Integrand size = 19, antiderivative size = 193 \[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\frac {d (b c+a d) x}{a c (b c-a d)^2 n \left (c+d x^n\right )}+\frac {b x}{a (b c-a d) n \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac {b^2 (a d (1-3 n)-b (c-c n)) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^3 n}-\frac {d^2 (b c (1-3 n)-a d (1-n)) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^2 (b c-a d)^3 n} \]
d*(a*d+b*c)*x/a/c/(-a*d+b*c)^2/n/(c+d*x^n)+b*x/a/(-a*d+b*c)/n/(a+b*x^n)/(c +d*x^n)+b^2*(a*d*(1-3*n)-b*(-c*n+c))*x*hypergeom([1, 1/n],[1+1/n],-b*x^n/a )/a^2/(-a*d+b*c)^3/n-d^2*(b*c*(1-3*n)-a*d*(1-n))*x*hypergeom([1, 1/n],[1+1 /n],-d*x^n/c)/c^2/(-a*d+b*c)^3/n
Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\frac {x \left (\frac {b^2 (b c-a d)}{a \left (a+b x^n\right )}+\frac {d^2 (b c-a d)}{c \left (c+d x^n\right )}+\frac {b^2 (a d (1-3 n)+b c (-1+n)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2}+\frac {d^2 (-a d (-1+n)+b c (-1+3 n)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c^2}\right )}{(b c-a d)^3 n} \]
(x*((b^2*(b*c - a*d))/(a*(a + b*x^n)) + (d^2*(b*c - a*d))/(c*(c + d*x^n)) + (b^2*(a*d*(1 - 3*n) + b*c*(-1 + n))*Hypergeometric2F1[1, n^(-1), 1 + n^( -1), -((b*x^n)/a)])/a^2 + (d^2*(-(a*d*(-1 + n)) + b*c*(-1 + 3*n))*Hypergeo metric2F1[1, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/c^2))/((b*c - a*d)^3*n)
Time = 0.46 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {931, 1024, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle \frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}-\frac {\int \frac {b d (1-2 n) x^n+a d n+b (c-c n)}{\left (b x^n+a\right ) \left (d x^n+c\right )^2}dx}{a n (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}-\frac {\frac {\int \frac {b d (b c+a d) (1-n) n x^n+n \left (b^2 (1-n) c^2+2 a b d n c+a^2 d^2 (1-n)\right )}{\left (b x^n+a\right ) \left (d x^n+c\right )}dx}{c n (b c-a d)}-\frac {d x (a d+b c)}{c (b c-a d) \left (c+d x^n\right )}}{a n (b c-a d)}\) |
| \(\Big \downarrow \) 1020 |
| \(\displaystyle \frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}-\frac {\frac {-\frac {b^2 c n (a d (1-3 n)-b c (1-n)) \int \frac {1}{b x^n+a}dx}{b c-a d}-\frac {a d^2 n (a d (1-n)-b (c-3 c n)) \int \frac {1}{d x^n+c}dx}{b c-a d}}{c n (b c-a d)}-\frac {d x (a d+b c)}{c (b c-a d) \left (c+d x^n\right )}}{a n (b c-a d)}\) |
| \(\Big \downarrow \) 778 |
| \(\displaystyle \frac {b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}-\frac {\frac {-\frac {b^2 c n x (a d (1-3 n)-b c (1-n)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a (b c-a d)}-\frac {a d^2 n x (a d (1-n)-b (c-3 c n)) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )}{c (b c-a d)}}{c n (b c-a d)}-\frac {d x (a d+b c)}{c (b c-a d) \left (c+d x^n\right )}}{a n (b c-a d)}\) |
(b*x)/(a*(b*c - a*d)*n*(a + b*x^n)*(c + d*x^n)) - (-((d*(b*c + a*d)*x)/(c* (b*c - a*d)*(c + d*x^n))) + (-((b^2*c*(a*d*(1 - 3*n) - b*c*(1 - n))*n*x*Hy pergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*(b*c - a*d))) - ( a*d^2*n*(a*d*(1 - n) - b*(c - 3*c*n))*x*Hypergeometric2F1[1, n^(-1), 1 + n ^(-1), -((d*x^n)/c)])/(c*(b*c - a*d)))/(c*(b*c - a*d)*n))/(a*(b*c - a*d)*n )
3.4.10.3.1 Defintions of rubi rules used
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])
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]
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]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f _.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
\[\int \frac {1}{\left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )^{2}}d x\]
\[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{2}} \,d x } \]
integral(1/(b^2*d^2*x^(4*n) + a^2*c^2 + 2*(b^2*c*d + a*b*d^2)*x^(3*n) + (b ^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(2*n) + 2*(a*b*c^2 + a^2*c*d)*x^n), x)
Exception generated. \[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{2}} \,d x } \]
(a*b^2*d*(3*n - 1) - b^3*c*(n - 1))*integrate(-1/(a^2*b^3*c^3*n - 3*a^3*b^ 2*c^2*d*n + 3*a^4*b*c*d^2*n - a^5*d^3*n + (a*b^4*c^3*n - 3*a^2*b^3*c^2*d*n + 3*a^3*b^2*c*d^2*n - a^4*b*d^3*n)*x^n), x) - (b*c*d^2*(3*n - 1) - a*d^3* (n - 1))*integrate(-1/(b^3*c^5*n - 3*a*b^2*c^4*d*n + 3*a^2*b*c^3*d^2*n - a ^3*c^2*d^3*n + (b^3*c^4*d*n - 3*a*b^2*c^3*d^2*n + 3*a^2*b*c^2*d^3*n - a^3* c*d^4*n)*x^n), x) + ((b^2*c*d + a*b*d^2)*x*x^n + (b^2*c^2 + a^2*d^2)*x)/(a ^2*b^2*c^4*n - 2*a^3*b*c^3*d*n + a^4*c^2*d^2*n + (a*b^3*c^3*d*n - 2*a^2*b^ 2*c^2*d^2*n + a^3*b*c*d^3*n)*x^(2*n) + (a*b^3*c^4*n - a^2*b^2*c^3*d*n - a^ 3*b*c^2*d^2*n + a^4*c*d^3*n)*x^n)
\[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\int \frac {1}{{\left (a+b\,x^n\right )}^2\,{\left (c+d\,x^n\right )}^2} \,d x \]