3.3023 \(\int \left (a+b \left (c x^n\right )^{2/n}\right ) \, dx\)

Optimal. Leaf size=21 \[ a x+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]

[Out]

a*x + (b*x*(c*x^n)^(2/n))/3

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Rubi [A]  time = 0.0152194, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ a x+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]

Antiderivative was successfully verified.

[In]  Int[a + b*(c*x^n)^(2/n),x]

[Out]

a*x + (b*x*(c*x^n)^(2/n))/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b x \left (c x^{n}\right )^{\frac{2}{n}}}{3} + \int a\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a+b*(c*x**n)**(2/n),x)

[Out]

b*x*(c*x**n)**(2/n)/3 + Integral(a, x)

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Mathematica [A]  time = 0.00339694, size = 21, normalized size = 1. \[ a x+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]

Antiderivative was successfully verified.

[In]  Integrate[a + b*(c*x^n)^(2/n),x]

[Out]

a*x + (b*x*(c*x^n)^(2/n))/3

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Maple [A]  time = 0.029, size = 23, normalized size = 1.1 \[ ax+{\frac{bx}{3}{{\rm e}^{2\,{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a+b*(c*x^n)^(2/n),x)

[Out]

a*x+1/3*x*b*exp(2/n*ln(c*exp(n*ln(x))))

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Maxima [A]  time = 1.41796, size = 23, normalized size = 1.1 \[ \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^n)^(2/n)*b + a,x, algorithm="maxima")

[Out]

1/3*b*c^(2/n)*x^3 + a*x

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Fricas [A]  time = 0.232092, size = 23, normalized size = 1.1 \[ \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^n)^(2/n)*b + a,x, algorithm="fricas")

[Out]

1/3*b*c^(2/n)*x^3 + a*x

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Sympy [A]  time = 0.607316, size = 19, normalized size = 0.9 \[ a x + \frac{b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a+b*(c*x**n)**(2/n),x)

[Out]

a*x + b*c**(2/n)*x*(x**n)**(2/n)/3

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GIAC/XCAS [A]  time = 0.220826, size = 24, normalized size = 1.14 \[ \frac{1}{3} \, b x^{3} e^{\left (\frac{2 \,{\rm ln}\left (c\right )}{n}\right )} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^n)^(2/n)*b + a,x, algorithm="giac")

[Out]

1/3*b*x^3*e^(2*ln(c)/n) + a*x