\(\int \frac {(c+d x)^2 (g+h x)}{(a+b x)^5 (e+f x)^{3/2}} \, dx\) [167]

Optimal result

Integrand size = 29, antiderivative size = 695 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^5 (e+f x)^{3/2}} \, dx=\frac {2 f (d e-c f)^2 (f g-e h)}{(b e-a f)^5 \sqrt {e+f x}}-\frac {(b c-a d)^2 (b g-a h) \sqrt {e+f x}}{4 b^2 (b e-a f)^2 (a+b x)^4}-\frac {(b c-a d) \left (9 a^2 d f h+b^2 (16 d e g-15 c f g+8 c e h)-a b (d f g+24 d e h-7 c f h)\right ) \sqrt {e+f x}}{24 b^2 (b e-a f)^3 (a+b x)^3}+\frac {\left (3 a^3 d^2 f^2 h+a^2 b d f (5 d f g-24 d e h+10 c f h)+a b^2 \left (35 c^2 f^2 h+10 c d f (7 f g-16 e h)-16 d^2 e (5 f g-9 e h)\right )-b^3 \left (48 d^2 e^2 g+c^2 f (123 f g-88 e h)-16 c d e (11 f g-6 e h)\right )\right ) \sqrt {e+f x}}{96 b^2 (b e-a f)^4 (a+b x)^2}-\frac {\left (3 a^3 d^2 f^3 h+a^2 b d f^2 (5 d f g-24 d e h+10 c f h)+a b^2 f \left (35 c^2 f^2 h+10 c d f (7 f g-16 e h)-16 d^2 e (5 f g-9 e h)\right )-b^3 \left (c^2 f^2 (187 f g-152 e h)-16 c d e f (19 f g-14 e h)+16 d^2 e^2 (7 f g-4 e h)\right )\right ) \sqrt {e+f x}}{64 b^2 (b e-a f)^5 (a+b x)}+\frac {f \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (5 d f g-24 d e h+10 c f h)+a b^2 f \left (35 c^2 f^2 h+10 c d f (7 f g-16 e h)-16 d^2 e (5 f g-9 e h)\right )-b^3 \left (35 c^2 f^2 (9 f g-8 e h)-80 c d e f (7 f g-6 e h)+48 d^2 e^2 (5 f g-4 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{64 b^{5/2} (b e-a f)^{11/2}} \] Output:

2*f*(-c*f+d*e)^2*(-e*h+f*g)/(-a*f+b*e)^5/(f*x+e)^(1/2)-1/4*(-a*d+b*c)^2*(- 
a*h+b*g)*(f*x+e)^(1/2)/b^2/(-a*f+b*e)^2/(b*x+a)^4-1/24*(-a*d+b*c)*(9*a^2*d 
*f*h+b^2*(8*c*e*h-15*c*f*g+16*d*e*g)-a*b*(-7*c*f*h+24*d*e*h+d*f*g))*(f*x+e 
)^(1/2)/b^2/(-a*f+b*e)^3/(b*x+a)^3+1/96*(3*a^3*d^2*f^2*h+a^2*b*d*f*(10*c*f 
*h-24*d*e*h+5*d*f*g)+a*b^2*(35*c^2*f^2*h+10*c*d*f*(-16*e*h+7*f*g)-16*d^2*e 
*(-9*e*h+5*f*g))-b^3*(48*d^2*e^2*g+c^2*f*(-88*e*h+123*f*g)-16*c*d*e*(-6*e* 
h+11*f*g)))*(f*x+e)^(1/2)/b^2/(-a*f+b*e)^4/(b*x+a)^2-1/64*(3*a^3*d^2*f^3*h 
+a^2*b*d*f^2*(10*c*f*h-24*d*e*h+5*d*f*g)+a*b^2*f*(35*c^2*f^2*h+10*c*d*f*(- 
16*e*h+7*f*g)-16*d^2*e*(-9*e*h+5*f*g))-b^3*(c^2*f^2*(-152*e*h+187*f*g)-16* 
c*d*e*f*(-14*e*h+19*f*g)+16*d^2*e^2*(-4*e*h+7*f*g)))*(f*x+e)^(1/2)/b^2/(-a 
*f+b*e)^5/(b*x+a)+1/64*f*(3*a^3*d^2*f^3*h+a^2*b*d*f^2*(10*c*f*h-24*d*e*h+5 
*d*f*g)+a*b^2*f*(35*c^2*f^2*h+10*c*d*f*(-16*e*h+7*f*g)-16*d^2*e*(-9*e*h+5* 
f*g))-b^3*(35*c^2*f^2*(-8*e*h+9*f*g)-80*c*d*e*f*(-6*e*h+7*f*g)+48*d^2*e^2* 
(-4*e*h+5*f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(5/2)/( 
-a*f+b*e)^(11/2)
 

Mathematica [A] (verified)

Time = 9.84 (sec) , antiderivative size = 1191, normalized size of antiderivative = 1.71 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^5 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[((c + d*x)^2*(g + h*x))/((a + b*x)^5*(e + f*x)^(3/2)),x]
 

Output:

((Sqrt[b]*(-9*a^6*d^2*f^3*h*(e + f*x) - 3*a^5*b*d*f^2*(e + f*x)*(10*c*f*h 
+ d*(5*f*g - 22*e*h + 11*f*h*x)) + b^6*(48*d^2*e^2*x^2*(-15*f^2*g*x^2 + 2* 
e^2*(g + 2*h*x) + e*f*x*(-5*g + 12*h*x)) + 16*c*d*e*x*(105*f^3*g*x^3 + 5*e 
*f^2*x^2*(7*g - 18*h*x) + 4*e^3*(2*g + 3*h*x) - 2*e^2*f*x*(7*g + 15*h*x)) 
+ c^2*(-945*f^4*g*x^4 + 16*e^4*(3*g + 4*h*x) + 105*e*f^3*x^3*(-3*g + 8*h*x 
) - 8*e^3*f*x*(9*g + 14*h*x) + 14*e^2*f^2*x^2*(9*g + 20*h*x))) + a*b^5*(2* 
c*d*(105*f^4*g*x^4 + 10*e^2*f^2*x^2*(105*g - 272*h*x) + 5*e*f^3*x^3*(623*g 
 - 48*h*x) + 16*e^4*(g + 4*h*x) - 8*e^3*f*x*(51*g + 110*h*x)) + 16*d^2*e*x 
*(-15*f^3*g*x^3 + 2*e^3*(2*g + 9*h*x) + e*f^2*x^2*(-170*g + 27*h*x) + e^2* 
f*x*(-55*g + 141*h*x)) + c^2*(16*e^4*h + 105*f^4*x^3*(-33*g + h*x) - 24*e^ 
3*f*(11*g + 17*h*x) + 6*e^2*f^2*x*(78*g + 175*h*x) + 7*e*f^3*x^2*(-171*g + 
 445*h*x))) + a^4*b^2*f*(3*c^2*f^2*(-128*f*g + 221*e*h + 93*f*h*x) + 2*c*d 
*f*(-794*e^2*h + e*f*(663*g - 337*h*x) + f^2*x*(279*g + 73*h*x)) + d^2*(84 
0*e^3*h + f^3*x^2*(73*g + 33*h*x) + e*f^2*x*(-337*g + 45*h*x) + e^2*f*(-79 
4*g + 468*h*x))) + a^2*b^4*(2*c*d*(16*e^4*h + 6*e^2*f^2*x*(238*g - 635*h*x 
) + 7*e*f^3*x^2*(603*g - 125*h*x) + 5*f^4*x^3*(77*g + 3*h*x) - 8*e^3*f*(13 
*g + 75*h*x)) + d^2*(15*f^4*g*x^4 + 16*e^4*(g + 12*h*x) + 30*e^2*f^2*x^2*( 
-127*g + 52*h*x) - e*f^3*x^3*(875*g + 72*h*x) + 24*e^3*f*x*(-25*g + 172*h* 
x)) + c^2*f*(-104*e^3*h + 42*e^2*f*(15*g + 34*h*x) + 7*f^3*x^2*(-657*g + 5 
5*h*x) + 9*e*f^2*x*(-185*g + 469*h*x))) + a^3*b^3*(2*c*d*f*(-152*e^3*h ...
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 654, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {166, 27, 161, 52, 52, 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.

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\int -\frac {(c+d x) (a (4 d e-c f) h-b (4 d e g-9 c f g+8 c e h)+d (5 b f g-8 b e h+3 a f h) x)}{2 (a+b x)^4 (e+f x)^{3/2}}dx}{4 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{4 b (a+b x)^4 \sqrt {e+f x} (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(c+d x) (a (4 d e-c f) h-b (4 d e g-9 c f g+8 c e h)+d (5 b f g-8 b e h+3 a f h) x)}{(a+b x)^4 (e+f x)^{3/2}}dx}{8 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{4 b (a+b x)^4 \sqrt {e+f x} (b e-a f)}\)

\(\Big \downarrow \) 161

\(\displaystyle -\frac {\frac {\left (3 a^3 d^2 f^3 h+a^2 b d f^2 (10 c f h-24 d e h+5 d f g)+a b^2 f \left (35 c^2 f^2 h+10 c d f (7 f g-16 e h)-16 d^2 e (5 f g-9 e h)\right )-\left (b^3 \left (35 c^2 f^2 (9 f g-8 e h)-80 c d e f (7 f g-6 e h)+48 d^2 e^2 (5 f g-4 e h)\right )\right )\right ) \int \frac {1}{(a+b x)^3 \sqrt {e+f x}}dx}{6 b f (b e-a f)^2}-\frac {3 a^3 d^2 e f^2 h+a^2 b f \left (-6 c^2 f^2 h+10 c d e f h+d^2 e (5 f g-18 e h)\right )+x \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (-2 c f h-12 d e h+5 d f g)-a b^2 f \left (7 c^2 f^2 h+2 c d f (7 f g-16 e h)-2 d^2 e (2 f g-3 e h)\right )+b^3 \left (7 c^2 f^2 (9 f g-8 e h)-16 c d e f (7 f g-6 e h)+6 d^2 e^2 (9 f g-8 e h)\right )\right )-a b^2 \left (-c^2 f^2 (54 f g-49 e h)+2 c d e f (61 f g-58 e h)-2 d^2 e^2 (29 f g-24 e h)\right )-b^3 c e f (8 c e h-9 c f g+4 d e g)}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}}{8 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{4 b (a+b x)^4 \sqrt {e+f x} (b e-a f)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {\frac {\left (3 a^3 d^2 f^3 h+a^2 b d f^2 (10 c f h-24 d e h+5 d f g)+a b^2 f \left (35 c^2 f^2 h+10 c d f (7 f g-16 e h)-16 d^2 e (5 f g-9 e h)\right )-\left (b^3 \left (35 c^2 f^2 (9 f g-8 e h)-80 c d e f (7 f g-6 e h)+48 d^2 e^2 (5 f g-4 e h)\right )\right )\right ) \left (-\frac {3 f \int \frac {1}{(a+b x)^2 \sqrt {e+f x}}dx}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right )}{6 b f (b e-a f)^2}-\frac {3 a^3 d^2 e f^2 h+a^2 b f \left (-6 c^2 f^2 h+10 c d e f h+d^2 e (5 f g-18 e h)\right )+x \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (-2 c f h-12 d e h+5 d f g)-a b^2 f \left (7 c^2 f^2 h+2 c d f (7 f g-16 e h)-2 d^2 e (2 f g-3 e h)\right )+b^3 \left (7 c^2 f^2 (9 f g-8 e h)-16 c d e f (7 f g-6 e h)+6 d^2 e^2 (9 f g-8 e h)\right )\right )-a b^2 \left (-c^2 f^2 (54 f g-49 e h)+2 c d e f (61 f g-58 e h)-2 d^2 e^2 (29 f g-24 e h)\right )-b^3 c e f (8 c e h-9 c f g+4 d e g)}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}}{8 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{4 b (a+b x)^4 \sqrt {e+f x} (b e-a f)}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {\frac {\left (3 a^3 d^2 f^3 h+a^2 b d f^2 (10 c f h-24 d e h+5 d f g)+a b^2 f \left (35 c^2 f^2 h+10 c d f (7 f g-16 e h)-16 d^2 e (5 f g-9 e h)\right )-\left (b^3 \left (35 c^2 f^2 (9 f g-8 e h)-80 c d e f (7 f g-6 e h)+48 d^2 e^2 (5 f g-4 e h)\right )\right )\right ) \left (-\frac {3 f \left (-\frac {f \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{2 (b e-a f)}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right )}{6 b f (b e-a f)^2}-\frac {3 a^3 d^2 e f^2 h+a^2 b f \left (-6 c^2 f^2 h+10 c d e f h+d^2 e (5 f g-18 e h)\right )+x \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (-2 c f h-12 d e h+5 d f g)-a b^2 f \left (7 c^2 f^2 h+2 c d f (7 f g-16 e h)-2 d^2 e (2 f g-3 e h)\right )+b^3 \left (7 c^2 f^2 (9 f g-8 e h)-16 c d e f (7 f g-6 e h)+6 d^2 e^2 (9 f g-8 e h)\right )\right )-a b^2 \left (-c^2 f^2 (54 f g-49 e h)+2 c d e f (61 f g-58 e h)-2 d^2 e^2 (29 f g-24 e h)\right )-b^3 c e f (8 c e h-9 c f g+4 d e g)}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}}{8 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{4 b (a+b x)^4 \sqrt {e+f x} (b e-a f)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {\left (3 a^3 d^2 f^3 h+a^2 b d f^2 (10 c f h-24 d e h+5 d f g)+a b^2 f \left (35 c^2 f^2 h+10 c d f (7 f g-16 e h)-16 d^2 e (5 f g-9 e h)\right )-\left (b^3 \left (35 c^2 f^2 (9 f g-8 e h)-80 c d e f (7 f g-6 e h)+48 d^2 e^2 (5 f g-4 e h)\right )\right )\right ) \left (-\frac {3 f \left (-\frac {\int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b e-a f}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right )}{6 b f (b e-a f)^2}-\frac {3 a^3 d^2 e f^2 h+a^2 b f \left (-6 c^2 f^2 h+10 c d e f h+d^2 e (5 f g-18 e h)\right )+x \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (-2 c f h-12 d e h+5 d f g)-a b^2 f \left (7 c^2 f^2 h+2 c d f (7 f g-16 e h)-2 d^2 e (2 f g-3 e h)\right )+b^3 \left (7 c^2 f^2 (9 f g-8 e h)-16 c d e f (7 f g-6 e h)+6 d^2 e^2 (9 f g-8 e h)\right )\right )-a b^2 \left (-c^2 f^2 (54 f g-49 e h)+2 c d e f (61 f g-58 e h)-2 d^2 e^2 (29 f g-24 e h)\right )-b^3 c e f (8 c e h-9 c f g+4 d e g)}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}}{8 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{4 b (a+b x)^4 \sqrt {e+f x} (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\left (-\frac {3 f \left (\frac {f \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b e-a f)^{3/2}}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right ) \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (10 c f h-24 d e h+5 d f g)+a b^2 f \left (35 c^2 f^2 h+10 c d f (7 f g-16 e h)-16 d^2 e (5 f g-9 e h)\right )-\left (b^3 \left (35 c^2 f^2 (9 f g-8 e h)-80 c d e f (7 f g-6 e h)+48 d^2 e^2 (5 f g-4 e h)\right )\right )\right )}{6 b f (b e-a f)^2}-\frac {3 a^3 d^2 e f^2 h+a^2 b f \left (-6 c^2 f^2 h+10 c d e f h+d^2 e (5 f g-18 e h)\right )+x \left (3 a^3 d^2 f^3 h+a^2 b d f^2 (-2 c f h-12 d e h+5 d f g)-a b^2 f \left (7 c^2 f^2 h+2 c d f (7 f g-16 e h)-2 d^2 e (2 f g-3 e h)\right )+b^3 \left (7 c^2 f^2 (9 f g-8 e h)-16 c d e f (7 f g-6 e h)+6 d^2 e^2 (9 f g-8 e h)\right )\right )-a b^2 \left (-c^2 f^2 (54 f g-49 e h)+2 c d e f (61 f g-58 e h)-2 d^2 e^2 (29 f g-24 e h)\right )-b^3 c e f (8 c e h-9 c f g+4 d e g)}{3 b f (a+b x)^3 \sqrt {e+f x} (b e-a f)^2}}{8 b (b e-a f)}-\frac {(c+d x)^2 (b g-a h)}{4 b (a+b x)^4 \sqrt {e+f x} (b e-a f)}\)

Input:

Int[((c + d*x)^2*(g + h*x))/((a + b*x)^5*(e + f*x)^(3/2)),x]
 

Output:

-1/4*((b*g - a*h)*(c + d*x)^2)/(b*(b*e - a*f)*(a + b*x)^4*Sqrt[e + f*x]) - 
 (-1/3*(3*a^3*d^2*e*f^2*h - b^3*c*e*f*(4*d*e*g - 9*c*f*g + 8*c*e*h) - a*b^ 
2*(2*c*d*e*f*(61*f*g - 58*e*h) - c^2*f^2*(54*f*g - 49*e*h) - 2*d^2*e^2*(29 
*f*g - 24*e*h)) + a^2*b*f*(10*c*d*e*f*h - 6*c^2*f^2*h + d^2*e*(5*f*g - 18* 
e*h)) + (3*a^3*d^2*f^3*h + a^2*b*d*f^2*(5*d*f*g - 12*d*e*h - 2*c*f*h) + b^ 
3*(6*d^2*e^2*(9*f*g - 8*e*h) + 7*c^2*f^2*(9*f*g - 8*e*h) - 16*c*d*e*f*(7*f 
*g - 6*e*h)) - a*b^2*f*(7*c^2*f^2*h + 2*c*d*f*(7*f*g - 16*e*h) - 2*d^2*e*( 
2*f*g - 3*e*h)))*x)/(b*f*(b*e - a*f)^2*(a + b*x)^3*Sqrt[e + f*x]) + ((3*a^ 
3*d^2*f^3*h + a^2*b*d*f^2*(5*d*f*g - 24*d*e*h + 10*c*f*h) + a*b^2*f*(35*c^ 
2*f^2*h + 10*c*d*f*(7*f*g - 16*e*h) - 16*d^2*e*(5*f*g - 9*e*h)) - b^3*(35* 
c^2*f^2*(9*f*g - 8*e*h) - 80*c*d*e*f*(7*f*g - 6*e*h) + 48*d^2*e^2*(5*f*g - 
 4*e*h)))*(-1/2*Sqrt[e + f*x]/((b*e - a*f)*(a + b*x)^2) - (3*f*(-(Sqrt[e + 
 f*x]/((b*e - a*f)*(a + b*x))) + (f*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b 
*e - a*f]])/(Sqrt[b]*(b*e - a*f)^(3/2))))/(4*(b*e - a*f))))/(6*b*f*(b*e - 
a*f)^2))/(8*b*(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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

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 161

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_)) 
*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 
 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) 
 + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1 
) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*( 
m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f* 
h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
 3)) + b^2*(c^2*f*h*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
 d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*( 
n + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]
 

rule 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 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 = 1.00 (sec) , antiderivative size = 1179, normalized size of antiderivative = 1.70

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1179\)
derivativedivides	\(\text {Expression too large to display}\)	\(1780\)
default	\(\text {Expression too large to display}\)	\(1780\)

Input:

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

Output:

3/64/(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2)*((b*x+a)^4*(f*x+e)^(1/2)*((-105*c^2 
*g*f^3+280/3*c*e*(c*h+2*d*g)*f^2-160*(c*h+1/2*d*g)*d*e^2*f+64*d^2*e^3*h)*b 
^3+35/3*a*((c^2*h+2*c*d*g)*f^2-32/7*(c*h+1/2*d*g)*d*e*f+144/35*d^2*e^2*h)* 
f*b^2+10/3*a^2*((c*h+1/2*d*g)*f-12/5*d*e*h)*d*f^2*b+a^3*d^2*f^3*h)*f*arcta 
n(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))-((a*f-b*e)*b)^(1/2)*((105*c^2*f^4*g 
*x^4+35*x^3*c*(8/3*(-c*h-2*d*g)*x+c*g)*e*f^3-14*x^2*e^2*(40/7*(-2*c*d*h-d^ 
2*g)*x^2+20/9*c*(c*h+2*d*g)*x+c^2*g)*f^2+8*x*(-8*d^2*h*x^3+10/3*(2*c*d*h+d 
^2*g)*x^2+14/9*c*(c*h+2*d*g)*x+c^2*g)*e^3*f-16/3*(4*d^2*h*x^3+2*(2*c*d*h+d 
^2*g)*x^2+4/3*c*(c*h+2*d*g)*x+c^2*g)*e^4)*b^6-16/9*a*(-3465/16*x^3*c*(1/33 
*(-c*h-2*d*g)*x+c*g)*f^4-1197/16*x^2*e*(160/399*(c*h+1/2*d*g)*d*x^2-445/17 
1*c*(c*h+2*d*g)*x+c^2*g)*f^3+117/4*x*(12/13*d^2*h*x^3-1360/117*(c*h+1/2*d* 
g)*d*x^2+175/78*c*(c*h+2*d*g)*x+c^2*g)*e^2*f^2-33/2*(-94/11*d^2*h*x^3+10/3 
*(2*c*d*h+d^2*g)*x^2+17/11*c*(c*h+2*d*g)*x+c^2*g)*e^3*f+(18*d^2*h*x^2+4*(2 
*c*d*h+d^2*g)*x+h*c^2+2*c*d*g)*e^4)*b^5+104/9*a^2*(4599/104*x^2*(-10/1533* 
(c*h+1/2*d*g)*d*x^2-55/657*c*(c*h+2*d*g)*x+c^2*g)*f^4+1665/104*x*(8/185*d^ 
2*h*x^3+350/333*(c*h+1/2*d*g)*d*x^2-469/185*c*(c*h+2*d*g)*x+c^2*g)*e*f^3-3 
15/52*(52/21*d^2*h*x^3-254/21*(c*h+1/2*d*g)*d*x^2+34/15*c*(c*h+2*d*g)*x+c^ 
2*g)*e^2*f^2+(-516/13*d^2*h*x^2+75/13*(2*c*d*h+d^2*g)*x+h*c^2+2*c*d*g)*e^3 
*f-4/13*d*e^4*(6*d*h*x+c*h+1/2*d*g))*b^4-370/9*a^3*(-2511/370*x*(-1/279*d^ 
2*h*x^3-110/2511*(c*h+1/2*d*g)*d*x^2-511/2511*c*(c*h+2*d*g)*x+c^2*g)*f^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4078 vs. \(2 (665) = 1330\).

Time = 1.24 (sec) , antiderivative size = 8170, normalized size of antiderivative = 11.76 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^5 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^5 (e+f x)^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^5 (e+f x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^2*(h*x+g)/(b*x+a)^5/(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. 2648 vs. \(2 (665) = 1330\).

Time = 0.20 (sec) , antiderivative size = 2648, normalized size of antiderivative = 3.81 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^5 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/64*(240*b^3*d^2*e^2*f^2*g - 560*b^3*c*d*e*f^3*g + 80*a*b^2*d^2*e*f^3*g + 
 315*b^3*c^2*f^4*g - 70*a*b^2*c*d*f^4*g - 5*a^2*b*d^2*f^4*g - 192*b^3*d^2* 
e^3*f*h + 480*b^3*c*d*e^2*f^2*h - 144*a*b^2*d^2*e^2*f^2*h - 280*b^3*c^2*e* 
f^3*h + 160*a*b^2*c*d*e*f^3*h + 24*a^2*b*d^2*e*f^3*h - 35*a*b^2*c^2*f^4*h 
- 10*a^2*b*c*d*f^4*h - 3*a^3*d^2*f^4*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e 
 + a*b*f))/((b^7*e^5 - 5*a*b^6*e^4*f + 10*a^2*b^5*e^3*f^2 - 10*a^3*b^4*e^2 
*f^3 + 5*a^4*b^3*e*f^4 - a^5*b^2*f^5)*sqrt(-b^2*e + a*b*f)) + 2*(d^2*e^2*f 
^2*g - 2*c*d*e*f^3*g + c^2*f^4*g - d^2*e^3*f*h + 2*c*d*e^2*f^2*h - c^2*e*f 
^3*h)/((b^5*e^5 - 5*a*b^4*e^4*f + 10*a^2*b^3*e^3*f^2 - 10*a^3*b^2*e^2*f^3 
+ 5*a^4*b*e*f^4 - a^5*f^5)*sqrt(f*x + e)) + 1/192*(336*(f*x + e)^(7/2)*b^6 
*d^2*e^2*f^2*g - 1104*(f*x + e)^(5/2)*b^6*d^2*e^3*f^2*g + 1200*(f*x + e)^( 
3/2)*b^6*d^2*e^4*f^2*g - 432*sqrt(f*x + e)*b^6*d^2*e^5*f^2*g - 912*(f*x + 
e)^(7/2)*b^6*c*d*e*f^3*g + 240*(f*x + e)^(7/2)*a*b^5*d^2*e*f^3*g + 3088*(f 
*x + e)^(5/2)*b^6*c*d*e^2*f^3*g + 224*(f*x + e)^(5/2)*a*b^5*d^2*e^2*f^3*g 
- 3568*(f*x + e)^(3/2)*b^6*c*d*e^3*f^3*g - 1232*(f*x + e)^(3/2)*a*b^5*d^2* 
e^3*f^3*g + 1392*sqrt(f*x + e)*b^6*c*d*e^4*f^3*g + 768*sqrt(f*x + e)*a*b^5 
*d^2*e^4*f^3*g + 561*(f*x + e)^(7/2)*b^6*c^2*f^4*g - 210*(f*x + e)^(7/2)*a 
*b^5*c*d*f^4*g - 15*(f*x + e)^(7/2)*a^2*b^4*d^2*f^4*g - 1929*(f*x + e)^(5/ 
2)*b^6*c^2*e*f^4*g - 2318*(f*x + e)^(5/2)*a*b^5*c*d*e*f^4*g + 935*(f*x + e 
)^(5/2)*a^2*b^4*d^2*e*f^4*g + 2295*(f*x + e)^(3/2)*b^6*c^2*e^2*f^4*g + ...
 

Mupad [B] (verification not implemented)

Time = 4.02 (sec) , antiderivative size = 1824, normalized size of antiderivative = 2.62 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^5 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

((11*(e + f*x)^3*(3*a^3*d^2*f^4*h - 315*b^3*c^2*f^4*g + 35*a*b^2*c^2*f^4*h 
 + 5*a^2*b*d^2*f^4*g + 280*b^3*c^2*e*f^3*h + 192*b^3*d^2*e^3*f*h - 240*b^3 
*d^2*e^2*f^2*g + 144*a*b^2*d^2*e^2*f^2*h + 70*a*b^2*c*d*f^4*g + 10*a^2*b*c 
*d*f^4*h + 560*b^3*c*d*e*f^3*g - 80*a*b^2*d^2*e*f^3*g - 24*a^2*b*d^2*e*f^3 
*h - 480*b^3*c*d*e^2*f^2*h - 160*a*b^2*c*d*e*f^3*h))/(192*(a*f - b*e)^4) - 
 (2*(c^2*f^4*g + d^2*e^2*f^2*g - c^2*e*f^3*h - d^2*e^3*f*h + 2*c*d*e^2*f^2 
*h - 2*c*d*e*f^3*g))/(a*f - b*e) + (b*(e + f*x)^4*(3*a^3*d^2*f^4*h - 315*b 
^3*c^2*f^4*g + 35*a*b^2*c^2*f^4*h + 5*a^2*b*d^2*f^4*g + 280*b^3*c^2*e*f^3* 
h + 192*b^3*d^2*e^3*f*h - 240*b^3*d^2*e^2*f^2*g + 144*a*b^2*d^2*e^2*f^2*h 
+ 70*a*b^2*c*d*f^4*g + 10*a^2*b*c*d*f^4*h + 560*b^3*c*d*e*f^3*g - 80*a*b^2 
*d^2*e*f^3*g - 24*a^2*b*d^2*e*f^3*h - 480*b^3*c*d*e^2*f^2*h - 160*a*b^2*c* 
d*e*f^3*h))/(64*(a*f - b*e)^5) - ((e + f*x)*(837*b^3*c^2*f^4*g + 3*a^3*d^2 
*f^4*h - 93*a*b^2*c^2*f^4*h + 5*a^2*b*d^2*f^4*g - 744*b^3*c^2*e*f^3*h - 57 
6*b^3*d^2*e^3*f*h + 656*b^3*d^2*e^2*f^2*g - 240*a*b^2*d^2*e^2*f^2*h - 186* 
a*b^2*c*d*f^4*g + 10*a^2*b*c*d*f^4*h - 1488*b^3*c*d*e*f^3*g + 176*a*b^2*d^ 
2*e*f^3*g - 24*a^2*b*d^2*e*f^3*h + 1312*b^3*c*d*e^2*f^2*h + 352*a*b^2*c*d* 
e*f^3*h))/(64*b^2*(a*f - b*e)^2) + ((e + f*x)^2*(511*a*b^2*c^2*f^4*h - 33* 
a^3*d^2*f^4*h - 4599*b^3*c^2*f^4*g + 73*a^2*b*d^2*f^4*g + 4088*b^3*c^2*e*f 
^3*h + 2880*b^3*d^2*e^3*f*h - 3504*b^3*d^2*e^2*f^2*g + 1872*a*b^2*d^2*e^2* 
f^2*h + 1022*a*b^2*c*d*f^4*g + 146*a^2*b*c*d*f^4*h + 8176*b^3*c*d*e*f^3...
 

Reduce [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 7371, normalized size of antiderivative = 10.61 \[ \int \frac {(c+d x)^2 (g+h x)}{(a+b x)^5 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(9*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**7*d**2*f**4*h + 30*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*f**4*h - 7 
2*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sq 
rt(a*f - b*e)))*a**6*b*d**2*e*f**3*h + 15*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**2*f**4*g 
 + 36*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**2*f**4*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**5*b**2*c** 
2*f**4*h - 480*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**2*c*d*e*f**3*h + 210*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**2*c*d*f**4*g + 120*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**2*c*d*f**4*h*x + 432*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**2*d**2*e**2*f**2*h - 240*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**2*d**2*e*f* 
*3*g - 288*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(s 
qrt(b)*sqrt(a*f - b*e)))*a**5*b**2*d**2*e*f**3*h*x + 60*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...