\(\int (b x^n)^{2/3} \, dx\) [109]

Optimal result

Integrand size = 9, antiderivative size = 19 \[ \int \left (b x^n\right )^{2/3} \, dx=\frac {3 x \left (b x^n\right )^{2/3}}{3+2 n} \]

[Out]

3*x*(b*x^n)^(2/3)/(3+2*n)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, 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} \[ \int \left (b x^n\right )^{2/3} \, dx=\frac {3 x \left (b x^n\right )^{2/3}}{2 n+3} \]

[In]

Int[(b*x^n)^(2/3),x]

[Out]

(3*x*(b*x^n)^(2/3))/(3 + 2*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^{-2 n/3} \left (b x^n\right )^{2/3}\right ) \int x^{2 n/3} \, dx \\ & = \frac {3 x \left (b x^n\right )^{2/3}}{3+2 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \left (b x^n\right )^{2/3} \, dx=\frac {x \left (b x^n\right )^{2/3}}{1+\frac {2 n}{3}} \]

[In]

Integrate[(b*x^n)^(2/3),x]

[Out]

(x*(b*x^n)^(2/3))/(1 + (2*n)/3)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
gosper	\(\frac {3 x \left (b \,x^{n}\right )^{\frac {2}{3}}}{3+2 n}\)	\(18\)
risch	\(\frac {3 b x \,x^{n}}{\left (3+2 n \right ) \left (b \,x^{n}\right )^{\frac {1}{3}}}\)	\(22\)

[In]

int((b*x^n)^(2/3),x,method=_RETURNVERBOSE)

[Out]

3*x*(b*x^n)^(2/3)/(3+2*n)

Fricas [F(-2)]

Exception generated. \[ \int \left (b x^n\right )^{2/3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((b*x^n)^(2/3),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. 39 vs. \(2 (15) = 30\).

Time = 1.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \left (b x^n\right )^{2/3} \, dx=\begin {cases} \frac {3 x \left (b x^{n}\right )^{\frac {2}{3}}}{2 n + 3} & \text {for}\: n \neq - \frac {3}{2} \\2 x \left (\frac {b}{x^{\frac {3}{2}}}\right )^{\frac {2}{3}} \log {\left (\sqrt {x} \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((b*x**n)**(2/3),x)

[Out]

Piecewise((3*x*(b*x**n)**(2/3)/(2*n + 3), Ne(n, -3/2)), (2*x*(b/x**(3/2))**(2/3)*log(sqrt(x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \left (b x^n\right )^{2/3} \, dx=\frac {3 \, \left (b x^{n}\right )^{\frac {2}{3}} x}{2 \, n + 3} \]

[In]

integrate((b*x^n)^(2/3),x, algorithm="maxima")

[Out]

3*(b*x^n)^(2/3)*x/(2*n + 3)

Giac [F]

\[ \int \left (b x^n\right )^{2/3} \, dx=\int { \left (b x^{n}\right )^{\frac {2}{3}} \,d x } \]

[In]

integrate((b*x^n)^(2/3),x, algorithm="giac")

[Out]

integrate((b*x^n)^(2/3), x)

Mupad [B] (verification not implemented)

Time = 5.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \left (b x^n\right )^{2/3} \, dx=\frac {3\,x\,{\left (b\,x^n\right )}^{2/3}}{2\,n+3} \]

[In]

int((b*x^n)^(2/3),x)

[Out]

(3*x*(b*x^n)^(2/3))/(2*n + 3)