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

Optimal. Leaf size=176 \[ -\frac{c^{4/3} \log \left (\sqrt [3]{b} x^{-n/3}+\sqrt [3]{c}\right )}{b^{7/3} n}+\frac{c^{4/3} \log \left (b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}+c^{2/3}\right )}{2 b^{7/3} n}+\frac{\sqrt{3} c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{b} x^{-n/3}}{\sqrt{3} \sqrt [3]{c}}\right )}{b^{7/3} n}+\frac{3 c x^{-n/3}}{b^2 n}-\frac{3 x^{-4 n/3}}{4 b n} \]

[Out]

-3/(4*b*n*x^((4*n)/3)) + (3*c)/(b^2*n*x^(n/3)) + (Sqrt[3]*c^(4/3)*ArcTan[(c^(1/3
) - (2*b^(1/3))/x^(n/3))/(Sqrt[3]*c^(1/3))])/(b^(7/3)*n) - (c^(4/3)*Log[c^(1/3)
+ b^(1/3)/x^(n/3)])/(b^(7/3)*n) + (c^(4/3)*Log[c^(2/3) + b^(2/3)/x^((2*n)/3) - (
b^(1/3)*c^(1/3))/x^(n/3)])/(2*b^(7/3)*n)

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Rubi [A]  time = 0.271777, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44 \[ -\frac{c^{4/3} \log \left (\sqrt [3]{b} x^{-n/3}+\sqrt [3]{c}\right )}{b^{7/3} n}+\frac{c^{4/3} \log \left (b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}+c^{2/3}\right )}{2 b^{7/3} n}+\frac{\sqrt{3} c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{b} x^{-n/3}}{\sqrt{3} \sqrt [3]{c}}\right )}{b^{7/3} n}+\frac{3 c x^{-n/3}}{b^2 n}-\frac{3 x^{-4 n/3}}{4 b n} \]

Antiderivative was successfully verified.

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

[Out]

-3/(4*b*n*x^((4*n)/3)) + (3*c)/(b^2*n*x^(n/3)) + (Sqrt[3]*c^(4/3)*ArcTan[(c^(1/3
) - (2*b^(1/3))/x^(n/3))/(Sqrt[3]*c^(1/3))])/(b^(7/3)*n) - (c^(4/3)*Log[c^(1/3)
+ b^(1/3)/x^(n/3)])/(b^(7/3)*n) + (c^(4/3)*Log[c^(2/3) + b^(2/3)/x^((2*n)/3) - (
b^(1/3)*c^(1/3))/x^(n/3)])/(2*b^(7/3)*n)

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Rubi in Sympy [A]  time = 48.8427, size = 151, normalized size = 0.86 \[ - \frac{3 x^{- \frac{4 n}{3}}}{4 b n} + \frac{3 c x^{- \frac{n}{3}}}{b^{2} n} - \frac{c^{\frac{4}{3}} \log{\left (\sqrt [3]{b} x^{- \frac{n}{3}} + \sqrt [3]{c} \right )}}{b^{\frac{7}{3}} n} + \frac{c^{\frac{4}{3}} \log{\left (b^{\frac{2}{3}} x^{- \frac{2 n}{3}} - \sqrt [3]{b} \sqrt [3]{c} x^{- \frac{n}{3}} + c^{\frac{2}{3}} \right )}}{2 b^{\frac{7}{3}} n} + \frac{\sqrt{3} c^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{b} x^{- \frac{n}{3}}}{3} + \frac{\sqrt [3]{c}}{3}\right )}{\sqrt [3]{c}} \right )}}{b^{\frac{7}{3}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x**(-4*n/3)/(4*b*n) + 3*c*x**(-n/3)/(b**2*n) - c**(4/3)*log(b**(1/3)*x**(-n/3
) + c**(1/3))/(b**(7/3)*n) + c**(4/3)*log(b**(2/3)*x**(-2*n/3) - b**(1/3)*c**(1/
3)*x**(-n/3) + c**(2/3))/(2*b**(7/3)*n) + sqrt(3)*c**(4/3)*atan(sqrt(3)*(-2*b**(
1/3)*x**(-n/3)/3 + c**(1/3)/3)/c**(1/3))/(b**(7/3)*n)

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Mathematica [C]  time = 0.0772193, size = 70, normalized size = 0.4 \[ -\frac{4 c^2 \text{RootSum}\left [\text{$\#$1}^3 b+c\&,\frac{3 \log \left (x^{-n/3}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^2}\&\right ]+9 b x^{-4 n/3} \left (b-4 c x^n\right )}{12 b^3 n} \]

Antiderivative was successfully verified.

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

[Out]

-((9*b*(b - 4*c*x^n))/x^((4*n)/3) + 4*c^2*RootSum[c + b*#1^3 & , (n*Log[x] + 3*L
og[x^(-n/3) - #1])/#1^2 & ])/(12*b^3*n)

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Maple [C]  time = 0.106, size = 73, normalized size = 0.4 \[ 3\,{\frac{c}{{b}^{2}n{x}^{n/3}}}-{\frac{3}{4\,bn} \left ({x}^{{\frac{n}{3}}} \right ) ^{-4}}+\sum _{{\it \_R}={\it RootOf} \left ({b}^{7}{n}^{3}{{\it \_Z}}^{3}+{c}^{4} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}+{\frac{{b}^{5}{n}^{2}{{\it \_R}}^{2}}{{c}^{3}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*c/b^2/n/(x^(1/3*n))-3/4/b/n/(x^(1/3*n))^4+sum(_R*ln(x^(1/3*n)+b^5*n^2/c^3*_R^2
),_R=RootOf(_Z^3*b^7*n^3+c^4))

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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)/(c*x^(2*n) + b*x^n),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301893, size = 228, normalized size = 1.3 \[ -\frac{3 \, b x^{4} x^{-\frac{4}{3} \, n - 4} - 12 \, c x x^{-\frac{1}{3} \, n - 1} + 4 \, \sqrt{3} c \left (-\frac{c}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x x^{-\frac{1}{3} \, n - 1} + \left (-\frac{c}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{b}\right )^{\frac{1}{3}}}\right ) - 4 \, c \left (-\frac{c}{b}\right )^{\frac{1}{3}} \log \left (\frac{x x^{-\frac{1}{3} \, n - 1} - \left (-\frac{c}{b}\right )^{\frac{1}{3}}}{x}\right ) + 2 \, c \left (-\frac{c}{b}\right )^{\frac{1}{3}} \log \left (\frac{x^{2} x^{-\frac{2}{3} \, n - 2} + x x^{-\frac{1}{3} \, n - 1} \left (-\frac{c}{b}\right )^{\frac{1}{3}} + \left (-\frac{c}{b}\right )^{\frac{2}{3}}}{x^{2}}\right )}{4 \, b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(3*b*x^4*x^(-4/3*n - 4) - 12*c*x*x^(-1/3*n - 1) + 4*sqrt(3)*c*(-c/b)^(1/3)*
arctan(1/3*sqrt(3)*(2*x*x^(-1/3*n - 1) + (-c/b)^(1/3))/(-c/b)^(1/3)) - 4*c*(-c/b
)^(1/3)*log((x*x^(-1/3*n - 1) - (-c/b)^(1/3))/x) + 2*c*(-c/b)^(1/3)*log((x^2*x^(
-2/3*n - 2) + x*x^(-1/3*n - 1)*(-c/b)^(1/3) + (-c/b)^(2/3))/x^2))/(b^2*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)/(b*x**n+c*x**(2*n)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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