3.1.68 \(\int x^{-1-2 p} (x^2)^p \, dx\)

Optimal. Leaf size=13 \[ x^{-2 p} \left (x^2\right )^p \log (x) \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 29} \begin {gather*} x^{-2 p} \left (x^2\right )^p \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*p)*(x^2)^p,x]

[Out]

((x^2)^p*Log[x])/x^(2*p)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin {align*} \int x^{-1-2 p} \left (x^2\right )^p \, dx &=\left (x^{-2 p} \left (x^2\right )^p\right ) \int \frac {1}{x} \, dx\\ &=x^{-2 p} \left (x^2\right )^p \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} x^{-2 p} \left (x^2\right )^p \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 2*p)*(x^2)^p,x]

[Out]

((x^2)^p*Log[x])/x^(2*p)

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IntegrateAlgebraic [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} x^{-2 p} \left (x^2\right )^p \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(-1 - 2*p)*(x^2)^p,x]

[Out]

((x^2)^p*Log[x])/x^(2*p)

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fricas [A]  time = 0.76, size = 2, normalized size = 0.15 \begin {gather*} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*p)*(x^2)^p,x, algorithm="fricas")

[Out]

log(x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{2}\right )}^{p} x^{-2 \, p - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*p)*(x^2)^p,x, algorithm="giac")

[Out]

integrate((x^2)^p*x^(-2*p - 1), x)

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maple [A]  time = 0.03, size = 21, normalized size = 1.62 \begin {gather*} x \,{\mathrm e}^{p \ln \left (x^{2}\right )} {\mathrm e}^{\left (-2 p -1\right ) \ln \relax (x )} \ln \relax (x ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-2*p)*(x^2)^p,x)

[Out]

x*ln(x)*exp(p*ln(x^2))*exp((-1-2*p)*ln(x))

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maxima [A]  time = 1.28, size = 2, normalized size = 0.15 \begin {gather*} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*p)*(x^2)^p,x, algorithm="maxima")

[Out]

log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \frac {{\left (x^2\right )}^p}{x^{2\,p+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2)^p/x^(2*p + 1),x)

[Out]

int((x^2)^p/x^(2*p + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- 2 p - 1} \left (x^{2}\right )^{p}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-2*p)*(x**2)**p,x)

[Out]

Integral(x**(-2*p - 1)*(x**2)**p, x)

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