3.2756 \(\int \frac{(c x)^{-1-\frac{2 n}{3}}}{a+b x^n} \, dx\)

Optimal. Leaf size=222 \[ -\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{5/3} c n}+\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 a^{5/3} c n}+\frac{\sqrt{3} b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3} c n}-\frac{3 (c x)^{-2 n/3}}{2 a c n} \]

[Out]

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

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Rubi [A]  time = 0.324833, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^{n/3}\right )}{a^{5/3} c n}+\frac{b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^{n/3}+b^{2/3} x^{2 n/3}\right )}{2 a^{5/3} c n}+\frac{\sqrt{3} b^{2/3} x^{2 n/3} (c x)^{-2 n/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^{n/3}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3} c n}-\frac{3 (c x)^{-2 n/3}}{2 a c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.8695, size = 192, normalized size = 0.86 \[ - \frac{3 \left (c x\right )^{- \frac{2 n}{3}}}{2 a c n} - \frac{b^{\frac{2}{3}} x^{\frac{2 n}{3}} \left (c x\right )^{- \frac{2 n}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x^{\frac{n}{3}} \right )}}{a^{\frac{5}{3}} c n} + \frac{b^{\frac{2}{3}} x^{\frac{2 n}{3}} \left (c x\right )^{- \frac{2 n}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{\frac{n}{3}} + b^{\frac{2}{3}} x^{\frac{2 n}{3}} \right )}}{2 a^{\frac{5}{3}} c n} + \frac{\sqrt{3} b^{\frac{2}{3}} x^{\frac{2 n}{3}} \left (c x\right )^{- \frac{2 n}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x^{\frac{n}{3}}}{3}\right )}{\sqrt [3]{a}} \right )}}{a^{\frac{5}{3}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0483066, size = 72, normalized size = 0.32 \[ \frac{(c x)^{-2 n/3} \left (2 b x^{2 n/3} \text{RootSum}\left [\text{$\#$1}^3 a+b\&,\frac{3 \log \left (x^{-n/3}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}}\&\right ]-9 a\right )}{6 a^2 c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{2\,n}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.460795, size = 451, normalized size = 2.03 \[ -\frac{3 \, x e^{\left (-\frac{1}{3} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} - 2 \, \sqrt{3} \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left (2 \, b c^{-n - \frac{3}{2}} \sqrt{x} e^{\left (-\frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} - a \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{2}{3}}\right )}}{3 \, a \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{2}{3}}}\right ) - 2 \, \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{1}{3}} \log \left (\frac{b c^{-n - \frac{3}{2}} \sqrt{x} e^{\left (-\frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} + a \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{2}{3}}}{\sqrt{x}}\right ) + \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{1}{3}} \log \left (\frac{b c^{-n - \frac{3}{2}} x e^{\left (-\frac{1}{3} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} - a \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{2}{3}} \sqrt{x} e^{\left (-\frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (2 \, n + 3\right )} \log \left (x\right )\right )} - b c^{-n - \frac{3}{2}} \left (-\frac{b^{2} c^{-2 \, n - 3}}{a^{2}}\right )^{\frac{1}{3}}}{x}\right )}{2 \, a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(3*x*e^(-1/3*(2*n + 3)*log(c) - 1/3*(2*n + 3)*log(x)) - 2*sqrt(3)*(-b^2*c^(
-2*n - 3)/a^2)^(1/3)*arctan(-1/3*sqrt(3)*(2*b*c^(-n - 3/2)*sqrt(x)*e^(-1/6*(2*n
+ 3)*log(c) - 1/6*(2*n + 3)*log(x)) - a*(-b^2*c^(-2*n - 3)/a^2)^(2/3))/(a*(-b^2*
c^(-2*n - 3)/a^2)^(2/3))) - 2*(-b^2*c^(-2*n - 3)/a^2)^(1/3)*log((b*c^(-n - 3/2)*
sqrt(x)*e^(-1/6*(2*n + 3)*log(c) - 1/6*(2*n + 3)*log(x)) + a*(-b^2*c^(-2*n - 3)/
a^2)^(2/3))/sqrt(x)) + (-b^2*c^(-2*n - 3)/a^2)^(1/3)*log((b*c^(-n - 3/2)*x*e^(-1
/3*(2*n + 3)*log(c) - 1/3*(2*n + 3)*log(x)) - a*(-b^2*c^(-2*n - 3)/a^2)^(2/3)*sq
rt(x)*e^(-1/6*(2*n + 3)*log(c) - 1/6*(2*n + 3)*log(x)) - b*c^(-n - 3/2)*(-b^2*c^
(-2*n - 3)/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((c*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. \[ \int \frac{\left (c x\right )^{-\frac{2}{3} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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