3.3036 \(\int \frac{1}{a+b \left (c x^n\right )^{3/n}} \, dx\)

Optimal. Leaf size=183 \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]

[Out]

-((x*ArcTan[(a^(1/3) - 2*b^(1/3)*(c*x^n)^n^(-1))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^
(2/3)*b^(1/3)*(c*x^n)^n^(-1))) + (x*Log[a^(1/3) + b^(1/3)*(c*x^n)^n^(-1)])/(3*a^
(2/3)*b^(1/3)*(c*x^n)^n^(-1)) - (x*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n)^n^(-1)
+ b^(2/3)*(c*x^n)^(2/n)])/(6*a^(2/3)*b^(1/3)*(c*x^n)^n^(-1))

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Rubi [A]  time = 0.170534, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{6 a^{2/3} \sqrt [3]{b}}+\frac{x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^n)^(3/n))^(-1),x]

[Out]

-((x*ArcTan[(a^(1/3) - 2*b^(1/3)*(c*x^n)^n^(-1))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^
(2/3)*b^(1/3)*(c*x^n)^n^(-1))) + (x*Log[a^(1/3) + b^(1/3)*(c*x^n)^n^(-1)])/(3*a^
(2/3)*b^(1/3)*(c*x^n)^n^(-1)) - (x*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n)^n^(-1)
+ b^(2/3)*(c*x^n)^(2/n)])/(6*a^(2/3)*b^(1/3)*(c*x^n)^n^(-1))

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Rubi in Sympy [A]  time = 28.9547, size = 165, normalized size = 0.9 \[ \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b}} - \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}} + b^{\frac{2}{3}} \left (c x^{n}\right )^{\frac{2}{n}} \right )}}{6 a^{\frac{2}{3}} \sqrt [3]{b}} - \frac{\sqrt{3} x \left (c x^{n}\right )^{- \frac{1}{n}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*(c*x**n)**(3/n)),x)

[Out]

x*(c*x**n)**(-1/n)*log(a**(1/3) + b**(1/3)*(c*x**n)**(1/n))/(3*a**(2/3)*b**(1/3)
) - x*(c*x**n)**(-1/n)*log(a**(2/3) - a**(1/3)*b**(1/3)*(c*x**n)**(1/n) + b**(2/
3)*(c*x**n)**(2/n))/(6*a**(2/3)*b**(1/3)) - sqrt(3)*x*(c*x**n)**(-1/n)*atan(sqrt
(3)*(a**(1/3)/3 - 2*b**(1/3)*(c*x**n)**(1/n)/3)/a**(1/3))/(3*a**(2/3)*b**(1/3))

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Mathematica [A]  time = 4.69656, size = 0, normalized size = 0. \[ \int \frac{1}{a+b \left (c x^n\right )^{3/n}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a + b*(c*x^n)^(3/n))^(-1),x]

[Out]

Integrate[(a + b*(c*x^n)^(3/n))^(-1), x]

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Maple [F]  time = 0.203, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{-1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(c*x^n)^(3/n)),x)

[Out]

int(1/(a+b*(c*x^n)^(3/n)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^n)^(3/n)*b + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237283, size = 167, normalized size = 0.91 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3} \log \left (\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x + a\right ) - 6 \, \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right )\right )}}{18 \, \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^n)^(3/n)*b + a),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(sqrt(3)*log((a^2*b*c^(3/n))^(2/3)*x^2 - (a^2*b*c^(3/n))^(1/3)*a*x
+ a^2) - 2*sqrt(3)*log((a^2*b*c^(3/n))^(1/3)*x + a) - 6*arctan(1/3*(2*sqrt(3)*(
a^2*b*c^(3/n))^(1/3)*x - sqrt(3)*a)/a))/(a^2*b*c^(3/n))^(1/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{a + b \left (c x^{n}\right )^{\frac{3}{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*(c*x**n)**(3/n)),x)

[Out]

Integral(1/(a + b*(c*x**n)**(3/n)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c x^{n}\right )^{\frac{3}{n}} b + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^n)^(3/n)*b + a),x, algorithm="giac")

[Out]

integrate(1/((c*x^n)^(3/n)*b + a), x)