3.265 \(\int x^{5/2} (a+b x^2) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2} \]

[Out]

(2*a*x^(7/2))/7 + (2*b*x^(11/2))/11

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Rubi [A]  time = 0.0043558, 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} \[ \frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^2),x]

[Out]

(2*a*x^(7/2))/7 + (2*b*x^(11/2))/11

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

Mathematica [A]  time = 0.0045701, size = 21, normalized size = 1. \[ \frac{2}{7} a x^{7/2}+\frac{2}{11} b x^{11/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x^2),x]

[Out]

(2*a*x^(7/2))/7 + (2*b*x^(11/2))/11

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Maple [A]  time = 0.002, size = 16, normalized size = 0.8 \begin{align*}{\frac{14\,b{x}^{2}+22\,a}{77}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x^2+a),x)

[Out]

2/77*x^(7/2)*(7*b*x^2+11*a)

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Maxima [A]  time = 1.25353, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{11} \, b x^{\frac{11}{2}} + \frac{2}{7} \, a x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a),x, algorithm="maxima")

[Out]

2/11*b*x^(11/2) + 2/7*a*x^(7/2)

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Fricas [A]  time = 1.47502, size = 47, normalized size = 2.24 \begin{align*} \frac{2}{77} \,{\left (7 \, b x^{5} + 11 \, a x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a),x, algorithm="fricas")

[Out]

2/77*(7*b*x^5 + 11*a*x^3)*sqrt(x)

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Sympy [A]  time = 2.72535, size = 19, normalized size = 0.9 \begin{align*} \frac{2 a x^{\frac{7}{2}}}{7} + \frac{2 b x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x**2+a),x)

[Out]

2*a*x**(7/2)/7 + 2*b*x**(11/2)/11

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Giac [A]  time = 2.77619, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{11} \, b x^{\frac{11}{2}} + \frac{2}{7} \, a x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a),x, algorithm="giac")

[Out]

2/11*b*x^(11/2) + 2/7*a*x^(7/2)