3.2637 \(\int \frac{x^{-1-\frac{n}{3}}}{a+b x^n} \, dx\)

Optimal. Leaf size=158 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} n}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} n}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{4/3} n}-\frac{3 x^{-n/3}}{a n} \]

[Out]

-3/(a*n*x^(n/3)) - (Sqrt[3]*b^(1/3)*ArcTan[(b^(1/3) - (2*a^(1/3))/x^(n/3))/(Sqrt
[3]*b^(1/3))])/(a^(4/3)*n) + (b^(1/3)*Log[b^(1/3) + a^(1/3)/x^(n/3)])/(a^(4/3)*n
) - (b^(1/3)*Log[b^(2/3) + a^(2/3)/x^((2*n)/3) - (a^(1/3)*b^(1/3))/x^(n/3)])/(2*
a^(4/3)*n)

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Rubi [A]  time = 0.253491, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} n}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} n}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{4/3} n}-\frac{3 x^{-n/3}}{a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n/3)/(a + b*x^n),x]

[Out]

-3/(a*n*x^(n/3)) - (Sqrt[3]*b^(1/3)*ArcTan[(b^(1/3) - (2*a^(1/3))/x^(n/3))/(Sqrt
[3]*b^(1/3))])/(a^(4/3)*n) + (b^(1/3)*Log[b^(1/3) + a^(1/3)/x^(n/3)])/(a^(4/3)*n
) - (b^(1/3)*Log[b^(2/3) + a^(2/3)/x^((2*n)/3) - (a^(1/3)*b^(1/3))/x^(n/3)])/(2*
a^(4/3)*n)

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Rubi in Sympy [A]  time = 38.6411, size = 134, normalized size = 0.85 \[ - \frac{3 x^{- \frac{n}{3}}}{a n} + \frac{\sqrt [3]{b} \log{\left (\sqrt [3]{a} x^{- \frac{n}{3}} + \sqrt [3]{b} \right )}}{a^{\frac{4}{3}} n} - \frac{\sqrt [3]{b} \log{\left (a^{\frac{2}{3}} x^{- \frac{2 n}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{- \frac{n}{3}} + b^{\frac{2}{3}} \right )}}{2 a^{\frac{4}{3}} n} - \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x^{- \frac{n}{3}}}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{a^{\frac{4}{3}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x**(-n/3)/(a*n) + b**(1/3)*log(a**(1/3)*x**(-n/3) + b**(1/3))/(a**(4/3)*n) -
b**(1/3)*log(a**(2/3)*x**(-2*n/3) - a**(1/3)*b**(1/3)*x**(-n/3) + b**(2/3))/(2*a
**(4/3)*n) - sqrt(3)*b**(1/3)*atan(sqrt(3)*(-2*a**(1/3)*x**(-n/3)/3 + b**(1/3)/3
)/b**(1/3))/(a**(4/3)*n)

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Mathematica [C]  time = 0.0367305, size = 59, normalized size = 0.37 \[ \frac{b \text{RootSum}\left [\text{$\#$1}^3 a+b\&,\frac{3 \log \left (x^{-n/3}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^2}\&\right ]-9 a x^{-n/3}}{3 a^2 n} \]

Antiderivative was successfully verified.

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

[Out]

((-9*a)/x^(n/3) + b*RootSum[b + a*#1^3 & , (n*Log[x] + 3*Log[x^(-n/3) - #1])/#1^
2 & ])/(3*a^2*n)

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Maple [C]  time = 0.079, size = 57, normalized size = 0.4 \[ -3\,{\frac{1}{an{x}^{n/3}}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{4}{n}^{3}{{\it \_Z}}^{3}-b \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}+{\frac{{a}^{3}{n}^{2}{{\it \_R}}^{2}}{b}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-1/3*n)/(a+b*x^n),x)

[Out]

-3/a/n/(x^(1/3*n))+sum(_R*ln(x^(1/3*n)+a^3*n^2/b*_R^2),_R=RootOf(_Z^3*a^4*n^3-b)
)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237047, size = 194, normalized size = 1.23 \[ -\frac{6 \, x x^{-\frac{1}{3} \, n - 1} - 2 \, \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x x^{-\frac{1}{3} \, n - 1} - \left (\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{b}{a}\right )^{\frac{1}{3}}}\right ) - 2 \, \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (\frac{x x^{-\frac{1}{3} \, n - 1} + \left (\frac{b}{a}\right )^{\frac{1}{3}}}{x}\right ) + \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (\frac{x^{2} x^{-\frac{2}{3} \, n - 2} - x x^{-\frac{1}{3} \, n - 1} \left (\frac{b}{a}\right )^{\frac{1}{3}} + \left (\frac{b}{a}\right )^{\frac{2}{3}}}{x^{2}}\right )}{2 \, a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(-1/3*n - 1)/(b*x^n + a), x)