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

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 = 0.00830932, 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 \[ x^{-2 p} \left (x^2\right )^p \log (x) \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 2*p)*(x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 1.93873, size = 12, normalized size = 0.92 \[ x^{- 2 p} \left (x^{2}\right )^{p} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-2*p)*(x**2)**p,x)

[Out]

x**(-2*p)*(x**2)**p*log(x)

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Mathematica [A]  time = 0.00957261, 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 = 0.035, size = 21, normalized size = 1.6 \[ x\ln \left ( x \right ){{\rm e}^{p\ln \left ({x}^{2} \right ) }}{{\rm e}^{ \left ( -1-2\,p \right ) \ln \left ( x \right ) }} \]

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 = 1.44193, size = 3, normalized size = 0.23 \[ \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)

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Fricas [A]  time = 0.260038, size = 3, normalized size = 0.23 \[ \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{2}\right )}^{p} x^{-2 \, p - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^2)^p*x^(-2*p - 1), x)