3.293 \(\int x^{5/2} (b x^2+c x^4) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2} \]

[Out]

(2*b*x^(11/2))/11 + (2*c*x^(15/2))/15

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Rubi [A]  time = 0.0053335, 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} \[ \frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(11/2))/11 + (2*c*x^(15/2))/15

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 (b x^2+c x^4\right ) \, dx &=\int \left (b x^{9/2}+c x^{13/2}\right ) \, dx\\ &=\frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0047712, size = 21, normalized size = 1. \[ \frac{2}{11} b x^{11/2}+\frac{2}{15} c x^{15/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(11/2))/11 + (2*c*x^(15/2))/15

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Maple [A]  time = 0.042, size = 16, normalized size = 0.8 \begin{align*}{\frac{22\,c{x}^{2}+30\,b}{165}{x}^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(c*x^4+b*x^2),x)

[Out]

2/165*x^(11/2)*(11*c*x^2+15*b)

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Maxima [A]  time = 0.998516, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{15} \, c x^{\frac{15}{2}} + \frac{2}{11} \, b x^{\frac{11}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

2/15*c*x^(15/2) + 2/11*b*x^(11/2)

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Fricas [A]  time = 1.23168, size = 50, normalized size = 2.38 \begin{align*} \frac{2}{165} \,{\left (11 \, c x^{7} + 15 \, b x^{5}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

2/165*(11*c*x^7 + 15*b*x^5)*sqrt(x)

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Sympy [A]  time = 7.91652, size = 19, normalized size = 0.9 \begin{align*} \frac{2 b x^{\frac{11}{2}}}{11} + \frac{2 c x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(c*x**4+b*x**2),x)

[Out]

2*b*x**(11/2)/11 + 2*c*x**(15/2)/15

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Giac [A]  time = 1.13242, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{15} \, c x^{\frac{15}{2}} + \frac{2}{11} \, b x^{\frac{11}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+b*x^2),x, algorithm="giac")

[Out]

2/15*c*x^(15/2) + 2/11*b*x^(11/2)