3.187 \(\int x^m (a x^n)^{-\frac {1+m}{n}} \, dx\)

Optimal. Leaf size=22 \[ x^{m+1} \log (x) \left (a x^n\right )^{-\frac {m+1}{n}} \]

[Out]

x^(1+m)*ln(x)/((a*x^n)^((1+m)/n))

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 29} \[ x^{m+1} \log (x) \left (a x^n\right )^{-\frac {m+1}{n}} \]

Antiderivative was successfully verified.

[In]

Int[x^m/(a*x^n)^((1 + m)/n),x]

[Out]

(x^(1 + m)*Log[x])/(a*x^n)^((1 + m)/n)

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^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx &=\left (x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}}\right ) \int \frac {1}{x} \, dx\\ &=x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ x^{m+1} \log (x) \left (a x^n\right )^{-\frac {m+1}{n}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/(a*x^n)^((1 + m)/n),x]

[Out]

(x^(1 + m)*Log[x])/(a*x^n)^((1 + m)/n)

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fricas [A]  time = 0.99, size = 14, normalized size = 0.64 \[ \frac {\log \relax (x)}{a^{\frac {m + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/((a*x^n)^((1+m)/n)),x, algorithm="fricas")

[Out]

log(x)/a^((m + 1)/n)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (a x^{n}\right )^{\frac {m + 1}{n}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/((a*x^n)^((1+m)/n)),x, algorithm="giac")

[Out]

integrate(x^m/(a*x^n)^((m + 1)/n), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a \,x^{n}\right )^{-\frac {m +1}{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/((a*x^n)^((m+1)/n)),x)

[Out]

int(x^m/((a*x^n)^((m+1)/n)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (a x^{n}\right )^{\frac {m + 1}{n}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/((a*x^n)^((1+m)/n)),x, algorithm="maxima")

[Out]

integrate(x^m/(a*x^n)^((m + 1)/n), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {x^m}{{\left (a\,x^n\right )}^{\frac {m+1}{n}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(a*x^n)^((m + 1)/n),x)

[Out]

int(x^m/(a*x^n)^((m + 1)/n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a x^{n}\right )^{- \frac {m + 1}{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/((a*x**n)**((1+m)/n)),x)

[Out]

Integral(x**m*(a*x**n)**(-(m + 1)/n), x)

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