\(\int \sqrt [3]{b x} \, dx\) [104]

Optimal result

Integrand size = 7, antiderivative size = 14 \[ \int \sqrt [3]{b x} \, dx=\frac {3 (b x)^{4/3}}{4 b} \]

[Out]

3/4*(b*x)^(4/3)/b

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int \sqrt [3]{b x} \, dx=\frac {3 (b x)^{4/3}}{4 b} \]

[In]

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

[Out]

(3*(b*x)^(4/3))/(4*b)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {3 (b x)^{4/3}}{4 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \sqrt [3]{b x} \, dx=\frac {3}{4} x \sqrt [3]{b x} \]

[In]

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

[Out]

(3*x*(b*x)^(1/3))/4

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64

method	result	size
gosper	\(\frac {3 x \left (b x \right )^{\frac {1}{3}}}{4}\)	\(9\)
trager	\(\frac {3 x \left (b x \right )^{\frac {1}{3}}}{4}\)	\(9\)
risch	\(\frac {3 x \left (b x \right )^{\frac {1}{3}}}{4}\)	\(9\)
pseudoelliptic	\(\frac {3 x \left (b x \right )^{\frac {1}{3}}}{4}\)	\(9\)
derivativedivides	\(\frac {3 \left (b x \right )^{\frac {4}{3}}}{4 b}\)	\(11\)
default	\(\frac {3 \left (b x \right )^{\frac {4}{3}}}{4 b}\)	\(11\)

[In]

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

[Out]

3/4*x*(b*x)^(1/3)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \sqrt [3]{b x} \, dx=\frac {3}{4} \, \left (b x\right )^{\frac {1}{3}} x \]

[In]

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

[Out]

3/4*(b*x)^(1/3)*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt [3]{b x} \, dx=\frac {3 \left (b x\right )^{\frac {4}{3}}}{4 b} \]

[In]

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

[Out]

3*(b*x)**(4/3)/(4*b)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt [3]{b x} \, dx=\frac {3 \, \left (b x\right )^{\frac {4}{3}}}{4 \, b} \]

[In]

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

[Out]

3/4*(b*x)^(4/3)/b

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \sqrt [3]{b x} \, dx=\frac {3}{4} \, \left (b x\right )^{\frac {1}{3}} x \]

[In]

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

[Out]

3/4*(b*x)^(1/3)*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt [3]{b x} \, dx=\frac {3\,{\left (b\,x\right )}^{4/3}}{4\,b} \]

[In]

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

[Out]

(3*(b*x)^(4/3))/(4*b)