 f g-2 e h) c^3+d e f (12 f g+11 e h) c^2-d^2 e^2 (27 f g+4 e h) c+12 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}+\frac {\frac {4 b^3 (d e-c f)^3 \left (7 d f h a^2-b (9 d f g+4 d e h+c f h) a+b^2 (6 d e g+3 c f g-2 c e h)\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {d^2 (b e-a f)^2 \left (\left (-35 f^2 h c^3+7 d f (9 f g+4 e h) c^2-8 d^2 e (9 f g+e h) c+24 d^3 e^2 g\right ) b^2+2 a d \left (4 e (3 f g-2 e h) d^2-c f (27 f g-16 e h) d+7 c^2 f^2 h\right ) b+3 a^2 d^2 f (5 d f g-4 d e h-c f h)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}}{(b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}-\frac {d \left (d f (5 d f g-4 d e h-c f h) a^2+b \left (e (3 f g+8 e h) d^2-c f (13 f g+11 e h) d+13 c^2 f^2 h\right ) a-b^2 \left (f (4 f g+9 e h) c^2-d e (21 f g+4 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}}{2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b g-a h}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {d (b (3 d e g-2 c f g-c e h)-a (d f g+2 d e h-3 c f h))}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {\frac {2 f \left (3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (-2 e (3 f g-4 e h) d^2-13 c f (3 f g-2 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (-e^2 (9 f g+8 e h) d^3+c e f (30 f g+17 e h) d^2+2 c^2 f^2 (12 f g-29 e h) d+4 c^3 f^3 h\right ) a+b^3 \left (-4 f^2 (3 f g-2 e h) c^3+d e f (12 f g+11 e h) c^2-d^2 e^2 (27 f g+4 e h) c+12 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}+\frac {\frac {8 b^3 (d e-c f)^3 \left (7 d f h a^2-b (9 d f g+4 d e h+c f h) a+b^2 (6 d e g+3 c f g-2 c e h)\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{(b c-a d) f}-\frac {2 d^2 (b e-a f)^2 \left (\left (-35 f^2 h c^3+7 d f (9 f g+4 e h) c^2-8 d^2 e (9 f g+e h) c+24 d^3 e^2 g\right ) b^2+2 a d \left (4 e (3 f g-2 e h) d^2-c f (27 f g-16 e h) d+7 c^2 f^2 h\right ) b+3 a^2 d^2 f (5 d f g-4 d e h-c f h)\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{(b c-a d) f}}{(b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}-\frac {d \left (d f (5 d f g-4 d e h-c f h) a^2+b \left (e (3 f g+8 e h) d^2-c f (13 f g+11 e h) d+13 c^2 f^2 h\right ) a-b^2 \left (f (4 f g+9 e h) c^2-d e (21 f g+4 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}}{2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b g-a h}{(b c-a d) (b e-a f) (a+b x) (c+d x)^2 \sqrt {e+f x}}-\frac {\frac {d (b (3 d e g-2 c f g-c e h)-a (d f g+2 d e h-3 c f h))}{(b c-a d) (d e-c f) (c+d x)^2 \sqrt {e+f x}}+\frac {\frac {\frac {2 f \left (3 d^2 f^2 (5 d f g-4 d e h-c f h) a^3+b d f \left (-2 e (3 f g-4 e h) d^2-13 c f (3 f g-2 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (-e^2 (9 f g+8 e h) d^3+c e f (30 f g+17 e h) d^2+2 c^2 f^2 (12 f g-29 e h) d+4 c^3 f^3 h\right ) a+b^3 \left (-4 f^2 (3 f g-2 e h) c^3+d e f (12 f g+11 e h) c^2-d^2 e^2 (27 f g+4 e h) c+12 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}+\frac {\frac {2 d^{3/2} (b e-a f)^2 \left (\left (-35 f^2 h c^3+7 d f (9 f g+4 e h) c^2-8 d^2 e (9 f g+e h) c+24 d^3 e^2 g\right ) b^2+2 a d \left (4 e (3 f g-2 e h) d^2-c f (27 f g-16 e h) d+7 c^2 f^2 h\right ) b+3 a^2 d^2 f (5 d f g-4 d e h-c f h)\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d) \sqrt {d e-c f}}-\frac {8 b^{5/2} (d e-c f)^3 \left (7 d f h a^2-b (9 d f g+4 d e h+c f h) a+b^2 (6 d e g+3 c f g-2 c e h)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) \sqrt {b e-a f}}}{(b e-a f) (d e-c f)}}{2 (b c-a d) (d e-c f)}-\frac {d \left (d f (5 d f g-4 d e h-c f h) a^2+b \left (e (3 f g+8 e h) d^2-c f (13 f g+11 e h) d+13 c^2 f^2 h\right ) a-b^2 \left (f (4 f g+9 e h) c^2-d e (21 f g+4 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}}{2 (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}\)

Input:

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

Output:

-((b*g - a*h)/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)^2*Sqrt[e + f*x] 
)) - ((d*(b*(3*d*e*g - 2*c*f*g - c*e*h) - a*(d*f*g + 2*d*e*h - 3*c*f*h)))/ 
((b*c - a*d)*(d*e - c*f)*(c + d*x)^2*Sqrt[e + f*x]) + (-((d*(a^2*d*f*(5*d* 
f*g - 4*d*e*h - c*f*h) - b^2*(12*d^2*e^2*g - c*d*e*(21*f*g + 4*e*h) + c^2* 
f*(4*f*g + 9*e*h)) + a*b*(13*c^2*f^2*h + d^2*e*(3*f*g + 8*e*h) - c*d*f*(13 
*f*g + 11*e*h))))/((b*c - a*d)*(d*e - c*f)*(c + d*x)*Sqrt[e + f*x])) + ((2 
*f*(3*a^3*d^2*f^2*(5*d*f*g - 4*d*e*h - c*f*h) + a^2*b*d*f*(11*c^2*f^2*h - 
2*d^2*e*(3*f*g - 4*e*h) - 13*c*d*f*(3*f*g - 2*e*h)) + b^3*(12*d^3*e^3*g - 
4*c^3*f^2*(3*f*g - 2*e*h) - c*d^2*e^2*(27*f*g + 4*e*h) + c^2*d*e*f*(12*f*g 
 + 11*e*h)) + a*b^2*(4*c^3*f^3*h + 2*c^2*d*f^2*(12*f*g - 29*e*h) - d^3*e^2 
*(9*f*g + 8*e*h) + c*d^2*e*f*(30*f*g + 17*e*h))))/((b*e - a*f)*(d*e - c*f) 
*Sqrt[e + f*x]) + ((-8*b^(5/2)*(d*e - c*f)^3*(7*a^2*d*f*h + b^2*(6*d*e*g + 
 3*c*f*g - 2*c*e*h) - a*b*(9*d*f*g + 4*d*e*h + c*f*h))*ArcTanh[(Sqrt[b]*Sq 
rt[e + f*x])/Sqrt[b*e - a*f]])/((b*c - a*d)*Sqrt[b*e - a*f]) + (2*d^(3/2)* 
(b*e - a*f)^2*(3*a^2*d^2*f*(5*d*f*g - 4*d*e*h - c*f*h) + 2*a*b*d*(7*c^2*f^ 
2*h - c*d*f*(27*f*g - 16*e*h) + 4*d^2*e*(3*f*g - 2*e*h)) + b^2*(24*d^3*e^2 
*g - 35*c^3*f^2*h - 8*c*d^2*e*(9*f*g + e*h) + 7*c^2*d*f*(9*f*g + 4*e*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)*(d*e - c*f)))/(2*(b*c - a 
*d)*(d*e - c*f)))/(2*(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 = 126.09 (sec) , antiderivative size = 817, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(2 f^{3} \left (-\frac {-e h +f g}{\left (c f -d e \right )^{3} \left (a f -b e \right )^{2} \sqrt {f x +e}}+\frac {d^{2} \left (\frac {\left (\frac {3}{8} a^{2} c \,d^{3} f^{2} h +\frac {1}{2} a^{2} d^{4} e f h -\frac {7}{8} a^{2} d^{4} f^{2} g -\frac {7}{4} a b \,c^{2} d^{2} f^{2} h +\frac {11}{4} a b c \,d^{3} f^{2} g -a b \,d^{4} e f g +\frac {11}{8} b^{2} c^{3} d \,f^{2} h -\frac {1}{2} b^{2} c^{2} d^{2} e f h -\frac {15}{8} b^{2} c^{2} d^{2} f^{2} g +b^{2} c \,d^{3} e f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\frac {f \left (5 a^{2} c^{2} d^{2} f^{2} h -a^{2} c \,d^{3} e f h -9 a^{2} c \,d^{3} f^{2} g -4 a^{2} d^{4} e^{2} h +9 a^{2} d^{4} e f g -18 a b \,c^{3} d \,f^{2} h +18 a b \,c^{2} d^{2} e f h +26 a b \,c^{2} d^{2} f^{2} g -34 a b c \,d^{3} e f g +8 a b \,d^{4} e^{2} g +13 b^{2} c^{4} f^{2} h -17 b^{2} c^{3} d e f h -17 b^{2} c^{3} d \,f^{2} g +4 b^{2} c^{2} d^{2} e^{2} h +25 b^{2} c^{2} d^{2} e f g -8 b^{2} c \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (3 a^{2} c \,d^{2} f^{2} h +12 a^{2} d^{3} e f h -15 a^{2} d^{3} f^{2} g -14 a b \,c^{2} d \,f^{2} h -32 a b c \,d^{2} e f h +54 a b c \,d^{2} f^{2} g +16 e^{2} h b a \,d^{3}-24 a b \,d^{3} e f g +35 b^{2} c^{3} f^{2} h -28 b^{2} c^{2} d e f h -63 b^{2} c^{2} d \,f^{2} g +8 b^{2} c \,d^{2} e^{2} h +72 b^{2} c \,d^{2} e f g -24 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{4} f^{3} \left (c f -d e \right )^{3}}-\frac {b^{3} \left (\frac {\left (\frac {1}{2} a^{2} d f h -\frac {1}{2} a b c f h -\frac {1}{2} a b d f g +\frac {1}{2} b^{2} c f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (7 a^{2} d f h -a b c f h -4 a b d e h -9 a b d f g -2 b^{2} c e h +3 b^{2} c f g +6 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{4} f^{3} \left (a f -b e \right )^{2}}\right )\)	\(817\)
default	\(2 f^{3} \left (-\frac {-e h +f g}{\left (c f -d e \right )^{3} \left (a f -b e \right )^{2} \sqrt {f x +e}}+\frac {d^{2} \left (\frac {\left (\frac {3}{8} a^{2} c \,d^{3} f^{2} h +\frac {1}{2} a^{2} d^{4} e f h -\frac {7}{8} a^{2} d^{4} f^{2} g -\frac {7}{4} a b \,c^{2} d^{2} f^{2} h +\frac {11}{4} a b c \,d^{3} f^{2} g -a b \,d^{4} e f g +\frac {11}{8} b^{2} c^{3} d \,f^{2} h -\frac {1}{2} b^{2} c^{2} d^{2} e f h -\frac {15}{8} b^{2} c^{2} d^{2} f^{2} g +b^{2} c \,d^{3} e f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\frac {f \left (5 a^{2} c^{2} d^{2} f^{2} h -a^{2} c \,d^{3} e f h -9 a^{2} c \,d^{3} f^{2} g -4 a^{2} d^{4} e^{2} h +9 a^{2} d^{4} e f g -18 a b \,c^{3} d \,f^{2} h +18 a b \,c^{2} d^{2} e f h +26 a b \,c^{2} d^{2} f^{2} g -34 a b c \,d^{3} e f g +8 a b \,d^{4} e^{2} g +13 b^{2} c^{4} f^{2} h -17 b^{2} c^{3} d e f h -17 b^{2} c^{3} d \,f^{2} g +4 b^{2} c^{2} d^{2} e^{2} h +25 b^{2} c^{2} d^{2} e f g -8 b^{2} c \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (3 a^{2} c \,d^{2} f^{2} h +12 a^{2} d^{3} e f h -15 a^{2} d^{3} f^{2} g -14 a b \,c^{2} d \,f^{2} h -32 a b c \,d^{2} e f h +54 a b c \,d^{2} f^{2} g +16 e^{2} h b a \,d^{3}-24 a b \,d^{3} e f g +35 b^{2} c^{3} f^{2} h -28 b^{2} c^{2} d e f h -63 b^{2} c^{2} d \,f^{2} g +8 b^{2} c \,d^{2} e^{2} h +72 b^{2} c \,d^{2} e f g -24 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{4} f^{3} \left (c f -d e \right )^{3}}-\frac {b^{3} \left (\frac {\left (\frac {1}{2} a^{2} d f h -\frac {1}{2} a b c f h -\frac {1}{2} a b d f g +\frac {1}{2} b^{2} c f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (7 a^{2} d f h -a b c f h -4 a b d e h -9 a b d f g -2 b^{2} c e h +3 b^{2} c f g +6 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{4} f^{3} \left (a f -b e \right )^{2}}\right )\)	\(817\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(1368\)

Input:

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

Output:

2*f^3*(-(-e*h+f*g)/(c*f-d*e)^3/(a*f-b*e)^2/(f*x+e)^(1/2)+d^2/(a*d-b*c)^4/f 
^3/(c*f-d*e)^3*(((3/8*a^2*c*d^3*f^2*h+1/2*a^2*d^4*e*f*h-7/8*a^2*d^4*f^2*g- 
7/4*a*b*c^2*d^2*f^2*h+11/4*a*b*c*d^3*f^2*g-a*b*d^4*e*f*g+11/8*b^2*c^3*d*f^ 
2*h-1/2*b^2*c^2*d^2*e*f*h-15/8*b^2*c^2*d^2*f^2*g+b^2*c*d^3*e*f*g)*(f*x+e)^ 
(3/2)+1/8*f*(5*a^2*c^2*d^2*f^2*h-a^2*c*d^3*e*f*h-9*a^2*c*d^3*f^2*g-4*a^2*d 
^4*e^2*h+9*a^2*d^4*e*f*g-18*a*b*c^3*d*f^2*h+18*a*b*c^2*d^2*e*f*h+26*a*b*c^ 
2*d^2*f^2*g-34*a*b*c*d^3*e*f*g+8*a*b*d^4*e^2*g+13*b^2*c^4*f^2*h-17*b^2*c^3 
*d*e*f*h-17*b^2*c^3*d*f^2*g+4*b^2*c^2*d^2*e^2*h+25*b^2*c^2*d^2*e*f*g-8*b^2 
*c*d^3*e^2*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+ 
12*a^2*d^3*e*f*h-15*a^2*d^3*f^2*g-14*a*b*c^2*d*f^2*h-32*a*b*c*d^2*e*f*h+54 
*a*b*c*d^2*f^2*g+16*a*b*d^3*e^2*h-24*a*b*d^3*e*f*g+35*b^2*c^3*f^2*h-28*b^2 
*c^2*d*e*f*h-63*b^2*c^2*d*f^2*g+8*b^2*c*d^2*e^2*h+72*b^2*c*d^2*e*f*g-24*b^ 
2*d^3*e^2*g)/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2 
)))-b^3/(a*d-b*c)^4/f^3/(a*f-b*e)^2*((1/2*a^2*d*f*h-1/2*a*b*c*f*h-1/2*a*b* 
d*f*g+1/2*b^2*c*f*g)*(f*x+e)^(1/2)/((f*x+e)*b+a*f-b*e)+1/2*(7*a^2*d*f*h-a* 
b*c*f*h-4*a*b*d*e*h-9*a*b*d*f*g-2*b^2*c*e*h+3*b^2*c*f*g+6*b^2*d*e*g)/((a*f 
-b*e)*b)^(1/2)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((h*x+g)/(b*x+a)^2/(d*x+c)^3/(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(a*f-b*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. 2420 vs. \(2 (802) = 1604\).

Time = 0.37 (sec) , antiderivative size = 2420, normalized size of antiderivative = 2.89 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-(6*b^5*d*e*g + 3*b^5*c*f*g - 9*a*b^4*d*f*g - 2*b^5*c*e*h - 4*a*b^4*d*e*h 
- a*b^4*c*f*h + 7*a^2*b^3*d*f*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b* 
f))/((b^6*c^4*e^2 - 4*a*b^5*c^3*d*e^2 + 6*a^2*b^4*c^2*d^2*e^2 - 4*a^3*b^3* 
c*d^3*e^2 + a^4*b^2*d^4*e^2 - 2*a*b^5*c^4*e*f + 8*a^2*b^4*c^3*d*e*f - 12*a 
^3*b^3*c^2*d^2*e*f + 8*a^4*b^2*c*d^3*e*f - 2*a^5*b*d^4*e*f + a^2*b^4*c^4*f 
^2 - 4*a^3*b^3*c^3*d*f^2 + 6*a^4*b^2*c^2*d^2*f^2 - 4*a^5*b*c*d^3*f^2 + a^6 
*d^4*f^2)*sqrt(-b^2*e + a*b*f)) + 1/4*(24*b^2*d^5*e^2*g - 72*b^2*c*d^4*e*f 
*g + 24*a*b*d^5*e*f*g + 63*b^2*c^2*d^3*f^2*g - 54*a*b*c*d^4*f^2*g + 15*a^2 
*d^5*f^2*g - 8*b^2*c*d^4*e^2*h - 16*a*b*d^5*e^2*h + 28*b^2*c^2*d^3*e*f*h + 
 32*a*b*c*d^4*e*f*h - 12*a^2*d^5*e*f*h - 35*b^2*c^3*d^2*f^2*h + 14*a*b*c^2 
*d^3*f^2*h - 3*a^2*c*d^4*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f 
))/((b^4*c^4*d^3*e^3 - 4*a*b^3*c^3*d^4*e^3 + 6*a^2*b^2*c^2*d^5*e^3 - 4*a^3 
*b*c*d^6*e^3 + a^4*d^7*e^3 - 3*b^4*c^5*d^2*e^2*f + 12*a*b^3*c^4*d^3*e^2*f 
- 18*a^2*b^2*c^3*d^4*e^2*f + 12*a^3*b*c^2*d^5*e^2*f - 3*a^4*c*d^6*e^2*f + 
3*b^4*c^6*d*e*f^2 - 12*a*b^3*c^5*d^2*e*f^2 + 18*a^2*b^2*c^4*d^3*e*f^2 - 12 
*a^3*b*c^3*d^4*e*f^2 + 3*a^4*c^2*d^5*e*f^2 - b^4*c^7*f^3 + 4*a*b^3*c^6*d*f 
^3 - 6*a^2*b^2*c^5*d^2*f^3 + 4*a^3*b*c^4*d^3*f^3 - a^4*c^3*d^4*f^3)*sqrt(- 
d^2*e + c*d*f)) - ((f*x + e)*b^4*d^3*e^3*f*g - 3*(f*x + e)*b^4*c*d^2*e^2*f 
^2*g + 3*(f*x + e)*b^4*c^2*d*e*f^3*g - 3*(f*x + e)*b^4*c^3*f^4*g + 6*(f*x 
+ e)*a*b^3*c^2*d*f^4*g - 6*(f*x + e)*a^2*b^2*c*d^2*f^4*g + 2*(f*x + e)*...
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 26977, normalized size of antiderivative = 32.19 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x)^3 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 28*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**3*b**2*c**6*d*f**5*h + 112*sqrt(b)*sqrt(e + f*x)*s 
qrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2 
*c**5*d**2*e*f**4*h - 56*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**3*b**2*c**5*d**2*f**5*h*x - 168* 
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**3*b**2*c**4*d**3*e**2*f**3*h + 224*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**3*b** 
2*c**4*d**3*e*f**4*h*x - 28*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sq 
rt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c**4*d**3*f**5*h*x**2 
+ 112*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**3*b**2*c**3*d**4*e**3*f**2*h - 336*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* 
*3*b**2*c**3*d**4*e**2*f**3*h*x + 112*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**3*b**2*c**3*d**4*e* 
f**4*h*x**2 - 28*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**3*b**2*c**2*d**5*e**4*f*h + 224*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**3*b**2*c**2*d**5*e**3*f**2*h*x - 168*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**3*b**2*c...