\(\int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^2} \, dx\) [87]

Optimal result

Integrand size = 29, antiderivative size = 424 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^2} \, dx=\frac {(b c-a d)^2 \left (a d (3 d f g+2 d e h-5 c f h)+b \left (6 d^2 e g+11 c^2 f h-c d (9 f g+8 e h)\right )\right ) \sqrt {e+f x}}{d^6}+\frac {2 (b c-a d)^2 (3 b d g-4 b c h+a d h) (e+f x)^{3/2}}{3 d^5}+\frac {(b c-a d)^3 (d g-c h) (e+f x)^{3/2}}{d^5 (c+d x)}+\frac {2 b \left (3 a^2 d^2 f^2 h+3 a b d f (d f g-d e h-2 c f h)+b^2 \left (3 c^2 f^2 h-d^2 e (f g-e h)-2 c d f (f g-e h)\right )\right ) (e+f x)^{5/2}}{5 d^4 f^3}+\frac {2 b^2 (3 a d f h+b (d f g-2 d e h-2 c f h)) (e+f x)^{7/2}}{7 d^3 f^3}+\frac {2 b^3 h (e+f x)^{9/2}}{9 d^2 f^3}-\frac {(b c-a d)^2 \sqrt {d e-c f} \left (a d (3 d f g+2 d e h-5 c f h)+b \left (6 d^2 e g+11 c^2 f h-c d (9 f g+8 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{13/2}} \] Output:

(-a*d+b*c)^2*(a*d*(-5*c*f*h+2*d*e*h+3*d*f*g)+b*(6*d^2*e*g+11*c^2*f*h-c*d*( 
8*e*h+9*f*g)))*(f*x+e)^(1/2)/d^6+2/3*(-a*d+b*c)^2*(a*d*h-4*b*c*h+3*b*d*g)* 
(f*x+e)^(3/2)/d^5+(-a*d+b*c)^3*(-c*h+d*g)*(f*x+e)^(3/2)/d^5/(d*x+c)+2/5*b* 
(3*a^2*d^2*f^2*h+3*a*b*d*f*(-2*c*f*h-d*e*h+d*f*g)+b^2*(3*c^2*f^2*h-d^2*e*( 
-e*h+f*g)-2*c*d*f*(-e*h+f*g)))*(f*x+e)^(5/2)/d^4/f^3+2/7*b^2*(3*a*d*f*h+b* 
(-2*c*f*h-2*d*e*h+d*f*g))*(f*x+e)^(7/2)/d^3/f^3+2/9*b^3*h*(f*x+e)^(9/2)/d^ 
2/f^3-(-a*d+b*c)^2*(-c*f+d*e)^(1/2)*(a*d*(-5*c*f*h+2*d*e*h+3*d*f*g)+b*(6*d 
^2*e*g+11*c^2*f*h-c*d*(8*e*h+9*f*g)))*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+ 
d*e)^(1/2))/d^(13/2)
 

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 710, normalized size of antiderivative = 1.67 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^2} \, dx=\frac {\sqrt {e+f x} \left (63 a^2 b d^2 f^2 \left (105 c^3 f^2 h-5 c^2 d f (15 f g+19 e h-14 f h x)+2 d^3 x \left (3 e^2 h+f^2 x (5 g+3 h x)+e f (20 g+6 h x)\right )+c d^2 \left (6 e^2 h+e f (55 g-68 h x)-2 f^2 x (25 g+7 h x)\right )\right )+105 a^3 d^3 f^3 \left (-15 c^2 f h+c d (9 f g+11 e h-10 f h x)+d^2 (2 f x (3 g+h x)+e (-3 g+8 h x))\right )-9 a b^2 d f \left (945 c^4 f^3 h-105 c^3 d f^2 (7 f g+9 e h-6 f h x)-6 d^4 x (e+f x)^2 (7 f g-2 e h+5 f h x)+7 c^2 d^2 f \left (12 e^2 h+e f (95 g-96 h x)-2 f^2 x (35 g+9 h x)\right )+2 c d^3 \left (6 e^3 h+f^3 x^2 (49 g+27 h x)+2 e f^2 x (119 g+30 h x)+e^2 f (-21 g+39 h x)\right )\right )+b^3 \left (3465 c^5 f^4 h-105 c^4 d f^3 (27 f g+35 e h-22 f h x)+2 d^5 x (e+f x)^2 \left (8 e^2 h+5 f^2 x (9 g+7 h x)-2 e f (9 g+10 h x)\right )+21 c^3 d^2 f^2 \left (18 e^2 h+e f (135 g-124 h x)-2 f^2 x (45 g+11 h x)\right )+18 c^2 d^3 f \left (4 e^3 h+f^3 x^2 (21 g+11 h x)+2 e f^2 x (56 g+13 h x)+e^2 f (-14 g+19 h x)\right )+2 c d^4 (e+f x)^2 \left (8 e^2 h-2 e f (9 g-8 h x)-f^2 x (81 g+55 h x)\right )\right )\right )}{315 d^6 f^3 (c+d x)}-\frac {(b c-a d)^2 \sqrt {-d e+c f} \left (a d (3 d f g+2 d e h-5 c f h)+b \left (6 d^2 e g+11 c^2 f h-c d (9 f g+8 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{13/2}} \] Input:

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

Output:

(Sqrt[e + f*x]*(63*a^2*b*d^2*f^2*(105*c^3*f^2*h - 5*c^2*d*f*(15*f*g + 19*e 
*h - 14*f*h*x) + 2*d^3*x*(3*e^2*h + f^2*x*(5*g + 3*h*x) + e*f*(20*g + 6*h* 
x)) + c*d^2*(6*e^2*h + e*f*(55*g - 68*h*x) - 2*f^2*x*(25*g + 7*h*x))) + 10 
5*a^3*d^3*f^3*(-15*c^2*f*h + c*d*(9*f*g + 11*e*h - 10*f*h*x) + d^2*(2*f*x* 
(3*g + h*x) + e*(-3*g + 8*h*x))) - 9*a*b^2*d*f*(945*c^4*f^3*h - 105*c^3*d* 
f^2*(7*f*g + 9*e*h - 6*f*h*x) - 6*d^4*x*(e + f*x)^2*(7*f*g - 2*e*h + 5*f*h 
*x) + 7*c^2*d^2*f*(12*e^2*h + e*f*(95*g - 96*h*x) - 2*f^2*x*(35*g + 9*h*x) 
) + 2*c*d^3*(6*e^3*h + f^3*x^2*(49*g + 27*h*x) + 2*e*f^2*x*(119*g + 30*h*x 
) + e^2*f*(-21*g + 39*h*x))) + b^3*(3465*c^5*f^4*h - 105*c^4*d*f^3*(27*f*g 
 + 35*e*h - 22*f*h*x) + 2*d^5*x*(e + f*x)^2*(8*e^2*h + 5*f^2*x*(9*g + 7*h* 
x) - 2*e*f*(9*g + 10*h*x)) + 21*c^3*d^2*f^2*(18*e^2*h + e*f*(135*g - 124*h 
*x) - 2*f^2*x*(45*g + 11*h*x)) + 18*c^2*d^3*f*(4*e^3*h + f^3*x^2*(21*g + 1 
1*h*x) + 2*e*f^2*x*(56*g + 13*h*x) + e^2*f*(-14*g + 19*h*x)) + 2*c*d^4*(e 
+ f*x)^2*(8*e^2*h - 2*e*f*(9*g - 8*h*x) - f^2*x*(81*g + 55*h*x)))))/(315*d 
^6*f^3*(c + d*x)) - ((b*c - a*d)^2*Sqrt[-(d*e) + c*f]*(a*d*(3*d*f*g + 2*d* 
e*h - 5*c*f*h) + b*(6*d^2*e*g + 11*c^2*f*h - c*d*(9*f*g + 8*e*h)))*ArcTan[ 
(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(13/2)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.19, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {166, 27, 170, 27, 164, 60, 60, 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.

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\int \frac {(a+b x)^2 (e+f x)^{3/2} (6 b e (d g-c h)+a (3 d f g+2 d e h-5 c f h)+b (9 d f g+2 d e h-11 c f h) x)}{2 (c+d x)}dx}{d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{d (c+d x) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b x)^2 (e+f x)^{3/2} (6 b e (d g-c h)+a (3 d f g+2 d e h-5 c f h)+b (9 d f g+2 d e h-11 c f h) x)}{c+d x}dx}{2 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{d (c+d x) (d e-c f)}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {\frac {2 \int -\frac {(a+b x) (e+f x)^{3/2} \left (b c (4 b e+5 a f) (9 d f g+2 d e h-11 c f h)-9 a d f (6 b e (d g-c h)+a (3 d f g+2 d e h-5 c f h))-b \left (a d f (63 d f g+26 d e h-89 c f h)+b \left (2 e (9 f g-4 e h) d^2-c f (81 f g+28 e h) d+99 c^2 f^2 h\right )\right ) x\right )}{2 (c+d x)}dx}{9 d f}+\frac {2 b (a+b x)^2 (e+f x)^{5/2} (-11 c f h+2 d e h+9 d f g)}{9 d f}}{2 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{d (c+d x) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 b (a+b x)^2 (e+f x)^{5/2} (-11 c f h+2 d e h+9 d f g)}{9 d f}-\frac {\int \frac {(a+b x) (e+f x)^{3/2} \left (b c (4 b e+5 a f) (9 d f g+2 d e h-11 c f h)-9 a d f (6 b e (d g-c h)+a (3 d f g+2 d e h-5 c f h))-b \left (a d f (63 d f g+26 d e h-89 c f h)+b \left (2 e (9 f g-4 e h) d^2-c f (81 f g+28 e h) d+99 c^2 f^2 h\right )\right ) x\right )}{c+d x}dx}{9 d f}}{2 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{d (c+d x) (d e-c f)}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {\frac {2 b (a+b x)^2 (e+f x)^{5/2} (-11 c f h+2 d e h+9 d f g)}{9 d f}-\frac {-\frac {9 f (b c-a d)^2 \left (a d (-5 c f h+2 d e h+3 d f g)+b \left (11 c^2 f h-c d (8 e h+9 f g)+6 d^2 e g\right )\right ) \int \frac {(e+f x)^{3/2}}{c+d x}dx}{d^2}-\frac {2 b (e+f x)^{5/2} \left (14 a^2 d^2 f^2 (-67 c f h+22 d e h+45 d f g)+5 b d f x \left (a d f (-89 c f h+26 d e h+63 d f g)+b \left (99 c^2 f^2 h-c d f (28 e h+81 f g)+2 d^2 e (9 f g-4 e h)\right )\right )+27 a b d f \left (63 c^2 f^2 h-c d f (24 e h+49 f g)+2 d^2 e (7 f g-2 e h)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (34 e h+63 f g)+8 c d^2 e f (27 f g-7 e h)+4 d^3 e^2 (9 f g-4 e h)\right )\right )\right )}{35 d^2 f^2}}{9 d f}}{2 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{d (c+d x) (d e-c f)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {2 b (a+b x)^2 (e+f x)^{5/2} (-11 c f h+2 d e h+9 d f g)}{9 d f}-\frac {-\frac {9 f (b c-a d)^2 \left (a d (-5 c f h+2 d e h+3 d f g)+b \left (11 c^2 f h-c d (8 e h+9 f g)+6 d^2 e g\right )\right ) \left (\frac {(d e-c f) \int \frac {\sqrt {e+f x}}{c+d x}dx}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}-\frac {2 b (e+f x)^{5/2} \left (14 a^2 d^2 f^2 (-67 c f h+22 d e h+45 d f g)+5 b d f x \left (a d f (-89 c f h+26 d e h+63 d f g)+b \left (99 c^2 f^2 h-c d f (28 e h+81 f g)+2 d^2 e (9 f g-4 e h)\right )\right )+27 a b d f \left (63 c^2 f^2 h-c d f (24 e h+49 f g)+2 d^2 e (7 f g-2 e h)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (34 e h+63 f g)+8 c d^2 e f (27 f g-7 e h)+4 d^3 e^2 (9 f g-4 e h)\right )\right )\right )}{35 d^2 f^2}}{9 d f}}{2 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{d (c+d x) (d e-c f)}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {2 b (a+b x)^2 (e+f x)^{5/2} (-11 c f h+2 d e h+9 d f g)}{9 d f}-\frac {-\frac {9 f (b c-a d)^2 \left (a d (-5 c f h+2 d e h+3 d f g)+b \left (11 c^2 f h-c d (8 e h+9 f g)+6 d^2 e g\right )\right ) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}-\frac {2 b (e+f x)^{5/2} \left (14 a^2 d^2 f^2 (-67 c f h+22 d e h+45 d f g)+5 b d f x \left (a d f (-89 c f h+26 d e h+63 d f g)+b \left (99 c^2 f^2 h-c d f (28 e h+81 f g)+2 d^2 e (9 f g-4 e h)\right )\right )+27 a b d f \left (63 c^2 f^2 h-c d f (24 e h+49 f g)+2 d^2 e (7 f g-2 e h)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (34 e h+63 f g)+8 c d^2 e f (27 f g-7 e h)+4 d^3 e^2 (9 f g-4 e h)\right )\right )\right )}{35 d^2 f^2}}{9 d f}}{2 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{d (c+d x) (d e-c f)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {2 b (a+b x)^2 (e+f x)^{5/2} (-11 c f h+2 d e h+9 d f g)}{9 d f}-\frac {-\frac {9 f (b c-a d)^2 \left (a d (-5 c f h+2 d e h+3 d f g)+b \left (11 c^2 f h-c d (8 e h+9 f g)+6 d^2 e g\right )\right ) \left (\frac {(d e-c f) \left (\frac {2 (d e-c f) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d f}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{d^2}-\frac {2 b (e+f x)^{5/2} \left (14 a^2 d^2 f^2 (-67 c f h+22 d e h+45 d f g)+5 b d f x \left (a d f (-89 c f h+26 d e h+63 d f g)+b \left (99 c^2 f^2 h-c d f (28 e h+81 f g)+2 d^2 e (9 f g-4 e h)\right )\right )+27 a b d f \left (63 c^2 f^2 h-c d f (24 e h+49 f g)+2 d^2 e (7 f g-2 e h)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (34 e h+63 f g)+8 c d^2 e f (27 f g-7 e h)+4 d^3 e^2 (9 f g-4 e h)\right )\right )\right )}{35 d^2 f^2}}{9 d f}}{2 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{d (c+d x) (d e-c f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {2 b (a+b x)^2 (e+f x)^{5/2} (-11 c f h+2 d e h+9 d f g)}{9 d f}-\frac {-\frac {2 b (e+f x)^{5/2} \left (14 a^2 d^2 f^2 (-67 c f h+22 d e h+45 d f g)+5 b d f x \left (a d f (-89 c f h+26 d e h+63 d f g)+b \left (99 c^2 f^2 h-c d f (28 e h+81 f g)+2 d^2 e (9 f g-4 e h)\right )\right )+27 a b d f \left (63 c^2 f^2 h-c d f (24 e h+49 f g)+2 d^2 e (7 f g-2 e h)\right )-\left (b^2 \left (693 c^3 f^3 h-9 c^2 d f^2 (34 e h+63 f g)+8 c d^2 e f (27 f g-7 e h)+4 d^3 e^2 (9 f g-4 e h)\right )\right )\right )}{35 d^2 f^2}-\frac {9 f (b c-a d)^2 \left (\frac {(d e-c f) \left (\frac {2 \sqrt {e+f x}}{d}-\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2}}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right ) \left (a d (-5 c f h+2 d e h+3 d f g)+b \left (11 c^2 f h-c d (8 e h+9 f g)+6 d^2 e g\right )\right )}{d^2}}{9 d f}}{2 d (d e-c f)}-\frac {(a+b x)^3 (e+f x)^{5/2} (d g-c h)}{d (c+d x) (d e-c f)}\)

Input:

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

Output:

-(((d*g - c*h)*(a + b*x)^3*(e + f*x)^(5/2))/(d*(d*e - c*f)*(c + d*x))) + ( 
(2*b*(9*d*f*g + 2*d*e*h - 11*c*f*h)*(a + b*x)^2*(e + f*x)^(5/2))/(9*d*f) - 
 ((-2*b*(e + f*x)^(5/2)*(14*a^2*d^2*f^2*(45*d*f*g + 22*d*e*h - 67*c*f*h) + 
 27*a*b*d*f*(63*c^2*f^2*h + 2*d^2*e*(7*f*g - 2*e*h) - c*d*f*(49*f*g + 24*e 
*h)) - b^2*(693*c^3*f^3*h + 8*c*d^2*e*f*(27*f*g - 7*e*h) + 4*d^3*e^2*(9*f* 
g - 4*e*h) - 9*c^2*d*f^2*(63*f*g + 34*e*h)) + 5*b*d*f*(a*d*f*(63*d*f*g + 2 
6*d*e*h - 89*c*f*h) + b*(99*c^2*f^2*h + 2*d^2*e*(9*f*g - 4*e*h) - c*d*f*(8 
1*f*g + 28*e*h)))*x))/(35*d^2*f^2) - (9*(b*c - a*d)^2*f*(a*d*(3*d*f*g + 2* 
d*e*h - 5*c*f*h) + b*(6*d^2*e*g + 11*c^2*f*h - c*d*(9*f*g + 8*e*h)))*((2*( 
e + f*x)^(3/2))/(3*d) + ((d*e - c*f)*((2*Sqrt[e + f*x])/d - (2*Sqrt[d*e - 
c*f]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(3/2)))/d))/d^2)/ 
(9*d*f))/(2*d*(d*e - c*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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(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*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

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 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

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 = 0.76 (sec) , antiderivative size = 767, normalized size of antiderivative = 1.81

method	result	size
pseudoelliptic	\(-\frac {5 \left (-\left (a d -b c \right )^{2} \left (c f -d e \right ) \left (x d +c \right ) \left (\frac {\left (-3 a f g -2 e \left (a h +3 b g \right )\right ) d^{2}}{5}+c \left (\left (a h +\frac {9 b g}{5}\right ) f +\frac {8 e h b}{5}\right ) d -\frac {11 b \,c^{2} f h}{5}\right ) f^{3} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\left (\frac {\left (-2 x \left (\frac {x^{3} \left (\frac {7 h x}{9}+g \right ) b^{3}}{7}+\frac {3 a \,x^{2} \left (\frac {5 h x}{7}+g \right ) b^{2}}{5}+a^{2} x \left (\frac {3 h x}{5}+g \right ) b +a^{3} \left (\frac {h x}{3}+g \right )\right ) f^{4}+\left (-\frac {16 x^{3} \left (\frac {25 h x}{36}+g \right ) b^{3}}{35}-\frac {12 a \,x^{2} \left (\frac {4 h x}{7}+g \right ) b^{2}}{5}-8 a^{2} x \left (\frac {3 h x}{10}+g \right ) b +a^{3} \left (-\frac {8 h x}{3}+g \right )\right ) e \,f^{3}-\frac {6 \left (\frac {x \left (\frac {h x}{3}+g \right ) b^{2}}{21}+a \left (\frac {h x}{7}+g \right ) b +a^{2} h \right ) x b \,e^{2} f^{2}}{5}+\frac {12 \left (\frac {\left (\frac {2 h x}{9}+g \right ) b}{3}+a h \right ) x \,b^{2} e^{3} f}{35}-\frac {16 b^{3} e^{4} h x}{315}\right ) d^{5}}{5}-\frac {11 c \left (\left (\frac {2 \left (-\frac {1}{3} h \,x^{4}-\frac {27}{55} g \,x^{3}\right ) b^{3}}{7}-\frac {42 \left (\frac {27 h x}{49}+g \right ) a \,x^{2} b^{2}}{55}-\frac {30 a^{2} x \left (\frac {7 h x}{25}+g \right ) b}{11}+\frac {9 \left (-\frac {10 h x}{9}+g \right ) a^{3}}{11}\right ) f^{4}+\left (-\frac {24 x^{2} \left (\frac {47 h x}{90}+g \right ) b^{3}}{77}-\frac {204 \left (\frac {30 h x}{119}+g \right ) a x \,b^{2}}{55}+3 a^{2} \left (-\frac {68 h x}{55}+g \right ) b +h \,a^{3}\right ) e \,f^{3}+\frac {18 \left (-\frac {13 x \left (\frac {5 h x}{39}+g \right ) b^{2}}{21}+a \left (-\frac {13 h x}{7}+g \right ) b +a^{2} h \right ) b \,e^{2} f^{2}}{55}-\frac {36 \left (\frac {\left (-\frac {16 h x}{9}+g \right ) b}{3}+a h \right ) b^{2} e^{3} f}{385}+\frac {16 b^{3} e^{4} h}{1155}\right ) d^{4}}{15}+c^{2} \left (\left (-\frac {6 x^{2} \left (\frac {11 h x}{21}+g \right ) b^{3}}{25}-\frac {14 a x \left (\frac {9 h x}{35}+g \right ) b^{2}}{5}+3 a^{2} \left (-\frac {14 h x}{15}+g \right ) b +h \,a^{3}\right ) f^{3}+\frac {19 b e \left (-\frac {32 x \left (\frac {13 h x}{56}+g \right ) b^{2}}{95}+a \left (-\frac {96 h x}{95}+g \right ) b +a^{2} h \right ) f^{2}}{5}+\frac {12 \left (\frac {\left (-\frac {19 h x}{14}+g \right ) b}{3}+a h \right ) b^{2} e^{2} f}{25}-\frac {8 b^{3} e^{3} h}{175}\right ) f \,d^{3}-\frac {21 c^{3} \left (\left (-\frac {2 x \left (\frac {11 h x}{45}+g \right ) b^{2}}{7}+a \left (-\frac {6 h x}{7}+g \right ) b +a^{2} h \right ) f^{2}+\frac {9 \left (\frac {\left (-\frac {124 h x}{135}+g \right ) b}{3}+a h \right ) b e f}{7}+\frac {2 b^{2} e^{2} h}{35}\right ) b \,f^{2} d^{2}}{5}+\frac {27 c^{4} b^{2} f^{3} \left (\left (\frac {\left (-\frac {22 h x}{27}+g \right ) b}{3}+a h \right ) f +\frac {35 e h b}{81}\right ) d}{5}-\frac {11 b^{3} c^{5} f^{4} h}{5}\right ) \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}\right )}{\sqrt {\left (c f -d e \right ) d}\, f^{3} \left (x d +c \right ) d^{6}}\)	\(767\)
risch	\(\text {Expression too large to display}\)	\(1051\)
derivativedivides	\(\text {Expression too large to display}\)	\(1356\)
default	\(\text {Expression too large to display}\)	\(1356\)

Input:

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

Output:

-5/((c*f-d*e)*d)^(1/2)*(-(a*d-b*c)^2*(c*f-d*e)*(d*x+c)*(1/5*(-3*a*f*g-2*e* 
(a*h+3*b*g))*d^2+c*((a*h+9/5*b*g)*f+8/5*e*h*b)*d-11/5*b*c^2*f*h)*f^3*arcta 
n(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+(1/5*(-2*x*(1/7*x^3*(7/9*h*x+g)*b^3 
+3/5*a*x^2*(5/7*h*x+g)*b^2+a^2*x*(3/5*h*x+g)*b+a^3*(1/3*h*x+g))*f^4+(-16/3 
5*x^3*(25/36*h*x+g)*b^3-12/5*a*x^2*(4/7*h*x+g)*b^2-8*a^2*x*(3/10*h*x+g)*b+ 
a^3*(-8/3*h*x+g))*e*f^3-6/5*(1/21*x*(1/3*h*x+g)*b^2+a*(1/7*h*x+g)*b+a^2*h) 
*x*b*e^2*f^2+12/35*(1/3*(2/9*h*x+g)*b+a*h)*x*b^2*e^3*f-16/315*b^3*e^4*h*x) 
*d^5-11/15*c*((2/7*(-1/3*h*x^4-27/55*g*x^3)*b^3-42/55*(27/49*h*x+g)*a*x^2* 
b^2-30/11*a^2*x*(7/25*h*x+g)*b+9/11*(-10/9*h*x+g)*a^3)*f^4+(-24/77*x^2*(47 
/90*h*x+g)*b^3-204/55*(30/119*h*x+g)*a*x*b^2+3*a^2*(-68/55*h*x+g)*b+h*a^3) 
*e*f^3+18/55*(-13/21*x*(5/39*h*x+g)*b^2+a*(-13/7*h*x+g)*b+a^2*h)*b*e^2*f^2 
-36/385*(1/3*(-16/9*h*x+g)*b+a*h)*b^2*e^3*f+16/1155*b^3*e^4*h)*d^4+c^2*((- 
6/25*x^2*(11/21*h*x+g)*b^3-14/5*a*x*(9/35*h*x+g)*b^2+3*a^2*(-14/15*h*x+g)* 
b+h*a^3)*f^3+19/5*b*e*(-32/95*x*(13/56*h*x+g)*b^2+a*(-96/95*h*x+g)*b+a^2*h 
)*f^2+12/25*(1/3*(-19/14*h*x+g)*b+a*h)*b^2*e^2*f-8/175*b^3*e^3*h)*f*d^3-21 
/5*c^3*((-2/7*x*(11/45*h*x+g)*b^2+a*(-6/7*h*x+g)*b+a^2*h)*f^2+9/7*(1/3*(-1 
24/135*h*x+g)*b+a*h)*b*e*f+2/35*b^2*e^2*h)*b*f^2*d^2+27/5*c^4*b^2*f^3*((1/ 
3*(-22/27*h*x+g)*b+a*h)*f+35/81*e*h*b)*d-11/5*b^3*c^5*f^4*h)*((c*f-d*e)*d) 
^(1/2)*(f*x+e)^(1/2))/f^3/(d*x+c)/d^6
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1499 vs. \(2 (394) = 788\).

Time = 0.22 (sec) , antiderivative size = 3008, normalized size of antiderivative = 7.09 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x+a)^3*(f*x+e)^(3/2)*(h*x+g)/(d*x+c)^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. 1485 vs. \(2 (394) = 788\).

Time = 0.17 (sec) , antiderivative size = 1485, normalized size of antiderivative = 3.50 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(6*b^3*c^2*d^3*e^2*g - 12*a*b^2*c*d^4*e^2*g + 6*a^2*b*d^5*e^2*g - 15*b^3*c 
^3*d^2*e*f*g + 33*a*b^2*c^2*d^3*e*f*g - 21*a^2*b*c*d^4*e*f*g + 3*a^3*d^5*e 
*f*g + 9*b^3*c^4*d*f^2*g - 21*a*b^2*c^3*d^2*f^2*g + 15*a^2*b*c^2*d^3*f^2*g 
 - 3*a^3*c*d^4*f^2*g - 8*b^3*c^3*d^2*e^2*h + 18*a*b^2*c^2*d^3*e^2*h - 12*a 
^2*b*c*d^4*e^2*h + 2*a^3*d^5*e^2*h + 19*b^3*c^4*d*e*f*h - 45*a*b^2*c^3*d^2 
*e*f*h + 33*a^2*b*c^2*d^3*e*f*h - 7*a^3*c*d^4*e*f*h - 11*b^3*c^5*f^2*h + 2 
7*a*b^2*c^4*d*f^2*h - 21*a^2*b*c^3*d^2*f^2*h + 5*a^3*c^2*d^3*f^2*h)*arctan 
(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)*d^6) + (sqrt( 
f*x + e)*b^3*c^3*d^2*e*f*g - 3*sqrt(f*x + e)*a*b^2*c^2*d^3*e*f*g + 3*sqrt( 
f*x + e)*a^2*b*c*d^4*e*f*g - sqrt(f*x + e)*a^3*d^5*e*f*g - sqrt(f*x + e)*b 
^3*c^4*d*f^2*g + 3*sqrt(f*x + e)*a*b^2*c^3*d^2*f^2*g - 3*sqrt(f*x + e)*a^2 
*b*c^2*d^3*f^2*g + sqrt(f*x + e)*a^3*c*d^4*f^2*g - sqrt(f*x + e)*b^3*c^4*d 
*e*f*h + 3*sqrt(f*x + e)*a*b^2*c^3*d^2*e*f*h - 3*sqrt(f*x + e)*a^2*b*c^2*d 
^3*e*f*h + sqrt(f*x + e)*a^3*c*d^4*e*f*h + sqrt(f*x + e)*b^3*c^5*f^2*h - 3 
*sqrt(f*x + e)*a*b^2*c^4*d*f^2*h + 3*sqrt(f*x + e)*a^2*b*c^3*d^2*f^2*h - s 
qrt(f*x + e)*a^3*c^2*d^3*f^2*h)/(((f*x + e)*d - d*e + c*f)*d^6) + 2/315*(4 
5*(f*x + e)^(7/2)*b^3*d^16*f^25*g - 63*(f*x + e)^(5/2)*b^3*d^16*e*f^25*g - 
 126*(f*x + e)^(5/2)*b^3*c*d^15*f^26*g + 189*(f*x + e)^(5/2)*a*b^2*d^16*f^ 
26*g + 315*(f*x + e)^(3/2)*b^3*c^2*d^14*f^27*g - 630*(f*x + e)^(3/2)*a*b^2 
*c*d^15*f^27*g + 315*(f*x + e)^(3/2)*a^2*b*d^16*f^27*g + 945*sqrt(f*x +...
 

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 1602, normalized size of antiderivative = 3.78 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(e + f*x)^(7/2)*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(7*d^2*f^3) - (4*b^ 
3*h*(c*f - d*e))/(7*d^3*f^3)) - (e + f*x)^(1/2)*((2*(c*f - d*e)*((2*(c*f - 
 d*e)*((2*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d^2*f^3) - (4*b^3*h*(c*f 
 - d*e))/(d^3*f^3))*(c*f - d*e))/d - (6*b*(a*f - b*e)*(a*f*h - 2*b*e*h + b 
*f*g))/(d^2*f^3) + (2*b^3*h*(c*f - d*e)^2)/(d^4*f^3)))/d - (((2*b^3*f*g - 
8*b^3*e*h + 6*a*b^2*f*h)/(d^2*f^3) - (4*b^3*h*(c*f - d*e))/(d^3*f^3))*(c*f 
 - d*e)^2)/d^2 + (2*(a*f - b*e)^2*(a*f*h - 4*b*e*h + 3*b*f*g))/(d^2*f^3))) 
/d - ((c*f - d*e)^2*((2*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d^2*f^3) - 
 (4*b^3*h*(c*f - d*e))/(d^3*f^3))*(c*f - d*e))/d - (6*b*(a*f - b*e)*(a*f*h 
 - 2*b*e*h + b*f*g))/(d^2*f^3) + (2*b^3*h*(c*f - d*e)^2)/(d^4*f^3)))/d^2 + 
 (2*(a*f - b*e)^3*(e*h - f*g))/(d^2*f^3)) + (e + f*x)^(3/2)*((2*(c*f - d*e 
)*((2*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d^2*f^3) - (4*b^3*h*(c*f - d 
*e))/(d^3*f^3))*(c*f - d*e))/d - (6*b*(a*f - b*e)*(a*f*h - 2*b*e*h + b*f*g 
))/(d^2*f^3) + (2*b^3*h*(c*f - d*e)^2)/(d^4*f^3)))/(3*d) - (((2*b^3*f*g - 
8*b^3*e*h + 6*a*b^2*f*h)/(d^2*f^3) - (4*b^3*h*(c*f - d*e))/(d^3*f^3))*(c*f 
 - d*e)^2)/(3*d^2) + (2*(a*f - b*e)^2*(a*f*h - 4*b*e*h + 3*b*f*g))/(3*d^2* 
f^3)) - (e + f*x)^(5/2)*((2*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d^2*f^ 
3) - (4*b^3*h*(c*f - d*e))/(d^3*f^3))*(c*f - d*e))/(5*d) - (6*b*(a*f - b*e 
)*(a*f*h - 2*b*e*h + b*f*g))/(5*d^2*f^3) + (2*b^3*h*(c*f - d*e)^2)/(5*d^4* 
f^3)) + ((e + f*x)^(1/2)*(b^3*c^5*f^2*h - a^3*d^5*e*f*g + a^3*c*d^4*f^2...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 3285, normalized size of antiderivative = 7.75 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2} (g+h x)}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(1575*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d 
*e)))*a**3*c**2*d**3*f**4*h - 630*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f 
*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c*d**4*e*f**3*h - 945*sqrt(d)*sqrt( 
c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c*d**4*f 
**4*g + 1575*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt( 
c*f - d*e)))*a**3*c*d**4*f**4*h*x - 630*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt 
(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*d**5*e*f**3*h*x - 945*sqrt(d) 
*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*d* 
*5*f**4*g*x - 6615*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d) 
*sqrt(c*f - d*e)))*a**2*b*c**3*d**2*f**4*h + 3780*sqrt(d)*sqrt(c*f - d*e)* 
atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**2*d**3*e*f**3* 
h + 4725*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f 
- d*e)))*a**2*b*c**2*d**3*f**4*g - 6615*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt 
(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**2*d**3*f**4*h*x - 1890*s 
qrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a 
**2*b*c*d**4*e*f**3*g + 3780*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d 
)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c*d**4*e*f**3*h*x + 4725*sqrt(d)*sqrt( 
c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c*d**4 
*f**4*g*x - 1890*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*s 
qrt(c*f - d*e)))*a**2*b*d**5*e*f**3*g*x + 8505*sqrt(d)*sqrt(c*f - d*e)*...