\(\int \frac {1}{x^2 (a+b x^n)^2 (c+d x^n)} \, dx\) [446]

Optimal result

Integrand size = 22, antiderivative size = 142 \[ \int \frac {1}{x^2 \left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\frac {b}{a (b c-a d) n x \left (a+b x^n\right )}-\frac {b (b c (1+n)-a d (1+2 n)) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^2 n x}-\frac {d^2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {d x^n}{c}\right )}{c (b c-a d)^2 x} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 \left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\frac {b c (-b c (1+n)+a d (1+2 n)) \left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},\frac {-1+n}{n},-\frac {b x^n}{a}\right )-a \left (b c (-b c+a d)+a d^2 n \left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},\frac {-1+n}{n},-\frac {d x^n}{c}\right )\right )}{a^2 c (b c-a d)^2 n x \left (a+b x^n\right )} \] Input:

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

Output:

(b*c*(-(b*c*(1 + n)) + a*d*(1 + 2*n))*(a + b*x^n)*Hypergeometric2F1[1, -n^ 
(-1), (-1 + n)/n, -((b*x^n)/a)] - a*(b*c*(-(b*c) + a*d) + a*d^2*n*(a + b*x 
^n)*Hypergeometric2F1[1, -n^(-1), (-1 + n)/n, -((d*x^n)/c)]))/(a^2*c*(b*c 
- a*d)^2*n*x*(a + b*x^n))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1006, 1067, 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.

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

\(\Big \downarrow \) 1006

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

\(\Big \downarrow \) 1067

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b}{a n x (b c-a d) \left (a+b x^n\right )}-\frac {\frac {a d^2 n \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {d x^n}{c}\right )}{c x (b c-a d)}+\frac {b (b c (n+1)-a d (2 n+1)) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{a x (b c-a d)}}{a n (b c-a d)}\)

Input:

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

Output:

b/(a*(b*c - a*d)*n*x*(a + b*x^n)) - ((b*(b*c*(1 + n) - a*d*(1 + 2*n))*Hype 
rgeometric2F1[1, -n^(-1), -((1 - n)/n), -((b*x^n)/a)])/(a*(b*c - a*d)*x) + 
 (a*d^2*n*Hypergeometric2F1[1, -n^(-1), -((1 - n)/n), -((d*x^n)/c)])/(c*(b 
*c - a*d)*x))/(a*(b*c - a*d)*n)
 

Defintions of rubi rules used

rule 1006

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x 
^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 
 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( 
b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ 
a, b, c, d, e, m, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomi 
alQ[a, b, c, d, e, m, n, p, q, x]
 

rule 1067

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, n, p}, x]
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(1/(b^2*d*x^2*x^(3*n) + a^2*c*x^2 + (b^2*c + 2*a*b*d)*x^2*x^(2*n) 
+ (2*a*b*c + a^2*d)*x^2*x^n), x)
 

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

d^2*integrate(1/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^2*x^n + (b^2*c^3 - 
2*a*b*c^2*d + a^2*c*d^2)*x^2), x) - (a*b*d*(2*n + 1) - b^2*c*(n + 1))*inte 
grate(1/((a*b^3*c^2*n - 2*a^2*b^2*c*d*n + a^3*b*d^2*n)*x^2*x^n + (a^2*b^2* 
c^2*n - 2*a^3*b*c*d*n + a^4*d^2*n)*x^2), x) + b/((a*b^2*c*n - a^2*b*d*n)*x 
*x^n + (a^2*b*c*n - a^3*d*n)*x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\int \frac {1}{x^{3 n} b^{2} d \,x^{2}+2 x^{2 n} a b d \,x^{2}+x^{2 n} b^{2} c \,x^{2}+x^{n} a^{2} d \,x^{2}+2 x^{n} a b c \,x^{2}+a^{2} c \,x^{2}}d x \] Input:

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

Output:

int(1/(x**(3*n)*b**2*d*x**2 + 2*x**(2*n)*a*b*d*x**2 + x**(2*n)*b**2*c*x**2 
 + x**n*a**2*d*x**2 + 2*x**n*a*b*c*x**2 + a**2*c*x**2),x)