Integrand size = 31, antiderivative size = 315 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\frac {d (A b c-2 a B c+a A d) (e x)^{1+m}}{a c (b c-a d)^2 e n \left (c+d x^n\right )}+\frac {(A b-a B) (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac {b (a B (b c (1+m)-a d (1+m-2 n))+A b (a d (1+m-3 n)-b c (1+m-n))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^3 e (1+m) n}-\frac {d (b c (A d (1+m-3 n)-B c (1+m-2 n))+a d (B c (1+m)-A d (1+m-n))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c^2 (b c-a d)^3 e (1+m) n} \]
d*(A*a*d+A*b*c-2*B*a*c)*(e*x)^(1+m)/a/c/(-a*d+b*c)^2/e/n/(c+d*x^n)+(A*b-B* a)*(e*x)^(1+m)/a/(-a*d+b*c)/e/n/(a+b*x^n)/(c+d*x^n)+b*(a*B*(b*c*(1+m)-a*d* (1+m-2*n))+A*b*(a*d*(1+m-3*n)-b*c*(1+m-n)))*(e*x)^(1+m)*hypergeom([1, (1+m )/n],[(1+m+n)/n],-b*x^n/a)/a^2/(-a*d+b*c)^3/e/(1+m)/n-d*(b*c*(A*d*(1+m-3*n )-B*c*(1+m-2*n))+a*d*(B*c*(1+m)-A*d*(1+m-n)))*(e*x)^(1+m)*hypergeom([1, (1 +m)/n],[(1+m+n)/n],-d*x^n/c)/c^2/(-a*d+b*c)^3/e/(1+m)/n
Time = 0.57 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.66 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\frac {x (e x)^m \left (\frac {b (b B c-2 A b d+a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a}-\frac {d (b B c-2 A b d+a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c}+\frac {b (-A b+a B) (-b c+a d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a^2}-\frac {d (b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c^2}\right )}{(b c-a d)^3 (1+m)} \]
(x*(e*x)^m*((b*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, ( 1 + m + n)/n, -((b*x^n)/a)])/a - (d*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometr ic2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c + (b*(-(A*b) + a*B)*(- (b*c) + a*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]) /a^2 - (d*(b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c^2))/((b*c - a*d)^3*(1 + m))
Time = 0.98 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1065, 25, 1065, 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.
\(\displaystyle \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx\) |
\(\Big \downarrow \) 1065 |
\(\displaystyle \frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}-\frac {\int -\frac {(e x)^m \left (-\left ((A b-a B) d (m-2 n+1) x^n\right )+a B c (m+1)-A b c (m-n+1)-a A d n\right )}{\left (b x^n+a\right ) \left (d x^n+c\right )^2}dx}{a n (b c-a d)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {(e x)^m \left (-\left ((A b-a B) d (m-2 n+1) x^n\right )+a B c (m+1)-A (b c (m-n+1)+a d n)\right )}{\left (b x^n+a\right ) \left (d x^n+c\right )^2}dx}{a n (b c-a d)}+\frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}\) |
\(\Big \downarrow \) 1065 |
\(\displaystyle \frac {\frac {\int \frac {(e x)^m \left (n \left (a B c (b c+a d) (m+1)-A \left (b^2 (m-n+1) c^2+2 a b d n c+a^2 d^2 (m-n+1)\right )\right )-b d (A b c-2 a B c+a A d) (m-n+1) n x^n\right )}{\left (b x^n+a\right ) \left (d x^n+c\right )}dx}{c n (b c-a d)}+\frac {d (e x)^{m+1} (a A d-2 a B c+A b c)}{c e (b c-a d) \left (c+d x^n\right )}}{a n (b c-a d)}+\frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}\) |
\(\Big \downarrow \) 1067 |
\(\displaystyle \frac {\frac {\int \left (\frac {b c (a B (b c (m+1)-a d (m-2 n+1))+A b (a d (m-3 n+1)-b c (m-n+1))) n (e x)^m}{(b c-a d) \left (b x^n+a\right )}+\frac {a d (-b c (A d (m-3 n+1)-B c (m-2 n+1))-a d (B c (m+1)-A d (m-n+1))) n (e x)^m}{(b c-a d) \left (d x^n+c\right )}\right )dx}{c n (b c-a d)}+\frac {d (e x)^{m+1} (a A d-2 a B c+A b c)}{c e (b c-a d) \left (c+d x^n\right )}}{a n (b c-a d)}+\frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\frac {b c n (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right ) (A b (a d (m-3 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-2 n+1)))}{a e (m+1) (b c-a d)}-\frac {a d n (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-3 n+1)-B c (m-2 n+1)))}{c e (m+1) (b c-a d)}}{c n (b c-a d)}+\frac {d (e x)^{m+1} (a A d-2 a B c+A b c)}{c e (b c-a d) \left (c+d x^n\right )}}{a n (b c-a d)}+\frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}\) |
((A*b - a*B)*(e*x)^(1 + m))/(a*(b*c - a*d)*e*n*(a + b*x^n)*(c + d*x^n)) + ((d*(A*b*c - 2*a*B*c + a*A*d)*(e*x)^(1 + m))/(c*(b*c - a*d)*e*(c + d*x^n)) + ((b*c*(a*B*(b*c*(1 + m) - a*d*(1 + m - 2*n)) + A*b*(a*d*(1 + m - 3*n) - b*c*(1 + m - n)))*n*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(b*c - a*d)*e*(1 + m)) - (a*d*(b*c*(A*d*(1 + m - 3*n) - B*c*(1 + m - 2*n)) + a*d*(B*c*(1 + m) - A*d*(1 + m - n)))*n*(e*x)^ (1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*( b*c - a*d)*e*(1 + m)))/(c*(b*c - a*d)*n))/(a*(b*c - a*d)*n)
3.1.34.3.1 Defintions of rubi rules used
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && LtQ[p, -1]
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]
\[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )^{2}}d x\]
\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{2}} \,d x } \]
integral((B*x^n + A)*(e*x)^m/(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. \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{2}} \,d x } \]
((b^3*c*e^m*(m - n + 1) - a*b^2*d*e^m*(m - 3*n + 1))*A + (a^2*b*d*e^m*(m - 2*n + 1) - a*b^2*c*e^m*(m + 1))*B)*integrate(-x^m/(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) - ((a*d^3*e^m*(m - n + 1) - b*c*d^2*e^m*(m - 3*n + 1))*A + (b*c^2*d*e^m*(m - 2*n + 1) - a*c*d^2*e^m*( m + 1))*B)*integrate(-x^m/(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^2*e^m + a^2*d^2*e^m)*A - (a*b*c^2*e^m + a ^2*c*d*e^m)*B)*x*x^m - (2*B*a*b*c*d*e^m - (b^2*c*d*e^m + a*b*d^2*e^m)*A)*x *e^(m*log(x) + n*log(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)
\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{2} {\left (d x^{n} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )}{{\left (a+b\,x^n\right )}^2\,{\left (c+d\,x^n\right )}^2} \,d x \]