3.181 \(\int (c x)^m (a x^n)^{-1/n} \, dx\)

Optimal. Leaf size=21 \[ \frac{x \left (a x^n\right )^{-1/n} (c x)^m}{m} \]

[Out]

(x*(c*x)^m)/(m*(a*x^n)^n^(-1))

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Rubi [A]  time = 0.0045587, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {15, 16, 32} \[ \frac{x \left (a x^n\right )^{-1/n} (c x)^m}{m} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m/(a*x^n)^n^(-1),x]

[Out]

(x*(c*x)^m)/(m*(a*x^n)^n^(-1))

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 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.002479, size = 21, normalized size = 1. \[ \frac{x \left (a x^n\right )^{-1/n} (c x)^m}{m} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m/(a*x^n)^n^(-1),x]

[Out]

(x*(c*x)^m)/(m*(a*x^n)^n^(-1))

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Maple [A]  time = 0.001, size = 22, normalized size = 1.1 \begin{align*}{\frac{x \left ( cx \right ) ^{m}}{m\sqrt [n]{a{x}^{n}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(c*x)^m/m/((a*x^n)^(1/n))

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Maxima [A]  time = 1.00879, size = 41, normalized size = 1.95 \begin{align*} \frac{c^{m} x e^{\left (m \log \left (x\right ) - \frac{\log \left (x^{n}\right )}{n}\right )}}{a^{\left (\frac{1}{n}\right )} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^m*x*e^(m*log(x) - log(x^n)/n)/(a^(1/n)*m)

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Fricas [A]  time = 1.81905, size = 50, normalized size = 2.38 \begin{align*} \frac{e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{a^{\left (\frac{1}{n}\right )} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(m*log(c) + m*log(x))/(a^(1/n)*m)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.10933, size = 28, normalized size = 1.33 \begin{align*} \frac{e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{a^{\left (\frac{1}{n}\right )} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(m*log(c) + m*log(x))/(a^(1/n)*m)