Optimal. Leaf size=210 \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )} \]
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Rubi [A] time = 0.191393, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ -\frac{x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac{2 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}+\frac{x}{3 a \left (a+b \left (c x^n\right )^{3/n}\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(c*x^n)^(3/n))^(-2),x]
[Out]
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Rubi in Sympy [A] time = 32.9159, size = 185, normalized size = 0.88 \[ \frac{x}{3 a \left (a + b \left (c x^{n}\right )^{\frac{3}{n}}\right )} + \frac{2 x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}} + b^{\frac{2}{3}} \left (c x^{n}\right )^{\frac{2}{n}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{2 \sqrt{3} x \left (c x^{n}\right )^{- \frac{1}{n}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \left (c x^{n}\right )^{\frac{1}{n}}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*(c*x**n)**(3/n))**2,x)
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Mathematica [A] time = 4.26296, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (c x^n\right )^{3/n}\right )^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(a + b*(c*x^n)^(3/n))^(-2),x]
[Out]
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Maple [F] time = 0.658, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{-2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*(c*x^n)^(3/n))^2,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^n)^(3/n)*b + a)^(-2),x, algorithm="maxima")
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Fricas [A] time = 0.238991, size = 294, normalized size = 1.4 \[ \frac{6 \,{\left (b c^{\frac{3}{n}} x^{3} + a\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) -{\left (\sqrt{3} b c^{\frac{3}{n}} x^{3} + \sqrt{3} a\right )} \log \left (\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} a x + a^{2}\right ) + 2 \,{\left (\sqrt{3} b c^{\frac{3}{n}} x^{3} + \sqrt{3} a\right )} \log \left (\left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x + a\right ) + 3 \, \sqrt{3} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}} x}{9 \,{\left (\sqrt{3} a b c^{\frac{3}{n}} x^{3} + \sqrt{3} a^{2}\right )} \left (a^{2} b c^{\frac{3}{n}}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(3/n)*b + a)^(-2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b \left (c x^{n}\right )^{\frac{3}{n}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*(c*x**n)**(3/n))**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x^{n}\right )^{\frac{3}{n}} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(3/n)*b + a)^(-2),x, algorithm="giac")
[Out]