3.2760 \(\int \frac{(c x)^{-1-\frac{n}{3}}}{a+b x^n} \, dx\)

Optimal. Leaf size=220 \[ -\frac{\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} c n}+\frac{\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} c n}-\frac{\sqrt{3} \sqrt [3]{b} x^{n/3} (c x)^{-n/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{4/3} c n}-\frac{3 (c x)^{-n/3}}{a c n} \]

[Out]

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

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Rubi [A]  time = 0.324649, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476 \[ -\frac{\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} c n}+\frac{\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} c n}-\frac{\sqrt{3} \sqrt [3]{b} x^{n/3} (c x)^{-n/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{4/3} c n}-\frac{3 (c x)^{-n/3}}{a c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.2296, size = 178, normalized size = 0.81 \[ - \frac{3 \left (c x\right )^{- \frac{n}{3}}}{a c n} + \frac{\sqrt [3]{b} x^{\frac{n}{3}} \left (c x\right )^{- \frac{n}{3}} \log{\left (\sqrt [3]{a} x^{- \frac{n}{3}} + \sqrt [3]{b} \right )}}{a^{\frac{4}{3}} c n} - \frac{\sqrt [3]{b} x^{\frac{n}{3}} \left (c x\right )^{- \frac{n}{3}} \log{\left (a^{\frac{2}{3}} x^{- \frac{2 n}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{- \frac{n}{3}} + b^{\frac{2}{3}} \right )}}{2 a^{\frac{4}{3}} c n} - \frac{\sqrt{3} \sqrt [3]{b} x^{\frac{n}{3}} \left (c x\right )^{- \frac{n}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x^{- \frac{n}{3}}}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{a^{\frac{4}{3}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0430835, size = 71, normalized size = 0.32 \[ \frac{(c x)^{-n/3} \left (b x^{n/3} \text{RootSum}\left [\text{$\#$1}^3 a+b\&,\frac{3 \log \left (x^{-n/3}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^2}\&\right ]-9 a\right )}{3 a^2 c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \[ \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{n}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249284, size = 331, normalized size = 1.5 \[ -\frac{6 \, x e^{\left (-\frac{1}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} + 2 \, \sqrt{3} \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left (2 \, x e^{\left (-\frac{1}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} - \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}}}\right ) - 2 \, \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}} \log \left (\frac{x e^{\left (-\frac{1}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} + \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}}}{x}\right ) + \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}} \log \left (\frac{x^{2} e^{\left (-\frac{2}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{2}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} - \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{1}{3}} x e^{\left (-\frac{1}{3} \,{\left (n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (n + 3\right )} \log \left (x\right )\right )} + \left (\frac{b c^{-n - 3}}{a}\right )^{\frac{2}{3}}}{x^{2}}\right )}{2 \, a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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