3.3003 \(\int \frac{1}{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \, dx\)

Optimal. Leaf size=26 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.0308096, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.01996, size = 26, normalized size = 1. \[ \frac{\log{\left (\left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a} - \frac{\log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0965878, size = 23, normalized size = 0.88 \[ \frac{\log (x)-\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 35, normalized size = 1.4 \[{\frac{\ln \left ( \sqrt [n]{c{x}^{n}} \right ) }{a}}-{\frac{\ln \left ( a+b\sqrt [n]{c{x}^{n}} \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*(c*x^n)^(1/n)),x)

[Out]

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

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Maxima [A]  time = 1.48653, size = 54, normalized size = 2.08 \[ -\frac{\log \left (\frac{{\left (b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a\right )} c^{-\frac{1}{n}}}{b}\right )}{a} + \frac{\log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233238, size = 28, normalized size = 1.08 \[ -\frac{\log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) - \log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(log(b*c^(1/n)*x + a) - log(x))/a

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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