3.2996 \(\int \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right ) \, dx\)

Optimal. Leaf size=19 \[ a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}} \]

[Out]

a*x + (b*x*(c*x^n)^n^(-1))/2

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Rubi [A]  time = 0.0147688, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}} \]

Antiderivative was successfully verified.

[In]  Int[a + b*(c*x^n)^n^(-1),x]

[Out]

a*x + (b*x*(c*x^n)^n^(-1))/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b \left (c x^{n}\right )^{\frac{1}{n}} \int x\, dx}{x} + \int a\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a+b*(c*x**n)**(1/n),x)

[Out]

b*(c*x**n)**(1/n)*Integral(x, x)/x + Integral(a, x)

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Mathematica [A]  time = 0.00965197, size = 19, normalized size = 1. \[ a x+\frac{1}{2} b x \left (c x^n\right )^{\frac{1}{n}} \]

Antiderivative was successfully verified.

[In]  Integrate[a + b*(c*x^n)^n^(-1),x]

[Out]

a*x + (b*x*(c*x^n)^n^(-1))/2

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Maple [A]  time = 0.029, size = 22, normalized size = 1.2 \[ ax+{\frac{bx}{2}{{\rm e}^{{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{n}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a+b*(c*x^n)^(1/n),x)

[Out]

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

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Maxima [A]  time = 1.42433, size = 20, normalized size = 1.05 \[ \frac{1}{2} \, b c^{\left (\frac{1}{n}\right )} x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^n)^(1/n)*b + a,x, algorithm="maxima")

[Out]

1/2*b*c^(1/n)*x^2 + a*x

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Fricas [A]  time = 0.232505, size = 20, normalized size = 1.05 \[ \frac{1}{2} \, b c^{\left (\frac{1}{n}\right )} x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^n)^(1/n)*b + a,x, algorithm="fricas")

[Out]

1/2*b*c^(1/n)*x^2 + a*x

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Sympy [A]  time = 0.614183, size = 19, normalized size = 1. \[ a x + \frac{b c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a+b*(c*x**n)**(1/n),x)

[Out]

a*x + b*c**(1/n)*x*(x**n)**(1/n)/2

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GIAC/XCAS [A]  time = 0.225456, size = 23, normalized size = 1.21 \[ \frac{1}{2} \, b x^{2} e^{\left (\frac{{\rm ln}\left (c\right )}{n}\right )} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^n)^(1/n)*b + a,x, algorithm="giac")

[Out]

1/2*b*x^2*e^(ln(c)/n) + a*x