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

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

[Out]

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

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Rubi [A]  time = 0.0024992, antiderivative size = 13, normalized size of antiderivative = 1., 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} \[ x^{-2 p} \left (x^2\right )^p \log (x) \]

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*}

Mathematica [A]  time = 0.0051337, size = 13, normalized size = 1. \[ x^{-2 p} \left (x^2\right )^p \log (x) \]

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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Maple [A]  time = 0.018, size = 21, normalized size = 1.6 \begin{align*} x\ln \left ( x \right ){{\rm e}^{p\ln \left ({x}^{2} \right ) }}{{\rm e}^{ \left ( -1-2\,p \right ) \ln \left ( x \right ) }} \end{align*}

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 = 0.996348, size = 3, normalized size = 0.23 \begin{align*} \log \left (x\right ) \end{align*}

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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Fricas [A]  time = 2.04296, size = 11, normalized size = 0.85 \begin{align*} \log \left (x\right ) \end{align*}

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

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

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)