3.3065 \(\int \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2 \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} a^2 x \left (c x^n\right )^{\frac{1}{n}}+\frac{2}{3} a b x \left (c x^n\right )^{2/n}+\frac{1}{4} b^2 x \left (c x^n\right )^{3/n} \]

[Out]

(a^2*x*(c*x^n)^n^(-1))/2 + (2*a*b*x*(c*x^n)^(2/n))/3 + (b^2*x*(c*x^n)^(3/n))/4

_______________________________________________________________________________________

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]

[Out]

(a^2*x*(c*x^n)^n^(-1))/2 + (2*a*b*x*(c*x^n)^(2/n))/3 + (b^2*x*(c*x^n)^(3/n))/4

_______________________________________________________________________________________

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)

[Out]

a**2*x*(c*x**n)**(-1/n)*Integral(x, (x, (c*x**n)**(1/n))) + 2*a*b*x*(c*x**n)**(2
/n)/3 + b**2*x*(c*x**n)**(3/n)/4

_______________________________________________________________________________________

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]

[Out]

(x*(c*x^n)^n^(-1)*(6*a^2 + 8*a*b*(c*x^n)^n^(-1) + 3*b^2*(c*x^n)^(2/n)))/12

_______________________________________________________________________________________

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)

[Out]

int((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^2,x)

_______________________________________________________________________________________

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")

[Out]

1/4*b^2*c^(3/n)*x^4 + 2/3*a*b*c^(2/n)*x^3 + 1/2*a^2*c^(1/n)*x^2

_______________________________________________________________________________________

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")

[Out]

1/4*b^2*c^(3/n)*x^4 + 2/3*a*b*c^(2/n)*x^3 + 1/2*a^2*c^(1/n)*x^2

_______________________________________________________________________________________

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]

a**2*c**(1/n)*x*(x**n)**(1/n)/2 + 2*a*b*c**(2/n)*x*(x**n)**(2/n)/3 + b**2*c**(3/
n)*x*(x**n)**(3/n)/4

_______________________________________________________________________________________

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")

[Out]

1/4*b^2*x^4*e^(3*ln(c)/n) + 2/3*a*b*x^3*e^(2*ln(c)/n) + 1/2*a^2*x^2*e^(ln(c)/n)