3.2773 \(\int \frac{(c x)^{-1+n}}{\left (a+b x^n\right )^2} \, dx\)

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 = 0.0256383, 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 \[ \frac{(c x)^n}{a c n \left (a+b x^n\right )} \]

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))

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Rubi in Sympy [A]  time = 3.68305, size = 15, normalized size = 0.62 \[ \frac{\left (c x\right )^{n}}{a c n \left (a + b x^{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x)**n/(a*c*n*(a + b*x**n))

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Mathematica [A]  time = 0.0264796, 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 = 0.038, size = 99, normalized size = 4.1 \[{\frac{x}{an \left ( a+b{x}^{n} \right ) }{{\rm e}^{{\frac{ \left ( -1+n \right ) \left ( -i\pi \, \left ({\it csgn} \left ( icx \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( icx \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( icx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \,{\it csgn} \left ( icx \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( x \right ) +2\,\ln \left ( c \right ) \right ) }{2}}}}} \]

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*Pi*csgn(I*c*x)^3+I*Pi*csgn(I*c*x)^2*csgn(I*
c)+I*Pi*csgn(I*c*x)^2*csgn(I*x)-I*Pi*csgn(I*c*x)*csgn(I*c)*csgn(I*x)+2*ln(x)+2*l
n(c)))

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Maxima [A]  time = 1.45415, size = 30, normalized size = 1.25 \[ -\frac{c^{n}}{b^{2} c n x^{n} + a b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c^n/(b^2*c*n*x^n + a*b*c*n)

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Fricas [A]  time = 0.223368, size = 30, normalized size = 1.25 \[ -\frac{c^{n - 1}}{b^{2} n x^{n} + a b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(n - 1)/(b*x^n + a)^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. \[ \text{Exception raised: TypeError} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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