\(\int x^{3/2} (a+b x^2)^2 (c+d x^2) \, dx\) [393]

Optimal result

Integrand size = 22, antiderivative size = 63 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {2}{5} a^2 c x^{5/2}+\frac {2}{9} a (2 b c+a d) x^{9/2}+\frac {2}{13} b (b c+2 a d) x^{13/2}+\frac {2}{17} b^2 d x^{17/2} \]

[Out]

2/5*a^2*c*x^(5/2)+2/9*a*(a*d+2*b*c)*x^(9/2)+2/13*b*(2*a*d+b*c)*x^(13/2)+2/17*b^2*d*x^(17/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {2}{5} a^2 c x^{5/2}+\frac {2}{13} b x^{13/2} (2 a d+b c)+\frac {2}{9} a x^{9/2} (a d+2 b c)+\frac {2}{17} b^2 d x^{17/2} \]

[In]

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

[Out]

(2*a^2*c*x^(5/2))/5 + (2*a*(2*b*c + a*d)*x^(9/2))/9 + (2*b*(b*c + 2*a*d)*x^(13/2))/13 + (2*b^2*d*x^(17/2))/17

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c x^{3/2}+a (2 b c+a d) x^{7/2}+b (b c+2 a d) x^{11/2}+b^2 d x^{15/2}\right ) \, dx \\ & = \frac {2}{5} a^2 c x^{5/2}+\frac {2}{9} a (2 b c+a d) x^{9/2}+\frac {2}{13} b (b c+2 a d) x^{13/2}+\frac {2}{17} b^2 d x^{17/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {2 x^{5/2} \left (221 a^2 \left (9 c+5 d x^2\right )+170 a b x^2 \left (13 c+9 d x^2\right )+45 b^2 x^4 \left (17 c+13 d x^2\right )\right )}{9945} \]

[In]

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

[Out]

(2*x^(5/2)*(221*a^2*(9*c + 5*d*x^2) + 170*a*b*x^2*(13*c + 9*d*x^2) + 45*b^2*x^4*(17*c + 13*d*x^2)))/9945

Maple [A] (verified)

Time = 2.67 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {2 b^{2} d \,x^{\frac {17}{2}}}{17}+\frac {2 \left (2 a b d +b^{2} c \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (a^{2} d +2 a b c \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} c \,x^{\frac {5}{2}}}{5}\)	\(52\)
default	\(\frac {2 b^{2} d \,x^{\frac {17}{2}}}{17}+\frac {2 \left (2 a b d +b^{2} c \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (a^{2} d +2 a b c \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} c \,x^{\frac {5}{2}}}{5}\)	\(52\)
gosper	\(\frac {2 x^{\frac {5}{2}} \left (585 b^{2} d \,x^{6}+1530 a b d \,x^{4}+765 b^{2} c \,x^{4}+1105 a^{2} d \,x^{2}+2210 a b c \,x^{2}+1989 a^{2} c \right )}{9945}\)	\(56\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (585 b^{2} d \,x^{6}+1530 a b d \,x^{4}+765 b^{2} c \,x^{4}+1105 a^{2} d \,x^{2}+2210 a b c \,x^{2}+1989 a^{2} c \right )}{9945}\)	\(56\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (585 b^{2} d \,x^{6}+1530 a b d \,x^{4}+765 b^{2} c \,x^{4}+1105 a^{2} d \,x^{2}+2210 a b c \,x^{2}+1989 a^{2} c \right )}{9945}\)	\(56\)

[In]

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

[Out]

2/17*b^2*d*x^(17/2)+2/13*(2*a*b*d+b^2*c)*x^(13/2)+2/9*(a^2*d+2*a*b*c)*x^(9/2)+2/5*a^2*c*x^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {2}{9945} \, {\left (585 \, b^{2} d x^{8} + 765 \, {\left (b^{2} c + 2 \, a b d\right )} x^{6} + 1989 \, a^{2} c x^{2} + 1105 \, {\left (2 \, a b c + a^{2} d\right )} x^{4}\right )} \sqrt {x} \]

[In]

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

[Out]

2/9945*(585*b^2*d*x^8 + 765*(b^2*c + 2*a*b*d)*x^6 + 1989*a^2*c*x^2 + 1105*(2*a*b*c + a^2*d)*x^4)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.27 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {2 a^{2} c x^{\frac {5}{2}}}{5} + \frac {2 a^{2} d x^{\frac {9}{2}}}{9} + \frac {4 a b c x^{\frac {9}{2}}}{9} + \frac {4 a b d x^{\frac {13}{2}}}{13} + \frac {2 b^{2} c x^{\frac {13}{2}}}{13} + \frac {2 b^{2} d x^{\frac {17}{2}}}{17} \]

[In]

integrate(x**(3/2)*(b*x**2+a)**2*(d*x**2+c),x)

[Out]

2*a**2*c*x**(5/2)/5 + 2*a**2*d*x**(9/2)/9 + 4*a*b*c*x**(9/2)/9 + 4*a*b*d*x**(13/2)/13 + 2*b**2*c*x**(13/2)/13
+ 2*b**2*d*x**(17/2)/17

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {2}{17} \, b^{2} d x^{\frac {17}{2}} + \frac {2}{13} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {13}{2}} + \frac {2}{5} \, a^{2} c x^{\frac {5}{2}} + \frac {2}{9} \, {\left (2 \, a b c + a^{2} d\right )} x^{\frac {9}{2}} \]

[In]

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

[Out]

2/17*b^2*d*x^(17/2) + 2/13*(b^2*c + 2*a*b*d)*x^(13/2) + 2/5*a^2*c*x^(5/2) + 2/9*(2*a*b*c + a^2*d)*x^(9/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {2}{17} \, b^{2} d x^{\frac {17}{2}} + \frac {2}{13} \, b^{2} c x^{\frac {13}{2}} + \frac {4}{13} \, a b d x^{\frac {13}{2}} + \frac {4}{9} \, a b c x^{\frac {9}{2}} + \frac {2}{9} \, a^{2} d x^{\frac {9}{2}} + \frac {2}{5} \, a^{2} c x^{\frac {5}{2}} \]

[In]

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

[Out]

2/17*b^2*d*x^(17/2) + 2/13*b^2*c*x^(13/2) + 4/13*a*b*d*x^(13/2) + 4/9*a*b*c*x^(9/2) + 2/9*a^2*d*x^(9/2) + 2/5*
a^2*c*x^(5/2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=x^{9/2}\,\left (\frac {2\,d\,a^2}{9}+\frac {4\,b\,c\,a}{9}\right )+x^{13/2}\,\left (\frac {2\,c\,b^2}{13}+\frac {4\,a\,d\,b}{13}\right )+\frac {2\,a^2\,c\,x^{5/2}}{5}+\frac {2\,b^2\,d\,x^{17/2}}{17} \]

[In]

int(x^(3/2)*(a + b*x^2)^2*(c + d*x^2),x)

[Out]

x^(9/2)*((2*a^2*d)/9 + (4*a*b*c)/9) + x^(13/2)*((2*b^2*c)/13 + (4*a*b*d)/13) + (2*a^2*c*x^(5/2))/5 + (2*b^2*d*
x^(17/2))/17