3.2758 \(\int \frac{(c x)^{-1-n}}{a+b x^n} \, dx\)

Optimal. Leaf size=69 \[ -\frac{b x^n \log (x) (c x)^{-n}}{a^2 c}+\frac{b x^n (c x)^{-n} \log \left (a+b x^n\right )}{a^2 c n}-\frac{(c x)^{-n}}{a c n} \]

[Out]

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

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Rubi [A]  time = 0.0990706, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{b x^n \log (x) (c x)^{-n}}{a^2 c}+\frac{b x^n (c x)^{-n} \log \left (a+b x^n\right )}{a^2 c n}-\frac{(c x)^{-n}}{a c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.4659, size = 58, normalized size = 0.84 \[ - \frac{\left (c x\right )^{- n}}{a c n} - \frac{b x^{n} \left (c x\right )^{- n} \log{\left (x^{n} \right )}}{a^{2} c n} + \frac{b x^{n} \left (c x\right )^{- n} \log{\left (a + b x^{n} \right )}}{a^{2} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c*x)**(-n)/(a*c*n) - b*x**n*(c*x)**(-n)*log(x**n)/(a**2*c*n) + b*x**n*(c*x)**(
-n)*log(a + b*x**n)/(a**2*c*n)

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Mathematica [A]  time = 0.0356842, size = 36, normalized size = 0.52 \[ \frac{(c x)^{-n} \left (b x^n \log \left (a x^{-n}+b\right )-a\right )}{a^2 c n} \]

Antiderivative was successfully verified.

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

[Out]

(-a + b*x^n*Log[b + a/x^n])/(a^2*c*n*(c*x)^n)

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{-1-n}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-1-n)/(a+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.35684, size = 85, normalized size = 1.23 \[ -\frac{b c^{-n - 1} \log \left (x\right )}{a^{2}} - \frac{c^{-n - 1} x^{-n}}{a n} + \frac{b c^{-n - 1} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.236272, size = 80, normalized size = 1.16 \[ -\frac{b c^{-n - 1} n x^{n} \log \left (x\right ) - b c^{-n - 1} x^{n} \log \left (b x^{n} + a\right ) + a c^{-n - 1}}{a^{2} n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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