\(\int \frac {g+h x}{(a+b x)^3 (c+d x)^2 (e+f x)^{3/2}} \, dx\) [183]

Optimal result

Integrand size = 29, antiderivative size = 848 \[ \int \frac {g+h x}{(a+b x)^3 (c+d x)^2 (e+f x)^{3/2}} \, dx=-\frac {f \left (4 a^3 d^2 f^2 (3 d f g-2 d e h-c f h)-b^3 \left (12 d^3 e^3 g-2 c^2 d e f (3 f g-4 e h)+3 c^3 f^2 (5 f g-4 e h)-c d^2 e^2 (9 f g+8 e h)\right )-a^2 b d f \left (11 c^2 f^2 h+2 c d f (12 f g-29 e h)+d^2 e (12 f g+11 e h)\right )+a b^2 \left (3 c^3 f^3 h+13 c^2 d f^2 (3 f g-2 e h)+d^3 e^2 (27 f g+4 e h)-c d^2 e f (30 f g+17 e h)\right )\right )}{4 (b c-a d)^3 (b e-a f)^3 (d e-c f)^2 \sqrt {e+f x}}+\frac {d \left (a^2 d f (4 d f g+9 d e h-13 c f h)+b^2 \left (12 d^2 e^2 g-c^2 f (5 f g-4 e h)-c d e (3 f g+8 e h)\right )+a b \left (c^2 f^2 h-d^2 e (21 f g+4 e h)+c d f (13 f g+11 e h)\right )\right )}{4 (b c-a d)^3 (b e-a f)^2 (d e-c f) (c+d x) \sqrt {e+f x}}-\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}+\frac {7 a^2 d f h+b^2 (6 d e g+5 c f g-4 c e h)-a b (11 d f g+2 d e h+c f h)}{4 (b c-a d)^2 (b e-a f)^2 (a+b x) (c+d x) \sqrt {e+f x}}+\frac {b^{3/2} \left (35 a^3 d^2 f^2 h-7 a^2 b d f (9 d f g+4 d e h+2 c f h)-b^3 \left (24 d^2 e^2 g+3 c^2 f (5 f g-4 e h)+8 c d e (3 f g-2 e h)\right )+a b^2 \left (3 c^2 f^2 h+2 c d f (27 f g-16 e h)+8 d^2 e (9 f g+e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{4 (b c-a d)^4 (b e-a f)^{7/2}}+\frac {d^{5/2} \left (a d (3 d f g-2 d e h-c f h)+b \left (6 d^2 e g+7 c^2 f h-c d (9 f g+4 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d)^4 (d e-c f)^{5/2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 10.89 (sec) , antiderivative size = 561, normalized size of antiderivative = 0.66 \[ \int \frac {g+h x}{(a+b x)^3 (c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {\frac {-b g+a h}{(a+b x)^2}+\frac {7 a^2 d f h+b^2 (6 d e g+5 c f g-4 c e h)-a b (11 d f g+2 d e h+c f h)}{2 (b c-a d) (b e-a f) (a+b x)}-\frac {d (b c-a d) (b e-a f) (-d e+c f) \left (a^2 d f (4 d f g+9 d e h-13 c f h)+b^2 \left (12 d^2 e^2 g+c^2 f (-5 f g+4 e h)-c d e (3 f g+8 e h)\right )+a b \left (c^2 f^2 h-d^2 e (21 f g+4 e h)+c d f (13 f g+11 e h)\right )\right )+(c+d x) \left (-b (d e-c f)^2 \left (-35 a^3 d^2 f^2 h+7 a^2 b d f (9 d f g+4 d e h+2 c f h)-a b^2 \left (3 c^2 f^2 h+2 c d f (27 f g-16 e h)+8 d^2 e (9 f g+e h)\right )+b^3 \left (24 d^2 e^2 g+3 c^2 f (5 f g-4 e h)-8 c d e (-3 f g+2 e h)\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {b (e+f x)}{b e-a f}\right )+4 d^2 (b e-a f)^3 \left (a d (3 d f g-2 d e h-c f h)+b \left (6 d^2 e g+7 c^2 f h-c d (9 f g+4 e h)\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {d (e+f x)}{d e-c f}\right )\right )}{2 (b c-a d)^3 (b e-a f)^2 (d e-c f)^2}}{2 (b c-a d) (b e-a f) (c+d x) \sqrt {e+f x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.21 (sec) , antiderivative size = 936, normalized size of antiderivative = 1.10, 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, 27, 168, 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.

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\int \frac {d f (8 d f g+18 d e h+9 c f h) a^2-b \left (2 e (21 f g+4 e h) d^2+c f (29 f g-12 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (3 f (5 f g-4 e h) c^2+8 d e (3 f g-2 e h) c+24 d^2 e^2 g\right )+5 d f \left (7 d f h a^2-b (11 d f g+2 d e h+c f h) a+b^2 (6 d e g+5 c f g-4 c e h)\right ) x}{2 (a+b x) (c+d x)^2 (e+f x)^{3/2}}dx}{(b c-a d) (b e-a f)}-\frac {7 a^2 d f h-a b (c f h+2 d e h+11 d f g)+b^2 (-4 c e h+5 c f g+6 d e g)}{(a+b x) (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {b g-a h}{2 (a+b x)^2 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {d f (8 d f g+18 d e h+9 c f h) a^2-b \left (2 e (21 f g+4 e h) d^2+c f (29 f g-12 e h) d+3 c^2 f^2 h\right ) a+b^2 \left (3 f (5 f g-4 e h) c^2+8 d e (3 f g-2 e h) c+24 d^2 e^2 g\right )+5 d f \left (7 d f h a^2-b (11 d f g+2 d e h+c f h) a+b^2 (6 d e g+5 c f g-4 c e h)\right ) x}{(a+b x) (c+d x)^2 (e+f x)^{3/2}}dx}{2 (b c-a d) (b e-a f)}-\frac {7 a^2 d f h-a b (c f h+2 d e h+11 d f g)+b^2 (-4 c e h+5 c f g+6 d e g)}{(a+b x) (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {b g-a h}{2 (a+b x)^2 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {\frac {\int \frac {4 d^2 f^2 (3 d f g-2 d e h-c f h) a^3+b d f \left (16 d^2 h e^2-11 c^2 f^2 h-c d f (24 f g-19 e h)\right ) a^2+b^2 \left (-4 e^2 (9 f g+2 e h) d^3+c e f (9 f g+16 e h) d^2+c^2 f^2 (39 f g-23 e h) d+3 c^3 f^3 h\right ) a+b^3 (d e-c f) \left (3 f (5 f g-4 e h) c^2+8 d e (3 f g-2 e h) c+24 d^2 e^2 g\right )+3 b d f \left (d f (4 d f g+9 d e h-13 c f h) a^2+b \left (-e (21 f g+4 e h) d^2+c f (13 f g+11 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (3 f g+8 e h) c+12 d^2 e^2 g\right )\right ) x}{(a+b x) (c+d x) (e+f x)^{3/2}}dx}{(b c-a d) (d e-c f)}+\frac {2 d \left (a^2 d f (-13 c f h+9 d e h+4 d f g)+a b \left (c^2 f^2 h+c d f (11 e h+13 f g)+d^2 (-e) (4 e h+21 f g)\right )+b^2 \left (c^2 (-f) (5 f g-4 e h)-c d e (8 e h+3 f g)+12 d^2 e^2 g\right )\right )}{(c+d x) \sqrt {e+f x} (b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}-\frac {7 a^2 d f h-a b (c f h+2 d e h+11 d f g)+b^2 (-4 c e h+5 c f g+6 d e g)}{(a+b x) (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}-\frac {b g-a h}{2 (a+b x)^2 (c+d x) \sqrt {e+f x} (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}-\frac {-\frac {7 d f h a^2-b (11 d f g+2 d e h+c f h) a+b^2 (6 d e g+5 c f g-4 c e h)}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}-\frac {\frac {2 d \left (d f (4 d f g+9 d e h-13 c f h) a^2+b \left (-e (21 f g+4 e h) d^2+c f (13 f g+11 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (3 f g+8 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {-\frac {2 f \left (4 d^2 f^2 (3 d f g-2 d e h-c f h) a^3-b d f \left (e (12 f g+11 e h) d^2+2 c f (12 f g-29 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (e^2 (27 f g+4 e h) d^3-c e f (30 f g+17 e h) d^2+13 c^2 f^2 (3 f g-2 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (3 f^2 (5 f g-4 e h) c^3-2 d e f (3 f g-4 e h) c^2-d^2 e^2 (9 f g+8 e h) c+12 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {2 \int \frac {4 d^3 f^3 (3 d f g-2 d e h-c f h) a^4+12 b d^2 f^2 \left (-e (f g-2 e h) d^2-c f (2 f g+e h) d+2 c^2 f^2 h\right ) a^3-b^2 d f \left (12 e^2 (3 f g+2 e h) d^3-c e f (96 f g+25 e h) d^2+2 c^2 f^2 (12 f g+13 e h) d+11 c^3 f^3 h\right ) a^2+b^3 \left (4 e^3 (15 f g+2 e h) d^4-c e^2 f (81 f g+40 e h) d^3-c^2 e f^2 (30 f g-67 e h) d^2+13 c^3 f^3 (3 f g-2 e h) d+3 c^4 f^4 h\right ) a-b^4 (d e-c f)^2 \left (3 f (5 f g-4 e h) c^2+8 d e (3 f g-2 e h) c+24 d^2 e^2 g\right )+b d f \left (4 d^2 f^2 (3 d f g-2 d e h-c f h) a^3-b d f \left (e (12 f g+11 e h) d^2+2 c f (12 f g-29 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (e^2 (27 f g+4 e h) d^3-c e f (30 f g+17 e h) d^2+13 c^2 f^2 (3 f g-2 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (3 f^2 (5 f g-4 e h) c^3-2 d e f (3 f g-4 e h) c^2-d^2 e^2 (9 f g+8 e h) c+12 d^3 e^3 g\right )\right ) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b g-a h}{2 (b c-a d) (b e-a f) (a+b x)^2 (c+d x) \sqrt {e+f x}}-\frac {-\frac {7 d f h a^2-b (11 d f g+2 d e h+c f h) a+b^2 (6 d e g+5 c f g-4 c e h)}{(b c-a d) (b e-a f) (a+b x) (c+d x) \sqrt {e+f x}}-\frac {\frac {2 d \left (d f (4 d f g+9 d e h-13 c f h) a^2+b \left (-e (21 f g+4 e h) d^2+c f (13 f g+11 e h) d+c^2 f^2 h\right ) a+b^2 \left (-f (5 f g-4 e h) c^2-d e (3 f g+8 e h) c+12 d^2 e^2 g\right )\right )}{(b c-a d) (d e-c f) (c+d x) \sqrt {e+f x}}+\frac {-\frac {2 f \left (4 d^2 f^2 (3 d f g-2 d e h-c f h) a^3-b d f \left (e (12 f g+11 e h) d^2+2 c f (12 f g-29 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (e^2 (27 f g+4 e h) d^3-c e f (30 f g+17 e h) d^2+13 c^2 f^2 (3 f g-2 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (3 f^2 (5 f g-4 e h) c^3-2 d e f (3 f g-4 e h) c^2-d^2 e^2 (9 f g+8 e h) c+12 d^3 e^3 g\right )\right )}{(b e-a f) (d e-c f) \sqrt {e+f x}}-\frac {\int \frac {4 d^3 f^3 (3 d f g-2 d e h-c f h) a^4+12 b d^2 f^2 \left (-e (f g-2 e h) d^2-c f (2 f g+e h) d+2 c^2 f^2 h\right ) a^3-b^2 d f \left (12 e^2 (3 f g+2 e h) d^3-c e f (96 f g+25 e h) d^2+2 c^2 f^2 (12 f g+13 e h) d+11 c^3 f^3 h\right ) a^2+b^3 \left (4 e^3 (15 f g+2 e h) d^4-c e^2 f (81 f g+40 e h) d^3-c^2 e f^2 (30 f g-67 e h) d^2+13 c^3 f^3 (3 f g-2 e h) d+3 c^4 f^4 h\right ) a-b^4 (d e-c f)^2 \left (3 f (5 f g-4 e h) c^2+8 d e (3 f g-2 e h) c+24 d^2 e^2 g\right )+b d f \left (4 d^2 f^2 (3 d f g-2 d e h-c f h) a^3-b d f \left (e (12 f g+11 e h) d^2+2 c f (12 f g-29 e h) d+11 c^2 f^2 h\right ) a^2+b^2 \left (e^2 (27 f g+4 e h) d^3-c e f (30 f g+17 e h) d^2+13 c^2 f^2 (3 f g-2 e h) d+3 c^3 f^3 h\right ) a-b^3 \left (3 f^2 (5 f g-4 e h) c^3-2 d e f (3 f g-4 e h) c^2-d^2 e^2 (9 f g+8 e h) c+12 d^3 e^3 g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{(b e-a f) (d e-c f)}}{(b c-a d) (d e-c f)}}{2 (b c-a d) (b e-a f)}}{4 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*(b*g - a*h)/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2*(c + d*x)*Sqrt[e + f 
*x]) - (-((7*a^2*d*f*h + b^2*(6*d*e*g + 5*c*f*g - 4*c*e*h) - a*b*(11*d*f*g 
 + 2*d*e*h + c*f*h))/((b*c - a*d)*(b*e - a*f)*(a + b*x)*(c + d*x)*Sqrt[e + 
 f*x])) - ((2*d*(a^2*d*f*(4*d*f*g + 9*d*e*h - 13*c*f*h) + b^2*(12*d^2*e^2* 
g - c^2*f*(5*f*g - 4*e*h) - c*d*e*(3*f*g + 8*e*h)) + a*b*(c^2*f^2*h - d^2* 
e*(21*f*g + 4*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*(4*a^3*d^2*f^2*(3*d*f*g - 2*d*e*h - c*f*h 
) - b^3*(12*d^3*e^3*g - 2*c^2*d*e*f*(3*f*g - 4*e*h) + 3*c^3*f^2*(5*f*g - 4 
*e*h) - c*d^2*e^2*(9*f*g + 8*e*h)) - a^2*b*d*f*(11*c^2*f^2*h + 2*c*d*f*(12 
*f*g - 29*e*h) + d^2*e*(12*f*g + 11*e*h)) + a*b^2*(3*c^3*f^3*h + 13*c^2*d* 
f^2*(3*f*g - 2*e*h) + d^3*e^2*(27*f*g + 4*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]) - ((-2*b^(3/2)*(d*e - c*f)^2 
*(35*a^3*d^2*f^2*h - 7*a^2*b*d*f*(9*d*f*g + 4*d*e*h + 2*c*f*h) - b^3*(24*d 
^2*e^2*g + 3*c^2*f*(5*f*g - 4*e*h) + 8*c*d*e*(3*f*g - 2*e*h)) + a*b^2*(3*c 
^2*f^2*h + 2*c*d*f*(27*f*g - 16*e*h) + 8*d^2*e*(9*f*g + e*h)))*ArcTanh[(Sq 
rt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/((b*c - a*d)*Sqrt[b*e - a*f]) - (8* 
d^(5/2)*(b*e - a*f)^3*(a*d*(3*d*f*g - 2*d*e*h - c*f*h) + b*(6*d^2*e*g + 7* 
c^2*f*h - c*d*(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)))/((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...
 

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 = 99.40 (sec) , antiderivative size = 815, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(2 f^{3} \left (-\frac {-e h +f g}{\left (c f -d e \right )^{2} \left (a f -b e \right )^{3} \sqrt {f x +e}}+\frac {b^{2} \left (\frac {\left (\frac {11}{8} a^{3} b \,d^{2} f^{2} h -\frac {7}{4} a^{2} b^{2} c d \,f^{2} h -\frac {1}{2} a^{2} b^{2} d^{2} e f h -\frac {15}{8} a^{2} b^{2} d^{2} f^{2} g +\frac {3}{8} a \,b^{3} c^{2} f^{2} h +\frac {11}{4} a \,b^{3} c d \,f^{2} g +a \,b^{3} d^{2} e f g +\frac {1}{2} b^{4} c^{2} e f h -\frac {7}{8} b^{4} c^{2} f^{2} g -b^{4} c d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\frac {f \left (13 a^{4} d^{2} f^{2} h -18 a^{3} b c d \,f^{2} h -17 a^{3} b \,d^{2} e f h -17 a^{3} b \,d^{2} f^{2} g +5 a^{2} b^{2} c^{2} f^{2} h +18 a^{2} b^{2} c d e f h +26 a^{2} b^{2} c d \,f^{2} g +4 a^{2} b^{2} d^{2} e^{2} h +25 a^{2} b^{2} d^{2} e f g -a \,b^{3} c^{2} e f h -9 a \,b^{3} c^{2} f^{2} g -34 a \,b^{3} c d e f g -8 a \,b^{3} d^{2} e^{2} g -4 b^{4} c^{2} e^{2} h +9 b^{4} c^{2} e f g +8 b^{4} c d \,e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (35 a^{3} d^{2} f^{2} h -14 a^{2} b c d \,f^{2} h -28 a^{2} b \,d^{2} e f h -63 a^{2} b \,d^{2} f^{2} g +3 a \,b^{2} c^{2} f^{2} h -32 a \,b^{2} c d e f h +54 a \,b^{2} c d \,f^{2} g +8 a \,b^{2} d^{2} e^{2} h +72 a \,b^{2} d^{2} e f g +12 b^{3} c^{2} e f h -15 b^{3} c^{2} f^{2} g +16 b^{3} c d \,e^{2} h -24 b^{3} c d e f g -24 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{4} f^{3} \left (a f -b e \right )^{3}}+\frac {d^{3} \left (\frac {\left (\frac {1}{2} a c d f h -\frac {1}{2} a \,d^{2} f g -\frac {1}{2} b \,c^{2} f h +\frac {1}{2} b c d f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (a c d f h +2 a \,d^{2} e h -3 a \,d^{2} f g -7 b \,c^{2} f h +4 b c d e h +9 b c d f g -6 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{4} f^{3} \left (c f -d e \right )^{2}}\right )\)	\(815\)
default	\(2 f^{3} \left (-\frac {-e h +f g}{\left (c f -d e \right )^{2} \left (a f -b e \right )^{3} \sqrt {f x +e}}+\frac {b^{2} \left (\frac {\left (\frac {11}{8} a^{3} b \,d^{2} f^{2} h -\frac {7}{4} a^{2} b^{2} c d \,f^{2} h -\frac {1}{2} a^{2} b^{2} d^{2} e f h -\frac {15}{8} a^{2} b^{2} d^{2} f^{2} g +\frac {3}{8} a \,b^{3} c^{2} f^{2} h +\frac {11}{4} a \,b^{3} c d \,f^{2} g +a \,b^{3} d^{2} e f g +\frac {1}{2} b^{4} c^{2} e f h -\frac {7}{8} b^{4} c^{2} f^{2} g -b^{4} c d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}+\frac {f \left (13 a^{4} d^{2} f^{2} h -18 a^{3} b c d \,f^{2} h -17 a^{3} b \,d^{2} e f h -17 a^{3} b \,d^{2} f^{2} g +5 a^{2} b^{2} c^{2} f^{2} h +18 a^{2} b^{2} c d e f h +26 a^{2} b^{2} c d \,f^{2} g +4 a^{2} b^{2} d^{2} e^{2} h +25 a^{2} b^{2} d^{2} e f g -a \,b^{3} c^{2} e f h -9 a \,b^{3} c^{2} f^{2} g -34 a \,b^{3} c d e f g -8 a \,b^{3} d^{2} e^{2} g -4 b^{4} c^{2} e^{2} h +9 b^{4} c^{2} e f g +8 b^{4} c d \,e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (35 a^{3} d^{2} f^{2} h -14 a^{2} b c d \,f^{2} h -28 a^{2} b \,d^{2} e f h -63 a^{2} b \,d^{2} f^{2} g +3 a \,b^{2} c^{2} f^{2} h -32 a \,b^{2} c d e f h +54 a \,b^{2} c d \,f^{2} g +8 a \,b^{2} d^{2} e^{2} h +72 a \,b^{2} d^{2} e f g +12 b^{3} c^{2} e f h -15 b^{3} c^{2} f^{2} g +16 b^{3} c d \,e^{2} h -24 b^{3} c d e f g -24 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{4} f^{3} \left (a f -b e \right )^{3}}+\frac {d^{3} \left (\frac {\left (\frac {1}{2} a c d f h -\frac {1}{2} a \,d^{2} f g -\frac {1}{2} b \,c^{2} f h +\frac {1}{2} b c d f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (a c d f h +2 a \,d^{2} e h -3 a \,d^{2} f g -7 b \,c^{2} f h +4 b c d e h +9 b c d f g -6 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{4} f^{3} \left (c f -d e \right )^{2}}\right )\)	\(815\)
pseudoelliptic	\(\text {Expression too large to display}\)	\(1362\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((h*x+g)/(b*x+a)^3/(d*x+c)^2/(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. 2417 vs. \(2 (810) = 1620\).

Time = 0.32 (sec) , antiderivative size = 2417, normalized size of antiderivative = 2.85 \[ \int \frac {g+h x}{(a+b x)^3 (c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 26977, normalized size of antiderivative = 31.81 \[ \int \frac {g+h x}{(a+b x)^3 (c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(35*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**5*b*c**4*d**2*f**5*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**5*b*c**3* 
d**3*e*f**4*h + 35*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**5*b*c**3*d**3*f**5*h*x + 105*sqrt(b)*s 
qrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* 
e)))*a**5*b*c**2*d**4*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**5*b*c**2*d**4*e*f 
**4*h*x - 35*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**5*b*c*d**5*e**3*f**2*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 
**5*b*c*d**5*e**2*f**3*h*x - 35*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**5*b*d**6*e**3*f**2*h*x - 
14*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**4*b**2*c**5*d*f**5*h + 14*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**4*b**2*c**4 
*d**2*e*f**4*h - 63*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**4*b**2*c**4*d**2*f**5*g + 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**4*b**2*c**4*d**2*f**5*h*x + 42*sqrt(b)*sqrt(e + f*x)*sqrt(a*f ...