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]

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

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Rubi [A]  time = 0.0087369, antiderivative size = 34, normalized size of antiderivative = 1., 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 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)]

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]

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 = 0.0133179, 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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sqrt [n]{c{x}^{n}} \right ) ^{2}\, dx \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 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 \end{align*}

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

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Fricas [A]  time = 1.25701, size = 63, normalized size = 1.85 \begin{align*} \frac{1}{3} \, b^{2} c^{\frac{2}{n}} x^{3} + a b c^{\left (\frac{1}{n}\right )} x^{2} + a^{2} x \end{align*}

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

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Sympy [A]  time = 0.441378, size = 39, normalized size = 1.15 \begin{align*} 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} \end{align*}

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

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Giac [A]  time = 1.10464, size = 43, normalized size = 1.26 \begin{align*} \frac{1}{3} \, b^{2} c^{\frac{2}{n}} x^{3} + a b c^{\left (\frac{1}{n}\right )} x^{2} + a^{2} x \end{align*}

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