\(\int \sqrt [3]{b x^n} \, dx\) [101]

Optimal result

Integrand size = 9, antiderivative size = 17 \[ \int \sqrt [3]{b x^n} \, dx=\frac {3 x \sqrt [3]{b x^n}}{3+n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \int \sqrt [3]{b x^n} \, dx=\frac {3 x \sqrt [3]{b x^n}}{n+3} \]

[In]

Int[(b*x^n)^(1/3),x]

[Out]

(3*x*(b*x^n)^(1/3))/(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^{-n/3} \sqrt [3]{b x^n}\right ) \int x^{n/3} \, dx \\ & = \frac {3 x \sqrt [3]{b x^n}}{3+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \sqrt [3]{b x^n} \, dx=\frac {3 x \sqrt [3]{b x^n}}{3+n} \]

[In]

Integrate[(b*x^n)^(1/3),x]

[Out]

(3*x*(b*x^n)^(1/3))/(3 + n)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
gosper	\(\frac {3 x \left (b \,x^{n}\right )^{\frac {1}{3}}}{3+n}\)	\(16\)
risch	\(\frac {3 x \left (b \,x^{n}\right )^{\frac {1}{3}}}{3+n}\)	\(16\)

[In]

int((b*x^n)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \sqrt [3]{b x^n} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((b*x^n)^(1/3),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (14) = 28\).

Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \sqrt [3]{b x^n} \, dx=\begin {cases} \frac {3 x \sqrt [3]{b x^{n}}}{n + 3} & \text {for}\: n \neq -3 \\x \sqrt [3]{\frac {b}{x^{3}}} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((b*x**n)**(1/3),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \sqrt [3]{b x^n} \, dx=\frac {3 \, \left (b x^{n}\right )^{\frac {1}{3}} x}{n + 3} \]

[In]

integrate((b*x^n)^(1/3),x, algorithm="maxima")

[Out]

3*(b*x^n)^(1/3)*x/(n + 3)

Giac [F]

\[ \int \sqrt [3]{b x^n} \, dx=\int { \left (b x^{n}\right )^{\frac {1}{3}} \,d x } \]

[In]

integrate((b*x^n)^(1/3),x, algorithm="giac")

[Out]

integrate((b*x^n)^(1/3), x)

Mupad [B] (verification not implemented)

Time = 5.66 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \sqrt [3]{b x^n} \, dx=\frac {3\,x\,{\left (b\,x^n\right )}^{1/3}}{n+3} \]

[In]

int((b*x^n)^(1/3),x)

[Out]

(3*x*(b*x^n)^(1/3))/(n + 3)