3.294 \(\int x^{3/2} (b x^2+c x^4) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{9} b x^{9/2}+\frac{2}{13} c x^{13/2} \]

[Out]

(2*b*x^(9/2))/9 + (2*c*x^(13/2))/13

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Rubi [A]  time = 0.0049, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {14} \[ \frac{2}{9} b x^{9/2}+\frac{2}{13} c x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(9/2))/9 + (2*c*x^(13/2))/13

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

Mathematica [A]  time = 0.0047868, size = 21, normalized size = 1. \[ \frac{2}{9} b x^{9/2}+\frac{2}{13} c x^{13/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(9/2))/9 + (2*c*x^(13/2))/13

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Maple [A]  time = 0.044, size = 16, normalized size = 0.8 \begin{align*}{\frac{18\,c{x}^{2}+26\,b}{117}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(c*x^4+b*x^2),x)

[Out]

2/117*x^(9/2)*(9*c*x^2+13*b)

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Maxima [A]  time = 0.969289, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{13} \, c x^{\frac{13}{2}} + \frac{2}{9} \, b x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

2/13*c*x^(13/2) + 2/9*b*x^(9/2)

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Fricas [A]  time = 1.2052, size = 49, normalized size = 2.33 \begin{align*} \frac{2}{117} \,{\left (9 \, c x^{6} + 13 \, b x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

2/117*(9*c*x^6 + 13*b*x^4)*sqrt(x)

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Sympy [A]  time = 3.25703, size = 19, normalized size = 0.9 \begin{align*} \frac{2 b x^{\frac{9}{2}}}{9} + \frac{2 c x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(c*x**4+b*x**2),x)

[Out]

2*b*x**(9/2)/9 + 2*c*x**(13/2)/13

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Giac [A]  time = 1.11468, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{13} \, c x^{\frac{13}{2}} + \frac{2}{9} \, b x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(c*x^4+b*x^2),x, algorithm="giac")

[Out]

2/13*c*x^(13/2) + 2/9*b*x^(9/2)