3.292 \(\int x^{7/2} (b x^2+c x^4) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{13} b x^{13/2}+\frac{2}{17} c x^{17/2} \]

[Out]

(2*b*x^(13/2))/13 + (2*c*x^(17/2))/17

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Rubi [A]  time = 0.0047629, 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} \[ \frac{2}{13} b x^{13/2}+\frac{2}{17} c x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(13/2))/13 + (2*c*x^(17/2))/17

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^{7/2} \left (b x^2+c x^4\right ) \, dx &=\int \left (b x^{11/2}+c x^{15/2}\right ) \, dx\\ &=\frac{2}{13} b x^{13/2}+\frac{2}{17} c x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0057258, size = 21, normalized size = 1. \[ \frac{2}{13} b x^{13/2}+\frac{2}{17} c x^{17/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(13/2))/13 + (2*c*x^(17/2))/17

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Maple [A]  time = 0.042, size = 16, normalized size = 0.8 \begin{align*}{\frac{26\,c{x}^{2}+34\,b}{221}{x}^{{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(c*x^4+b*x^2),x)

[Out]

2/221*x^(13/2)*(13*c*x^2+17*b)

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Maxima [A]  time = 0.972262, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{17} \, c x^{\frac{17}{2}} + \frac{2}{13} \, b x^{\frac{13}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

2/17*c*x^(17/2) + 2/13*b*x^(13/2)

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Fricas [A]  time = 1.24512, size = 50, normalized size = 2.38 \begin{align*} \frac{2}{221} \,{\left (13 \, c x^{8} + 17 \, b x^{6}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

2/221*(13*c*x^8 + 17*b*x^6)*sqrt(x)

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Sympy [A]  time = 14.1793, size = 19, normalized size = 0.9 \begin{align*} \frac{2 b x^{\frac{13}{2}}}{13} + \frac{2 c x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(c*x**4+b*x**2),x)

[Out]

2*b*x**(13/2)/13 + 2*c*x**(17/2)/17

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Giac [A]  time = 1.14206, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{17} \, c x^{\frac{17}{2}} + \frac{2}{13} \, b x^{\frac{13}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^4+b*x^2),x, algorithm="giac")

[Out]

2/17*c*x^(17/2) + 2/13*b*x^(13/2)