3.2770 \(\int \frac{(c x)^{-1+n}}{a+b x^n} \, dx\)

Optimal. Leaf size=28 \[ \frac{x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n} \]

[Out]

((c*x)^n*Log[a + b*x^n])/(b*c*n*x^n)

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Rubi [A]  time = 0.0411639, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^{-n} (c x)^n \log \left (a+b x^n\right )}{b c n} \]

Antiderivative was successfully verified.

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

[Out]

((c*x)^n*Log[a + b*x^n])/(b*c*n*x^n)

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Rubi in Sympy [A]  time = 4.76296, size = 20, normalized size = 0.71 \[ \frac{x^{- n} \left (c x\right )^{n} \log{\left (a + b x^{n} \right )}}{b c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0110343, size = 29, normalized size = 1.04 \[ \frac{x^{1-n} (c x)^{n-1} \log \left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 - n)*(c*x)^(-1 + n)*Log[a + b*x^n])/(b*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 = 1.3717, size = 32, normalized size = 1.14 \[ \frac{c^{n - 1} \log \left (\frac{b x^{n} + a}{b}\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^(n - 1)*log((b*x^n + a)/b)/(b*n)

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Fricas [A]  time = 0.233131, size = 27, normalized size = 0.96 \[ \frac{c^{n - 1} \log \left (b x^{n} + a\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^(n - 1)*log(b*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),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)