3.3002 \(\int \frac{1}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx\)

Optimal. Leaf size=30 \[ \frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0192271, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.27477, size = 24, normalized size = 0.8 \[ \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00714554, size = 30, normalized size = 1. \[ \frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.097, size = 214, normalized size = 7.1 \[{\frac{1}{\sqrt [n]{c}b}\ln \left ( b{{\rm e}^{{\frac{-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\pi +2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -2\,n\ln \left ( x \right ) }{2\,n}}}}x+a \right ){{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) \pi \,{\it csgn} \left ( ic \right ) -i \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\pi +i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(c*x^n)^(1/n)),x)

[Out]

1/(c^(1/n))/b*ln(b*exp(1/2*(-I*Pi*csgn(I*x^n)*csgn(I*c)*csgn(I*c*x^n)+I*Pi*csgn(
I*x^n)*csgn(I*c*x^n)^2+I*Pi*csgn(I*c)*csgn(I*c*x^n)^2-I*Pi*csgn(I*c*x^n)^3+2*ln(
c)+2*ln(x^n)-2*n*ln(x))/n)*x+a)*exp(-1/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*
csgn(I*x^n)*csgn(I*c)*csgn(I*c*x^n)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c)*csgn(I*c
*x^n)^2-2*n*ln(x)+2*ln(x^n))/n)

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Maxima [A]  time = 22.2525, size = 30, normalized size = 1. \[ \frac{c^{-\frac{1}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^(-1/n)*log(b*c^(1/n)*x + a)/b

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Fricas [A]  time = 0.252587, size = 30, normalized size = 1. \[ \frac{\log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b c^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*c^(1/n)*x + a)/(b*c^(1/n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^n)^(1/n)*b + a), x)