3.2510 \(\int \frac{\sqrt [3]{a+b x^n}}{x} \, dx\)

Optimal. Leaf size=106 \[ \frac{3 \sqrt [3]{a+b x^n}}{n}+\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )}{2 n}-\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^n}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{n}-\frac{1}{2} \sqrt [3]{a} \log (x) \]

[Out]

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

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Rubi [A]  time = 0.173095, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{3 \sqrt [3]{a+b x^n}}{n}+\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )}{2 n}-\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^n}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{n}-\frac{1}{2} \sqrt [3]{a} \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(1/3)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.4656, size = 97, normalized size = 0.92 \[ - \frac{\sqrt [3]{a} \log{\left (x^{n} \right )}}{2 n} + \frac{3 \sqrt [3]{a} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{n}} \right )}}{2 n} - \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{n}}}{3}\right )}{\sqrt [3]{a}} \right )}}{n} + \frac{3 \sqrt [3]{a + b x^{n}}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0673516, size = 68, normalized size = 0.64 \[ \frac{6 \left (a+b x^n\right )-3 a \left (\frac{a x^{-n}}{b}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a x^{-n}}{b}\right )}{2 n \left (a+b x^n\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(6*(a + b*x^n) - 3*a*(1 + a/(b*x^n))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, -(a/
(b*x^n))])/(2*n*(a + b*x^n)^(2/3))

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Maple [A]  time = 0.008, size = 107, normalized size = 1. \[ 3\,{\frac{\sqrt [3]{a+b{x}^{n}}}{n}}+{\frac{1}{n}\sqrt [3]{a}\ln \left ( \sqrt [3]{a+b{x}^{n}}-\sqrt [3]{a} \right ) }-{\frac{1}{2\,n}\sqrt [3]{a}\ln \left ( \left ( a+b{x}^{n} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a+b{x}^{n}}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ) }-{\frac{\sqrt{3}}{n}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{a+b{x}^{n}}}{\sqrt [3]{a}}}+1 \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(1/3)/x,x)

[Out]

3*(a+b*x^n)^(1/3)/n+1/n*a^(1/3)*ln((a+b*x^n)^(1/3)-a^(1/3))-1/2/n*a^(1/3)*ln((a+
b*x^n)^(2/3)+(a+b*x^n)^(1/3)*a^(1/3)+a^(2/3))-1/n*a^(1/3)*3^(1/2)*arctan(1/3*3^(
1/2)*(2/a^(1/3)*(a+b*x^n)^(1/3)+1))

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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((b*x^n + a)^(1/3)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227586, size = 136, normalized size = 1.28 \[ -\frac{2 \, \sqrt{3} a^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{n} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right ) + a^{\frac{1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac{2}{3}} +{\left (b x^{n} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) - 2 \, a^{\frac{1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) - 6 \,{\left (b x^{n} + a\right )}^{\frac{1}{3}}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^(1/3)/x, x)