Optimal. Leaf size=101 \[ -\frac{a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}+\frac{a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac{a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac{x^4 \left (c x^n\right )^{-1/n}}{3 b} \]
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Rubi [A] time = 0.0918268, antiderivative size = 101, 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 \[ -\frac{a^3 x^4 \left (c x^n\right )^{-4/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^4}+\frac{a^2 x^4 \left (c x^n\right )^{-3/n}}{b^3}-\frac{a x^4 \left (c x^n\right )^{-2/n}}{2 b^2}+\frac{x^4 \left (c x^n\right )^{-1/n}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a + b*(c*x^n)^n^(-1)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3} x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{4}} - \frac{a x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} x\, dx}{b^{2}} + \frac{x^{4} \left (c x^{n}\right )^{- \frac{1}{n}}}{3 b} + \frac{x^{4} \left (c x^{n}\right )^{- \frac{4}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} a^{2}\, dx}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(a+b*(c*x**n)**(1/n)),x)
[Out]
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Mathematica [A] time = 4.82499, size = 0, normalized size = 0. \[ \int \frac{x^3}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x^3/(a + b*(c*x^n)^n^(-1)),x]
[Out]
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Maple [C] time = 0.279, size = 553, normalized size = 5.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(a+b*(c*x^n)^(1/n)),x)
[Out]
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Maxima [A] time = 22.5987, size = 99, normalized size = 0.98 \[ -\frac{a^{3} c^{-\frac{4}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{4}} + \frac{{\left (2 \, b^{2} c^{\frac{2}{n}} x^{3} - 3 \, a b c^{\left (\frac{1}{n}\right )} x^{2} + 6 \, a^{2} x\right )} c^{-\frac{3}{n}}}{6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((c*x^n)^(1/n)*b + a),x, algorithm="maxima")
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Fricas [A] time = 0.256682, size = 100, normalized size = 0.99 \[ \frac{2 \, b^{3} c^{\frac{3}{n}} x^{3} - 3 \, a b^{2} c^{\frac{2}{n}} x^{2} + 6 \, a^{2} b c^{\left (\frac{1}{n}\right )} x - 6 \, a^{3} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{6 \, b^{4} c^{\frac{4}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((c*x^n)^(1/n)*b + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{a + b \left (c x^{n}\right )^{\frac{1}{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(a+b*(c*x**n)**(1/n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((c*x^n)^(1/n)*b + a),x, algorithm="giac")
[Out]