\(\int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{\sqrt {e+f x}} \, dx\) [115]

Optimal result

Integrand size = 29, antiderivative size = 409 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 (b e-a f)^2 (d e-c f)^2 (f g-e h) \sqrt {e+f x}}{f^6}-\frac {2 (b e-a f) (d e-c f) (b d e (4 f g-5 e h)-b c f (2 f g-3 e h)-a f (2 d f g-3 d e h+c f h)) (e+f x)^{3/2}}{3 f^6}+\frac {2 \left (a^2 d f^2 (d f g-3 d e h+2 c f h)+2 a b f \left (c^2 f^2 h+2 c d f (f g-3 e h)-3 d^2 e (f g-2 e h)\right )+b^2 \left (2 d^2 e^2 (3 f g-5 e h)+c^2 f^2 (f g-3 e h)-6 c d e f (f g-2 e h)\right )\right ) (e+f x)^{5/2}}{5 f^6}+\frac {2 \left (a^2 d^2 f^2 h+2 a b d f (d f g-4 d e h+2 c f h)+b^2 \left (c^2 f^2 h-2 d^2 e (2 f g-5 e h)+2 c d f (f g-4 e h)\right )\right ) (e+f x)^{7/2}}{7 f^6}+\frac {2 b d (2 a d f h+b (d f g-5 d e h+2 c f h)) (e+f x)^{9/2}}{9 f^6}+\frac {2 b^2 d^2 h (e+f x)^{11/2}}{11 f^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{\sqrt {e+f x}} \, dx=\frac {2 \sqrt {e+f x} \left (33 a^2 f^2 \left (35 c^2 f^2 (3 f g-2 e h+f h x)+14 c d f \left (8 e^2 h-2 e f (5 g+2 h x)+f^2 x (5 g+3 h x)\right )+d^2 \left (-48 e^3 h+8 e^2 f (7 g+3 h x)+3 f^3 x^2 (7 g+5 h x)-2 e f^2 x (14 g+9 h x)\right )\right )+22 a b f \left (21 c^2 f^2 \left (8 e^2 h-2 e f (5 g+2 h x)+f^2 x (5 g+3 h x)\right )+6 c d f \left (-48 e^3 h+8 e^2 f (7 g+3 h x)+3 f^3 x^2 (7 g+5 h x)-2 e f^2 x (14 g+9 h x)\right )+d^2 \left (128 e^4 h+24 e^2 f^2 x (3 g+2 h x)-16 e^3 f (9 g+4 h x)+5 f^4 x^3 (9 g+7 h x)-2 e f^3 x^2 (27 g+20 h x)\right )\right )+b^2 \left (33 c^2 f^2 \left (-48 e^3 h+8 e^2 f (7 g+3 h x)+3 f^3 x^2 (7 g+5 h x)-2 e f^2 x (14 g+9 h x)\right )+22 c d f \left (128 e^4 h+24 e^2 f^2 x (3 g+2 h x)-16 e^3 f (9 g+4 h x)+5 f^4 x^3 (9 g+7 h x)-2 e f^3 x^2 (27 g+20 h x)\right )+d^2 \left (-1280 e^5 h+128 e^4 f (11 g+5 h x)+35 f^5 x^4 (11 g+9 h x)-32 e^3 f^2 x (22 g+15 h x)+16 e^2 f^3 x^2 (33 g+25 h x)-10 e f^4 x^3 (44 g+35 h x)\right )\right )\right )}{3465 f^6} \] Input:

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

Output:

(2*Sqrt[e + f*x]*(33*a^2*f^2*(35*c^2*f^2*(3*f*g - 2*e*h + f*h*x) + 14*c*d* 
f*(8*e^2*h - 2*e*f*(5*g + 2*h*x) + f^2*x*(5*g + 3*h*x)) + d^2*(-48*e^3*h + 
 8*e^2*f*(7*g + 3*h*x) + 3*f^3*x^2*(7*g + 5*h*x) - 2*e*f^2*x*(14*g + 9*h*x 
))) + 22*a*b*f*(21*c^2*f^2*(8*e^2*h - 2*e*f*(5*g + 2*h*x) + f^2*x*(5*g + 3 
*h*x)) + 6*c*d*f*(-48*e^3*h + 8*e^2*f*(7*g + 3*h*x) + 3*f^3*x^2*(7*g + 5*h 
*x) - 2*e*f^2*x*(14*g + 9*h*x)) + d^2*(128*e^4*h + 24*e^2*f^2*x*(3*g + 2*h 
*x) - 16*e^3*f*(9*g + 4*h*x) + 5*f^4*x^3*(9*g + 7*h*x) - 2*e*f^3*x^2*(27*g 
 + 20*h*x))) + b^2*(33*c^2*f^2*(-48*e^3*h + 8*e^2*f*(7*g + 3*h*x) + 3*f^3* 
x^2*(7*g + 5*h*x) - 2*e*f^2*x*(14*g + 9*h*x)) + 22*c*d*f*(128*e^4*h + 24*e 
^2*f^2*x*(3*g + 2*h*x) - 16*e^3*f*(9*g + 4*h*x) + 5*f^4*x^3*(9*g + 7*h*x) 
- 2*e*f^3*x^2*(27*g + 20*h*x)) + d^2*(-1280*e^5*h + 128*e^4*f*(11*g + 5*h* 
x) + 35*f^5*x^4*(11*g + 9*h*x) - 32*e^3*f^2*x*(22*g + 15*h*x) + 16*e^2*f^3 
*x^2*(33*g + 25*h*x) - 10*e*f^4*x^3*(44*g + 35*h*x)))))/(3465*f^6)
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 409, 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.

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

\(\Big \downarrow \) 165

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*(b*e - a*f)^2*(d*e - c*f)^2*(f*g - e*h)*Sqrt[e + f*x])/f^6 - (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))*(e + f*x)^(3/2))/(3*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)^(5/2))/(5*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)^(7/2))/(7*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)^(9/2))/(9*f^6) + (2*b^2*d^2 
*h*(e + f*x)^(11/2))/(11*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 = 1.02 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {\frac {2 h \,b^{2} d^{2} \left (f x +e \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (2 b \left (a f -b e \right ) d^{2}+2 b^{2} d \left (c f -d e \right )\right ) h +b^{2} d^{2} \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (\left (a f -b e \right )^{2} d^{2}+4 b \left (a f -b e \right ) d \left (c f -d e \right )+b^{2} \left (c f -d e \right )^{2}\right ) h +\left (2 b \left (a f -b e \right ) d^{2}+2 b^{2} d \left (c f -d e \right )\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (2 \left (a f -b e \right )^{2} d \left (c f -d e \right )+2 b \left (a f -b e \right ) \left (c f -d e \right )^{2}\right ) h +\left (\left (a f -b e \right )^{2} d^{2}+4 b \left (a f -b e \right ) d \left (c f -d e \right )+b^{2} \left (c f -d e \right )^{2}\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a f -b e \right )^{2} \left (c f -d e \right )^{2} h +\left (2 \left (a f -b e \right )^{2} d \left (c f -d e \right )+2 b \left (a f -b e \right ) \left (c f -d e \right )^{2}\right ) \left (-e h +f g \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+2 \left (a f -b e \right )^{2} \left (c f -d e \right )^{2} \left (-e h +f g \right ) \sqrt {f x +e}}{f^{6}}\)	\(411\)
default	\(\frac {\frac {2 h \,b^{2} d^{2} \left (f x +e \right )^{\frac {11}{2}}}{11}-\frac {2 \left (-\left (2 b \left (a f -b e \right ) d^{2}+2 b^{2} d \left (c f -d e \right )\right ) h +b^{2} d^{2} \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}-\frac {2 \left (-\left (\left (a f -b e \right )^{2} d^{2}+4 b \left (a f -b e \right ) d \left (c f -d e \right )+b^{2} \left (c f -d e \right )^{2}\right ) h +\left (2 b \left (a f -b e \right ) d^{2}+2 b^{2} d \left (c f -d e \right )\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {2 \left (-\left (2 \left (a f -b e \right )^{2} d \left (c f -d e \right )+2 b \left (a f -b e \right ) \left (c f -d e \right )^{2}\right ) h +\left (\left (a f -b e \right )^{2} d^{2}+4 b \left (a f -b e \right ) d \left (c f -d e \right )+b^{2} \left (c f -d e \right )^{2}\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 \left (-\left (a f -b e \right )^{2} \left (c f -d e \right )^{2} h +\left (2 \left (a f -b e \right )^{2} d \left (c f -d e \right )+2 b \left (a f -b e \right ) \left (c f -d e \right )^{2}\right ) \left (e h -f g \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}-2 \left (a f -b e \right )^{2} \left (c f -d e \right )^{2} \left (e h -f g \right ) \sqrt {f x +e}}{f^{6}}\)	\(416\)
pseudoelliptic	\(-\frac {4 \left (\left (-\frac {3 x^{2} \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 ) b^{2}}{10}-a x \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 -\frac {3 \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 ) a^{2}}{2}\right ) f^{5}+\left (\frac {2 x \left (\frac {10 x^{2} \left (\frac {35 h x}{44}+g \right ) d^{2}}{21}+\frac {9 \left (\frac {20 h x}{27}+g \right ) x c d}{7}+c^{2} \left (\frac {9 h x}{14}+g \right )\right ) b^{2}}{5}+2 a \left (\left (\frac {4}{21} h \,x^{3}+\frac {9}{35} g \,x^{2}\right ) d^{2}+\frac {4 x c \left (\frac {9 h x}{14}+g \right ) d}{5}+c^{2} \left (\frac {2 h x}{5}+g \right )\right ) b +a^{2} \left (\frac {2 x \left (\frac {9 h x}{14}+g \right ) d^{2}}{5}+2 c \left (\frac {2 h x}{5}+g \right ) d +h \,c^{2}\right )\right ) e \,f^{4}-\frac {8 \left (\left (\frac {x^{2} \left (\frac {25 h x}{33}+g \right ) d^{2}}{7}+\frac {3 x c \left (\frac {2 h x}{3}+g \right ) d}{7}+\frac {c^{2} \left (\frac {3 h x}{7}+g \right )}{2}\right ) b^{2}+a \left (\frac {3 x \left (\frac {2 h x}{3}+g \right ) d^{2}}{7}+2 c \left (\frac {3 h x}{7}+g \right ) d +h \,c^{2}\right ) b +a^{2} d \left (\left (\frac {3 h x}{14}+\frac {g}{2}\right ) d +c h \right )\right ) e^{2} f^{3}}{5}+\frac {24 \left (\left (\frac {4 x \left (\frac {15 h x}{22}+g \right ) d^{2}}{9}+2 \left (\frac {4 h x}{9}+g \right ) c d +h \,c^{2}\right ) b^{2}+4 a d \left (\left (\frac {2 h x}{9}+\frac {g}{2}\right ) d +c h \right ) b +a^{2} d^{2} h \right ) e^{3} f^{2}}{35}-\frac {128 d \left (\left (\left (\frac {5 h x}{22}+\frac {g}{2}\right ) d +c h \right ) b +a d h \right ) b \,e^{4} f}{105}+\frac {128 b^{2} d^{2} e^{5} h}{231}\right ) \sqrt {f x +e}}{3 f^{6}}\)	\(477\)
gosper	\(-\frac {2 \sqrt {f x +e}\, \left (-315 h \,b^{2} d^{2} x^{5} f^{5}-770 a b \,d^{2} f^{5} h \,x^{4}-770 b^{2} c d \,f^{5} h \,x^{4}+350 b^{2} d^{2} e \,f^{4} h \,x^{4}-385 b^{2} d^{2} f^{5} g \,x^{4}-495 a^{2} d^{2} f^{5} h \,x^{3}-1980 a b c d \,f^{5} h \,x^{3}+880 a b \,d^{2} e \,f^{4} h \,x^{3}-990 a b \,d^{2} f^{5} g \,x^{3}-495 b^{2} c^{2} f^{5} h \,x^{3}+880 b^{2} c d e \,f^{4} h \,x^{3}-990 b^{2} c d \,f^{5} g \,x^{3}-400 b^{2} d^{2} e^{2} f^{3} h \,x^{3}+440 b^{2} d^{2} e \,f^{4} g \,x^{3}-1386 a^{2} c d \,f^{5} h \,x^{2}+594 a^{2} d^{2} e \,f^{4} h \,x^{2}-693 a^{2} d^{2} f^{5} g \,x^{2}-1386 a b \,c^{2} f^{5} h \,x^{2}+2376 a b c d e \,f^{4} h \,x^{2}-2772 a b c d \,f^{5} g \,x^{2}-1056 a b \,d^{2} e^{2} f^{3} h \,x^{2}+1188 a b \,d^{2} e \,f^{4} g \,x^{2}+594 b^{2} c^{2} e \,f^{4} h \,x^{2}-693 b^{2} c^{2} f^{5} g \,x^{2}-1056 b^{2} c d \,e^{2} f^{3} h \,x^{2}+1188 b^{2} c d e \,f^{4} g \,x^{2}+480 b^{2} d^{2} e^{3} f^{2} h \,x^{2}-528 b^{2} d^{2} e^{2} f^{3} g \,x^{2}-1155 a^{2} c^{2} f^{5} h x +1848 a^{2} c d e \,f^{4} h x -2310 a^{2} c d \,f^{5} g x -792 a^{2} d^{2} e^{2} f^{3} h x +924 a^{2} d^{2} e \,f^{4} g x +1848 a b \,c^{2} e \,f^{4} h x -2310 a b \,c^{2} f^{5} g x -3168 a b c d \,e^{2} f^{3} h x +3696 a b c d e \,f^{4} g x +1408 a b \,d^{2} e^{3} f^{2} h x -1584 a b \,d^{2} e^{2} f^{3} g x -792 b^{2} c^{2} e^{2} f^{3} h x +924 b^{2} c^{2} e \,f^{4} g x +1408 b^{2} c d \,e^{3} f^{2} h x -1584 b^{2} c d \,e^{2} f^{3} g x -640 b^{2} d^{2} e^{4} f h x +704 b^{2} d^{2} e^{3} f^{2} g x +2310 a^{2} c^{2} e \,f^{4} h -3465 g \,a^{2} c^{2} f^{5}-3696 a^{2} c d \,e^{2} f^{3} h +4620 a^{2} c d e \,f^{4} g +1584 a^{2} d^{2} e^{3} f^{2} h -1848 a^{2} d^{2} e^{2} f^{3} g -3696 a b \,c^{2} e^{2} f^{3} h +4620 a b \,c^{2} e \,f^{4} g +6336 a b c d \,e^{3} f^{2} h -7392 a b c d \,e^{2} f^{3} g -2816 a b \,d^{2} e^{4} f h +3168 a b \,d^{2} e^{3} f^{2} g +1584 b^{2} c^{2} e^{3} f^{2} h -1848 b^{2} c^{2} e^{2} f^{3} g -2816 b^{2} c d \,e^{4} f h +3168 b^{2} c d \,e^{3} f^{2} g +1280 b^{2} d^{2} e^{5} h -1408 b^{2} d^{2} e^{4} f g \right )}{3465 f^{6}}\)	\(919\)
trager	\(-\frac {2 \sqrt {f x +e}\, \left (-315 h \,b^{2} d^{2} x^{5} f^{5}-770 a b \,d^{2} f^{5} h \,x^{4}-770 b^{2} c d \,f^{5} h \,x^{4}+350 b^{2} d^{2} e \,f^{4} h \,x^{4}-385 b^{2} d^{2} f^{5} g \,x^{4}-495 a^{2} d^{2} f^{5} h \,x^{3}-1980 a b c d \,f^{5} h \,x^{3}+880 a b \,d^{2} e \,f^{4} h \,x^{3}-990 a b \,d^{2} f^{5} g \,x^{3}-495 b^{2} c^{2} f^{5} h \,x^{3}+880 b^{2} c d e \,f^{4} h \,x^{3}-990 b^{2} c d \,f^{5} g \,x^{3}-400 b^{2} d^{2} e^{2} f^{3} h \,x^{3}+440 b^{2} d^{2} e \,f^{4} g \,x^{3}-1386 a^{2} c d \,f^{5} h \,x^{2}+594 a^{2} d^{2} e \,f^{4} h \,x^{2}-693 a^{2} d^{2} f^{5} g \,x^{2}-1386 a b \,c^{2} f^{5} h \,x^{2}+2376 a b c d e \,f^{4} h \,x^{2}-2772 a b c d \,f^{5} g \,x^{2}-1056 a b \,d^{2} e^{2} f^{3} h \,x^{2}+1188 a b \,d^{2} e \,f^{4} g \,x^{2}+594 b^{2} c^{2} e \,f^{4} h \,x^{2}-693 b^{2} c^{2} f^{5} g \,x^{2}-1056 b^{2} c d \,e^{2} f^{3} h \,x^{2}+1188 b^{2} c d e \,f^{4} g \,x^{2}+480 b^{2} d^{2} e^{3} f^{2} h \,x^{2}-528 b^{2} d^{2} e^{2} f^{3} g \,x^{2}-1155 a^{2} c^{2} f^{5} h x +1848 a^{2} c d e \,f^{4} h x -2310 a^{2} c d \,f^{5} g x -792 a^{2} d^{2} e^{2} f^{3} h x +924 a^{2} d^{2} e \,f^{4} g x +1848 a b \,c^{2} e \,f^{4} h x -2310 a b \,c^{2} f^{5} g x -3168 a b c d \,e^{2} f^{3} h x +3696 a b c d e \,f^{4} g x +1408 a b \,d^{2} e^{3} f^{2} h x -1584 a b \,d^{2} e^{2} f^{3} g x -792 b^{2} c^{2} e^{2} f^{3} h x +924 b^{2} c^{2} e \,f^{4} g x +1408 b^{2} c d \,e^{3} f^{2} h x -1584 b^{2} c d \,e^{2} f^{3} g x -640 b^{2} d^{2} e^{4} f h x +704 b^{2} d^{2} e^{3} f^{2} g x +2310 a^{2} c^{2} e \,f^{4} h -3465 g \,a^{2} c^{2} f^{5}-3696 a^{2} c d \,e^{2} f^{3} h +4620 a^{2} c d e \,f^{4} g +1584 a^{2} d^{2} e^{3} f^{2} h -1848 a^{2} d^{2} e^{2} f^{3} g -3696 a b \,c^{2} e^{2} f^{3} h +4620 a b \,c^{2} e \,f^{4} g +6336 a b c d \,e^{3} f^{2} h -7392 a b c d \,e^{2} f^{3} g -2816 a b \,d^{2} e^{4} f h +3168 a b \,d^{2} e^{3} f^{2} g +1584 b^{2} c^{2} e^{3} f^{2} h -1848 b^{2} c^{2} e^{2} f^{3} g -2816 b^{2} c d \,e^{4} f h +3168 b^{2} c d \,e^{3} f^{2} g +1280 b^{2} d^{2} e^{5} h -1408 b^{2} d^{2} e^{4} f g \right )}{3465 f^{6}}\)	\(919\)
risch	\(-\frac {2 \sqrt {f x +e}\, \left (-315 h \,b^{2} d^{2} x^{5} f^{5}-770 a b \,d^{2} f^{5} h \,x^{4}-770 b^{2} c d \,f^{5} h \,x^{4}+350 b^{2} d^{2} e \,f^{4} h \,x^{4}-385 b^{2} d^{2} f^{5} g \,x^{4}-495 a^{2} d^{2} f^{5} h \,x^{3}-1980 a b c d \,f^{5} h \,x^{3}+880 a b \,d^{2} e \,f^{4} h \,x^{3}-990 a b \,d^{2} f^{5} g \,x^{3}-495 b^{2} c^{2} f^{5} h \,x^{3}+880 b^{2} c d e \,f^{4} h \,x^{3}-990 b^{2} c d \,f^{5} g \,x^{3}-400 b^{2} d^{2} e^{2} f^{3} h \,x^{3}+440 b^{2} d^{2} e \,f^{4} g \,x^{3}-1386 a^{2} c d \,f^{5} h \,x^{2}+594 a^{2} d^{2} e \,f^{4} h \,x^{2}-693 a^{2} d^{2} f^{5} g \,x^{2}-1386 a b \,c^{2} f^{5} h \,x^{2}+2376 a b c d e \,f^{4} h \,x^{2}-2772 a b c d \,f^{5} g \,x^{2}-1056 a b \,d^{2} e^{2} f^{3} h \,x^{2}+1188 a b \,d^{2} e \,f^{4} g \,x^{2}+594 b^{2} c^{2} e \,f^{4} h \,x^{2}-693 b^{2} c^{2} f^{5} g \,x^{2}-1056 b^{2} c d \,e^{2} f^{3} h \,x^{2}+1188 b^{2} c d e \,f^{4} g \,x^{2}+480 b^{2} d^{2} e^{3} f^{2} h \,x^{2}-528 b^{2} d^{2} e^{2} f^{3} g \,x^{2}-1155 a^{2} c^{2} f^{5} h x +1848 a^{2} c d e \,f^{4} h x -2310 a^{2} c d \,f^{5} g x -792 a^{2} d^{2} e^{2} f^{3} h x +924 a^{2} d^{2} e \,f^{4} g x +1848 a b \,c^{2} e \,f^{4} h x -2310 a b \,c^{2} f^{5} g x -3168 a b c d \,e^{2} f^{3} h x +3696 a b c d e \,f^{4} g x +1408 a b \,d^{2} e^{3} f^{2} h x -1584 a b \,d^{2} e^{2} f^{3} g x -792 b^{2} c^{2} e^{2} f^{3} h x +924 b^{2} c^{2} e \,f^{4} g x +1408 b^{2} c d \,e^{3} f^{2} h x -1584 b^{2} c d \,e^{2} f^{3} g x -640 b^{2} d^{2} e^{4} f h x +704 b^{2} d^{2} e^{3} f^{2} g x +2310 a^{2} c^{2} e \,f^{4} h -3465 g \,a^{2} c^{2} f^{5}-3696 a^{2} c d \,e^{2} f^{3} h +4620 a^{2} c d e \,f^{4} g +1584 a^{2} d^{2} e^{3} f^{2} h -1848 a^{2} d^{2} e^{2} f^{3} g -3696 a b \,c^{2} e^{2} f^{3} h +4620 a b \,c^{2} e \,f^{4} g +6336 a b c d \,e^{3} f^{2} h -7392 a b c d \,e^{2} f^{3} g -2816 a b \,d^{2} e^{4} f h +3168 a b \,d^{2} e^{3} f^{2} g +1584 b^{2} c^{2} e^{3} f^{2} h -1848 b^{2} c^{2} e^{2} f^{3} g -2816 b^{2} c d \,e^{4} f h +3168 b^{2} c d \,e^{3} f^{2} g +1280 b^{2} d^{2} e^{5} h -1408 b^{2} d^{2} e^{4} f g \right )}{3465 f^{6}}\)	\(919\)
orering	\(-\frac {2 \sqrt {f x +e}\, \left (-315 h \,b^{2} d^{2} x^{5} f^{5}-770 a b \,d^{2} f^{5} h \,x^{4}-770 b^{2} c d \,f^{5} h \,x^{4}+350 b^{2} d^{2} e \,f^{4} h \,x^{4}-385 b^{2} d^{2} f^{5} g \,x^{4}-495 a^{2} d^{2} f^{5} h \,x^{3}-1980 a b c d \,f^{5} h \,x^{3}+880 a b \,d^{2} e \,f^{4} h \,x^{3}-990 a b \,d^{2} f^{5} g \,x^{3}-495 b^{2} c^{2} f^{5} h \,x^{3}+880 b^{2} c d e \,f^{4} h \,x^{3}-990 b^{2} c d \,f^{5} g \,x^{3}-400 b^{2} d^{2} e^{2} f^{3} h \,x^{3}+440 b^{2} d^{2} e \,f^{4} g \,x^{3}-1386 a^{2} c d \,f^{5} h \,x^{2}+594 a^{2} d^{2} e \,f^{4} h \,x^{2}-693 a^{2} d^{2} f^{5} g \,x^{2}-1386 a b \,c^{2} f^{5} h \,x^{2}+2376 a b c d e \,f^{4} h \,x^{2}-2772 a b c d \,f^{5} g \,x^{2}-1056 a b \,d^{2} e^{2} f^{3} h \,x^{2}+1188 a b \,d^{2} e \,f^{4} g \,x^{2}+594 b^{2} c^{2} e \,f^{4} h \,x^{2}-693 b^{2} c^{2} f^{5} g \,x^{2}-1056 b^{2} c d \,e^{2} f^{3} h \,x^{2}+1188 b^{2} c d e \,f^{4} g \,x^{2}+480 b^{2} d^{2} e^{3} f^{2} h \,x^{2}-528 b^{2} d^{2} e^{2} f^{3} g \,x^{2}-1155 a^{2} c^{2} f^{5} h x +1848 a^{2} c d e \,f^{4} h x -2310 a^{2} c d \,f^{5} g x -792 a^{2} d^{2} e^{2} f^{3} h x +924 a^{2} d^{2} e \,f^{4} g x +1848 a b \,c^{2} e \,f^{4} h x -2310 a b \,c^{2} f^{5} g x -3168 a b c d \,e^{2} f^{3} h x +3696 a b c d e \,f^{4} g x +1408 a b \,d^{2} e^{3} f^{2} h x -1584 a b \,d^{2} e^{2} f^{3} g x -792 b^{2} c^{2} e^{2} f^{3} h x +924 b^{2} c^{2} e \,f^{4} g x +1408 b^{2} c d \,e^{3} f^{2} h x -1584 b^{2} c d \,e^{2} f^{3} g x -640 b^{2} d^{2} e^{4} f h x +704 b^{2} d^{2} e^{3} f^{2} g x +2310 a^{2} c^{2} e \,f^{4} h -3465 g \,a^{2} c^{2} f^{5}-3696 a^{2} c d \,e^{2} f^{3} h +4620 a^{2} c d e \,f^{4} g +1584 a^{2} d^{2} e^{3} f^{2} h -1848 a^{2} d^{2} e^{2} f^{3} g -3696 a b \,c^{2} e^{2} f^{3} h +4620 a b \,c^{2} e \,f^{4} g +6336 a b c d \,e^{3} f^{2} h -7392 a b c d \,e^{2} f^{3} g -2816 a b \,d^{2} e^{4} f h +3168 a b \,d^{2} e^{3} f^{2} g +1584 b^{2} c^{2} e^{3} f^{2} h -1848 b^{2} c^{2} e^{2} f^{3} g -2816 b^{2} c d \,e^{4} f h +3168 b^{2} c d \,e^{3} f^{2} g +1280 b^{2} d^{2} e^{5} h -1408 b^{2} d^{2} e^{4} f g \right )}{3465 f^{6}}\)	\(919\)

Input:

int((b*x+a)^2*(d*x+c)^2*(h*x+g)/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 712, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{\sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^2*(d*x+c)^2*(h*x+g)/(f*x+e)^(1/2),x, algorithm="fricas")
 

Output:

2/3465*(315*b^2*d^2*f^5*h*x^5 + 35*(11*b^2*d^2*f^5*g - 2*(5*b^2*d^2*e*f^4 
- 11*(b^2*c*d + a*b*d^2)*f^5)*h)*x^4 - 5*(22*(4*b^2*d^2*e*f^4 - 9*(b^2*c*d 
 + a*b*d^2)*f^5)*g - (80*b^2*d^2*e^2*f^3 - 176*(b^2*c*d + a*b*d^2)*e*f^4 + 
 99*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^5)*h)*x^3 + 3*(11*(16*b^2*d^2*e^2*f^ 
3 - 36*(b^2*c*d + a*b*d^2)*e*f^4 + 21*(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 - 176*(b^2*c*d + a*b*d^2)*e^2*f^3 + 99*(b^2*c^2 
 + 4*a*b*c*d + a^2*d^2)*e*f^4 - 231*(a*b*c^2 + a^2*c*d)*f^5)*h)*x^2 + 11*( 
128*b^2*d^2*e^4*f + 315*a^2*c^2*f^5 - 288*(b^2*c*d + a*b*d^2)*e^3*f^2 + 16 
8*(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 + 1155*a^2*c^2*e*f^4 - 1408*(b^2*c*d + a*b*d^2)*e^ 
4*f + 792*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3*f^2 - 1848*(a*b*c^2 + a^2*c* 
d)*e^2*f^3)*h - (22*(32*b^2*d^2*e^3*f^2 - 72*(b^2*c*d + a*b*d^2)*e^2*f^3 + 
 42*(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 + 1155*a^2*c^2*f^5 - 1408*(b^2*c*d + a*b*d^2)*e^3*f^ 
2 + 792*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f^3 - 1848*(a*b*c^2 + a^2*c*d) 
*e*f^4)*h)*x)*sqrt(f*x + e)/f^6
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (444) = 888\).

Time = 1.76 (sec) , antiderivative size = 1202, normalized size of antiderivative = 2.94 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{\sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)**2*(d*x+c)**2*(h*x+g)/(f*x+e)**(1/2),x)
 

Output:

Piecewise((2*(b**2*d**2*h*(e + f*x)**(11/2)/(11*f**5) + (e + f*x)**(9/2)*( 
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)/(9*f**5 
) + (e + f*x)**(7/2)*(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)/(7*f**5) + (e + f*x)**(5 
/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)/(5*f**5) + (e + f*x)**(3/2)*(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)/(3*f**5) + sqrt(e + f*x 
)*(-a**2*c**2*e*f**4*h + a**2*c**2*f**5*g + 2*a**2*c*d*e**2*f**3*h - 2*a** 
2*c*d*e*f**4*g - a**2*d**2*e**3*f**2*h + a**2*d**2*e**2*f**3*g + 2*a*b*c** 
2*e**2*f**3*h - 2*a*b*c**2*e*f**4*g - 4*a*b*c*d*e**3*f**2*h + 4*a*b*c*d*e* 
*2*f**3*g + 2*a*b*d**2*e**4*f*h - 2*a*b*d**2*e**3*f**2*g - b**2*c**2*e**3* 
f**2*h + b**2*c**2*e**2*f**3*g + 2*b**2*c*d*e**4*f*h - 2*b**2*c*d*e**3*...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.70 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{\sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^2*(d*x+c)^2*(h*x+g)/(f*x+e)^(1/2),x, algorithm="maxima")
 

Output:

2/3465*(315*(f*x + e)^(11/2)*b^2*d^2*h + 385*(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)^(9/2) - 495*(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)^(7/2) + 693*((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)^(5/2) - 1155*( 
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)*(f*x + e)^(3/2) + 3465*((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^6
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (387) = 774\).

Time = 0.14 (sec) , antiderivative size = 884, normalized size of antiderivative = 2.16 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{\sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^2*(d*x+c)^2*(h*x+g)/(f*x+e)^(1/2),x, algorithm="giac")
 

Output:

2/3465*(3465*sqrt(f*x + e)*a^2*c^2*g + 2310*((f*x + e)^(3/2) - 3*sqrt(f*x 
+ e)*e)*a*b*c^2*g/f + 2310*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^2*c*d*g 
/f + 1155*((f*x + e)^(3/2) - 3*sqrt(f*x + e)*e)*a^2*c^2*h/f + 231*(3*(f*x 
+ e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*b^2*c^2*g/f^2 + 
924*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + e)*e^2)*a*b* 
c*d*g/f^2 + 231*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15*sqrt(f*x + 
e)*e^2)*a^2*d^2*g/f^2 + 462*(3*(f*x + e)^(5/2) - 10*(f*x + e)^(3/2)*e + 15 
*sqrt(f*x + e)*e^2)*a*b*c^2*h/f^2 + 462*(3*(f*x + e)^(5/2) - 10*(f*x + e)^ 
(3/2)*e + 15*sqrt(f*x + e)*e^2)*a^2*c*d*h/f^2 + 198*(5*(f*x + e)^(7/2) - 2 
1*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*sqrt(f*x + e)*e^3)*b^2*c 
*d*g/f^3 + 198*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3 
/2)*e^2 - 35*sqrt(f*x + e)*e^3)*a*b*d^2*g/f^3 + 99*(5*(f*x + e)^(7/2) - 21 
*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*sqrt(f*x + e)*e^3)*b^2*c^ 
2*h/f^3 + 396*(5*(f*x + e)^(7/2) - 21*(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/ 
2)*e^2 - 35*sqrt(f*x + e)*e^3)*a*b*c*d*h/f^3 + 99*(5*(f*x + e)^(7/2) - 21* 
(f*x + e)^(5/2)*e + 35*(f*x + e)^(3/2)*e^2 - 35*sqrt(f*x + e)*e^3)*a^2*d^2 
*h/f^3 + 11*(35*(f*x + e)^(9/2) - 180*(f*x + e)^(7/2)*e + 378*(f*x + e)^(5 
/2)*e^2 - 420*(f*x + e)^(3/2)*e^3 + 315*sqrt(f*x + e)*e^4)*b^2*d^2*g/f^4 + 
 22*(35*(f*x + e)^(9/2) - 180*(f*x + e)^(7/2)*e + 378*(f*x + e)^(5/2)*e^2 
- 420*(f*x + e)^(3/2)*e^3 + 315*sqrt(f*x + e)*e^4)*b^2*c*d*h/f^4 + 22*(...
 

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{\sqrt {e+f x}} \, dx=\frac {{\left (e+f\,x\right )}^{5/2}\,\left (4\,h\,a^2\,c\,d\,f^3-6\,h\,a^2\,d^2\,e\,f^2+2\,g\,a^2\,d^2\,f^3+4\,h\,a\,b\,c^2\,f^3-24\,h\,a\,b\,c\,d\,e\,f^2+8\,g\,a\,b\,c\,d\,f^3+24\,h\,a\,b\,d^2\,e^2\,f-12\,g\,a\,b\,d^2\,e\,f^2-6\,h\,b^2\,c^2\,e\,f^2+2\,g\,b^2\,c^2\,f^3+24\,h\,b^2\,c\,d\,e^2\,f-12\,g\,b^2\,c\,d\,e\,f^2-20\,h\,b^2\,d^2\,e^3+12\,g\,b^2\,d^2\,e^2\,f\right )}{5\,f^6}+\frac {{\left (e+f\,x\right )}^{7/2}\,\left (2\,h\,a^2\,d^2\,f^2+8\,h\,a\,b\,c\,d\,f^2-16\,h\,a\,b\,d^2\,e\,f+4\,g\,a\,b\,d^2\,f^2+2\,h\,b^2\,c^2\,f^2-16\,h\,b^2\,c\,d\,e\,f+4\,g\,b^2\,c\,d\,f^2+20\,h\,b^2\,d^2\,e^2-8\,g\,b^2\,d^2\,e\,f\right )}{7\,f^6}-\frac {2\,\sqrt {e+f\,x}\,{\left (a\,f-b\,e\right )}^2\,{\left (c\,f-d\,e\right )}^2\,\left (e\,h-f\,g\right )}{f^6}+\frac {2\,b^2\,d^2\,h\,{\left (e+f\,x\right )}^{11/2}}{11\,f^6}+\frac {2\,b\,d\,{\left (e+f\,x\right )}^{9/2}\,\left (2\,a\,d\,f\,h+2\,b\,c\,f\,h-5\,b\,d\,e\,h+b\,d\,f\,g\right )}{9\,f^6}+\frac {2\,{\left (e+f\,x\right )}^{3/2}\,\left (a\,f-b\,e\right )\,\left (c\,f-d\,e\right )\,\left (a\,c\,f^2\,h+2\,a\,d\,f^2\,g+2\,b\,c\,f^2\,g+5\,b\,d\,e^2\,h-3\,a\,d\,e\,f\,h-3\,b\,c\,e\,f\,h-4\,b\,d\,e\,f\,g\right )}{3\,f^6} \] Input:

int(((g + h*x)*(a + b*x)^2*(c + d*x)^2)/(e + f*x)^(1/2),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 917, normalized size of antiderivative = 2.24 \[ \int \frac {(a+b x)^2 (c+d x)^2 (g+h x)}{\sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^2*(d*x+c)^2*(h*x+g)/(f*x+e)^(1/2),x)
 

Output:

(2*sqrt(e + f*x)*( - 2310*a**2*c**2*e*f**4*h + 3465*a**2*c**2*f**5*g + 115 
5*a**2*c**2*f**5*h*x + 3696*a**2*c*d*e**2*f**3*h - 4620*a**2*c*d*e*f**4*g 
- 1848*a**2*c*d*e*f**4*h*x + 2310*a**2*c*d*f**5*g*x + 1386*a**2*c*d*f**5*h 
*x**2 - 1584*a**2*d**2*e**3*f**2*h + 1848*a**2*d**2*e**2*f**3*g + 792*a**2 
*d**2*e**2*f**3*h*x - 924*a**2*d**2*e*f**4*g*x - 594*a**2*d**2*e*f**4*h*x* 
*2 + 693*a**2*d**2*f**5*g*x**2 + 495*a**2*d**2*f**5*h*x**3 + 3696*a*b*c**2 
*e**2*f**3*h - 4620*a*b*c**2*e*f**4*g - 1848*a*b*c**2*e*f**4*h*x + 2310*a* 
b*c**2*f**5*g*x + 1386*a*b*c**2*f**5*h*x**2 - 6336*a*b*c*d*e**3*f**2*h + 7 
392*a*b*c*d*e**2*f**3*g + 3168*a*b*c*d*e**2*f**3*h*x - 3696*a*b*c*d*e*f**4 
*g*x - 2376*a*b*c*d*e*f**4*h*x**2 + 2772*a*b*c*d*f**5*g*x**2 + 1980*a*b*c* 
d*f**5*h*x**3 + 2816*a*b*d**2*e**4*f*h - 3168*a*b*d**2*e**3*f**2*g - 1408* 
a*b*d**2*e**3*f**2*h*x + 1584*a*b*d**2*e**2*f**3*g*x + 1056*a*b*d**2*e**2* 
f**3*h*x**2 - 1188*a*b*d**2*e*f**4*g*x**2 - 880*a*b*d**2*e*f**4*h*x**3 + 9 
90*a*b*d**2*f**5*g*x**3 + 770*a*b*d**2*f**5*h*x**4 - 1584*b**2*c**2*e**3*f 
**2*h + 1848*b**2*c**2*e**2*f**3*g + 792*b**2*c**2*e**2*f**3*h*x - 924*b** 
2*c**2*e*f**4*g*x - 594*b**2*c**2*e*f**4*h*x**2 + 693*b**2*c**2*f**5*g*x** 
2 + 495*b**2*c**2*f**5*h*x**3 + 2816*b**2*c*d*e**4*f*h - 3168*b**2*c*d*e** 
3*f**2*g - 1408*b**2*c*d*e**3*f**2*h*x + 1584*b**2*c*d*e**2*f**3*g*x + 105 
6*b**2*c*d*e**2*f**3*h*x**2 - 1188*b**2*c*d*e*f**4*g*x**2 - 880*b**2*c*d*e 
*f**4*h*x**3 + 990*b**2*c*d*f**5*g*x**3 + 770*b**2*c*d*f**5*h*x**4 - 12...