\(\int (d x^3)^n \, dx\) [28]

Optimal result

Integrand size = 7, antiderivative size = 16 \[ \int \left (d x^3\right )^n \, dx=\frac {x \left (d x^3\right )^n}{1+3 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {15, 30} \[ \int \left (d x^3\right )^n \, dx=\frac {x \left (d x^3\right )^n}{3 n+1} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (d x^3\right )^n \, dx=\frac {x \left (d x^3\right )^n}{1+3 n} \]

[In]

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

[Out]

(x*(d*x^3)^n)/(1 + 3*n)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
gosper	\(\frac {x \left (x^{3} d \right )^{n}}{1+3 n}\)	\(17\)
risch	\(\frac {x \left (x^{3} d \right )^{n}}{1+3 n}\)	\(17\)
parallelrisch	\(\frac {x \left (x^{3} d \right )^{n}}{1+3 n}\)	\(17\)
norman	\(\frac {x \,{\mathrm e}^{n \ln \left (x^{3} d \right )}}{1+3 n}\)	\(19\)

[In]

int((x^3*d)^n,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (d x^3\right )^n \, dx=\frac {\left (d x^{3}\right )^{n} x}{3 \, n + 1} \]

[In]

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

[Out]

(d*x^3)^n*x/(3*n + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \left (d x^3\right )^n \, dx=\begin {cases} \frac {x \left (d x^{3}\right )^{n}}{3 n + 1} & \text {for}\: n \neq - \frac {1}{3} \\\frac {x \log {\left (x \right )}}{\sqrt [3]{d x^{3}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x*(d*x**3)**n/(3*n + 1), Ne(n, -1/3)), (x*log(x)/(d*x**3)**(1/3), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (d x^3\right )^n \, dx=\frac {d^{n} x x^{3 \, n}}{3 \, n + 1} \]

[In]

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

[Out]

d^n*x*x^(3*n)/(3*n + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (d x^3\right )^n \, dx=\frac {\left (d x^{3}\right )^{n} x}{3 \, n + 1} \]

[In]

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

[Out]

(d*x^3)^n*x/(3*n + 1)

Mupad [B] (verification not implemented)

Time = 9.84 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (d x^3\right )^n \, dx=\frac {x\,{\left (d\,x^3\right )}^n}{3\,n+1} \]

[In]

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

[Out]

(x*(d*x^3)^n)/(3*n + 1)