\(\int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx\) [160]

Optimal result

Integrand size = 29, antiderivative size = 407 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=-\frac {2 (b e-a f)^2 (d e-c f)^2 (f g-e h)}{f^6 \sqrt {e+f x}}-\frac {2 (b e-a f) (d e-c f) (b d e (4 f g-5 e h)-b c f (2 f g-3 e h)-a f (2 d f g-3 d e h+c f h)) \sqrt {e+f x}}{f^6}+\frac {2 \left (a^2 d f^2 (d f g-3 d e h+2 c f h)+2 a b f \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )+b^2 \left (2 d^2 e^2 (3 f g-5 e h)+c^2 f^2 (f g-3 e h)-6 c d e f (f g-2 e h)\right )\right ) (e+f x)^{3/2}}{3 f^6}+\frac {2 \left (a^2 d^2 f^2 h+2 a b d f (d f g-4 d e h+2 c f h)+b^2 \left (c^2 f^2 h-2 d^2 e (2 f g-5 e h)+2 c d f (f g-4 e h)\right )\right ) (e+f x)^{5/2}}{5 f^6}+\frac {2 b d (2 a d f h+b (d f g-5 d e h+2 c f h)) (e+f x)^{7/2}}{7 f^6}+\frac {2 b^2 d^2 h (e+f x)^{9/2}}{9 f^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {2 \left (21 a^2 f^2 \left (15 c^2 f^2 (-f g+2 e h+f h x)+10 c d f \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )+d^2 \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )\right )+6 a b f \left (35 c^2 f^2 \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )+14 c d f \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )-3 d^2 \left (128 e^4 h-16 e^3 f (7 g-4 h x)-8 e^2 f^2 x (7 g+2 h x)+2 e f^3 x^2 (7 g+4 h x)-f^4 x^3 (7 g+5 h x)\right )\right )+b^2 \left (21 c^2 f^2 \left (48 e^3 h-8 e^2 f (5 g-3 h x)+f^3 x^2 (5 g+3 h x)-2 e f^2 x (10 g+3 h x)\right )+18 c d f \left (-128 e^4 h+16 e^3 f (7 g-4 h x)+8 e^2 f^2 x (7 g+2 h x)-2 e f^3 x^2 (7 g+4 h x)+f^4 x^3 (7 g+5 h x)\right )+d^2 \left (1280 e^5 h-128 e^4 f (9 g-5 h x)+16 e^2 f^3 x^2 (9 g+5 h x)-32 e^3 f^2 x (18 g+5 h x)+5 f^5 x^4 (9 g+7 h x)-2 e f^4 x^3 (36 g+25 h x)\right )\right )\right )}{315 f^6 \sqrt {e+f x}} \] Input:

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

Output:

(2*(21*a^2*f^2*(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))) + 6*a*b*f*(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))) + b^2*(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) - 3 
2*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)))))/(315*f^6*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 407, 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.

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

\(\Big \downarrow \) 165

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 0.44 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(\frac {4 \left (\frac {x^{2} \left (\frac {3 x^{2} \left (\frac {7 h x}{9}+g \right ) d^{2}}{7}+\frac {6 x c \left (\frac {5 h x}{7}+g \right ) d}{5}+c^{2} \left (\frac {3 h x}{5}+g \right )\right ) b^{2}}{6}+a x \left (\frac {x^{2} \left (\frac {5 h x}{7}+g \right ) d^{2}}{5}+\frac {2 x c \left (\frac {3 h x}{5}+g \right ) d}{3}+c^{2} \left (\frac {h x}{3}+g \right )\right ) b -\frac {a^{2} \left (-\frac {x^{2} \left (\frac {3 h x}{5}+g \right ) d^{2}}{3}-2 x c \left (\frac {h x}{3}+g \right ) d +c^{2} \left (-h x +g \right )\right )}{2}\right ) f^{5}+4 \left (-\frac {2 x \left (\left (\frac {5}{42} h \,x^{3}+\frac {6}{35} g \,x^{2}\right ) d^{2}+\frac {3 x c \left (\frac {4 h x}{7}+g \right ) d}{5}+\left (\frac {3 h x}{10}+g \right ) c^{2}\right ) b^{2}}{3}+2 a \left (-\frac {x^{2} \left (\frac {4 h x}{7}+g \right ) d^{2}}{5}-\frac {4 \left (\frac {3 h x}{10}+g \right ) x c d}{3}+c^{2} \left (-\frac {2 h x}{3}+g \right )\right ) b +a^{2} \left (\left (-\frac {1}{5} h \,x^{2}-\frac {2}{3} g x \right ) d^{2}+2 c \left (-\frac {2 h x}{3}+g \right ) d +h \,c^{2}\right )\right ) e \,f^{4}-\frac {32 \left (\left (-\frac {3 x^{2} \left (\frac {5 h x}{9}+g \right ) d^{2}}{35}-\frac {3 x c \left (\frac {2 h x}{7}+g \right ) d}{5}+\frac {c^{2} \left (-\frac {3 h x}{5}+g \right )}{2}\right ) b^{2}+a \left (-\frac {3 x \left (\frac {2 h x}{7}+g \right ) d^{2}}{5}+2 c \left (-\frac {3 h x}{5}+g \right ) d +h \,c^{2}\right ) b +a^{2} d \left (\left (-\frac {3 h x}{10}+\frac {g}{2}\right ) d +c h \right )\right ) e^{2} f^{3}}{3}+\frac {32 \left (\left (-\frac {4 x \left (\frac {5 h x}{18}+g \right ) d^{2}}{7}+2 c \left (-\frac {4 h x}{7}+g \right ) d +h \,c^{2}\right ) b^{2}+4 a \left (\left (-\frac {2 h x}{7}+\frac {g}{2}\right ) d +c h \right ) d b +a^{2} d^{2} h \right ) e^{3} f^{2}}{5}-\frac {512 d \left (\left (\left (-\frac {5 h x}{18}+\frac {g}{2}\right ) d +c h \right ) b +a d h \right ) b \,e^{4} f}{35}+\frac {512 b^{2} d^{2} e^{5} h}{63}}{\sqrt {f x +e}\, f^{6}}\)	\(479\)
risch	\(\frac {2 \left (35 h \,b^{2} d^{2} x^{4} f^{4}+90 a b \,d^{2} f^{4} h \,x^{3}+90 b^{2} c d \,f^{4} h \,x^{3}-85 b^{2} d^{2} e \,f^{3} h \,x^{3}+45 b^{2} d^{2} f^{4} g \,x^{3}+63 a^{2} d^{2} f^{4} h \,x^{2}+252 a b c d \,f^{4} h \,x^{2}-234 a b \,d^{2} e \,f^{3} h \,x^{2}+126 a b \,d^{2} f^{4} g \,x^{2}+63 b^{2} c^{2} f^{4} h \,x^{2}-234 b^{2} c d e \,f^{3} h \,x^{2}+126 b^{2} c d \,f^{4} g \,x^{2}+165 b^{2} d^{2} e^{2} f^{2} h \,x^{2}-117 b^{2} d^{2} e \,f^{3} g \,x^{2}+210 a^{2} c d \,f^{4} h x -189 a^{2} d^{2} e \,f^{3} h x +105 a^{2} d^{2} f^{4} g x +210 a b \,c^{2} f^{4} h x -756 a b c d e \,f^{3} h x +420 a b c d \,f^{4} g x +522 a b \,d^{2} e^{2} f^{2} h x -378 a b \,d^{2} e \,f^{3} g x -189 b^{2} c^{2} e \,f^{3} h x +105 b^{2} c^{2} f^{4} g x +522 b^{2} c d \,e^{2} f^{2} h x -378 b^{2} c d e \,f^{3} g x -325 b^{2} d^{2} e^{3} f h x +261 b^{2} d^{2} e^{2} f^{2} g x +315 a^{2} c^{2} h \,f^{4}-1050 a^{2} c d e \,f^{3} h +630 g \,a^{2} c d \,f^{4}+693 a^{2} d^{2} e^{2} f^{2} h -525 a^{2} d^{2} e \,f^{3} g -1050 a b \,c^{2} e \,f^{3} h +630 a b \,c^{2} g \,f^{4}+2772 a b c d \,e^{2} f^{2} h -2100 a b c d e \,f^{3} g -1674 a b \,d^{2} e^{3} f h +1386 a b \,d^{2} e^{2} f^{2} g +693 b^{2} c^{2} e^{2} f^{2} h -525 b^{2} c^{2} e \,f^{3} g -1674 b^{2} c d \,e^{3} f h +1386 b^{2} c d \,e^{2} f^{2} g +965 b^{2} d^{2} e^{4} h -837 b^{2} d^{2} e^{3} f g \right ) \sqrt {f x +e}}{315 f^{6}}+\frac {2 a^{2} c^{2} e \,f^{4} h -2 g \,a^{2} c^{2} f^{5}-4 a^{2} c d \,e^{2} f^{3} h +4 a^{2} c d e \,f^{4} g +2 a^{2} d^{2} e^{3} f^{2} h -2 a^{2} d^{2} e^{2} f^{3} g -4 a b \,c^{2} e^{2} f^{3} h +4 a b \,c^{2} e \,f^{4} g +8 a b c d \,e^{3} f^{2} h -8 a b c d \,e^{2} f^{3} g -4 a b \,d^{2} e^{4} f h +4 a b \,d^{2} e^{3} f^{2} g +2 b^{2} c^{2} e^{3} f^{2} h -2 b^{2} c^{2} e^{2} f^{3} g -4 b^{2} c d \,e^{4} f h +4 b^{2} c d \,e^{3} f^{2} g +2 b^{2} d^{2} e^{5} h -2 b^{2} d^{2} e^{4} f g}{\sqrt {f x +e}\, f^{6}}\)	\(875\)
gosper	\(\frac {-\frac {16}{5} a b c d e \,f^{4} h \,x^{2}+\frac {64}{5} a b c d \,e^{2} f^{3} h x -\frac {32}{3} a b c d e \,f^{4} g x -2 g \,a^{2} c^{2} f^{5}+\frac {32}{5} b^{2} c^{2} e^{3} f^{2} h -\frac {16}{3} b^{2} c^{2} e^{2} f^{3} g -\frac {16}{3} a^{2} d^{2} e^{2} f^{3} g +\frac {128}{5} a b c d \,e^{3} f^{2} h -\frac {64}{3} a b c d \,e^{2} f^{3} g +\frac {8}{5} a b c d \,f^{5} h \,x^{3}-\frac {32}{35} a b \,d^{2} e \,f^{4} h \,x^{3}-\frac {32}{35} b^{2} c d e \,f^{4} h \,x^{3}+\frac {8}{3} a b c d \,f^{5} g \,x^{2}+\frac {64}{35} a b \,d^{2} e^{2} f^{3} h \,x^{2}-\frac {8}{5} a b \,d^{2} e \,f^{4} g \,x^{2}-\frac {8}{5} b^{2} c d e \,f^{4} g \,x^{2}-\frac {16}{3} a^{2} c d e \,f^{4} h x -\frac {16}{3} a b \,c^{2} e \,f^{4} h x +\frac {2}{3} a^{2} d^{2} f^{5} g \,x^{2}+4 a^{2} c^{2} e \,f^{4} h +\frac {32}{5} a^{2} d^{2} e^{3} f^{2} h +\frac {512}{63} b^{2} d^{2} e^{5} h +\frac {2}{9} h \,b^{2} d^{2} x^{5} f^{5}+\frac {4}{7} a b \,d^{2} f^{5} h \,x^{4}+\frac {4}{7} b^{2} c d \,f^{5} h \,x^{4}-\frac {20}{63} b^{2} d^{2} e \,f^{4} h \,x^{4}+\frac {4}{5} a b \,d^{2} f^{5} g \,x^{3}+\frac {4}{5} b^{2} c d \,f^{5} g \,x^{3}+\frac {32}{63} b^{2} d^{2} e^{2} f^{3} h \,x^{3}-\frac {16}{35} b^{2} d^{2} e \,f^{4} g \,x^{3}+\frac {4}{3} a^{2} c d \,f^{5} h \,x^{2}-\frac {4}{5} a^{2} d^{2} e \,f^{4} h \,x^{2}+\frac {4}{3} a b \,c^{2} f^{5} h \,x^{2}-\frac {4}{5} b^{2} c^{2} e \,f^{4} h \,x^{2}-\frac {64}{63} b^{2} d^{2} e^{3} f^{2} h \,x^{2}+\frac {32}{35} b^{2} d^{2} e^{2} f^{3} g \,x^{2}+4 a^{2} c d \,f^{5} g x +\frac {16}{5} a^{2} d^{2} e^{2} f^{3} h x -\frac {8}{3} a^{2} d^{2} e \,f^{4} g x +4 a b \,c^{2} f^{5} g x +\frac {16}{5} b^{2} c^{2} e^{2} f^{3} h x -\frac {8}{3} b^{2} c^{2} e \,f^{4} g x +\frac {256}{63} b^{2} d^{2} e^{4} f h x -\frac {128}{35} b^{2} d^{2} e^{3} f^{2} g x +\frac {64}{5} a b \,d^{2} e^{3} f^{2} g +8 a b \,c^{2} e \,f^{4} g -\frac {512}{35} a b \,d^{2} e^{4} f h -\frac {32}{3} a b \,c^{2} e^{2} f^{3} h +8 a^{2} c d e \,f^{4} g +\frac {2}{3} b^{2} c^{2} f^{5} g \,x^{2}+2 a^{2} c^{2} f^{5} h x +\frac {2}{7} b^{2} d^{2} f^{5} g \,x^{4}+\frac {2}{5} a^{2} d^{2} f^{5} h \,x^{3}+\frac {2}{5} b^{2} c^{2} f^{5} h \,x^{3}-\frac {32}{3} a^{2} c d \,e^{2} f^{3} h -\frac {512}{35} b^{2} c d \,e^{4} f h +\frac {64}{5} b^{2} c d \,e^{3} f^{2} g -\frac {256}{35} b^{2} d^{2} e^{4} f g +\frac {32}{5} b^{2} c d \,e^{2} f^{3} g x -\frac {256}{35} a b \,d^{2} e^{3} f^{2} h x +\frac {32}{5} a b \,d^{2} e^{2} f^{3} g x -\frac {256}{35} b^{2} c d \,e^{3} f^{2} h x +\frac {64}{35} b^{2} c d \,e^{2} f^{3} h \,x^{2}}{\sqrt {f x +e}\, f^{6}}\)	\(919\)
trager	\(\frac {-\frac {16}{5} a b c d e \,f^{4} h \,x^{2}+\frac {64}{5} a b c d \,e^{2} f^{3} h x -\frac {32}{3} a b c d e \,f^{4} g x -2 g \,a^{2} c^{2} f^{5}+\frac {32}{5} b^{2} c^{2} e^{3} f^{2} h -\frac {16}{3} b^{2} c^{2} e^{2} f^{3} g -\frac {16}{3} a^{2} d^{2} e^{2} f^{3} g +\frac {128}{5} a b c d \,e^{3} f^{2} h -\frac {64}{3} a b c d \,e^{2} f^{3} g +\frac {8}{5} a b c d \,f^{5} h \,x^{3}-\frac {32}{35} a b \,d^{2} e \,f^{4} h \,x^{3}-\frac {32}{35} b^{2} c d e \,f^{4} h \,x^{3}+\frac {8}{3} a b c d \,f^{5} g \,x^{2}+\frac {64}{35} a b \,d^{2} e^{2} f^{3} h \,x^{2}-\frac {8}{5} a b \,d^{2} e \,f^{4} g \,x^{2}-\frac {8}{5} b^{2} c d e \,f^{4} g \,x^{2}-\frac {16}{3} a^{2} c d e \,f^{4} h x -\frac {16}{3} a b \,c^{2} e \,f^{4} h x +\frac {2}{3} a^{2} d^{2} f^{5} g \,x^{2}+4 a^{2} c^{2} e \,f^{4} h +\frac {32}{5} a^{2} d^{2} e^{3} f^{2} h +\frac {512}{63} b^{2} d^{2} e^{5} h +\frac {2}{9} h \,b^{2} d^{2} x^{5} f^{5}+\frac {4}{7} a b \,d^{2} f^{5} h \,x^{4}+\frac {4}{7} b^{2} c d \,f^{5} h \,x^{4}-\frac {20}{63} b^{2} d^{2} e \,f^{4} h \,x^{4}+\frac {4}{5} a b \,d^{2} f^{5} g \,x^{3}+\frac {4}{5} b^{2} c d \,f^{5} g \,x^{3}+\frac {32}{63} b^{2} d^{2} e^{2} f^{3} h \,x^{3}-\frac {16}{35} b^{2} d^{2} e \,f^{4} g \,x^{3}+\frac {4}{3} a^{2} c d \,f^{5} h \,x^{2}-\frac {4}{5} a^{2} d^{2} e \,f^{4} h \,x^{2}+\frac {4}{3} a b \,c^{2} f^{5} h \,x^{2}-\frac {4}{5} b^{2} c^{2} e \,f^{4} h \,x^{2}-\frac {64}{63} b^{2} d^{2} e^{3} f^{2} h \,x^{2}+\frac {32}{35} b^{2} d^{2} e^{2} f^{3} g \,x^{2}+4 a^{2} c d \,f^{5} g x +\frac {16}{5} a^{2} d^{2} e^{2} f^{3} h x -\frac {8}{3} a^{2} d^{2} e \,f^{4} g x +4 a b \,c^{2} f^{5} g x +\frac {16}{5} b^{2} c^{2} e^{2} f^{3} h x -\frac {8}{3} b^{2} c^{2} e \,f^{4} g x +\frac {256}{63} b^{2} d^{2} e^{4} f h x -\frac {128}{35} b^{2} d^{2} e^{3} f^{2} g x +\frac {64}{5} a b \,d^{2} e^{3} f^{2} g +8 a b \,c^{2} e \,f^{4} g -\frac {512}{35} a b \,d^{2} e^{4} f h -\frac {32}{3} a b \,c^{2} e^{2} f^{3} h +8 a^{2} c d e \,f^{4} g +\frac {2}{3} b^{2} c^{2} f^{5} g \,x^{2}+2 a^{2} c^{2} f^{5} h x +\frac {2}{7} b^{2} d^{2} f^{5} g \,x^{4}+\frac {2}{5} a^{2} d^{2} f^{5} h \,x^{3}+\frac {2}{5} b^{2} c^{2} f^{5} h \,x^{3}-\frac {32}{3} a^{2} c d \,e^{2} f^{3} h -\frac {512}{35} b^{2} c d \,e^{4} f h +\frac {64}{5} b^{2} c d \,e^{3} f^{2} g -\frac {256}{35} b^{2} d^{2} e^{4} f g +\frac {32}{5} b^{2} c d \,e^{2} f^{3} g x -\frac {256}{35} a b \,d^{2} e^{3} f^{2} h x +\frac {32}{5} a b \,d^{2} e^{2} f^{3} g x -\frac {256}{35} b^{2} c d \,e^{3} f^{2} h x +\frac {64}{35} b^{2} c d \,e^{2} f^{3} h \,x^{2}}{\sqrt {f x +e}\, f^{6}}\)	\(919\)
orering	\(\frac {-\frac {16}{5} a b c d e \,f^{4} h \,x^{2}+\frac {64}{5} a b c d \,e^{2} f^{3} h x -\frac {32}{3} a b c d e \,f^{4} g x -2 g \,a^{2} c^{2} f^{5}+\frac {32}{5} b^{2} c^{2} e^{3} f^{2} h -\frac {16}{3} b^{2} c^{2} e^{2} f^{3} g -\frac {16}{3} a^{2} d^{2} e^{2} f^{3} g +\frac {128}{5} a b c d \,e^{3} f^{2} h -\frac {64}{3} a b c d \,e^{2} f^{3} g +\frac {8}{5} a b c d \,f^{5} h \,x^{3}-\frac {32}{35} a b \,d^{2} e \,f^{4} h \,x^{3}-\frac {32}{35} b^{2} c d e \,f^{4} h \,x^{3}+\frac {8}{3} a b c d \,f^{5} g \,x^{2}+\frac {64}{35} a b \,d^{2} e^{2} f^{3} h \,x^{2}-\frac {8}{5} a b \,d^{2} e \,f^{4} g \,x^{2}-\frac {8}{5} b^{2} c d e \,f^{4} g \,x^{2}-\frac {16}{3} a^{2} c d e \,f^{4} h x -\frac {16}{3} a b \,c^{2} e \,f^{4} h x +\frac {2}{3} a^{2} d^{2} f^{5} g \,x^{2}+4 a^{2} c^{2} e \,f^{4} h +\frac {32}{5} a^{2} d^{2} e^{3} f^{2} h +\frac {512}{63} b^{2} d^{2} e^{5} h +\frac {2}{9} h \,b^{2} d^{2} x^{5} f^{5}+\frac {4}{7} a b \,d^{2} f^{5} h \,x^{4}+\frac {4}{7} b^{2} c d \,f^{5} h \,x^{4}-\frac {20}{63} b^{2} d^{2} e \,f^{4} h \,x^{4}+\frac {4}{5} a b \,d^{2} f^{5} g \,x^{3}+\frac {4}{5} b^{2} c d \,f^{5} g \,x^{3}+\frac {32}{63} b^{2} d^{2} e^{2} f^{3} h \,x^{3}-\frac {16}{35} b^{2} d^{2} e \,f^{4} g \,x^{3}+\frac {4}{3} a^{2} c d \,f^{5} h \,x^{2}-\frac {4}{5} a^{2} d^{2} e \,f^{4} h \,x^{2}+\frac {4}{3} a b \,c^{2} f^{5} h \,x^{2}-\frac {4}{5} b^{2} c^{2} e \,f^{4} h \,x^{2}-\frac {64}{63} b^{2} d^{2} e^{3} f^{2} h \,x^{2}+\frac {32}{35} b^{2} d^{2} e^{2} f^{3} g \,x^{2}+4 a^{2} c d \,f^{5} g x +\frac {16}{5} a^{2} d^{2} e^{2} f^{3} h x -\frac {8}{3} a^{2} d^{2} e \,f^{4} g x +4 a b \,c^{2} f^{5} g x +\frac {16}{5} b^{2} c^{2} e^{2} f^{3} h x -\frac {8}{3} b^{2} c^{2} e \,f^{4} g x +\frac {256}{63} b^{2} d^{2} e^{4} f h x -\frac {128}{35} b^{2} d^{2} e^{3} f^{2} g x +\frac {64}{5} a b \,d^{2} e^{3} f^{2} g +8 a b \,c^{2} e \,f^{4} g -\frac {512}{35} a b \,d^{2} e^{4} f h -\frac {32}{3} a b \,c^{2} e^{2} f^{3} h +8 a^{2} c d e \,f^{4} g +\frac {2}{3} b^{2} c^{2} f^{5} g \,x^{2}+2 a^{2} c^{2} f^{5} h x +\frac {2}{7} b^{2} d^{2} f^{5} g \,x^{4}+\frac {2}{5} a^{2} d^{2} f^{5} h \,x^{3}+\frac {2}{5} b^{2} c^{2} f^{5} h \,x^{3}-\frac {32}{3} a^{2} c d \,e^{2} f^{3} h -\frac {512}{35} b^{2} c d \,e^{4} f h +\frac {64}{5} b^{2} c d \,e^{3} f^{2} g -\frac {256}{35} b^{2} d^{2} e^{4} f g +\frac {32}{5} b^{2} c d \,e^{2} f^{3} g x -\frac {256}{35} a b \,d^{2} e^{3} f^{2} h x +\frac {32}{5} a b \,d^{2} e^{2} f^{3} g x -\frac {256}{35} b^{2} c d \,e^{3} f^{2} h x +\frac {64}{35} b^{2} c d \,e^{2} f^{3} h \,x^{2}}{\sqrt {f x +e}\, f^{6}}\)	\(919\)
derivativedivides	\(\text {Expression too large to display}\)	\(1094\)
default	\(\text {Expression too large to display}\)	\(1094\)

Input:

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

Output:

4/(f*x+e)^(1/2)*((1/6*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+a*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-1/2*a^2*(-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^5+(-2/3*x*((5/42*h*x^3+6/35*g*x^2)*d^2+3/5*x*c*(4/7*h*x+g)*d+ 
(3/10*h*x+g)*c^2)*b^2+2*a*(-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^2*((-1/5*h*x^2-2/3*g*x)*d^2+2*c*(-2/3*h*x+g)*d+h*c^ 
2))*e*f^4-8/3*((-3/35*x^2*(5/9*h*x+g)*d^2-3/5*x*c*(2/7*h*x+g)*d+1/2*c^2*(- 
3/5*h*x+g))*b^2+a*(-3/5*x*(2/7*h*x+g)*d^2+2*c*(-3/5*h*x+g)*d+h*c^2)*b+a^2* 
d*((-3/10*h*x+1/2*g)*d+c*h))*e^2*f^3+8/5*((-4/7*x*(5/18*h*x+g)*d^2+2*c*(-4 
/7*h*x+g)*d+h*c^2)*b^2+4*a*((-2/7*h*x+1/2*g)*d+c*h)*d*b+a^2*d^2*h)*e^3*f^2 
-128/35*d*(((-5/18*h*x+1/2*g)*d+c*h)*b+a*d*h)*b*e^4*f+128/63*b^2*d^2*e^5*h 
)/f^6
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/315*(35*b^2*d^2*f^5*h*x^5 + 5*(9*b^2*d^2*f^5*g - 2*(5*b^2*d^2*e*f^4 - 9* 
(b^2*c*d + a*b*d^2)*f^5)*h)*x^4 - (18*(4*b^2*d^2*e*f^4 - 7*(b^2*c*d + a*b* 
d^2)*f^5)*g - (80*b^2*d^2*e^2*f^3 - 144*(b^2*c*d + a*b*d^2)*e*f^4 + 63*(b^ 
2*c^2 + 4*a*b*c*d + a^2*d^2)*f^5)*h)*x^3 + (3*(48*b^2*d^2*e^2*f^3 - 84*(b^ 
2*c*d + a*b*d^2)*e*f^4 + 35*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^5)*g - 2*(80 
*b^2*d^2*e^3*f^2 - 144*(b^2*c*d + a*b*d^2)*e^2*f^3 + 63*(b^2*c^2 + 4*a*b*c 
*d + a^2*d^2)*e*f^4 - 105*(a*b*c^2 + a^2*c*d)*f^5)*h)*x^2 - 3*(384*b^2*d^2 
*e^4*f + 105*a^2*c^2*f^5 - 672*(b^2*c*d + a*b*d^2)*e^3*f^2 + 280*(b^2*c^2 
+ 4*a*b*c*d + a^2*d^2)*e^2*f^3 - 420*(a*b*c^2 + a^2*c*d)*e*f^4)*g + 2*(640 
*b^2*d^2*e^5 + 315*a^2*c^2*e*f^4 - 1152*(b^2*c*d + a*b*d^2)*e^4*f + 504*(b 
^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^2 - 840*(a*b*c^2 + a^2*c*d)*e^2*f^3)*h 
 - (6*(96*b^2*d^2*e^3*f^2 - 168*(b^2*c*d + a*b*d^2)*e^2*f^3 + 70*(b^2*c^2 
+ 4*a*b*c*d + a^2*d^2)*e*f^4 - 105*(a*b*c^2 + a^2*c*d)*f^5)*g - (640*b^2*d 
^2*e^4*f + 315*a^2*c^2*f^5 - 1152*(b^2*c*d + a*b*d^2)*e^3*f^2 + 504*(b^2*c 
^2 + 4*a*b*c*d + a^2*d^2)*e^2*f^3 - 840*(a*b*c^2 + a^2*c*d)*e*f^4)*h)*x)*s 
qrt(f*x + e)/(f^7*x + e*f^6)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 949 vs. \(2 (440) = 880\).

Time = 88.52 (sec) , antiderivative size = 949, normalized size of antiderivative = 2.33 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{2} d^{2} h \left (e + f x\right )^{\frac {9}{2}}}{9 f^{5}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (2 a b d^{2} f h + 2 b^{2} c d f h - 5 b^{2} d^{2} e h + b^{2} d^{2} f g\right )}{7 f^{5}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (a^{2} d^{2} f^{2} h + 4 a b c d f^{2} h - 8 a b d^{2} e f h + 2 a b d^{2} f^{2} g + b^{2} c^{2} f^{2} h - 8 b^{2} c d e f h + 2 b^{2} c d f^{2} g + 10 b^{2} d^{2} e^{2} h - 4 b^{2} d^{2} e f g\right )}{5 f^{5}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \cdot \left (2 a^{2} c d f^{3} h - 3 a^{2} d^{2} e f^{2} h + a^{2} d^{2} f^{3} g + 2 a b c^{2} f^{3} h - 12 a b c d e f^{2} h + 4 a b c d f^{3} g + 12 a b d^{2} e^{2} f h - 6 a b d^{2} e f^{2} g - 3 b^{2} c^{2} e f^{2} h + b^{2} c^{2} f^{3} g + 12 b^{2} c d e^{2} f h - 6 b^{2} c d e f^{2} g - 10 b^{2} d^{2} e^{3} h + 6 b^{2} d^{2} e^{2} f g\right )}{3 f^{5}} + \frac {\sqrt {e + f x} \left (a^{2} c^{2} f^{4} h - 4 a^{2} c d e f^{3} h + 2 a^{2} c d f^{4} g + 3 a^{2} d^{2} e^{2} f^{2} h - 2 a^{2} d^{2} e f^{3} g - 4 a b c^{2} e f^{3} h + 2 a b c^{2} f^{4} g + 12 a b c d e^{2} f^{2} h - 8 a b c d e f^{3} g - 8 a b d^{2} e^{3} f h + 6 a b d^{2} e^{2} f^{2} g + 3 b^{2} c^{2} e^{2} f^{2} h - 2 b^{2} c^{2} e f^{3} g - 8 b^{2} c d e^{3} f h + 6 b^{2} c d e^{2} f^{2} g + 5 b^{2} d^{2} e^{4} h - 4 b^{2} d^{2} e^{3} f g\right )}{f^{5}} + \frac {\left (a f - b e\right )^{2} \left (c f - d e\right )^{2} \left (e h - f g\right )}{f^{5} \sqrt {e + f x}}\right )}{f} & \text {for}\: f \neq 0 \\\frac {a^{2} c^{2} g x + \frac {b^{2} d^{2} h x^{6}}{6} + \frac {x^{5} \cdot \left (2 a b d^{2} h + 2 b^{2} c d h + b^{2} d^{2} g\right )}{5} + \frac {x^{4} \left (a^{2} d^{2} h + 4 a b c d h + 2 a b d^{2} g + b^{2} c^{2} h + 2 b^{2} c d g\right )}{4} + \frac {x^{3} \cdot \left (2 a^{2} c d h + a^{2} d^{2} g + 2 a b c^{2} h + 4 a b c d g + b^{2} c^{2} g\right )}{3} + \frac {x^{2} \left (a^{2} c^{2} h + 2 a^{2} c d g + 2 a b c^{2} g\right )}{2}}{e^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*(b**2*d**2*h*(e + f*x)**(9/2)/(9*f**5) + (e + f*x)**(7/2)*(2* 
a*b*d**2*f*h + 2*b**2*c*d*f*h - 5*b**2*d**2*e*h + b**2*d**2*f*g)/(7*f**5) 
+ (e + f*x)**(5/2)*(a**2*d**2*f**2*h + 4*a*b*c*d*f**2*h - 8*a*b*d**2*e*f*h 
 + 2*a*b*d**2*f**2*g + b**2*c**2*f**2*h - 8*b**2*c*d*e*f*h + 2*b**2*c*d*f* 
*2*g + 10*b**2*d**2*e**2*h - 4*b**2*d**2*e*f*g)/(5*f**5) + (e + f*x)**(3/2 
)*(2*a**2*c*d*f**3*h - 3*a**2*d**2*e*f**2*h + a**2*d**2*f**3*g + 2*a*b*c** 
2*f**3*h - 12*a*b*c*d*e*f**2*h + 4*a*b*c*d*f**3*g + 12*a*b*d**2*e**2*f*h - 
 6*a*b*d**2*e*f**2*g - 3*b**2*c**2*e*f**2*h + b**2*c**2*f**3*g + 12*b**2*c 
*d*e**2*f*h - 6*b**2*c*d*e*f**2*g - 10*b**2*d**2*e**3*h + 6*b**2*d**2*e**2 
*f*g)/(3*f**5) + sqrt(e + f*x)*(a**2*c**2*f**4*h - 4*a**2*c*d*e*f**3*h + 2 
*a**2*c*d*f**4*g + 3*a**2*d**2*e**2*f**2*h - 2*a**2*d**2*e*f**3*g - 4*a*b* 
c**2*e*f**3*h + 2*a*b*c**2*f**4*g + 12*a*b*c*d*e**2*f**2*h - 8*a*b*c*d*e*f 
**3*g - 8*a*b*d**2*e**3*f*h + 6*a*b*d**2*e**2*f**2*g + 3*b**2*c**2*e**2*f* 
*2*h - 2*b**2*c**2*e*f**3*g - 8*b**2*c*d*e**3*f*h + 6*b**2*c*d*e**2*f**2*g 
 + 5*b**2*d**2*e**4*h - 4*b**2*d**2*e**3*f*g)/f**5 + (a*f - b*e)**2*(c*f - 
 d*e)**2*(e*h - f*g)/(f**5*sqrt(e + f*x)))/f, Ne(f, 0)), ((a**2*c**2*g*x + 
 b**2*d**2*h*x**6/6 + x**5*(2*a*b*d**2*h + 2*b**2*c*d*h + b**2*d**2*g)/5 + 
 x**4*(a**2*d**2*h + 4*a*b*c*d*h + 2*a*b*d**2*g + b**2*c**2*h + 2*b**2*c*d 
*g)/4 + x**3*(2*a**2*c*d*h + a**2*d**2*g + 2*a*b*c**2*h + 4*a*b*c*d*g + b* 
*2*c**2*g)/3 + x**2*(a**2*c**2*h + 2*a**2*c*d*g + 2*a*b*c**2*g)/2)/e**(...
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/315*((35*(f*x + e)^(9/2)*b^2*d^2*h + 45*(b^2*d^2*f*g - (5*b^2*d^2*e - 2* 
(b^2*c*d + a*b*d^2)*f)*h)*(f*x + e)^(7/2) - 63*(2*(2*b^2*d^2*e*f - (b^2*c* 
d + a*b*d^2)*f^2)*g - (10*b^2*d^2*e^2 - 8*(b^2*c*d + a*b*d^2)*e*f + (b^2*c 
^2 + 4*a*b*c*d + a^2*d^2)*f^2)*h)*(f*x + e)^(5/2) + 105*((6*b^2*d^2*e^2*f 
- 6*(b^2*c*d + a*b*d^2)*e*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^3)*g - ( 
10*b^2*d^2*e^3 - 12*(b^2*c*d + a*b*d^2)*e^2*f + 3*(b^2*c^2 + 4*a*b*c*d + a 
^2*d^2)*e*f^2 - 2*(a*b*c^2 + a^2*c*d)*f^3)*h)*(f*x + e)^(3/2) - 315*(2*(2* 
b^2*d^2*e^3*f - 3*(b^2*c*d + a*b*d^2)*e^2*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2 
*d^2)*e*f^3 - (a*b*c^2 + a^2*c*d)*f^4)*g - (5*b^2*d^2*e^4 + a^2*c^2*f^4 - 
8*(b^2*c*d + a*b*d^2)*e^3*f + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f^2 - 
4*(a*b*c^2 + a^2*c*d)*e*f^3)*h)*sqrt(f*x + e))/f^5 - 315*((b^2*d^2*e^4*f + 
 a^2*c^2*f^5 - 2*(b^2*c*d + a*b*d^2)*e^3*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2* 
d^2)*e^2*f^3 - 2*(a*b*c^2 + a^2*c*d)*e*f^4)*g - (b^2*d^2*e^5 + a^2*c^2*e*f 
^4 - 2*(b^2*c*d + a*b*d^2)*e^4*f + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^2 
 - 2*(a*b*c^2 + a^2*c*d)*e^2*f^3)*h)/(sqrt(f*x + e)*f^5))/f
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1137 vs. \(2 (387) = 774\).

Time = 0.15 (sec) , antiderivative size = 1137, normalized size of antiderivative = 2.79 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2*(b^2*d^2*e^4*f*g - 2*b^2*c*d*e^3*f^2*g - 2*a*b*d^2*e^3*f^2*g + b^2*c^2* 
e^2*f^3*g + 4*a*b*c*d*e^2*f^3*g + a^2*d^2*e^2*f^3*g - 2*a*b*c^2*e*f^4*g - 
2*a^2*c*d*e*f^4*g + a^2*c^2*f^5*g - b^2*d^2*e^5*h + 2*b^2*c*d*e^4*f*h + 2* 
a*b*d^2*e^4*f*h - b^2*c^2*e^3*f^2*h - 4*a*b*c*d*e^3*f^2*h - a^2*d^2*e^3*f^ 
2*h + 2*a*b*c^2*e^2*f^3*h + 2*a^2*c*d*e^2*f^3*h - a^2*c^2*e*f^4*h)/(sqrt(f 
*x + e)*f^6) + 2/315*(45*(f*x + e)^(7/2)*b^2*d^2*f^49*g - 252*(f*x + e)^(5 
/2)*b^2*d^2*e*f^49*g + 630*(f*x + e)^(3/2)*b^2*d^2*e^2*f^49*g - 1260*sqrt( 
f*x + e)*b^2*d^2*e^3*f^49*g + 126*(f*x + e)^(5/2)*b^2*c*d*f^50*g + 126*(f* 
x + e)^(5/2)*a*b*d^2*f^50*g - 630*(f*x + e)^(3/2)*b^2*c*d*e*f^50*g - 630*( 
f*x + e)^(3/2)*a*b*d^2*e*f^50*g + 1890*sqrt(f*x + e)*b^2*c*d*e^2*f^50*g + 
1890*sqrt(f*x + e)*a*b*d^2*e^2*f^50*g + 105*(f*x + e)^(3/2)*b^2*c^2*f^51*g 
 + 420*(f*x + e)^(3/2)*a*b*c*d*f^51*g + 105*(f*x + e)^(3/2)*a^2*d^2*f^51*g 
 - 630*sqrt(f*x + e)*b^2*c^2*e*f^51*g - 2520*sqrt(f*x + e)*a*b*c*d*e*f^51* 
g - 630*sqrt(f*x + e)*a^2*d^2*e*f^51*g + 630*sqrt(f*x + e)*a*b*c^2*f^52*g 
+ 630*sqrt(f*x + e)*a^2*c*d*f^52*g + 35*(f*x + e)^(9/2)*b^2*d^2*f^48*h - 2 
25*(f*x + e)^(7/2)*b^2*d^2*e*f^48*h + 630*(f*x + e)^(5/2)*b^2*d^2*e^2*f^48 
*h - 1050*(f*x + e)^(3/2)*b^2*d^2*e^3*f^48*h + 1575*sqrt(f*x + e)*b^2*d^2* 
e^4*f^48*h + 90*(f*x + e)^(7/2)*b^2*c*d*f^49*h + 90*(f*x + e)^(7/2)*a*b*d^ 
2*f^49*h - 504*(f*x + e)^(5/2)*b^2*c*d*e*f^49*h - 504*(f*x + e)^(5/2)*a*b* 
d^2*e*f^49*h + 1260*(f*x + e)^(3/2)*b^2*c*d*e^2*f^49*h + 1260*(f*x + e)...
 

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx=\frac {{\left (e+f\,x\right )}^{3/2}\,\left (4\,h\,a^2\,c\,d\,f^3-6\,h\,a^2\,d^2\,e\,f^2+2\,g\,a^2\,d^2\,f^3+4\,h\,a\,b\,c^2\,f^3-24\,h\,a\,b\,c\,d\,e\,f^2+8\,g\,a\,b\,c\,d\,f^3+24\,h\,a\,b\,d^2\,e^2\,f-12\,g\,a\,b\,d^2\,e\,f^2-6\,h\,b^2\,c^2\,e\,f^2+2\,g\,b^2\,c^2\,f^3+24\,h\,b^2\,c\,d\,e^2\,f-12\,g\,b^2\,c\,d\,e\,f^2-20\,h\,b^2\,d^2\,e^3+12\,g\,b^2\,d^2\,e^2\,f\right )}{3\,f^6}-\frac {-2\,h\,a^2\,c^2\,e\,f^4+2\,g\,a^2\,c^2\,f^5+4\,h\,a^2\,c\,d\,e^2\,f^3-4\,g\,a^2\,c\,d\,e\,f^4-2\,h\,a^2\,d^2\,e^3\,f^2+2\,g\,a^2\,d^2\,e^2\,f^3+4\,h\,a\,b\,c^2\,e^2\,f^3-4\,g\,a\,b\,c^2\,e\,f^4-8\,h\,a\,b\,c\,d\,e^3\,f^2+8\,g\,a\,b\,c\,d\,e^2\,f^3+4\,h\,a\,b\,d^2\,e^4\,f-4\,g\,a\,b\,d^2\,e^3\,f^2-2\,h\,b^2\,c^2\,e^3\,f^2+2\,g\,b^2\,c^2\,e^2\,f^3+4\,h\,b^2\,c\,d\,e^4\,f-4\,g\,b^2\,c\,d\,e^3\,f^2-2\,h\,b^2\,d^2\,e^5+2\,g\,b^2\,d^2\,e^4\,f}{f^6\,\sqrt {e+f\,x}}+\frac {{\left (e+f\,x\right )}^{5/2}\,\left (2\,h\,a^2\,d^2\,f^2+8\,h\,a\,b\,c\,d\,f^2-16\,h\,a\,b\,d^2\,e\,f+4\,g\,a\,b\,d^2\,f^2+2\,h\,b^2\,c^2\,f^2-16\,h\,b^2\,c\,d\,e\,f+4\,g\,b^2\,c\,d\,f^2+20\,h\,b^2\,d^2\,e^2-8\,g\,b^2\,d^2\,e\,f\right )}{5\,f^6}+\frac {2\,b^2\,d^2\,h\,{\left (e+f\,x\right )}^{9/2}}{9\,f^6}+\frac {2\,b\,d\,{\left (e+f\,x\right )}^{7/2}\,\left (2\,a\,d\,f\,h+2\,b\,c\,f\,h-5\,b\,d\,e\,h+b\,d\,f\,g\right )}{7\,f^6}+\frac {2\,\sqrt {e+f\,x}\,\left (a\,f-b\,e\right )\,\left (c\,f-d\,e\right )\,\left (a\,c\,f^2\,h+2\,a\,d\,f^2\,g+2\,b\,c\,f^2\,g+5\,b\,d\,e^2\,h-3\,a\,d\,e\,f\,h-3\,b\,c\,e\,f\,h-4\,b\,d\,e\,f\,g\right )}{f^6} \] Input:

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

Output:

((e + f*x)^(3/2)*(2*a^2*d^2*f^3*g + 2*b^2*c^2*f^3*g - 20*b^2*d^2*e^3*h + 4 
*a*b*c^2*f^3*h + 4*a^2*c*d*f^3*h - 6*a^2*d^2*e*f^2*h - 6*b^2*c^2*e*f^2*h + 
 12*b^2*d^2*e^2*f*g + 8*a*b*c*d*f^3*g - 12*a*b*d^2*e*f^2*g + 24*a*b*d^2*e^ 
2*f*h - 12*b^2*c*d*e*f^2*g + 24*b^2*c*d*e^2*f*h - 24*a*b*c*d*e*f^2*h))/(3* 
f^6) - (2*a^2*c^2*f^5*g - 2*b^2*d^2*e^5*h - 2*a^2*c^2*e*f^4*h + 2*b^2*d^2* 
e^4*f*g + 2*a^2*d^2*e^2*f^3*g + 2*b^2*c^2*e^2*f^3*g - 2*a^2*d^2*e^3*f^2*h 
- 2*b^2*c^2*e^3*f^2*h - 4*a*b*c^2*e*f^4*g + 4*a*b*d^2*e^4*f*h - 4*a^2*c*d* 
e*f^4*g + 4*b^2*c*d*e^4*f*h + 4*a*b*c^2*e^2*f^3*h - 4*a*b*d^2*e^3*f^2*g + 
4*a^2*c*d*e^2*f^3*h - 4*b^2*c*d*e^3*f^2*g + 8*a*b*c*d*e^2*f^3*g - 8*a*b*c* 
d*e^3*f^2*h)/(f^6*(e + f*x)^(1/2)) + ((e + f*x)^(5/2)*(2*a^2*d^2*f^2*h + 2 
*b^2*c^2*f^2*h + 20*b^2*d^2*e^2*h + 4*a*b*d^2*f^2*g + 4*b^2*c*d*f^2*g - 8* 
b^2*d^2*e*f*g + 8*a*b*c*d*f^2*h - 16*a*b*d^2*e*f*h - 16*b^2*c*d*e*f*h))/(5 
*f^6) + (2*b^2*d^2*h*(e + f*x)^(9/2))/(9*f^6) + (2*b*d*(e + f*x)^(7/2)*(2* 
a*d*f*h + 2*b*c*f*h - 5*b*d*e*h + b*d*f*g))/(7*f^6) + (2*(e + f*x)^(1/2)*( 
a*f - b*e)*(c*f - d*e)*(a*c*f^2*h + 2*a*d*f^2*g + 2*b*c*f^2*g + 5*b*d*e^2* 
h - 3*a*d*e*f*h - 3*b*c*e*f*h - 4*b*d*e*f*g))/f^6
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 919, normalized size of antiderivative = 2.26 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{(e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(630*a**2*c**2*e*f**4*h - 315*a**2*c**2*f**5*g + 315*a**2*c**2*f**5*h*x 
 - 1680*a**2*c*d*e**2*f**3*h + 1260*a**2*c*d*e*f**4*g - 840*a**2*c*d*e*f** 
4*h*x + 630*a**2*c*d*f**5*g*x + 210*a**2*c*d*f**5*h*x**2 + 1008*a**2*d**2* 
e**3*f**2*h - 840*a**2*d**2*e**2*f**3*g + 504*a**2*d**2*e**2*f**3*h*x - 42 
0*a**2*d**2*e*f**4*g*x - 126*a**2*d**2*e*f**4*h*x**2 + 105*a**2*d**2*f**5* 
g*x**2 + 63*a**2*d**2*f**5*h*x**3 - 1680*a*b*c**2*e**2*f**3*h + 1260*a*b*c 
**2*e*f**4*g - 840*a*b*c**2*e*f**4*h*x + 630*a*b*c**2*f**5*g*x + 210*a*b*c 
**2*f**5*h*x**2 + 4032*a*b*c*d*e**3*f**2*h - 3360*a*b*c*d*e**2*f**3*g + 20 
16*a*b*c*d*e**2*f**3*h*x - 1680*a*b*c*d*e*f**4*g*x - 504*a*b*c*d*e*f**4*h* 
x**2 + 420*a*b*c*d*f**5*g*x**2 + 252*a*b*c*d*f**5*h*x**3 - 2304*a*b*d**2*e 
**4*f*h + 2016*a*b*d**2*e**3*f**2*g - 1152*a*b*d**2*e**3*f**2*h*x + 1008*a 
*b*d**2*e**2*f**3*g*x + 288*a*b*d**2*e**2*f**3*h*x**2 - 252*a*b*d**2*e*f** 
4*g*x**2 - 144*a*b*d**2*e*f**4*h*x**3 + 126*a*b*d**2*f**5*g*x**3 + 90*a*b* 
d**2*f**5*h*x**4 + 1008*b**2*c**2*e**3*f**2*h - 840*b**2*c**2*e**2*f**3*g 
+ 504*b**2*c**2*e**2*f**3*h*x - 420*b**2*c**2*e*f**4*g*x - 126*b**2*c**2*e 
*f**4*h*x**2 + 105*b**2*c**2*f**5*g*x**2 + 63*b**2*c**2*f**5*h*x**3 - 2304 
*b**2*c*d*e**4*f*h + 2016*b**2*c*d*e**3*f**2*g - 1152*b**2*c*d*e**3*f**2*h 
*x + 1008*b**2*c*d*e**2*f**3*g*x + 288*b**2*c*d*e**2*f**3*h*x**2 - 252*b** 
2*c*d*e*f**4*g*x**2 - 144*b**2*c*d*e*f**4*h*x**3 + 126*b**2*c*d*f**5*g*x** 
3 + 90*b**2*c*d*f**5*h*x**4 + 1280*b**2*d**2*e**5*h - 1152*b**2*d**2*e*...