3.2763 \(\int \frac{(c x)^{-1-\frac{4 n}{3}}}{a+b x^n} \, dx\)

Optimal. Leaf size=246 \[ -\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} c n}+\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} c n}+\frac{\sqrt{3} b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{7/3} c n}+\frac{3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac{3 (c x)^{-4 n/3}}{4 a c n} \]

[Out]

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

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Rubi [A]  time = 0.36578, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524 \[ -\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} c n}+\frac{b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} c n}+\frac{\sqrt{3} b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt{3} \sqrt [3]{b}}\right )}{a^{7/3} c n}+\frac{3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac{3 (c x)^{-4 n/3}}{4 a c n} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(-1 - (4*n)/3)/(a + b*x^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 57.4544, size = 214, normalized size = 0.87 \[ - \frac{3 \left (c x\right )^{- \frac{4 n}{3}}}{4 a c n} + \frac{3 b x^{n} \left (c x\right )^{- \frac{4 n}{3}}}{a^{2} c n} - \frac{b^{\frac{4}{3}} x^{\frac{4 n}{3}} \left (c x\right )^{- \frac{4 n}{3}} \log{\left (\sqrt [3]{a} x^{- \frac{n}{3}} + \sqrt [3]{b} \right )}}{a^{\frac{7}{3}} c n} + \frac{b^{\frac{4}{3}} x^{\frac{4 n}{3}} \left (c x\right )^{- \frac{4 n}{3}} \log{\left (a^{\frac{2}{3}} x^{- \frac{2 n}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{- \frac{n}{3}} + b^{\frac{2}{3}} \right )}}{2 a^{\frac{7}{3}} c n} + \frac{\sqrt{3} b^{\frac{4}{3}} x^{\frac{4 n}{3}} \left (c x\right )^{- \frac{4 n}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x^{- \frac{n}{3}}}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{a^{\frac{7}{3}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(-1-4/3*n)/(a+b*x**n),x)

[Out]

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

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Mathematica [C]  time = 0.0534554, size = 82, normalized size = 0.33 \[ \frac{(c x)^{-4 n/3} \left (-4 b^2 x^{4 n/3} \text{RootSum}\left [\text{$\#$1}^3 a+b\&,\frac{3 \log \left (x^{-n/3}-\text{$\#$1}\right )+n \log (x)}{\text{$\#$1}^2}\&\right ]-9 a \left (a-4 b x^n\right )\right )}{12 a^3 c n} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(-1 - (4*n)/3)/(a + b*x^n),x]

[Out]

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

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{4\,n}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-1-4/3*n)/(a+b*x^n),x)

[Out]

int((c*x)^(-1-4/3*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)^(-4/3*n - 1)/(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.33591, size = 455, normalized size = 1.85 \[ -\frac{4 \, \sqrt{3} b c^{-n - \frac{3}{4}} \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{4}} e^{\left (-\frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} + \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}}}\right ) - 4 \, b c^{-n - \frac{3}{4}} \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}} \log \left (\frac{x^{\frac{1}{4}} e^{\left (-\frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} - \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}}}{x^{\frac{1}{4}}}\right ) + 2 \, b c^{-n - \frac{3}{4}} \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}} \log \left (\frac{\left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{1}{3}} x^{\frac{1}{4}} e^{\left (-\frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} + \sqrt{x} e^{\left (-\frac{1}{6} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} + \left (-\frac{b c^{-n - \frac{3}{4}}}{a}\right )^{\frac{2}{3}}}{\sqrt{x}}\right ) - 12 \, b c^{-n - \frac{3}{4}} x^{\frac{1}{4}} e^{\left (-\frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{12} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )} + 3 \, a x e^{\left (-\frac{1}{3} \,{\left (4 \, n + 3\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (4 \, n + 3\right )} \log \left (x\right )\right )}}{4 \, a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(-4/3*n - 1)/(b*x^n + a),x, algorithm="fricas")

[Out]

-1/4*(4*sqrt(3)*b*c^(-n - 3/4)*(-b*c^(-n - 3/4)/a)^(1/3)*arctan(1/3*sqrt(3)*(2*x
^(1/4)*e^(-1/12*(4*n + 3)*log(c) - 1/12*(4*n + 3)*log(x)) + (-b*c^(-n - 3/4)/a)^
(1/3))/(-b*c^(-n - 3/4)/a)^(1/3)) - 4*b*c^(-n - 3/4)*(-b*c^(-n - 3/4)/a)^(1/3)*l
og((x^(1/4)*e^(-1/12*(4*n + 3)*log(c) - 1/12*(4*n + 3)*log(x)) - (-b*c^(-n - 3/4
)/a)^(1/3))/x^(1/4)) + 2*b*c^(-n - 3/4)*(-b*c^(-n - 3/4)/a)^(1/3)*log(((-b*c^(-n
- 3/4)/a)^(1/3)*x^(1/4)*e^(-1/12*(4*n + 3)*log(c) - 1/12*(4*n + 3)*log(x)) + sq
rt(x)*e^(-1/6*(4*n + 3)*log(c) - 1/6*(4*n + 3)*log(x)) + (-b*c^(-n - 3/4)/a)^(2/
3))/sqrt(x)) - 12*b*c^(-n - 3/4)*x^(1/4)*e^(-1/12*(4*n + 3)*log(c) - 1/12*(4*n +
3)*log(x)) + 3*a*x*e^(-1/3*(4*n + 3)*log(c) - 1/3*(4*n + 3)*log(x)))/(a^2*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)**(-1-4/3*n)/(a+b*x**n),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{-\frac{4}{3} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(-4/3*n - 1)/(b*x^n + a),x, algorithm="giac")

[Out]

integrate((c*x)^(-4/3*n - 1)/(b*x^n + a), x)