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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x \left (d x^3\right )^n}{3 n+1} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x \left (d x^3\right )^n}{1+3 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(d*x^3)^n)/(1 + 3*n)

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Maple [A]
time = 0.01, size = 17, normalized size = 1.06

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 17, normalized size = 1.06 \begin {gather*} \frac {d^{n} x x^{3 \, n}}{3 \, n + 1} \end {gather*}

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)

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Fricas [A]
time = 0.39, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (d x^{3}\right )^{n} x}{3 \, n + 1} \end {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x \left (d x^{3}\right )^{n}}{3 n + 1} & \text {for}\: n \neq - \frac {1}{3} \\\int \frac {1}{\sqrt [3]{d x^{3}}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.42, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (d x^{3}\right )^{n} x}{3 \, n + 1} \end {gather*}

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)

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Mupad [B]
time = 2.50, size = 16, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (d\,x^3\right )}^n}{3\,n+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(d*x^3)^n)/(3*n + 1)

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