\(\int x^m (a x^n)^{-\frac {1+m}{n}} \, dx\) [187]

Optimal result

Integrand size = 18, antiderivative size = 22 \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \log (x) \]

[Out]

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

Rubi [A] (verified)

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

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \log (x) \]

[In]

Integrate[x^m/(a*x^n)^((1 + m)/n),x]

[Out]

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

Maple [F]

\[\int x^{m} \left (a \,x^{n}\right )^{-\frac {1+m}{n}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\frac {\log \left (x\right )}{a^{\frac {m + 1}{n}}} \]

[In]

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

[Out]

log(x)/a^((m + 1)/n)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (17) = 34\).

Time = 1.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.95 \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\begin {cases} \frac {x x^{m} \left (a x^{n}\right )^{- \frac {m}{n} - \frac {1}{n}}}{m + n \left (- \frac {m}{n} - \frac {1}{n}\right ) + 1} & \text {for}\: m + n \left (- \frac {m}{n} - \frac {1}{n}\right ) \neq -1 \\x x^{m} \left (a x^{n}\right )^{- \frac {m}{n} - \frac {1}{n}} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x*x**m*(a*x**n)**(-m/n - 1/n)/(m + n*(-m/n - 1/n) + 1), Ne(m + n*(-m/n - 1/n), -1)), (x*x**m*(a*x**
n)**(-m/n - 1/n)*log(x), True))

Maxima [F]

\[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\int { \frac {x^{m}}{\left (a x^{n}\right )^{\frac {m + 1}{n}}} \,d x } \]

[In]

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

[Out]

integrate(x^m/(a*x^n)^((m + 1)/n), x)

Giac [F]

\[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\int { \frac {x^{m}}{\left (a x^{n}\right )^{\frac {m + 1}{n}}} \,d x } \]

[In]

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

[Out]

integrate(x^m/(a*x^n)^((m + 1)/n), x)

Mupad [F(-1)]

Timed out. \[ \int x^m \left (a x^n\right )^{-\frac {1+m}{n}} \, dx=\int \frac {x^m}{{\left (a\,x^n\right )}^{\frac {m+1}{n}}} \,d x \]

[In]

int(x^m/(a*x^n)^((m + 1)/n),x)

[Out]

int(x^m/(a*x^n)^((m + 1)/n), x)