Optimal. Leaf size=55 \[ \frac{1}{2} a^2 x \left (c x^n\right )^{\frac{1}{n}}+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{4} b^2 x \left (c x^n\right )^{3/n} \]
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Rubi [A] time = 0.058905, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{1}{2} a^2 x \left (c x^n\right )^{\frac{1}{n}}+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{4} b^2 x \left (c x^n\right )^{3/n} \]
Antiderivative was successfully verified.
[In] Int[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^2,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ a^{2} x \left (c x^{n}\right )^{- \frac{1}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} x\, dx + \frac{2 a b x \left (c x^{n}\right )^{\frac{2}{n}}}{3} + \frac{b^{2} x \left (c x^{n}\right )^{\frac{3}{n}}}{4} \]
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))**2,x)
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Mathematica [A] time = 0.125991, size = 49, normalized size = 0.89 \[ \frac{1}{12} x \left (c x^n\right )^{\frac{1}{n}} \left (6 a^2+8 a b \left (c x^n\right )^{\frac{1}{n}}+3 b^2 \left (c x^n\right )^{2/n}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^2,x]
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Maple [F] time = 0.04, size = 0, normalized size = 0. \[ \int \sqrt [n]{c{x}^{n}} \left ( a+b\sqrt [n]{c{x}^{n}} \right ) ^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^2,x)
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Maxima [A] time = 1.47739, size = 58, normalized size = 1.05 \[ \frac{1}{4} \, b^{2} c^{\frac{3}{n}} x^{4} + \frac{2}{3} \, a b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{2} c^{\left (\frac{1}{n}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^2*(c*x^n)^(1/n),x, algorithm="maxima")
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Fricas [A] time = 0.216965, size = 58, normalized size = 1.05 \[ \frac{1}{4} \, b^{2} c^{\frac{3}{n}} x^{4} + \frac{2}{3} \, a b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{2} c^{\left (\frac{1}{n}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^2*(c*x^n)^(1/n),x, algorithm="fricas")
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Sympy [A] time = 2.25347, size = 56, normalized size = 1.02 \[ \frac{a^{2} c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} + \frac{2 a b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}}}{3} + \frac{b^{2} c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**n)**(1/n)*(a+b*(c*x**n)**(1/n))**2,x)
[Out]
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GIAC/XCAS [A] time = 0.226952, size = 63, normalized size = 1.15 \[ \frac{1}{4} \, b^{2} x^{4} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{2}{3} \, a b x^{3} e^{\left (\frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{1}{2} \, a^{2} x^{2} e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^2*(c*x^n)^(1/n),x, algorithm="giac")
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