3.3010 \(\int \frac{1}{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx\)

Optimal. Leaf size=45 \[ -\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{1}{a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \]

[Out]

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

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Rubi [A]  time = 0.0539213, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{1}{a \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.57037, size = 44, normalized size = 0.98 \[ \frac{1}{a \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} + \frac{\log{\left (\left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{2}} - \frac{\log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.10613, size = 40, normalized size = 0.89 \[ \frac{\frac{a}{a+b \left (c x^n\right )^{\frac{1}{n}}}-\log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+\log (x)}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 54, normalized size = 1.2 \[{\frac{\ln \left ( \sqrt [n]{c{x}^{n}} \right ) }{{a}^{2}}}-{\frac{\ln \left ( a+b\sqrt [n]{c{x}^{n}} \right ) }{{a}^{2}}}+{\frac{1}{a \left ( a+b\sqrt [n]{c{x}^{n}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37171, size = 82, normalized size = 1.82 \[ \frac{1}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} - \frac{\log \left (\frac{{\left (b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a\right )} c^{-\frac{1}{n}}}{b}\right )}{a^{2}} + \frac{\log \left (x\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237304, size = 77, normalized size = 1.71 \[ \frac{b c^{\left (\frac{1}{n}\right )} x \log \left (x\right ) -{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) + a \log \left (x\right ) + a}{a^{2} b c^{\left (\frac{1}{n}\right )} x + a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*c^(1/n)*x*log(x) - (b*c^(1/n)*x + a)*log(b*c^(1/n)*x + a) + a*log(x) + a)/(a^
2*b*c^(1/n)*x + a^3)

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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))**2,x)

[Out]

Exception raised: RecursionError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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