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

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

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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IntegrateAlgebraic [A]  time = 0.01, size = 28, normalized size = 1.00 \[ \frac {3 x^{2/3}}{2}-\frac {15}{2 x^{4/3}}-\frac {15}{2 x^{10/3}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(5 + x^2)^2/x^(13/3),x]

[Out]

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

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fricas [A]  time = 0.89, size = 15, normalized size = 0.54 \[ \frac {3 \, {\left (x^{4} - 5 \, x^{2} - 5\right )}}{2 \, x^{\frac {10}{3}}} \]

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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giac [A]  time = 0.63, size = 16, normalized size = 0.57 \[ \frac {3}{2} \, x^{\frac {2}{3}} - \frac {15 \, {\left (x^{2} + 1\right )}}{2 \, x^{\frac {10}{3}}} \]

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)

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maple [A]  time = 0.29, size = 16, normalized size = 0.57

method	result	size
gosper	\(\frac {\frac {3}{2} x^{4}-\frac {15}{2} x^{2}-\frac {15}{2}}{x^{\frac {10}{3}}}\)	\(16\)
trager	\(\frac {\frac {3}{2} x^{4}-\frac {15}{2} x^{2}-\frac {15}{2}}{x^{\frac {10}{3}}}\)	\(16\)
risch	\(\frac {\frac {3}{2} x^{4}-\frac {15}{2} x^{2}-\frac {15}{2}}{x^{\frac {10}{3}}}\)	\(16\)
derivativedivides	\(-\frac {15}{2 x^{\frac {10}{3}}}-\frac {15}{2 x^{\frac {4}{3}}}+\frac {3 x^{\frac {2}{3}}}{2}\)	\(17\)
default	\(-\frac {15}{2 x^{\frac {10}{3}}}-\frac {15}{2 x^{\frac {4}{3}}}+\frac {3 x^{\frac {2}{3}}}{2}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 16, normalized size = 0.57 \[ \frac {3}{2} \, x^{\frac {2}{3}} - \frac {15 \, {\left (x^{2} + 1\right )}}{2 \, x^{\frac {10}{3}}} \]

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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mupad [B]  time = 0.27, size = 17, normalized size = 0.61 \[ -\frac {-3\,x^4+15\,x^2+15}{2\,x^{10/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 5)^2/x^(13/3),x)

[Out]

-(15*x^2 - 3*x^4 + 15)/(2*x^(10/3))

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sympy [A]  time = 7.85, size = 24, normalized size = 0.86 \[ \frac {3 x^{\frac {2}{3}}}{2} - \frac {15}{2 x^{\frac {4}{3}}} - \frac {15}{2 x^{\frac {10}{3}}} \]

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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