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