3.2633 \(\int \frac{x^{-1-\frac{2 n}{3}}}{a+b x^n} \, dx\)

Optimal. Leaf size=160 \[ -\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{5/3} n}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 a^{5/3} n}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3} n}-\frac{3 x^{-2 n/3}}{2 a n} \]

[Out]

-3/(2*a*n*x^((2*n)/3)) + (Sqrt[3]*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(n/3))/(
Sqrt[3]*a^(1/3))])/(a^(5/3)*n) - (b^(2/3)*Log[a^(1/3) + b^(1/3)*x^(n/3)])/(a^(5/
3)*n) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(n/3) + b^(2/3)*x^((2*n)/3)])/(
2*a^(5/3)*n)

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Rubi [A]  time = 0.259612, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{5/3} n}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 a^{5/3} n}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3} n}-\frac{3 x^{-2 n/3}}{2 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - (2*n)/3)/(a + b*x^n),x]

[Out]

-3/(2*a*n*x^((2*n)/3)) + (Sqrt[3]*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(n/3))/(
Sqrt[3]*a^(1/3))])/(a^(5/3)*n) - (b^(2/3)*Log[a^(1/3) + b^(1/3)*x^(n/3)])/(a^(5/
3)*n) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(n/3) + b^(2/3)*x^((2*n)/3)])/(
2*a^(5/3)*n)

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Rubi in Sympy [A]  time = 38.1632, size = 138, normalized size = 0.86 \[ - \frac{3 x^{- \frac{2 n}{3}}}{2 a n} - \frac{b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x^{\frac{n}{3}} \right )}}{a^{\frac{5}{3}} n} + \frac{b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{\frac{n}{3}} + b^{\frac{2}{3}} x^{\frac{2 n}{3}} \right )}}{2 a^{\frac{5}{3}} n} + \frac{\sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x^{\frac{n}{3}}}{3}\right )}{\sqrt [3]{a}} \right )}}{a^{\frac{5}{3}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-2/3*n)/(a+b*x**n),x)

[Out]

-3*x**(-2*n/3)/(2*a*n) - b**(2/3)*log(a**(1/3) + b**(1/3)*x**(n/3))/(a**(5/3)*n)
+ b**(2/3)*log(a**(2/3) - a**(1/3)*b**(1/3)*x**(n/3) + b**(2/3)*x**(2*n/3))/(2*
a**(5/3)*n) + sqrt(3)*b**(2/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x**(n/3)/3)
/a**(1/3))/(a**(5/3)*n)

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Mathematica [C]  time = 0.0423757, size = 60, normalized size = 0.38 \[ \frac{2 b \text{RootSum}\left [\text{$\#$1}^3 a+b\&,\frac{3 \log \left (x^{-n/3}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}}\&\right ]-9 a x^{-2 n/3}}{6 a^2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - (2*n)/3)/(a + b*x^n),x]

[Out]

((-9*a)/x^((2*n)/3) + 2*b*RootSum[b + a*#1^3 & , (n*Log[x] + 3*Log[x^(-n/3) - #1
])/#1 & ])/(6*a^2*n)

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Maple [C]  time = 0.32, size = 54, normalized size = 0.3 \[ -{\frac{3}{2\,an} \left ({x}^{{\frac{n}{3}}} \right ) ^{-2}}+\sum _{{\it \_R}={\it RootOf} \left ({a}^{5}{n}^{3}{{\it \_Z}}^{3}+{b}^{2} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}-{\frac{{a}^{2}n{\it \_R}}{b}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-2/3*n)/(a+b*x^n),x)

[Out]

-3/2/a/n/(x^(1/3*n))^2+sum(_R*ln(x^(1/3*n)-a^2*n/b*_R),_R=RootOf(_Z^3*a^5*n^3+b^
2))

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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^(-2/3*n - 1)/(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301768, size = 250, normalized size = 1.56 \[ -\frac{3 \, x x^{-\frac{2}{3} \, n - 1} + 2 \, \sqrt{3} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, b \sqrt{x} x^{-\frac{1}{3} \, n - \frac{1}{2}} - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right )}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}\right ) - 2 \, \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (\frac{b \sqrt{x} x^{-\frac{1}{3} \, n - \frac{1}{2}} + a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}{\sqrt{x}}\right ) + \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (-\frac{a \sqrt{x} x^{-\frac{1}{3} \, n - \frac{1}{2}} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - b x x^{-\frac{2}{3} \, n - 1} + b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{x}\right )}{2 \, a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2/3*n - 1)/(b*x^n + a),x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{2}{3} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2/3*n - 1)/(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(-2/3*n - 1)/(b*x^n + a), x)