Integrand size = 22, antiderivative size = 142 \[ \int \frac {x^2}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\frac {b x^3}{a (b c-a d) n \left (a+b x^n\right )}+\frac {b (a d (3-2 n)-b c (3-n)) x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {b x^n}{a}\right )}{3 a^2 (b c-a d)^2 n}+\frac {d^2 x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {d x^n}{c}\right )}{3 c (b c-a d)^2} \]
b*x^3/a/(-a*d+b*c)/n/(a+b*x^n)+1/3*b*(a*d*(3-2*n)-b*c*(3-n))*x^3*hypergeom ([1, 3/n],[(3+n)/n],-b*x^n/a)/a^2/(-a*d+b*c)^2/n+1/3*d^2*x^3*hypergeom([1, 3/n],[(3+n)/n],-d*x^n/c)/c/(-a*d+b*c)^2
Time = 0.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.95 \[ \int \frac {x^2}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\frac {x^3 \left (b c (a d (3-2 n)+b c (-3+n)) \left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {b x^n}{a}\right )+a \left (3 b c (b c-a d)+a d^2 n \left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {3+n}{n},-\frac {d x^n}{c}\right )\right )\right )}{3 a^2 c (b c-a d)^2 n \left (a+b x^n\right )} \]
(x^3*(b*c*(a*d*(3 - 2*n) + b*c*(-3 + n))*(a + b*x^n)*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((b*x^n)/a)] + a*(3*b*c*(b*c - a*d) + a*d^2*n*(a + b*x^n) *Hypergeometric2F1[1, 3/n, (3 + n)/n, -((d*x^n)/c)])))/(3*a^2*c*(b*c - a*d )^2*n*(a + b*x^n))
Time = 0.39 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.13, 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.
\(\displaystyle \int \frac {x^2}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx\) |
\(\Big \downarrow \) 1006 |
\(\displaystyle \frac {b x^3}{a n (b c-a d) \left (a+b x^n\right )}-\frac {\int \frac {x^2 \left (b d (3-n) x^n+b c (3-n)+a d n\right )}{\left (b x^n+a\right ) \left (d x^n+c\right )}dx}{a n (b c-a d)}\) |
\(\Big \downarrow \) 1067 |
\(\displaystyle \frac {b x^3}{a n (b c-a d) \left (a+b x^n\right )}-\frac {\int \left (\frac {b (b c (3-n)-a d (3-2 n)) x^2}{(b c-a d) \left (b x^n+a\right )}+\frac {a d^2 n x^2}{(a d-b c) \left (d x^n+c\right )}\right )dx}{a n (b c-a d)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b x^3}{a n (b c-a d) \left (a+b x^n\right )}-\frac {-\frac {a d^2 n x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {d x^n}{c}\right )}{3 c (b c-a d)}-\frac {b x^3 (a d (3-2 n)-b c (3-n)) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{n},\frac {n+3}{n},-\frac {b x^n}{a}\right )}{3 a (b c-a d)}}{a n (b c-a d)}\) |
(b*x^3)/(a*(b*c - a*d)*n*(a + b*x^n)) - (-1/3*(b*(a*d*(3 - 2*n) - b*c*(3 - n))*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((b*x^n)/a)])/(a*(b*c - a*d )) - (a*d^2*n*x^3*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((d*x^n)/c)])/(3*c *(b*c - a*d)))/(a*(b*c - a*d)*n)
3.11.33.3.1 Defintions of rubi rules used
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]
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]
\[\int \frac {x^{2}}{\left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )}d x\]
\[ \int \frac {x^2}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\int { \frac {x^{2}}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}} \,d x } \]
Exception generated. \[ \int \frac {x^2}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int \frac {x^2}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\int { \frac {x^{2}}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}} \,d x } \]
b*x^3/(a^2*b*c*n - a^3*d*n + (a*b^2*c*n - a^2*b*d*n)*x^n) + d^2*integrate( x^2/(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 - 3) - b^2*c*(n - 3))*integrate(x^2/(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)
\[ \int \frac {x^2}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\int { \frac {x^{2}}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx=\int \frac {x^2}{{\left (a+b\,x^n\right )}^2\,\left (c+d\,x^n\right )} \,d x \]