\(\int \frac {1}{\sqrt [3]{-a+b x}} \, dx\) [402]

Optimal result

Integrand size = 11, antiderivative size = 18 \[ \int \frac {1}{\sqrt [3]{-a+b x}} \, dx=\frac {3 (-a+b x)^{2/3}}{2 b} \]

[Out]

3/2*(b*x-a)^(2/3)/b

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {32} \[ \int \frac {1}{\sqrt [3]{-a+b x}} \, dx=\frac {3 (b x-a)^{2/3}}{2 b} \]

[In]

Int[(-a + b*x)^(-1/3),x]

[Out]

(3*(-a + b*x)^(2/3))/(2*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 (-a+b x)^{2/3}}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt [3]{-a+b x}} \, dx=\frac {3 (-a+b x)^{2/3}}{2 b} \]

[In]

Integrate[(-a + b*x)^(-1/3),x]

[Out]

(3*(-a + b*x)^(2/3))/(2*b)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
gosper	\(\frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b}\)	\(15\)
derivativedivides	\(\frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b}\)	\(15\)
default	\(\frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b}\)	\(15\)
trager	\(\frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b}\)	\(15\)
pseudoelliptic	\(\frac {3 \left (b x -a \right )^{\frac {2}{3}}}{2 b}\)	\(15\)
risch	\(-\frac {3 \left (-b x +a \right )}{2 b \left (b x -a \right )^{\frac {1}{3}}}\)	\(21\)

[In]

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

[Out]

3/2*(b*x-a)^(2/3)/b

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt [3]{-a+b x}} \, dx=\frac {3 \, {\left (b x - a\right )}^{\frac {2}{3}}}{2 \, b} \]

[In]

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

[Out]

3/2*(b*x - a)^(2/3)/b

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt [3]{-a+b x}} \, dx=\frac {3 \left (- a + b x\right )^{\frac {2}{3}}}{2 b} \]

[In]

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

[Out]

3*(-a + b*x)**(2/3)/(2*b)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt [3]{-a+b x}} \, dx=\frac {3 \, {\left (b x - a\right )}^{\frac {2}{3}}}{2 \, b} \]

[In]

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

[Out]

3/2*(b*x - a)^(2/3)/b

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt [3]{-a+b x}} \, dx=\frac {3 \, {\left (b x - a\right )}^{\frac {2}{3}}}{2 \, b} \]

[In]

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

[Out]

3/2*(b*x - a)^(2/3)/b

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt [3]{-a+b x}} \, dx=\frac {3\,{\left (b\,x-a\right )}^{2/3}}{2\,b} \]

[In]

int(1/(b*x - a)^(1/3),x)

[Out]

(3*(b*x - a)^(2/3))/(2*b)