3.471 \(\int \frac{(5+x^2)^2}{x^{13/3}} \, dx\)

Optimal. Leaf size=28 \[ \frac{3 x^{2/3}}{2}-\frac{15}{2 x^{4/3}}-\frac{15}{2 x^{10/3}} \]

[Out]

-15/(2*x^(10/3)) - 15/(2*x^(4/3)) + (3*x^(2/3))/2

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Rubi [A]  time = 0.0055334, antiderivative size = 28, 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 = {270} \[ \frac{3 x^{2/3}}{2}-\frac{15}{2 x^{4/3}}-\frac{15}{2 x^{10/3}} \]

Antiderivative was successfully verified.

[In]

Int[(5 + x^2)^2/x^(13/3),x]

[Out]

-15/(2*x^(10/3)) - 15/(2*x^(4/3)) + (3*x^(2/3))/2

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (5+x^2\right )^2}{x^{13/3}} \, dx &=\int \left (\frac{25}{x^{13/3}}+\frac{10}{x^{7/3}}+\frac{1}{\sqrt [3]{x}}\right ) \, dx\\ &=-\frac{15}{2 x^{10/3}}-\frac{15}{2 x^{4/3}}+\frac{3 x^{2/3}}{2}\\ \end{align*}

Mathematica [A]  time = 0.0051637, size = 19, normalized size = 0.68 \[ \frac{3 \left (x^4-5 x^2-5\right )}{2 x^{10/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + x^2)^2/x^(13/3),x]

[Out]

(3*(-5 - 5*x^2 + x^4))/(2*x^(10/3))

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Maple [A]  time = 0.005, size = 16, normalized size = 0.6 \begin{align*}{\frac{3\,{x}^{4}-15\,{x}^{2}-15}{2}{x}^{-{\frac{10}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+5)^2/x^(13/3),x)

[Out]

3/2*(x^4-5*x^2-5)/x^(10/3)

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Maxima [A]  time = 0.932026, size = 22, normalized size = 0.79 \begin{align*} \frac{3}{2} \, x^{\frac{2}{3}} - \frac{15 \,{\left (x^{2} + 1\right )}}{2 \, x^{\frac{10}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5)^2/x^(13/3),x, algorithm="maxima")

[Out]

3/2*x^(2/3) - 15/2*(x^2 + 1)/x^(10/3)

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Fricas [A]  time = 2.0601, size = 43, normalized size = 1.54 \begin{align*} \frac{3 \,{\left (x^{4} - 5 \, x^{2} - 5\right )}}{2 \, x^{\frac{10}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5)^2/x^(13/3),x, algorithm="fricas")

[Out]

3/2*(x^4 - 5*x^2 - 5)/x^(10/3)

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Sympy [A]  time = 8.78522, size = 24, normalized size = 0.86 \begin{align*} \frac{3 x^{\frac{2}{3}}}{2} - \frac{15}{2 x^{\frac{4}{3}}} - \frac{15}{2 x^{\frac{10}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+5)**2/x**(13/3),x)

[Out]

3*x**(2/3)/2 - 15/(2*x**(4/3)) - 15/(2*x**(10/3))

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Giac [A]  time = 1.06838, size = 22, normalized size = 0.79 \begin{align*} \frac{3}{2} \, x^{\frac{2}{3}} - \frac{15 \,{\left (x^{2} + 1\right )}}{2 \, x^{\frac{10}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5)^2/x^(13/3),x, algorithm="giac")

[Out]

3/2*x^(2/3) - 15/2*(x^2 + 1)/x^(10/3)