3.2.81 \(\int (c x)^m (a x^n)^{-1/n} \, dx\) [181]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x \left (a x^n\right )^{-1/n} (c x)^m}{m} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {x (c x)^m \left (a x^n\right )^{-1/n}}{m} \end {gather*}

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 = 0.06, size = 22, normalized size = 1.05

method	result	size
gosper	\(\frac {x \left (c x \right )^{m} \left (a \,x^{n}\right )^{-\frac {1}{n}}}{m}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 30, normalized size = 1.43 \begin {gather*} \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 {gather*}

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 = 0.38, size = 21, normalized size = 1.00 \begin {gather*} \frac {e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{a^{\left (\frac {1}{n}\right )} m} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x \left (a x^{n}\right )^{- \frac {1}{n}} \left (c x\right )^{m}}{m} & \text {for}\: m \neq 0 \\\int \left (a x^{n}\right )^{- \frac {1}{n}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*(c*x)**m/(m*(a*x**n)**(1/n)), Ne(m, 0)), (Integral((a*x**n)**(-1/n), x), True))

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Giac [A]
time = 2.40, size = 21, normalized size = 1.00 \begin {gather*} \frac {e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{a^{\left (\frac {1}{n}\right )} m} \end {gather*}

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)

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Mupad [B]
time = 0.99, size = 21, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (c\,x\right )}^m}{m\,{\left (a\,x^n\right )}^{1/n}} \end {gather*}

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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