\(\int \frac {1}{(a+b x)^{2/3}} \, dx\) [409]

Optimal result

Integrand size = 9, antiderivative size = 14 \[ \int \frac {1}{(a+b x)^{2/3}} \, dx=\frac {3 \sqrt [3]{a+b x}}{b} \]

[Out]

3*(b*x+a)^(1/3)/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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \[ \int \frac {1}{(a+b x)^{2/3}} \, dx=\frac {3 \sqrt [3]{a+b x}}{b} \]

[In]

Int[(a + b*x)^(-2/3),x]

[Out]

(3*(a + b*x)^(1/3))/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 \sqrt [3]{a+b x}}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^{2/3}} \, dx=\frac {3 \sqrt [3]{a+b x}}{b} \]

[In]

Integrate[(a + b*x)^(-2/3),x]

[Out]

(3*(a + b*x)^(1/3))/b

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)
derivativedivides	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)
default	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)
trager	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)
risch	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)
pseudoelliptic	\(\frac {3 \left (b x +a \right )^{\frac {1}{3}}}{b}\)	\(13\)

[In]

int(1/(b*x+a)^(2/3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{2/3}} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}}}{b} \]

[In]

integrate(1/(b*x+a)^(2/3),x, algorithm="fricas")

[Out]

3*(b*x + a)^(1/3)/b

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(a+b x)^{2/3}} \, dx=\frac {3 \sqrt [3]{a + b x}}{b} \]

[In]

integrate(1/(b*x+a)**(2/3),x)

[Out]

3*(a + b*x)**(1/3)/b

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{2/3}} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}}}{b} \]

[In]

integrate(1/(b*x+a)^(2/3),x, algorithm="maxima")

[Out]

3*(b*x + a)^(1/3)/b

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{2/3}} \, dx=\frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}}}{b} \]

[In]

integrate(1/(b*x+a)^(2/3),x, algorithm="giac")

[Out]

3*(b*x + a)^(1/3)/b

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^{2/3}} \, dx=\frac {3\,{\left (a+b\,x\right )}^{1/3}}{b} \]

[In]

int(1/(a + b*x)^(2/3),x)

[Out]

(3*(a + b*x)^(1/3))/b