\(\int \frac {1}{x^2 (a+b (c x^n)^{\frac {1}{n}})^2} \, dx\) [116]

Optimal result

Integrand size = 19, antiderivative size = 94 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=-\frac {1}{a^2 x}-\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 b \left (c x^n\right )^{\frac {1}{n}} \log (x)}{a^3 x}+\frac {2 b \left (c x^n\right )^{\frac {1}{n}} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3 x} \] Output:

-1/a^2/x-b*(c*x^n)^(1/n)/a^2/x/(a+b*(c*x^n)^(1/n))-2*b*(c*x^n)^(1/n)*ln(x) 
/a^3/x+2*b*(c*x^n)^(1/n)*ln(a+b*(c*x^n)^(1/n))/a^3/x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=-\frac {\left (c x^n\right )^{\frac {1}{n}} \left (a \left (\left (c x^n\right )^{-1/n}+\frac {b}{a+b \left (c x^n\right )^{\frac {1}{n}}}\right )+2 b \log (x)-2 b \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{a^3 x} \] Input:

Integrate[1/(x^2*(a + b*(c*x^n)^n^(-1))^2),x]
 

Output:

-(((c*x^n)^n^(-1)*(a*((c*x^n)^(-n^(-1)) + b/(a + b*(c*x^n)^n^(-1))) + 2*b* 
Log[x] - 2*b*Log[a + b*(c*x^n)^n^(-1)]))/(a^3*x))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {892, 54, 2009}

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.

\(\displaystyle \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx\)

\(\Big \downarrow \) 892

\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \int \frac {\left (c x^n\right )^{-2/n}}{\left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^2}d\left (c x^n\right )^{\frac {1}{n}}}{x}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \int \left (\frac {\left (c x^n\right )^{-2/n}}{a^2}-\frac {2 b \left (c x^n\right )^{-1/n}}{a^3}+\frac {2 b^2}{a^3 \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )}+\frac {b^2}{a^2 \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^2}\right )d\left (c x^n\right )^{\frac {1}{n}}}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \left (-\frac {2 b \log \left (\left (c x^n\right )^{\frac {1}{n}}\right )}{a^3}+\frac {2 b \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{a^3}-\frac {b}{a^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {\left (c x^n\right )^{-1/n}}{a^2}\right )}{x}\)

Input:

Int[1/(x^2*(a + b*(c*x^n)^n^(-1))^2),x]
 

Output:

((c*x^n)^n^(-1)*(-(1/(a^2*(c*x^n)^n^(-1))) - b/(a^2*(a + b*(c*x^n)^n^(-1)) 
) - (2*b*Log[(c*x^n)^n^(-1)])/a^3 + (2*b*Log[a + b*(c*x^n)^n^(-1)])/a^3))/ 
x
 

Defintions of rubi rules used

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 892

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> Simp[(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1))   Subst[Int[x^m*(a + b 
*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x 
] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.19 (sec) , antiderivative size = 296, normalized size of antiderivative = 3.15

method	result	size
risch	\(\frac {1}{a x \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}-\frac {2}{a^{2} x}-\frac {2 b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}} \ln \left (x \right )}{a^{3} x}+\frac {2 \ln \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{a^{3} x}\)	\(296\)

Input:

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

Output:

1/a/x/(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn 
(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)+a)-2/a^2/x-2/a^3*b/x*(x^n)^(1/n)*c 
^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)- 
csgn(I*c*x^n))/n)*ln(x)+2/a^3*ln(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*I*Pi*csgn(I 
*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)+a)*c^(1/ 
n)*(x^n)^(1/n)/x*b*exp(1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n)) 
*(csgn(I*c)-csgn(I*c*x^n))/n)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=-\frac {2 \, b^{2} c^{\frac {2}{n}} x^{2} \log \left (x\right ) + a^{2} + 2 \, {\left (a b x \log \left (x\right ) + a b x\right )} c^{\left (\frac {1}{n}\right )} - 2 \, {\left (b^{2} c^{\frac {2}{n}} x^{2} + a b c^{\left (\frac {1}{n}\right )} x\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{a^{3} b c^{\left (\frac {1}{n}\right )} x^{2} + a^{4} x} \] Input:

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

Output:

-(2*b^2*c^(2/n)*x^2*log(x) + a^2 + 2*(a*b*x*log(x) + a*b*x)*c^(1/n) - 2*(b 
^2*c^(2/n)*x^2 + a*b*c^(1/n)*x)*log(b*c^(1/n)*x + a))/(a^3*b*c^(1/n)*x^2 + 
 a^4*x)
 

Sympy [F]

\[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {1}{x^{2} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \] Input:

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

Output:

Integral(1/(x**2*(a + b*(c*x**n)**(1/n))**2), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2} x^{2}} \,d x } \] Input:

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

Output:

1/(a*b*c^(1/n)*x*(x^n)^(1/n) + a^2*x) + 2*integrate(1/(a*b*c^(1/n)*x^2*(x^ 
n)^(1/n) + a^2*x^2), x)
 

Giac [F]

\[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2} x^{2}} \,d x } \] Input:

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

Output:

integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \] Input:

int(1/(x^2*(a + b*(c*x^n)^(1/n))^2),x)
 

Output:

int(1/(x^2*(a + b*(c*x^n)^(1/n))^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {2 c^{\frac {2}{n}} \mathrm {log}\left (c^{\frac {1}{n}} b x +a \right ) b^{2} x^{2}-2 c^{\frac {2}{n}} \mathrm {log}\left (x \right ) b^{2} x^{2}+2 c^{\frac {2}{n}} b^{2} x^{2}+2 c^{\frac {1}{n}} \mathrm {log}\left (c^{\frac {1}{n}} b x +a \right ) a b x -2 c^{\frac {1}{n}} \mathrm {log}\left (x \right ) a b x -a^{2}}{a^{3} x \left (c^{\frac {1}{n}} b x +a \right )} \] Input:

int(1/x^2/(a+b*(c*x^n)^(1/n))^2,x)
 

Output:

(2*c**(2/n)*log(c**(1/n)*b*x + a)*b**2*x**2 - 2*c**(2/n)*log(x)*b**2*x**2 
+ 2*c**(2/n)*b**2*x**2 + 2*c**(1/n)*log(c**(1/n)*b*x + a)*a*b*x - 2*c**(1/ 
n)*log(x)*a*b*x - a**2)/(a**3*x*(c**(1/n)*b*x + a))