Integrand size = 29, antiderivative size = 178 \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=-\frac {B (a d (1+m)-b c (1+m+n)) (e x)^{1+m}}{c d^2 e (1+m) n}-\frac {(b c-a d) (e x)^{1+m} \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}+\frac {(A d (b c (1+m)-a d (1+m-n))+B c (a d (1+m)-b c (1+m+n))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c^2 d^2 e (1+m) n} \]
-B*(a*d*(1+m)-b*c*(1+m+n))*(e*x)^(1+m)/c/d^2/e/(1+m)/n-(-a*d+b*c)*(e*x)^(1 +m)*(A+B*x^n)/c/d/e/n/(c+d*x^n)+(A*d*(b*c*(1+m)-a*d*(1+m-n))+B*c*(a*d*(1+m )-b*c*(1+m+n)))*(e*x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/n],-d*x^n/c)/c ^2/d^2/e/(1+m)/n
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.62 \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\frac {x (e x)^m \left (b B c^2+c (-2 b B c+A b d+a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )+(b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )\right )}{c^2 d^2 (1+m)} \]
(x*(e*x)^m*(b*B*c^2 + c*(-2*b*B*c + A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + (b*c - a*d)*(B*c - A*d)*Hypergeome tric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(c^2*d^2*(1 + m))
Time = 0.42 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1064, 25, 959, 888}
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.
| \(\displaystyle \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx\) |
| \(\Big \downarrow \) 1064 |
| \(\displaystyle -\frac {\int -\frac {(e x)^m \left (A (b c (m+1)-a d (m-n+1))-B (a d (m+1)-b c (m+n+1)) x^n\right )}{d x^n+c}dx}{c d n}-\frac {(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(e x)^m \left (A (b c (m+1)-a d (m-n+1))-B (a d (m+1)-b c (m+n+1)) x^n\right )}{d x^n+c}dx}{c d n}-\frac {(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {\frac {(a d (B c (m+1)-A d (m-n+1))+b c (A d (m+1)-B c (m+n+1))) \int \frac {(e x)^m}{d x^n+c}dx}{d}-\frac {B (e x)^{m+1} (a d (m+1)-b c (m+n+1))}{d e (m+1)}}{c d n}-\frac {(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {\frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m+1)-B c (m+n+1)))}{c d e (m+1)}-\frac {B (e x)^{m+1} (a d (m+1)-b c (m+n+1))}{d e (m+1)}}{c d n}-\frac {(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}\) |
-(((b*c - a*d)*(e*x)^(1 + m)*(A + B*x^n))/(c*d*e*n*(c + d*x^n))) + (-((B*( a*d*(1 + m) - b*c*(1 + m + n))*(e*x)^(1 + m))/(d*e*(1 + m))) + ((a*d*(B*c* (1 + m) - A*d*(1 + m - n)) + b*c*(A*d*(1 + m) - B*c*(1 + m + n)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*d*e *(1 + m)))/(c*d*n)
3.1.31.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
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/(a*b*g*n*(p + 1))), x] + Simp[1/( a*b*n*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c *(b*e*n*(p + 1) + (b*e - a*f)*(m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && LtQ [p, -1] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
\[\int \frac {\left (e x \right )^{m} \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right )}{\left (c +d \,x^{n}\right )^{2}}d x\]
\[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 25.68 (sec) , antiderivative size = 5176, normalized size of antiderivative = 29.08 \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\text {Too large to display} \]
A*a*(-c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*m**2*x**(m + 1)*lerchphi(d *x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n)) + c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*m*n*x**(m + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m /n + 1 + 1/n)) + c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*m*n*x**(m + 1)* gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n)) - 2*c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*m*x**(m + 1)*lerchph i(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m /n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n)) + c*c**(m/n + 1/n)*c**(- m/n - 2 - 1/n)*e**m*n*x**(m + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma( m/n + 1 + 1/n)) + c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*n*x**(m + 1)*g amma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n)) - c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*x**(m + 1)*lerchphi(d*x **n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n)) - c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*d*e**m*m**2*x**n*x**(m + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gam ma(m/n + 1 + 1/n)) + c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*d*e**m*m*n*x**n...
\[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}} \,d x } \]
-((a*d^2*e^m*(m - n + 1) - b*c*d*e^m*(m + 1))*A + (b*c^2*e^m*(m + n + 1) - a*c*d*e^m*(m + 1))*B)*integrate(x^m/(c*d^3*n*x^n + c^2*d^2*n), x) + (B*b* c*d*e^m*n*x*e^(m*log(x) + n*log(x)) - ((b*c*d*e^m*(m + 1) - a*d^2*e^m*(m + 1))*A - (b*c^2*e^m*(m + n + 1) - a*c*d*e^m*(m + 1))*B)*x*x^m)/((m*n + n)* c*d^3*x^n + (m*n + n)*c^2*d^2)
\[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e x)^m \left (a+b x^n\right ) \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,\left (a+b\,x^n\right )}{{\left (c+d\,x^n\right )}^2} \,d x \]