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

Optimal. Leaf size=21 \[ \frac{2}{5} a x^{5/2}+\frac{2}{9} b x^{9/2} \]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(9/2))/9

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Rubi [A]  time = 0.0044014, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{2}{5} a x^{5/2}+\frac{2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x^2),x]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(9/2))/9

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0044832, size = 21, normalized size = 1. \[ \frac{2}{5} a x^{5/2}+\frac{2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x^2),x]

[Out]

(2*a*x^(5/2))/5 + (2*b*x^(9/2))/9

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Maple [A]  time = 0.002, size = 16, normalized size = 0.8 \begin{align*}{\frac{10\,b{x}^{2}+18\,a}{45}{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),x)

[Out]

2/45*x^(5/2)*(5*b*x^2+9*a)

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Maxima [A]  time = 1.89795, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{9} \, b x^{\frac{9}{2}} + \frac{2}{5} \, a 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),x, algorithm="maxima")

[Out]

2/9*b*x^(9/2) + 2/5*a*x^(5/2)

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Fricas [A]  time = 1.4849, size = 46, normalized size = 2.19 \begin{align*} \frac{2}{45} \,{\left (5 \, b x^{4} + 9 \, a 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),x, algorithm="fricas")

[Out]

2/45*(5*b*x^4 + 9*a*x^2)*sqrt(x)

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Sympy [A]  time = 0.939886, size = 19, normalized size = 0.9 \begin{align*} \frac{2 a x^{\frac{5}{2}}}{5} + \frac{2 b x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(5/2)/5 + 2*b*x**(9/2)/9

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Giac [A]  time = 2.03944, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{9} \, b x^{\frac{9}{2}} + \frac{2}{5} \, a 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),x, algorithm="giac")

[Out]

2/9*b*x^(9/2) + 2/5*a*x^(5/2)