3.28 \(\int (d x^3)^n \, dx\)

Optimal. Leaf size=16 \[ \frac{x \left (d x^3\right )^n}{3 n+1} \]

[Out]

(x*(d*x^3)^n)/(1 + 3*n)

________________________________________________________________________________________

Rubi [A]  time = 0.0040014, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {15, 30} \[ \frac{x \left (d x^3\right )^n}{3 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(d*x^3)^n,x]

[Out]

(x*(d*x^3)^n)/(1 + 3*n)

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 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 (d x^3\right )^n \, dx &=\left (x^{-3 n} \left (d x^3\right )^n\right ) \int x^{3 n} \, dx\\ &=\frac{x \left (d x^3\right )^n}{1+3 n}\\ \end{align*}

Mathematica [A]  time = 0.0018816, size = 16, normalized size = 1. \[ \frac{x \left (d x^3\right )^n}{3 n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x^3)^n,x]

[Out]

(x*(d*x^3)^n)/(1 + 3*n)

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 17, normalized size = 1.1 \begin{align*}{\frac{x \left ( d{x}^{3} \right ) ^{n}}{1+3\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3)^n,x)

[Out]

x*(d*x^3)^n/(1+3*n)

________________________________________________________________________________________

Maxima [A]  time = 1.13058, size = 23, normalized size = 1.44 \begin{align*} \frac{d^{n} x x^{3 \, n}}{3 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3)^n,x, algorithm="maxima")

[Out]

d^n*x*x^(3*n)/(3*n + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.37066, size = 31, normalized size = 1.94 \begin{align*} \frac{\left (d x^{3}\right )^{n} x}{3 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3)^n,x, algorithm="fricas")

[Out]

(d*x^3)^n*x/(3*n + 1)

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3)**n,x)

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [A]  time = 1.1404, size = 22, normalized size = 1.38 \begin{align*} \frac{\left (d x^{3}\right )^{n} x}{3 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3)^n,x, algorithm="giac")

[Out]

(d*x^3)^n*x/(3*n + 1)