3.2780 \(\int \frac{(c x)^{-1+n}}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=24 \[ \frac{(c x)^n}{a c n \left (a+b x^n\right )} \]

[Out]

(c*x)^n/(a*c*n*(a + b*x^n))

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Rubi [A]  time = 0.0056073, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ \frac{(c x)^n}{a c n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 + n)/(a + b*x^n)^2,x]

[Out]

(c*x)^n/(a*c*n*(a + b*x^n))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(c x)^{-1+n}}{\left (a+b x^n\right )^2} \, dx &=\frac{(c x)^n}{a c n \left (a+b x^n\right )}\\ \end{align*}

Mathematica [A]  time = 0.013752, size = 31, normalized size = 1.29 \[ -\frac{x^{1-n} (c x)^{n-1}}{b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(-1 + n)/(a + b*x^n)^2,x]

[Out]

-((x^(1 - n)*(c*x)^(-1 + n))/(b*n*(a + b*x^n)))

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Maple [C]  time = 0.021, size = 99, normalized size = 4.1 \begin{align*}{\frac{x}{an \left ( a+b{x}^{n} \right ) }{{\rm e}^{{\frac{ \left ( -1+n \right ) \left ( -i \left ({\it csgn} \left ( icx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( icx \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \pi +i \left ({\it csgn} \left ( icx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( icx \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( x \right ) +2\,\ln \left ( c \right ) \right ) }{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(-1+n)/(a+b*x^n)^2,x)

[Out]

1/a/n*x/(a+b*x^n)*exp(1/2*(-1+n)*(-I*csgn(I*c*x)^3*Pi+I*csgn(I*c*x)^2*csgn(I*c)*Pi+I*csgn(I*c*x)^2*csgn(I*x)*P
i-I*csgn(I*c*x)*csgn(I*c)*csgn(I*x)*Pi+2*ln(x)+2*ln(c)))

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Maxima [A]  time = 1.37117, size = 30, normalized size = 1.25 \begin{align*} -\frac{c^{n}}{b^{2} c n x^{n} + a b c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-1+n)/(a+b*x^n)^2,x, algorithm="maxima")

[Out]

-c^n/(b^2*c*n*x^n + a*b*c*n)

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Fricas [A]  time = 1.275, size = 43, normalized size = 1.79 \begin{align*} -\frac{c^{n - 1}}{b^{2} n x^{n} + a b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-1+n)/(a+b*x^n)^2,x, algorithm="fricas")

[Out]

-c^(n - 1)/(b^2*n*x^n + a*b*n)

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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)**(-1+n)/(a+b*x**n)**2,x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-1+n)/(a+b*x^n)^2,x, algorithm="giac")

[Out]

integrate((c*x)^(n - 1)/(b*x^n + a)^2, x)