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

Optimal. Leaf size=63 \[ \frac{2}{5} a^2 c x^{5/2}+\frac{2}{13} b x^{13/2} (2 a d+b c)+\frac{2}{9} a x^{9/2} (a d+2 b c)+\frac{2}{17} b^2 d x^{17/2} \]

[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

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Rubi [A]  time = 0.0298885, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{5} a^2 c x^{5/2}+\frac{2}{13} b x^{13/2} (2 a d+b c)+\frac{2}{9} a x^{9/2} (a d+2 b c)+\frac{2}{17} b^2 d x^{17/2} \]

Antiderivative was successfully verified.

[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 448

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*} \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\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]  time = 0.0282503, size = 53, normalized size = 0.84 \[ \frac{2 x^{5/2} \left (1989 a^2 c+765 b x^4 (2 a d+b c)+1105 a x^2 (a d+2 b c)+585 b^2 d x^6\right )}{9945} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(1989*a^2*c + 1105*a*(2*b*c + a*d)*x^2 + 765*b*(b*c + 2*a*d)*x^4 + 585*b^2*d*x^6))/9945

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Maple [A]  time = 0.004, size = 56, normalized size = 0.9 \begin{align*}{\frac{1170\,{b}^{2}d{x}^{6}+3060\,{x}^{4}abd+1530\,{b}^{2}c{x}^{4}+2210\,{x}^{2}{a}^{2}d+4420\,abc{x}^{2}+3978\,{a}^{2}c}{9945}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9945*x^(5/2)*(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)

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Maxima [A]  time = 1.04525, size = 69, normalized size = 1.1 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0.744456, size = 143, normalized size = 2.27 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 5.93053, size = 80, normalized size = 1.27 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.15493, size = 72, normalized size = 1.14 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)