Integrand size = 29, antiderivative size = 123 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx=\frac {(b B c+A b d-a B d) (e x)^{1+m}}{b^2 e (1+m)}+\frac {B d x^n (e x)^{1+m}}{b e (1+m+n)}+\frac {(A b-a B) (b c-a d) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a b^2 e (1+m)} \] Output:
(A*b*d-B*a*d+B*b*c)*(e*x)^(1+m)/b^2/e/(1+m)+B*d*x^n*(e*x)^(1+m)/b/e/(1+m+n )+(A*b-B*a)*(-a*d+b*c)*(e*x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/n],-b*x ^n/a)/a/b^2/e/(1+m)
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx=\frac {x (e x)^m \left (\frac {b B c+A b d-a B d}{1+m}+\frac {b B d x^n}{1+m+n}+\frac {(-A b+a B) (-b c+a d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a (1+m)}\right )}{b^2} \] Input:
Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n),x]
Output:
(x*(e*x)^m*((b*B*c + A*b*d - a*B*d)/(1 + m) + (b*B*d*x^n)/(1 + m + n) + (( -(A*b) + a*B)*(-(b*c) + a*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n , -((b*x^n)/a)])/(a*(1 + m))))/b^2
Time = 0.49 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.
\(\displaystyle \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx\) |
\(\Big \downarrow \) 1040 |
\(\displaystyle \int \left (\frac {(e x)^m (A b-a B) (b c-a d)}{b^2 \left (a+b x^n\right )}+\frac {(e x)^m (-a B d+A b d+b B c)}{b^2}+\frac {B d x^n (e x)^m}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(e x)^{m+1} (A b-a B) (b c-a d) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{a b^2 e (m+1)}+\frac {(e x)^{m+1} (-a B d+A b d+b B c)}{b^2 e (m+1)}+\frac {B d x^{n+1} (e x)^m}{b (m+n+1)}\) |
Input:
Int[((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n),x]
Output:
(B*d*x^(1 + n)*(e*x)^m)/(b*(1 + m + n)) + ((b*B*c + A*b*d - a*B*d)*(e*x)^( 1 + m))/(b^2*e*(1 + m)) + ((A*b - a*B)*(b*c - a*d)*(e*x)^(1 + m)*Hypergeom etric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*b^2*e*(1 + m))
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]
\[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )}{a +b \,x^{n}}d x\]
Input:
int((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n),x)
Output:
int((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n),x)
\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )} \left (e x\right )^{m}}{b x^{n} + a} \,d x } \] Input:
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n),x, algorithm="fricas")
Output:
integral((B*d*x^(2*n) + A*c + (B*c + A*d)*x^n)*(e*x)^m/(b*x^n + a), x)
Result contains complex when optimal does not.
Time = 6.60 (sec) , antiderivative size = 872, normalized size of antiderivative = 7.09 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx=\text {Too large to display} \] Input:
integrate((e*x)**m*(A+B*x**n)*(c+d*x**n)/(a+b*x**n),x)
Output:
A*a**(m/n + 1/n)*a**(-m/n - 1 - 1/n)*c*e**m*m*x**(m + 1)*lerchphi(b*x**n*e xp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n )) + A*a**(m/n + 1/n)*a**(-m/n - 1 - 1/n)*c*e**m*x**(m + 1)*lerchphi(b*x** n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n)) + A*a**(-m/n - 2 - 1/n)*a**(m/n + 1 + 1/n)*d*e**m*m*x**(m + n + 1)*l erchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/( n**2*gamma(m/n + 2 + 1/n)) + A*a**(-m/n - 2 - 1/n)*a**(m/n + 1 + 1/n)*d*e* *m*x**(m + n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gam ma(m/n + 1 + 1/n)/(n*gamma(m/n + 2 + 1/n)) + A*a**(-m/n - 2 - 1/n)*a**(m/n + 1 + 1/n)*d*e**m*x**(m + n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/ n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)) + B*a**(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)*d*e**m*m*x**(m + 2*n + 1)*lerchphi(b*x**n*e xp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1/n)) + 2*B*a**(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)*d*e**m*x**(m + 2*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n*gamma(m/n + 3 + 1/n)) + B*a**(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)* d*e**m*x**(m + 2*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/ n)*gamma(m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1/n)) + B*a**(-m/n - 2 - 1/n )*a**(m/n + 1 + 1/n)*c*e**m*m*x**(m + n + 1)*lerchphi(b*x**n*exp_polar(I*p i)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)...
\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )} \left (e x\right )^{m}}{b x^{n} + a} \,d x } \] Input:
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n),x, algorithm="maxima")
Output:
((b^2*c*e^m - a*b*d*e^m)*A - (a*b*c*e^m - a^2*d*e^m)*B)*integrate(x^m/(b^3 *x^n + a*b^2), x) + (B*b*d*e^m*(m + 1)*x*e^(m*log(x) + n*log(x)) + (A*b*d* e^m*(m + n + 1) + (b*c*e^m*(m + n + 1) - a*d*e^m*(m + n + 1))*B)*x*x^m)/(( m^2 + m*(n + 2) + n + 1)*b^2)
\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )} \left (e x\right )^{m}}{b x^{n} + a} \,d x } \] Input:
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n),x, algorithm="giac")
Output:
integrate((B*x^n + A)*(d*x^n + c)*(e*x)^m/(b*x^n + a), x)
Timed out. \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,\left (c+d\,x^n\right )}{a+b\,x^n} \,d x \] Input:
int(((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n),x)
Output:
int(((e*x)^m*(A + B*x^n)*(c + d*x^n))/(a + b*x^n), x)
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.33 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx=\frac {x^{m} e^{m} x \left (x^{n} d m +x^{n} d +c m +c n +c \right )}{m^{2}+m n +2 m +n +1} \] Input:
int((e*x)^m*(A+B*x^n)*(c+d*x^n)/(a+b*x^n),x)
Output:
(x**m*e**m*x*(x**n*d*m + x**n*d + c*m + c*n + c))/(m**2 + m*n + 2*m + n + 1)