3.3034 \(\int \left (a+b \left (c x^n\right )^{3/n}\right )^2 \, dx\)

Optimal. Leaf size=43 \[ a^2 x+\frac{1}{2} a b x \left (c x^n\right )^{3/n}+\frac{1}{7} b^2 x \left (c x^n\right )^{6/n} \]

[Out]

a^2*x + (a*b*x*(c*x^n)^(3/n))/2 + (b^2*x*(c*x^n)^(6/n))/7

_______________________________________________________________________________________

Rubi [A]  time = 0.0351556, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ a^2 x+\frac{1}{2} a b x \left (c x^n\right )^{3/n}+\frac{1}{7} b^2 x \left (c x^n\right )^{6/n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^n)^(3/n))^2,x]

[Out]

a^2*x + (a*b*x*(c*x^n)^(3/n))/2 + (b^2*x*(c*x^n)^(6/n))/7

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a b x \left (c x^{n}\right )^{\frac{3}{n}}}{2} + \frac{b^{2} x \left (c x^{n}\right )^{\frac{6}{n}}}{7} + x \left (c x^{n}\right )^{- \frac{1}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**n)**(3/n))**2,x)

[Out]

a*b*x*(c*x**n)**(3/n)/2 + b**2*x*(c*x**n)**(6/n)/7 + x*(c*x**n)**(-1/n)*Integral
(a**2, (x, (c*x**n)**(1/n)))

_______________________________________________________________________________________

Mathematica [A]  time = 0.124356, size = 43, normalized size = 1. \[ a^2 x+\frac{1}{2} a b x \left (c x^n\right )^{3/n}+\frac{1}{7} b^2 x \left (c x^n\right )^{6/n} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*(c*x^n)^(3/n))^2,x]

[Out]

a^2*x + (a*b*x*(c*x^n)^(3/n))/2 + (b^2*x*(c*x^n)^(6/n))/7

_______________________________________________________________________________________

Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{3\,{n}^{-1}} \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^n)^(3/n))^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.44237, size = 47, normalized size = 1.09 \[ \frac{1}{7} \, b^{2} c^{\frac{6}{n}} x^{7} + \frac{1}{2} \, a b c^{\frac{3}{n}} x^{4} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(3/n)*b + a)^2,x, algorithm="maxima")

[Out]

1/7*b^2*c^(6/n)*x^7 + 1/2*a*b*c^(3/n)*x^4 + a^2*x

_______________________________________________________________________________________

Fricas [A]  time = 0.247586, size = 47, normalized size = 1.09 \[ \frac{1}{7} \, b^{2} c^{\frac{6}{n}} x^{7} + \frac{1}{2} \, a b c^{\frac{3}{n}} x^{4} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(3/n)*b + a)^2,x, algorithm="fricas")

[Out]

1/7*b^2*c^(6/n)*x^7 + 1/2*a*b*c^(3/n)*x^4 + a^2*x

_______________________________________________________________________________________

Sympy [A]  time = 1.24158, size = 41, normalized size = 0.95 \[ a^{2} x + \frac{a b c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{2} + \frac{b^{2} c^{\frac{6}{n}} x \left (x^{n}\right )^{\frac{6}{n}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**n)**(3/n))**2,x)

[Out]

a**2*x + a*b*c**(3/n)*x*(x**n)**(3/n)/2 + b**2*c**(6/n)*x*(x**n)**(6/n)/7

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222746, size = 50, normalized size = 1.16 \[ \frac{1}{7} \, b^{2} x^{7} e^{\left (\frac{6 \,{\rm ln}\left (c\right )}{n}\right )} + \frac{1}{2} \, a b x^{4} e^{\left (\frac{3 \,{\rm ln}\left (c\right )}{n}\right )} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(3/n)*b + a)^2,x, algorithm="giac")

[Out]

1/7*b^2*x^7*e^(6*ln(c)/n) + 1/2*a*b*x^4*e^(3*ln(c)/n) + a^2*x