\(\int \frac {(a+b x)^3 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx\) [159]

Optimal result

Integrand size = 29, antiderivative size = 575 \[ \int \frac {(a+b x)^3 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 (b e-a f)^3 (d e-c f)^2 (f g-e h)}{f^7 \sqrt {e+f x}}+\frac {2 (b e-a f)^2 (d e-c f) (b d e (5 f g-6 e h)-b c f (3 f g-4 e h)-a f (2 d f g-3 d e h+c f h)) \sqrt {e+f x}}{f^7}-\frac {2 (b e-a f) \left (a^2 d f^2 (d f g-3 d e h+2 c f h)+a b f \left (3 c^2 f^2 h-d^2 e (8 f g-15 e h)+2 c d f (3 f g-8 e h)\right )-b^2 \left (4 c d e f (3 f g-5 e h)-5 d^2 e^2 (2 f g-3 e h)-3 c^2 f^2 (f g-2 e h)\right )\right ) (e+f x)^{3/2}}{3 f^7}+\frac {2 \left (a^3 d^2 f^3 h+3 a^2 b d f^2 (d f g-4 d e h+2 c f h)+3 a b^2 f \left (c^2 f^2 h-2 d^2 e (2 f g-5 e h)+2 c d f (f g-4 e h)\right )-b^3 \left (4 c d e f (2 f g-5 e h)-c^2 f^2 (f g-4 e h)-10 d^2 e^2 (f g-2 e h)\right )\right ) (e+f x)^{5/2}}{5 f^7}+\frac {2 b \left (3 a^2 d^2 f^2 h+3 a b d f (d f g-5 d e h+2 c f h)+b^2 \left (c^2 f^2 h+2 c d f (f g-5 e h)-5 d^2 e (f g-3 e h)\right )\right ) (e+f x)^{7/2}}{7 f^7}+\frac {2 b^2 d (3 a d f h+b (d f g-6 d e h+2 c f h)) (e+f x)^{9/2}}{9 f^7}+\frac {2 b^3 d^2 h (e+f x)^{11/2}}{11 f^7} \] Output:

2*(-a*f+b*e)^3*(-c*f+d*e)^2*(-e*h+f*g)/f^7/(f*x+e)^(1/2)+2*(-a*f+b*e)^2*(- 
c*f+d*e)*(b*d*e*(-6*e*h+5*f*g)-b*c*f*(-4*e*h+3*f*g)-a*f*(c*f*h-3*d*e*h+2*d 
*f*g))*(f*x+e)^(1/2)/f^7-2/3*(-a*f+b*e)*(a^2*d*f^2*(2*c*f*h-3*d*e*h+d*f*g) 
+a*b*f*(3*c^2*f^2*h-d^2*e*(-15*e*h+8*f*g)+2*c*d*f*(-8*e*h+3*f*g))-b^2*(4*c 
*d*e*f*(-5*e*h+3*f*g)-5*d^2*e^2*(-3*e*h+2*f*g)-3*c^2*f^2*(-2*e*h+f*g)))*(f 
*x+e)^(3/2)/f^7+2/5*(a^3*d^2*f^3*h+3*a^2*b*d*f^2*(2*c*f*h-4*d*e*h+d*f*g)+3 
*a*b^2*f*(c^2*f^2*h-2*d^2*e*(-5*e*h+2*f*g)+2*c*d*f*(-4*e*h+f*g))-b^3*(4*c* 
d*e*f*(-5*e*h+2*f*g)-c^2*f^2*(-4*e*h+f*g)-10*d^2*e^2*(-2*e*h+f*g)))*(f*x+e 
)^(5/2)/f^7+2/7*b*(3*a^2*d^2*f^2*h+3*a*b*d*f*(2*c*f*h-5*d*e*h+d*f*g)+b^2*( 
c^2*f^2*h+2*c*d*f*(-5*e*h+f*g)-5*d^2*e*(-3*e*h+f*g)))*(f*x+e)^(7/2)/f^7+2/ 
9*b^2*d*(3*a*d*f*h+b*(2*c*f*h-6*d*e*h+d*f*g))*(f*x+e)^(9/2)/f^7+2/11*b^3*d 
^2*h*(f*x+e)^(11/2)/f^7
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 845, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^3 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 \left (231 a^3 f^3 \left (15 c^2 f^2 (-f g+2 e h+f h x)+10 c d f \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )+d^2 \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )\right )+99 a^2 b f^2 \left (35 c^2 f^2 \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )+14 c d f \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )-3 d^2 \left (128 e^4 h-16 e^3 f (7 g-4 h x)-8 e^2 f^2 x (7 g+2 h x)+2 e f^3 x^2 (7 g+4 h x)-f^4 x^3 (7 g+5 h x)\right )\right )+33 a b^2 f \left (21 c^2 f^2 \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )+18 c d f \left (-128 e^4 h+16 e^3 f (7 g-4 h x)+8 e^2 f^2 x (7 g+2 h x)-2 e f^3 x^2 (7 g+4 h x)+f^4 x^3 (7 g+5 h x)\right )+d^2 \left (1280 e^5 h-128 e^4 f (9 g-5 h x)+16 e^2 f^3 x^2 (9 g+5 h x)-32 e^3 f^2 x (18 g+5 h x)+5 f^5 x^4 (9 g+7 h x)-2 e f^4 x^3 (36 g+25 h x)\right )\right )+b^3 \left (99 c^2 f^2 \left (-128 e^4 h+16 e^3 f (7 g-4 h x)+8 e^2 f^2 x (7 g+2 h x)-2 e f^3 x^2 (7 g+4 h x)+f^4 x^3 (7 g+5 h x)\right )+22 c d f \left (1280 e^5 h-128 e^4 f (9 g-5 h x)+16 e^2 f^3 x^2 (9 g+5 h x)-32 e^3 f^2 x (18 g+5 h x)+5 f^5 x^4 (9 g+7 h x)-2 e f^4 x^3 (36 g+25 h x)\right )-5 d^2 \left (3072 e^6 h-256 e^5 f (11 g-6 h x)-128 e^4 f^2 x (11 g+3 h x)+32 e^3 f^3 x^2 (11 g+6 h x)-7 f^6 x^5 (11 g+9 h x)-8 e^2 f^4 x^3 (22 g+15 h x)+2 e f^5 x^4 (55 g+42 h x)\right )\right )\right )}{3465 f^7 \sqrt {e+f x}} \] Input:

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

Output:

(2*(231*a^3*f^3*(15*c^2*f^2*(-(f*g) + 2*e*h + f*h*x) + 10*c*d*f*(-8*e^2*h 
+ e*f*(6*g - 4*h*x) + f^2*x*(3*g + h*x)) + d^2*(48*e^3*h - 8*e^2*f*(5*g - 
3*h*x) + f^3*x^2*(5*g + 3*h*x) - 2*e*f^2*x*(10*g + 3*h*x))) + 99*a^2*b*f^2 
*(35*c^2*f^2*(-8*e^2*h + e*f*(6*g - 4*h*x) + f^2*x*(3*g + h*x)) + 14*c*d*f 
*(48*e^3*h - 8*e^2*f*(5*g - 3*h*x) + f^3*x^2*(5*g + 3*h*x) - 2*e*f^2*x*(10 
*g + 3*h*x)) - 3*d^2*(128*e^4*h - 16*e^3*f*(7*g - 4*h*x) - 8*e^2*f^2*x*(7* 
g + 2*h*x) + 2*e*f^3*x^2*(7*g + 4*h*x) - f^4*x^3*(7*g + 5*h*x))) + 33*a*b^ 
2*f*(21*c^2*f^2*(48*e^3*h - 8*e^2*f*(5*g - 3*h*x) + f^3*x^2*(5*g + 3*h*x) 
- 2*e*f^2*x*(10*g + 3*h*x)) + 18*c*d*f*(-128*e^4*h + 16*e^3*f*(7*g - 4*h*x 
) + 8*e^2*f^2*x*(7*g + 2*h*x) - 2*e*f^3*x^2*(7*g + 4*h*x) + f^4*x^3*(7*g + 
 5*h*x)) + d^2*(1280*e^5*h - 128*e^4*f*(9*g - 5*h*x) + 16*e^2*f^3*x^2*(9*g 
 + 5*h*x) - 32*e^3*f^2*x*(18*g + 5*h*x) + 5*f^5*x^4*(9*g + 7*h*x) - 2*e*f^ 
4*x^3*(36*g + 25*h*x))) + b^3*(99*c^2*f^2*(-128*e^4*h + 16*e^3*f*(7*g - 4* 
h*x) + 8*e^2*f^2*x*(7*g + 2*h*x) - 2*e*f^3*x^2*(7*g + 4*h*x) + f^4*x^3*(7* 
g + 5*h*x)) + 22*c*d*f*(1280*e^5*h - 128*e^4*f*(9*g - 5*h*x) + 16*e^2*f^3* 
x^2*(9*g + 5*h*x) - 32*e^3*f^2*x*(18*g + 5*h*x) + 5*f^5*x^4*(9*g + 7*h*x) 
- 2*e*f^4*x^3*(36*g + 25*h*x)) - 5*d^2*(3072*e^6*h - 256*e^5*f*(11*g - 6*h 
*x) - 128*e^4*f^2*x*(11*g + 3*h*x) + 32*e^3*f^3*x^2*(11*g + 6*h*x) - 7*f^6 
*x^5*(11*g + 9*h*x) - 8*e^2*f^4*x^3*(22*g + 15*h*x) + 2*e*f^5*x^4*(55*g + 
42*h*x)))))/(3465*f^7*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {165, 2009}

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.

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

\(\Big \downarrow \) 165

\(\displaystyle \int \left (\frac {\sqrt {e+f x} (b e-a f) \left (-a^2 d f^2 (2 c f h-3 d e h+d f g)-a b f \left (3 c^2 f^2 h+2 c d f (3 f g-8 e h)+d^2 (-e) (8 f g-15 e h)\right )+b^2 \left (-3 c^2 f^2 (f g-2 e h)+4 c d e f (3 f g-5 e h)-5 d^2 e^2 (2 f g-3 e h)\right )\right )}{f^6}+\frac {b (e+f x)^{5/2} \left (3 a^2 d^2 f^2 h+3 a b d f (2 c f h-5 d e h+d f g)+b^2 \left (c^2 f^2 h+2 c d f (f g-5 e h)-5 d^2 e (f g-3 e h)\right )\right )}{f^6}+\frac {(e+f x)^{3/2} \left (a^3 d^2 f^3 h+3 a^2 b d f^2 (2 c f h-4 d e h+d f g)+3 a b^2 f \left (c^2 f^2 h+2 c d f (f g-4 e h)-2 d^2 e (2 f g-5 e h)\right )-\left (b^3 \left (-c^2 f^2 (f g-4 e h)+4 c d e f (2 f g-5 e h)-10 d^2 e^2 (f g-2 e h)\right )\right )\right )}{f^6}+\frac {b^2 d (e+f x)^{7/2} (3 a d f h+b (2 c f h-6 d e h+d f g))}{f^6}+\frac {(b e-a f)^2 (d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (3 f g-4 e h)+b d e (5 f g-6 e h))}{f^6 \sqrt {e+f x}}+\frac {(a f-b e)^3 (c f-d e)^2 (f g-e h)}{f^6 (e+f x)^{3/2}}+\frac {b^3 d^2 h (e+f x)^{9/2}}{f^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 (e+f x)^{3/2} (b e-a f) \left (a^2 d f^2 (2 c f h-3 d e h+d f g)+a b f \left (3 c^2 f^2 h+2 c d f (3 f g-8 e h)+d^2 (-e) (8 f g-15 e h)\right )-\left (b^2 \left (-3 c^2 f^2 (f g-2 e h)+4 c d e f (3 f g-5 e h)-5 d^2 e^2 (2 f g-3 e h)\right )\right )\right )}{3 f^7}+\frac {2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2 h+3 a b d f (2 c f h-5 d e h+d f g)+b^2 \left (c^2 f^2 h+2 c d f (f g-5 e h)-5 d^2 e (f g-3 e h)\right )\right )}{7 f^7}+\frac {2 (e+f x)^{5/2} \left (a^3 d^2 f^3 h+3 a^2 b d f^2 (2 c f h-4 d e h+d f g)+3 a b^2 f \left (c^2 f^2 h+2 c d f (f g-4 e h)-2 d^2 e (2 f g-5 e h)\right )-\left (b^3 \left (-c^2 f^2 (f g-4 e h)+4 c d e f (2 f g-5 e h)-10 d^2 e^2 (f g-2 e h)\right )\right )\right )}{5 f^7}+\frac {2 b^2 d (e+f x)^{9/2} (3 a d f h+b (2 c f h-6 d e h+d f g))}{9 f^7}+\frac {2 \sqrt {e+f x} (b e-a f)^2 (d e-c f) (-a f (c f h-3 d e h+2 d f g)-b c f (3 f g-4 e h)+b d e (5 f g-6 e h))}{f^7}+\frac {2 (b e-a f)^3 (d e-c f)^2 (f g-e h)}{f^7 \sqrt {e+f x}}+\frac {2 b^3 d^2 h (e+f x)^{11/2}}{11 f^7}\)

Input:

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

Output:

(2*(b*e - a*f)^3*(d*e - c*f)^2*(f*g - e*h))/(f^7*Sqrt[e + f*x]) + (2*(b*e 
- a*f)^2*(d*e - c*f)*(b*d*e*(5*f*g - 6*e*h) - b*c*f*(3*f*g - 4*e*h) - a*f* 
(2*d*f*g - 3*d*e*h + c*f*h))*Sqrt[e + f*x])/f^7 - (2*(b*e - a*f)*(a^2*d*f^ 
2*(d*f*g - 3*d*e*h + 2*c*f*h) + a*b*f*(3*c^2*f^2*h - d^2*e*(8*f*g - 15*e*h 
) + 2*c*d*f*(3*f*g - 8*e*h)) - b^2*(4*c*d*e*f*(3*f*g - 5*e*h) - 5*d^2*e^2* 
(2*f*g - 3*e*h) - 3*c^2*f^2*(f*g - 2*e*h)))*(e + f*x)^(3/2))/(3*f^7) + (2* 
(a^3*d^2*f^3*h + 3*a^2*b*d*f^2*(d*f*g - 4*d*e*h + 2*c*f*h) + 3*a*b^2*f*(c^ 
2*f^2*h - 2*d^2*e*(2*f*g - 5*e*h) + 2*c*d*f*(f*g - 4*e*h)) - b^3*(4*c*d*e* 
f*(2*f*g - 5*e*h) - c^2*f^2*(f*g - 4*e*h) - 10*d^2*e^2*(f*g - 2*e*h)))*(e 
+ f*x)^(5/2))/(5*f^7) + (2*b*(3*a^2*d^2*f^2*h + 3*a*b*d*f*(d*f*g - 5*d*e*h 
 + 2*c*f*h) + b^2*(c^2*f^2*h + 2*c*d*f*(f*g - 5*e*h) - 5*d^2*e*(f*g - 3*e* 
h)))*(e + f*x)^(7/2))/(7*f^7) + (2*b^2*d*(3*a*d*f*h + b*(d*f*g - 6*d*e*h + 
 2*c*f*h))*(e + f*x)^(9/2))/(9*f^7) + (2*b^3*d^2*h*(e + f*x)^(11/2))/(11*f 
^7)
 

Defintions of rubi rules used

rule 165

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 736, normalized size of antiderivative = 1.28

method	result	size
pseudoelliptic	\(\frac {2 \left (\frac {\left (\frac {5 x^{2} \left (\frac {9 h x}{11}+g \right ) d^{2}}{9}+\frac {10 x c \left (\frac {7 h x}{9}+g \right ) d}{7}+c^{2} \left (\frac {5 h x}{7}+g \right )\right ) x^{3} b^{3}}{5}+a \,x^{2} \left (\frac {3 x^{2} \left (\frac {7 h x}{9}+g \right ) d^{2}}{7}+\frac {6 x c \left (\frac {5 h x}{7}+g \right ) d}{5}+c^{2} \left (\frac {3 h x}{5}+g \right )\right ) b^{2}+3 a^{2} x \left (\frac {x^{2} \left (\frac {5 h x}{7}+g \right ) d^{2}}{5}+\frac {2 x c \left (\frac {3 h x}{5}+g \right ) d}{3}+c^{2} \left (\frac {h x}{3}+g \right )\right ) b -a^{3} \left (-\frac {x^{2} \left (\frac {3 h x}{5}+g \right ) d^{2}}{3}-2 x c \left (\frac {h x}{3}+g \right ) d +c^{2} \left (-h x +g \right )\right )\right ) f^{6}+4 \left (-\frac {\left (\frac {25 \left (\frac {42 h x}{55}+g \right ) x^{2} d^{2}}{63}+\frac {8 x c \left (\frac {25 h x}{36}+g \right ) d}{7}+c^{2} \left (\frac {4 h x}{7}+g \right )\right ) x^{2} b^{3}}{5}-2 a x \left (\frac {\left (\frac {5}{6} h \,x^{3}+\frac {6}{5} g \,x^{2}\right ) d^{2}}{7}+\frac {3 x c \left (\frac {4 h x}{7}+g \right ) d}{5}+\left (\frac {3 h x}{10}+g \right ) c^{2}\right ) b^{2}+3 a^{2} \left (-\frac {x^{2} \left (\frac {4 h x}{7}+g \right ) d^{2}}{5}-\frac {4 \left (\frac {3 h x}{10}+g \right ) x c d}{3}+c^{2} \left (-\frac {2 h x}{3}+g \right )\right ) b +a^{3} \left (\left (-\frac {1}{5} h \,x^{2}-\frac {2}{3} g x \right ) d^{2}+2 c \left (-\frac {2 h x}{3}+g \right ) d +h \,c^{2}\right )\right ) e \,f^{5}-\frac {32 \left (-\frac {3 \left (\frac {10 x^{2} \left (\frac {15 h x}{22}+g \right ) d^{2}}{63}+\frac {4 x c \left (\frac {5 h x}{9}+g \right ) d}{7}+c^{2} \left (\frac {2 h x}{7}+g \right )\right ) x \,b^{3}}{10}+\frac {3 a \left (-\frac {6 x^{2} \left (\frac {5 h x}{9}+g \right ) d^{2}}{35}-\frac {6 x c \left (\frac {2 h x}{7}+g \right ) d}{5}+c^{2} \left (-\frac {3 h x}{5}+g \right )\right ) b^{2}}{2}+\frac {3 a^{2} \left (-\frac {3 x \left (\frac {2 h x}{7}+g \right ) d^{2}}{5}+2 c \left (-\frac {3 h x}{5}+g \right ) d +h \,c^{2}\right ) b}{2}+a^{3} d \left (\frac {\left (-\frac {3 h x}{5}+g \right ) d}{2}+c h \right )\right ) e^{2} f^{4}}{3}+\frac {32 e^{3} \left (\left (-\frac {10 x^{2} \left (\frac {6 h x}{11}+g \right ) d^{2}}{63}-\frac {8 x c \left (\frac {5 h x}{18}+g \right ) d}{7}+c^{2} \left (-\frac {4 h x}{7}+g \right )\right ) b^{3}+3 a \left (-\frac {4 x \left (\frac {5 h x}{18}+g \right ) d^{2}}{7}+2 c \left (-\frac {4 h x}{7}+g \right ) d +h \,c^{2}\right ) b^{2}+6 a^{2} d \left (\left (-\frac {2 h x}{7}+\frac {g}{2}\right ) d +c h \right ) b +h \,d^{2} a^{3}\right ) f^{3}}{5}-\frac {768 \left (\frac {\left (-\frac {5 x \left (\frac {3 h x}{11}+g \right ) d^{2}}{9}+2 \left (-\frac {5 h x}{9}+g \right ) c d +h \,c^{2}\right ) b^{2}}{3}+2 a d \left (\frac {\left (-\frac {5 h x}{9}+g \right ) d}{2}+c h \right ) b +a^{2} d^{2} h \right ) b \,e^{4} f^{2}}{35}+\frac {512 d \left (\left (\left (-\frac {2 h x}{11}+\frac {g}{3}\right ) d +\frac {2 c h}{3}\right ) b +a d h \right ) b^{2} e^{5} f}{21}-\frac {2048 b^{3} d^{2} e^{6} h}{231}}{\sqrt {f x +e}\, f^{7}}\)	\(736\)
risch	\(\text {Expression too large to display}\)	\(1449\)
gosper	\(\text {Expression too large to display}\)	\(1511\)
trager	\(\text {Expression too large to display}\)	\(1511\)
orering	\(\text {Expression too large to display}\)	\(1511\)
derivativedivides	\(\text {Expression too large to display}\)	\(1795\)
default	\(\text {Expression too large to display}\)	\(1795\)

Input:

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

Output:

4/(f*x+e)^(1/2)*(1/2*(1/5*(5/9*x^2*(9/11*h*x+g)*d^2+10/7*x*c*(7/9*h*x+g)*d 
+c^2*(5/7*h*x+g))*x^3*b^3+a*x^2*(3/7*x^2*(7/9*h*x+g)*d^2+6/5*x*c*(5/7*h*x+ 
g)*d+c^2*(3/5*h*x+g))*b^2+3*a^2*x*(1/5*x^2*(5/7*h*x+g)*d^2+2/3*x*c*(3/5*h* 
x+g)*d+c^2*(1/3*h*x+g))*b-a^3*(-1/3*x^2*(3/5*h*x+g)*d^2-2*x*c*(1/3*h*x+g)* 
d+c^2*(-h*x+g)))*f^6+(-1/5*(25/63*(42/55*h*x+g)*x^2*d^2+8/7*x*c*(25/36*h*x 
+g)*d+c^2*(4/7*h*x+g))*x^2*b^3-2*a*x*(1/7*(5/6*h*x^3+6/5*g*x^2)*d^2+3/5*x* 
c*(4/7*h*x+g)*d+(3/10*h*x+g)*c^2)*b^2+3*a^2*(-1/5*x^2*(4/7*h*x+g)*d^2-4/3* 
(3/10*h*x+g)*x*c*d+c^2*(-2/3*h*x+g))*b+a^3*((-1/5*h*x^2-2/3*g*x)*d^2+2*c*( 
-2/3*h*x+g)*d+h*c^2))*e*f^5-8/3*(-3/10*(10/63*x^2*(15/22*h*x+g)*d^2+4/7*x* 
c*(5/9*h*x+g)*d+c^2*(2/7*h*x+g))*x*b^3+3/2*a*(-6/35*x^2*(5/9*h*x+g)*d^2-6/ 
5*x*c*(2/7*h*x+g)*d+c^2*(-3/5*h*x+g))*b^2+3/2*a^2*(-3/5*x*(2/7*h*x+g)*d^2+ 
2*c*(-3/5*h*x+g)*d+h*c^2)*b+a^3*d*(1/2*(-3/5*h*x+g)*d+c*h))*e^2*f^4+8/5*e^ 
3*((-10/63*x^2*(6/11*h*x+g)*d^2-8/7*x*c*(5/18*h*x+g)*d+c^2*(-4/7*h*x+g))*b 
^3+3*a*(-4/7*x*(5/18*h*x+g)*d^2+2*c*(-4/7*h*x+g)*d+h*c^2)*b^2+6*a^2*d*((-2 
/7*h*x+1/2*g)*d+c*h)*b+h*d^2*a^3)*f^3-192/35*(1/3*(-5/9*x*(3/11*h*x+g)*d^2 
+2*(-5/9*h*x+g)*c*d+h*c^2)*b^2+2*a*d*(1/2*(-5/9*h*x+g)*d+c*h)*b+a^2*d^2*h) 
*b*e^4*f^2+128/21*d*(((-2/11*h*x+1/3*g)*d+2/3*c*h)*b+a*d*h)*b^2*e^5*f-512/ 
231*b^3*d^2*e^6*h)/f^7
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1183 vs. \(2 (548) = 1096\).

Time = 0.10 (sec) , antiderivative size = 1183, normalized size of antiderivative = 2.06 \[ \int \frac {(a+b x)^3 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/3465*(315*b^3*d^2*f^6*h*x^6 + 35*(11*b^3*d^2*f^6*g - (12*b^3*d^2*e*f^5 - 
 11*(2*b^3*c*d + 3*a*b^2*d^2)*f^6)*h)*x^5 - 5*(11*(10*b^3*d^2*e*f^5 - 9*(2 
*b^3*c*d + 3*a*b^2*d^2)*f^6)*g - (120*b^3*d^2*e^2*f^4 - 110*(2*b^3*c*d + 3 
*a*b^2*d^2)*e*f^5 + 99*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*f^6)*h)*x^4 + 
 (11*(80*b^3*d^2*e^2*f^4 - 72*(2*b^3*c*d + 3*a*b^2*d^2)*e*f^5 + 63*(b^3*c^ 
2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*f^6)*g - (960*b^3*d^2*e^3*f^3 - 880*(2*b^3* 
c*d + 3*a*b^2*d^2)*e^2*f^4 + 792*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e*f 
^5 - 693*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^6)*h)*x^3 - (11*(160*b^3* 
d^2*e^3*f^3 - 144*(2*b^3*c*d + 3*a*b^2*d^2)*e^2*f^4 + 126*(b^3*c^2 + 6*a*b 
^2*c*d + 3*a^2*b*d^2)*e*f^5 - 105*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^ 
6)*g - (1920*b^3*d^2*e^4*f^2 - 1760*(2*b^3*c*d + 3*a*b^2*d^2)*e^3*f^3 + 15 
84*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^2*f^4 - 1386*(3*a*b^2*c^2 + 6*a 
^2*b*c*d + a^3*d^2)*e*f^5 + 1155*(3*a^2*b*c^2 + 2*a^3*c*d)*f^6)*h)*x^2 + 1 
1*(1280*b^3*d^2*e^5*f - 315*a^3*c^2*f^6 - 1152*(2*b^3*c*d + 3*a*b^2*d^2)*e 
^4*f^2 + 1008*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^3*f^3 - 840*(3*a*b^2 
*c^2 + 6*a^2*b*c*d + a^3*d^2)*e^2*f^4 + 630*(3*a^2*b*c^2 + 2*a^3*c*d)*e*f^ 
5)*g - 2*(7680*b^3*d^2*e^6 - 3465*a^3*c^2*e*f^5 - 7040*(2*b^3*c*d + 3*a*b^ 
2*d^2)*e^5*f + 6336*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^4*f^2 - 5544*( 
3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*e^3*f^3 + 4620*(3*a^2*b*c^2 + 2*a^3*c 
*d)*e^2*f^4)*h + (11*(640*b^3*d^2*e^4*f^2 - 576*(2*b^3*c*d + 3*a*b^2*d^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1160 vs. \(2 (548) = 1096\).

Time = 0.04 (sec) , antiderivative size = 1160, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b x)^3 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/3465*((315*(f*x + e)^(11/2)*b^3*d^2*h + 385*(b^3*d^2*f*g - (6*b^3*d^2*e 
- (2*b^3*c*d + 3*a*b^2*d^2)*f)*h)*(f*x + e)^(9/2) - 495*((5*b^3*d^2*e*f - 
(2*b^3*c*d + 3*a*b^2*d^2)*f^2)*g - (15*b^3*d^2*e^2 - 5*(2*b^3*c*d + 3*a*b^ 
2*d^2)*e*f + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*f^2)*h)*(f*x + e)^(7/2) 
 + 693*((10*b^3*d^2*e^2*f - 4*(2*b^3*c*d + 3*a*b^2*d^2)*e*f^2 + (b^3*c^2 + 
 6*a*b^2*c*d + 3*a^2*b*d^2)*f^3)*g - (20*b^3*d^2*e^3 - 10*(2*b^3*c*d + 3*a 
*b^2*d^2)*e^2*f + 4*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e*f^2 - (3*a*b^2 
*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^3)*h)*(f*x + e)^(5/2) - 1155*((10*b^3*d^2* 
e^3*f - 6*(2*b^3*c*d + 3*a*b^2*d^2)*e^2*f^2 + 3*(b^3*c^2 + 6*a*b^2*c*d + 3 
*a^2*b*d^2)*e*f^3 - (3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f^4)*g - (15*b^3 
*d^2*e^4 - 10*(2*b^3*c*d + 3*a*b^2*d^2)*e^3*f + 6*(b^3*c^2 + 6*a*b^2*c*d + 
 3*a^2*b*d^2)*e^2*f^2 - 3*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*e*f^3 + (3 
*a^2*b*c^2 + 2*a^3*c*d)*f^4)*h)*(f*x + e)^(3/2) + 3465*((5*b^3*d^2*e^4*f - 
 4*(2*b^3*c*d + 3*a*b^2*d^2)*e^3*f^2 + 3*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b* 
d^2)*e^2*f^3 - 2*(3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*e*f^4 + (3*a^2*b*c^ 
2 + 2*a^3*c*d)*f^5)*g - (6*b^3*d^2*e^5 - a^3*c^2*f^5 - 5*(2*b^3*c*d + 3*a* 
b^2*d^2)*e^4*f + 4*(b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^3*f^2 - 3*(3*a* 
b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*e^2*f^3 + 2*(3*a^2*b*c^2 + 2*a^3*c*d)*e*f 
^4)*h)*sqrt(f*x + e))/f^6 + 3465*((b^3*d^2*e^5*f - a^3*c^2*f^6 - (2*b^3*c* 
d + 3*a*b^2*d^2)*e^4*f^2 + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*e^3*f^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1849 vs. \(2 (548) = 1096\).

Time = 0.17 (sec) , antiderivative size = 1849, normalized size of antiderivative = 3.22 \[ \int \frac {(a+b x)^3 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2*(b^3*d^2*e^5*f*g - 2*b^3*c*d*e^4*f^2*g - 3*a*b^2*d^2*e^4*f^2*g + b^3*c^2 
*e^3*f^3*g + 6*a*b^2*c*d*e^3*f^3*g + 3*a^2*b*d^2*e^3*f^3*g - 3*a*b^2*c^2*e 
^2*f^4*g - 6*a^2*b*c*d*e^2*f^4*g - a^3*d^2*e^2*f^4*g + 3*a^2*b*c^2*e*f^5*g 
 + 2*a^3*c*d*e*f^5*g - a^3*c^2*f^6*g - b^3*d^2*e^6*h + 2*b^3*c*d*e^5*f*h + 
 3*a*b^2*d^2*e^5*f*h - b^3*c^2*e^4*f^2*h - 6*a*b^2*c*d*e^4*f^2*h - 3*a^2*b 
*d^2*e^4*f^2*h + 3*a*b^2*c^2*e^3*f^3*h + 6*a^2*b*c*d*e^3*f^3*h + a^3*d^2*e 
^3*f^3*h - 3*a^2*b*c^2*e^2*f^4*h - 2*a^3*c*d*e^2*f^4*h + a^3*c^2*e*f^5*h)/ 
(sqrt(f*x + e)*f^7) + 2/3465*(385*(f*x + e)^(9/2)*b^3*d^2*f^71*g - 2475*(f 
*x + e)^(7/2)*b^3*d^2*e*f^71*g + 6930*(f*x + e)^(5/2)*b^3*d^2*e^2*f^71*g - 
 11550*(f*x + e)^(3/2)*b^3*d^2*e^3*f^71*g + 17325*sqrt(f*x + e)*b^3*d^2*e^ 
4*f^71*g + 990*(f*x + e)^(7/2)*b^3*c*d*f^72*g + 1485*(f*x + e)^(7/2)*a*b^2 
*d^2*f^72*g - 5544*(f*x + e)^(5/2)*b^3*c*d*e*f^72*g - 8316*(f*x + e)^(5/2) 
*a*b^2*d^2*e*f^72*g + 13860*(f*x + e)^(3/2)*b^3*c*d*e^2*f^72*g + 20790*(f* 
x + e)^(3/2)*a*b^2*d^2*e^2*f^72*g - 27720*sqrt(f*x + e)*b^3*c*d*e^3*f^72*g 
 - 41580*sqrt(f*x + e)*a*b^2*d^2*e^3*f^72*g + 693*(f*x + e)^(5/2)*b^3*c^2* 
f^73*g + 4158*(f*x + e)^(5/2)*a*b^2*c*d*f^73*g + 2079*(f*x + e)^(5/2)*a^2* 
b*d^2*f^73*g - 3465*(f*x + e)^(3/2)*b^3*c^2*e*f^73*g - 20790*(f*x + e)^(3/ 
2)*a*b^2*c*d*e*f^73*g - 10395*(f*x + e)^(3/2)*a^2*b*d^2*e*f^73*g + 10395*s 
qrt(f*x + e)*b^3*c^2*e^2*f^73*g + 62370*sqrt(f*x + e)*a*b^2*c*d*e^2*f^73*g 
 + 31185*sqrt(f*x + e)*a^2*b*d^2*e^2*f^73*g + 3465*(f*x + e)^(3/2)*a*b^...
 

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1011, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x)^3 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

((e + f*x)^(7/2)*(2*b^3*c^2*f^2*h + 30*b^3*d^2*e^2*h + 4*b^3*c*d*f^2*g - 1 
0*b^3*d^2*e*f*g + 6*a*b^2*d^2*f^2*g + 6*a^2*b*d^2*f^2*h - 20*b^3*c*d*e*f*h 
 + 12*a*b^2*c*d*f^2*h - 30*a*b^2*d^2*e*f*h))/(7*f^7) + ((e + f*x)^(5/2)*(2 
*b^3*c^2*f^3*g + 2*a^3*d^2*f^3*h - 40*b^3*d^2*e^3*h + 6*a*b^2*c^2*f^3*h + 
6*a^2*b*d^2*f^3*g - 8*b^3*c^2*e*f^2*h + 20*b^3*d^2*e^2*f*g + 12*a*b^2*c*d* 
f^3*g + 12*a^2*b*c*d*f^3*h - 16*b^3*c*d*e*f^2*g + 40*b^3*c*d*e^2*f*h - 24* 
a*b^2*d^2*e*f^2*g + 60*a*b^2*d^2*e^2*f*h - 24*a^2*b*d^2*e*f^2*h - 48*a*b^2 
*c*d*e*f^2*h))/(5*f^7) - (2*a^3*c^2*f^6*g + 2*b^3*d^2*e^6*h - 2*a^3*c^2*e* 
f^5*h - 2*b^3*d^2*e^5*f*g + 2*a^3*d^2*e^2*f^4*g - 2*b^3*c^2*e^3*f^3*g - 2* 
a^3*d^2*e^3*f^3*h + 2*b^3*c^2*e^4*f^2*h + 6*a*b^2*c^2*e^2*f^4*g - 6*a*b^2* 
c^2*e^3*f^3*h + 6*a*b^2*d^2*e^4*f^2*g + 6*a^2*b*c^2*e^2*f^4*h - 6*a^2*b*d^ 
2*e^3*f^3*g + 6*a^2*b*d^2*e^4*f^2*h - 4*a^3*c*d*e*f^5*g - 4*b^3*c*d*e^5*f* 
h - 6*a^2*b*c^2*e*f^5*g - 6*a*b^2*d^2*e^5*f*h + 4*a^3*c*d*e^2*f^4*h + 4*b^ 
3*c*d*e^4*f^2*g - 12*a*b^2*c*d*e^3*f^3*g + 12*a^2*b*c*d*e^2*f^4*g + 12*a*b 
^2*c*d*e^4*f^2*h - 12*a^2*b*c*d*e^3*f^3*h)/(f^7*(e + f*x)^(1/2)) + (2*(e + 
 f*x)^(3/2)*(a*f - b*e)*(a^2*d^2*f^3*g + 3*b^2*c^2*f^3*g - 15*b^2*d^2*e^3* 
h + 3*a*b*c^2*f^3*h + 2*a^2*c*d*f^3*h - 3*a^2*d^2*e*f^2*h - 6*b^2*c^2*e*f^ 
2*h + 10*b^2*d^2*e^2*f*g + 6*a*b*c*d*f^3*g - 8*a*b*d^2*e*f^2*g + 15*a*b*d^ 
2*e^2*f*h - 12*b^2*c*d*e*f^2*g + 20*b^2*c*d*e^2*f*h - 16*a*b*c*d*e*f^2*h)) 
/(3*f^7) + (2*b^3*d^2*h*(e + f*x)^(11/2))/(11*f^7) + (2*b^2*d*(e + f*x)...
 

Reduce [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 1511, normalized size of antiderivative = 2.63 \[ \int \frac {(a+b x)^3 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(6930*a**3*c**2*e*f**5*h - 3465*a**3*c**2*f**6*g + 3465*a**3*c**2*f**6* 
h*x - 18480*a**3*c*d*e**2*f**4*h + 13860*a**3*c*d*e*f**5*g - 9240*a**3*c*d 
*e*f**5*h*x + 6930*a**3*c*d*f**6*g*x + 2310*a**3*c*d*f**6*h*x**2 + 11088*a 
**3*d**2*e**3*f**3*h - 9240*a**3*d**2*e**2*f**4*g + 5544*a**3*d**2*e**2*f* 
*4*h*x - 4620*a**3*d**2*e*f**5*g*x - 1386*a**3*d**2*e*f**5*h*x**2 + 1155*a 
**3*d**2*f**6*g*x**2 + 693*a**3*d**2*f**6*h*x**3 - 27720*a**2*b*c**2*e**2* 
f**4*h + 20790*a**2*b*c**2*e*f**5*g - 13860*a**2*b*c**2*e*f**5*h*x + 10395 
*a**2*b*c**2*f**6*g*x + 3465*a**2*b*c**2*f**6*h*x**2 + 66528*a**2*b*c*d*e* 
*3*f**3*h - 55440*a**2*b*c*d*e**2*f**4*g + 33264*a**2*b*c*d*e**2*f**4*h*x 
- 27720*a**2*b*c*d*e*f**5*g*x - 8316*a**2*b*c*d*e*f**5*h*x**2 + 6930*a**2* 
b*c*d*f**6*g*x**2 + 4158*a**2*b*c*d*f**6*h*x**3 - 38016*a**2*b*d**2*e**4*f 
**2*h + 33264*a**2*b*d**2*e**3*f**3*g - 19008*a**2*b*d**2*e**3*f**3*h*x + 
16632*a**2*b*d**2*e**2*f**4*g*x + 4752*a**2*b*d**2*e**2*f**4*h*x**2 - 4158 
*a**2*b*d**2*e*f**5*g*x**2 - 2376*a**2*b*d**2*e*f**5*h*x**3 + 2079*a**2*b* 
d**2*f**6*g*x**3 + 1485*a**2*b*d**2*f**6*h*x**4 + 33264*a*b**2*c**2*e**3*f 
**3*h - 27720*a*b**2*c**2*e**2*f**4*g + 16632*a*b**2*c**2*e**2*f**4*h*x - 
13860*a*b**2*c**2*e*f**5*g*x - 4158*a*b**2*c**2*e*f**5*h*x**2 + 3465*a*b** 
2*c**2*f**6*g*x**2 + 2079*a*b**2*c**2*f**6*h*x**3 - 76032*a*b**2*c*d*e**4* 
f**2*h + 66528*a*b**2*c*d*e**3*f**3*g - 38016*a*b**2*c*d*e**3*f**3*h*x + 3 
3264*a*b**2*c*d*e**2*f**4*g*x + 9504*a*b**2*c*d*e**2*f**4*h*x**2 - 8316...