Optimal. Leaf size=114 \[ \frac{a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}-\frac{2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac{x^4 \left (c x^n\right )^{-2/n}}{2 b^2} \]
[Out]
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Rubi [A] time = 0.103353, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{a^3 x^4 \left (c x^n\right )^{-4/n}}{b^4 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{3 a^2 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}-\frac{2 a x^4 \left (c x^n\right )^{-3/n}}{b^3}+\frac{x^4 \left (c x^n\right )^{-2/n}}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a + b*(c*x^n)^n^(-1))^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{3} x^{4} \left (c x^{n}\right )^{- \frac{4}{n}}}{b^{4} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} + \frac{3 a^{2} x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{4}} - \frac{2 a x^{4} \left (c x^{n}\right )^{- \frac{3}{n}}}{b^{3}} + \frac{x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} x\, dx}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(a+b*(c*x**n)**(1/n))**2,x)
[Out]
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Mathematica [A] time = 4.38211, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x^3/(a + b*(c*x^n)^n^(-1))^2,x]
[Out]
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Maple [C] time = 0.06, size = 662, normalized size = 5.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(a+b*(c*x^n)^(1/n))^2,x)
[Out]
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Maxima [A] time = 22.7162, size = 136, normalized size = 1.19 \[ \frac{x^{4}}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} + \frac{3 \, a^{2} c^{-\frac{4}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{4}} - \frac{{\left (2 \, b^{2} c^{\frac{2}{n}} x^{3} - 3 \, a b c^{\left (\frac{1}{n}\right )} x^{2} + 6 \, a^{2} x\right )} c^{-\frac{3}{n}}}{2 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((c*x^n)^(1/n)*b + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232186, size = 142, normalized size = 1.25 \[ \frac{b^{3} c^{\frac{3}{n}} x^{3} - 3 \, a b^{2} c^{\frac{2}{n}} x^{2} - 4 \, a^{2} b c^{\left (\frac{1}{n}\right )} x + 2 \, a^{3} + 6 \,{\left (a^{2} b c^{\left (\frac{1}{n}\right )} x + a^{3}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \,{\left (b^{5} c^{\frac{5}{n}} x + a b^{4} c^{\frac{4}{n}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((c*x^n)^(1/n)*b + a)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(a+b*(c*x**n)**(1/n))**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((c*x^n)^(1/n)*b + a)^2,x, algorithm="giac")
[Out]