3.3073 \(\int (c x^n)^{\frac{1}{n}} (a+b (c x^n)^{\frac{1}{n}}) \, dx\)

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

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Rubi [A]  time = 0.0141981, 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, Rules used = {15, 14, 30} \[ \frac{1}{2} a x \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{3} b x \left (c x^n\right )^{2/n} \]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0110046, 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

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int \sqrt [n]{c{x}^{n}} \left ( a+b\sqrt [n]{c{x}^{n}} \right ) \, dx \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )} \left (c x^{n}\right )^{\left (\frac{1}{n}\right )}\,{d x} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.65355, size = 53, normalized size = 1.61 \begin{align*} \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a c^{\left (\frac{1}{n}\right )} x^{2} \end{align*}

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, algorithm="fricas")

[Out]

1/3*b*c^(2/n)*x^3 + 1/2*a*c^(1/n)*x^2

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Sympy [A]  time = 0.393404, size = 32, normalized size = 0.97 \begin{align*} \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} \end{align*}

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

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Giac [A]  time = 1.29521, size = 34, normalized size = 1.03 \begin{align*} \frac{1}{3} \, b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a c^{\left (\frac{1}{n}\right )} x^{2} \end{align*}

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, algorithm="giac")

[Out]

1/3*b*c^(2/n)*x^3 + 1/2*a*c^(1/n)*x^2