Optimal. Leaf size=33 \[ \frac{1}{2} a x \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]
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Rubi [A] time = 0.0308211, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{1}{2} a x \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]
Antiderivative was successfully verified.
[In] Int[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1)),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ a x \left (c x^{n}\right )^{- \frac{1}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} x\, dx + \frac{b x \left (c x^{n}\right )^{\frac{2}{n}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**n)**(1/n)*(a+b*(c*x**n)**(1/n)),x)
[Out]
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Mathematica [A] time = 0.0574741, size = 30, normalized size = 0.91 \[ \frac{1}{6} x \left (c x^n\right )^{\frac{1}{n}} \left (3 a+2 b \left (c x^n\right )^{\frac{1}{n}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1)),x]
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Maple [F] time = 0.026, size = 0, normalized size = 0. \[ \int \sqrt [n]{c{x}^{n}} \left ( a+b\sqrt [n]{c{x}^{n}} \right ) \, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n)),x)
[Out]
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Maxima [A] time = 1.50243, size = 34, normalized size = 1.03 \[ \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a c^{\left (\frac{1}{n}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)*(c*x^n)^(1/n),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215891, size = 34, normalized size = 1.03 \[ \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a c^{\left (\frac{1}{n}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)*(c*x^n)^(1/n),x, algorithm="fricas")
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Sympy [A] time = 1.16224, size = 32, normalized size = 0.97 \[ \frac{a c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} + \frac{b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**n)**(1/n)*(a+b*(c*x**n)**(1/n)),x)
[Out]
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GIAC/XCAS [A] time = 0.221561, size = 38, normalized size = 1.15 \[ \frac{1}{3} \, b x^{3} e^{\left (\frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{1}{2} \, a x^{2} e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)*(c*x^n)^(1/n),x, algorithm="giac")
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