3.2.24 \(\int \frac {1}{(b x^n)^{2/3}} \, dx\) [124]

3.2.24.1 Optimal result

Integrand size = 9, antiderivative size = 19 \[ \int \frac {1}{\left (b x^n\right )^{2/3}} \, dx=\frac {3 x}{(3-2 n) \left (b x^n\right )^{2/3}} \]

output

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

3.2.24.2 Mathematica [A] (verified)

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

input

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

output

x/((1 - (2*n)/3)*(b*x^n)^(2/3))
 

3.2.24.3 Rubi [A] (verified)

Time = 0.14 (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 = {20, 15}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\left (b x^n\right )^{2/3}} \, dx\)

\(\Big \downarrow \) 20

\(\displaystyle \frac {x^{2 n/3} \int x^{-2 n/3}dx}{\left (b x^n\right )^{2/3}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {3 x}{(3-2 n) \left (b x^n\right )^{2/3}}\)

input

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

output

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

3.2.24.3.1 Defintions of rubi rules used

rule 15

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

rule 20

Int[((a_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a*x^n)^p/x^(n*p)   Int[x^(n* 
p), x], x] /; FreeQ[{a, n, p}, x] &&  !IntegerQ[p]
 

3.2.24.4 Maple [A] (verified)

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

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

input

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

output

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

3.2.24.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (b x^n\right )^{2/3}} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

3.2.24.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (15) = 30\).

Time = 0.70 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.53 \[ \int \frac {1}{\left (b x^n\right )^{2/3}} \, dx=\begin {cases} - \frac {3 x}{2 n \left (b x^{n}\right )^{\frac {2}{3}} - 3 \left (b x^{n}\right )^{\frac {2}{3}}} & \text {for}\: n \neq \frac {3}{2} \\\frac {2 x \log {\left (\sqrt {x} \right )}}{\left (b x^{\frac {3}{2}}\right )^{\frac {2}{3}}} & \text {otherwise} \end {cases} \]

input

integrate(1/(b*x**n)**(2/3),x)
 

output

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

3.2.24.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (b x^n\right )^{2/3}} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(-(2*n)/3>0)', see `assume?` for 
more detai
 

3.2.24.8 Giac [F]

\[ \int \frac {1}{\left (b x^n\right )^{2/3}} \, dx=\int { \frac {1}{\left (b x^{n}\right )^{\frac {2}{3}}} \,d x } \]

input

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

output

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

3.2.24.9 Mupad [B] (verification not implemented)

Time = 5.77 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\left (b x^n\right )^{2/3}} \, dx=-\frac {3\,x^{1-n}\,{\left (b\,x^n\right )}^{1/3}}{b\,\left (2\,n-3\right )} \]

input

int(1/(b*x^n)^(2/3),x)
 

output

-(3*x^(1 - n)*(b*x^n)^(1/3))/(b*(2*n - 3))