\(\int \frac {1}{(a+b x^n)^2 (c+d x^n)} \, dx\) [92]

Optimal result

Integrand size = 19, antiderivative size = 122 \[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\frac {b x}{a (b c-a d) n \left (a+b x^n\right )}+\frac {b (a d (1-2 n)-b c (1-n)) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^2 n}+\frac {d^2 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c (b c-a d)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\frac {x \left (\frac {b^2 c-a b d}{a^2 n+a b n x^n}+\frac {b (a d (1-2 n)+b c (-1+n)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2 n}+\frac {d^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c}\right )}{(b c-a d)^2} \] Input:

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

Output:

(x*((b^2*c - a*b*d)/(a^2*n + a*b*n*x^n) + (b*(a*d*(1 - 2*n) + b*c*(-1 + n) 
)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a^2*n) + (d^2*H 
ypergeometric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c))/(b*c - a*d)^2
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {931, 1020, 778}

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.

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

\(\Big \downarrow \) 931

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

\(\Big \downarrow \) 1020

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

\(\Big \downarrow \) 778

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 931

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

rule 1020

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( 
n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^n), x], x 
] - Simp[(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b 
, c, d, e, f, n}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate(1/(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)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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