Integrand size = 31, antiderivative size = 187 \[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx=-\frac {b (b B c-A b d-2 a B d) x^{1+n} (e x)^m}{d^2 (1+m+n)}+\frac {b^2 B x^{1+2 n} (e x)^m}{d (1+m+2 n)}+\frac {\left (a^2 B d^2+b^2 c (B c-A d)-2 a b d (B c-A d)\right ) (e x)^{1+m}}{d^3 e (1+m)}-\frac {(b c-a d)^2 (B c-A d) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c d^3 e (1+m)} \]
-b*(-A*b*d-2*B*a*d+B*b*c)*x^(1+n)*(e*x)^m/d^2/(1+m+n)+b^2*B*x^(1+2*n)*(e*x )^m/d/(1+m+2*n)+(a^2*B*d^2+b^2*c*(-A*d+B*c)-2*a*b*d*(-A*d+B*c))*(e*x)^(1+m )/d^3/e/(1+m)-(-a*d+b*c)^2*(-A*d+B*c)*(e*x)^(1+m)*hypergeom([1, (1+m)/n],[ (1+m+n)/n],-d*x^n/c)/c/d^3/e/(1+m)
Time = 0.45 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.82 \[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {x (e x)^m \left (\frac {a^2 B d^2+b^2 c (B c-A d)+2 a b d (-B c+A d)}{1+m}+\frac {b d (-b B c+A b d+2 a B d) x^n}{1+m+n}+\frac {b^2 B d^2 x^{2 n}}{1+m+2 n}-\frac {(b c-a d)^2 (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c (1+m)}\right )}{d^3} \]
(x*(e*x)^m*((a^2*B*d^2 + b^2*c*(B*c - A*d) + 2*a*b*d*(-(B*c) + A*d))/(1 + m) + (b*d*(-(b*B*c) + A*b*d + 2*a*B*d)*x^n)/(1 + m + n) + (b^2*B*d^2*x^(2* n))/(1 + m + 2*n) - ((b*c - a*d)^2*(B*c - A*d)*Hypergeometric2F1[1, (1 + m )/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*(1 + m))))/d^3
Time = 0.41 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1040, 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.
\(\displaystyle \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx\) |
\(\Big \downarrow \) 1040 |
\(\displaystyle \int \left (\frac {(e x)^m \left (a^2 B d^2-2 a b d (B c-A d)+b^2 c (B c-A d)\right )}{d^3}+\frac {(e x)^m (a d-b c)^2 (A d-B c)}{d^3 \left (c+d x^n\right )}+\frac {b x^n (e x)^m (2 a B d+A b d-b B c)}{d^2}+\frac {b^2 B x^{2 n} (e x)^m}{d}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(e x)^{m+1} \left (a^2 B d^2-2 a b d (B c-A d)+b^2 c (B c-A d)\right )}{d^3 e (m+1)}-\frac {(e x)^{m+1} (b c-a d)^2 (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {d x^n}{c}\right )}{c d^3 e (m+1)}-\frac {b x^{n+1} (e x)^m (-2 a B d-A b d+b B c)}{d^2 (m+n+1)}+\frac {b^2 B x^{2 n+1} (e x)^m}{d (m+2 n+1)}\) |
-((b*(b*B*c - A*b*d - 2*a*B*d)*x^(1 + n)*(e*x)^m)/(d^2*(1 + m + n))) + (b^ 2*B*x^(1 + 2*n)*(e*x)^m)/(d*(1 + m + 2*n)) + ((a^2*B*d^2 + b^2*c*(B*c - A* d) - 2*a*b*d*(B*c - A*d))*(e*x)^(1 + m))/(d^3*e*(1 + m)) - ((b*c - a*d)^2* (B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, - ((d*x^n)/c)])/(c*d^3*e*(1 + m))
3.1.23.3.1 Defintions of rubi rules used
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[ (g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c , d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
\[\int \frac {\left (e x \right )^{m} \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{c +d \,x^{n}}d x\]
\[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{2} \left (e x\right )^{m}}{d x^{n} + c} \,d x } \]
integral((B*b^2*x^(3*n) + A*a^2 + (2*B*a*b + A*b^2)*x^(2*n) + (B*a^2 + 2*A *a*b)*x^n)*(e*x)^m/(d*x^n + c), x)
Result contains complex when optimal does not.
Time = 9.44 (sec) , antiderivative size = 1402, normalized size of antiderivative = 7.50 \[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx=\text {Too large to display} \]
A*a**2*c**(m/n + 1/n)*c**(-m/n - 1 - 1/n)*e**m*m*x**(m + 1)*lerchphi(d*x** n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n)) + A*a**2*c**(m/n + 1/n)*c**(-m/n - 1 - 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)/(n**2*gamma(m/n + 1 + 1/n)) + 2*A*a*b*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*m*x**(m + n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)) + 2*A*a*b*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*x**(m + n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n*gamma(m/n + 2 + 1/n)) + 2*A*a*b*c**(-m/ n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*x**(m + n + 1)*lerchphi(d*x**n*exp_po lar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)) + A*b**2*c**(-m/n - 3 - 1/n)*c**(m/n + 2 + 1/n)*e**m*m*x**(m + 2*n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1 /n)/(n**2*gamma(m/n + 3 + 1/n)) + 2*A*b**2*c**(-m/n - 3 - 1/n)*c**(m/n + 2 + 1/n)*e**m*x**(m + 2*n + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n*gamma(m/n + 3 + 1/n)) + A*b**2*c**(-m/n - 3 - 1/n)*c**(m/n + 2 + 1/n)*e**m*x**(m + 2*n + 1)*lerchphi(d*x**n*exp_pol ar(I*pi)/c, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1 /n)) + B*a**2*c**(-m/n - 2 - 1/n)*c**(m/n + 1 + 1/n)*e**m*m*x**(m + n + 1) *lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1...
\[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{2} \left (e x\right )^{m}}{d x^{n} + c} \,d x } \]
((b^2*c^2*d*e^m - 2*a*b*c*d^2*e^m + a^2*d^3*e^m)*A - (b^2*c^3*e^m - 2*a*b* c^2*d*e^m + a^2*c*d^2*e^m)*B)*integrate(x^m/(d^4*x^n + c*d^3), x) + ((m^2 + m*(n + 2) + n + 1)*B*b^2*d^2*e^m*x*e^(m*log(x) + 2*n*log(x)) - (((m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*b^2*c*d*e^m - 2*(m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*a*b*d^2*e^m)*A - ((m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*b^2*c^2 *e^m - 2*(m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*a*b*c*d*e^m + (m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*a^2*d^2*e^m)*B)*x*x^m + ((m^2 + 2*m*(n + 1) + 2*n + 1)*A*b^2*d^2*e^m - ((m^2 + 2*m*(n + 1) + 2*n + 1)*b^2*c*d*e^m - 2*(m^2 + 2*m*(n + 1) + 2*n + 1)*a*b*d^2*e^m)*B)*x*e^(m*log(x) + n*log(x)))/((m^3 + 3*m^2*(n + 1) + (2*n^2 + 6*n + 3)*m + 2*n^2 + 3*n + 1)*d^3)
\[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{2} \left (e x\right )^{m}}{d x^{n} + c} \,d x } \]
Timed out. \[ \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^2}{c+d\,x^n} \,d x \]