3.3066 \(\int \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right ) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{2} a x \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]

[Out]

(a*x*(c*x^n)^n^(-1))/2 + (b*x*(c*x^n)^(2/n))/3

_______________________________________________________________________________________

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]

[Out]

(a*x*(c*x^n)^n^(-1))/2 + (b*x*(c*x^n)^(2/n))/3

_______________________________________________________________________________________

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]

a*x*(c*x**n)**(-1/n)*Integral(x, (x, (c*x**n)**(1/n))) + b*x*(c*x**n)**(2/n)/3

_______________________________________________________________________________________

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]

[Out]

(x*(c*x^n)^n^(-1)*(3*a + 2*b*(c*x^n)^n^(-1)))/6

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

1/3*b*c^(2/n)*x^3 + 1/2*a*c^(1/n)*x^2

_______________________________________________________________________________________

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")

[Out]

1/3*b*c^(2/n)*x^3 + 1/2*a*c^(1/n)*x^2

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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")

[Out]

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