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

Optimal. Leaf size=36 \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{9} a b x^{9/2}+\frac{2}{13} b^2 x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0085225, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{9} a b x^{9/2}+\frac{2}{13} b^2 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x^2)^2,x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^{3/2} \left (a+b x^2\right )^2 \, dx &=\int \left (a^2 x^{3/2}+2 a b x^{7/2}+b^2 x^{11/2}\right ) \, dx\\ &=\frac{2}{5} a^2 x^{5/2}+\frac{4}{9} a b x^{9/2}+\frac{2}{13} b^2 x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0072432, size = 30, normalized size = 0.83 \[ \frac{2}{585} x^{5/2} \left (117 a^2+130 a b x^2+45 b^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x^2)^2,x]

[Out]

(2*x^(5/2)*(117*a^2 + 130*a*b*x^2 + 45*b^2*x^4))/585

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*}{\frac{90\,{b}^{2}{x}^{4}+260\,ab{x}^{2}+234\,{a}^{2}}{585}{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,x)

[Out]

2/585*x^(5/2)*(45*b^2*x^4+130*a*b*x^2+117*a^2)

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Maxima [A]  time = 1.98848, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{13} \, b^{2} x^{\frac{13}{2}} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{2}{5} \, a^{2} 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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.24236, size = 76, normalized size = 2.11 \begin{align*} \frac{2}{585} \,{\left (45 \, b^{2} x^{6} + 130 \, a b x^{4} + 117 \, a^{2} x^{2}\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,x, algorithm="fricas")

[Out]

2/585*(45*b^2*x^6 + 130*a*b*x^4 + 117*a^2*x^2)*sqrt(x)

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Sympy [A]  time = 2.56054, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{5}{2}}}{5} + \frac{4 a b x^{\frac{9}{2}}}{9} + \frac{2 b^{2} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.79306, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{13} \, b^{2} x^{\frac{13}{2}} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{2}{5} \, a^{2} 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,x, algorithm="giac")

[Out]

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