3.3004 \(\int (a+b (c x^n)^{\frac {1}{n}})^2 \, dx\)

Optimal. Leaf size=34 \[ \frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}{3 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {254, 32} \[ \frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 254

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps

\begin {align*} \int \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}{3 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 38, normalized size = 1.12 \[ a^2 x+a b x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{3} b^2 x \left (c x^n\right )^{2/n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.89, size = 32, normalized size = 0.94 \[ \frac {1}{3} \, b^{2} c^{\frac {2}{n}} x^{3} + a b c^{\left (\frac {1}{n}\right )} x^{2} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^n)^(1/n))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.19, size = 32, normalized size = 0.94 \[ \frac {1}{3} \, b^{2} c^{\frac {2}{n}} x^{3} + a b c^{\left (\frac {1}{n}\right )} x^{2} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^n)^(1/n))^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b^{2} c^{\frac {2}{n}} \int {\left (x^{n}\right )}^{\frac {2}{n}}\,{d x} + 2 \, a b c^{\left (\frac {1}{n}\right )} \int {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}\,{d x} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^n)^(1/n))^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.23, size = 36, normalized size = 1.06 \[ a^2\,x+\frac {b^2\,x\,{\left (c\,x^n\right )}^{2/n}}{3}+a\,b\,x\,{\left (c\,x^n\right )}^{1/n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c*x^n)^(1/n))^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.43, size = 39, normalized size = 1.15 \[ a^{2} x + a b c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}} + \frac {b^{2} c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________