\(\int \frac {(e x)^m (a+b x^2) (A+B x^2)}{c+d x^2} \, dx\) [25]

Optimal result

Integrand size = 29, antiderivative size = 120 \[ \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{c+d x^2} \, dx=-\frac {(b B c-A b d-a B d) (e x)^{1+m}}{d^2 e (1+m)}+\frac {b B (e x)^{3+m}}{d e^3 (3+m)}+\frac {(b c-a d) (B c-A d) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )}{c d^2 e (1+m)} \] Output:

-(-A*b*d-B*a*d+B*b*c)*(e*x)^(1+m)/d^2/e/(1+m)+b*B*(e*x)^(3+m)/d/e^3/(3+m)+ 
(-a*d+b*c)*(-A*d+B*c)*(e*x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],-d* 
x^2/c)/c/d^2/e/(1+m)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.78 \[ \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{c+d x^2} \, dx=\frac {x (e x)^m \left (\frac {-b B c+A b d+a B d}{1+m}+\frac {b B d x^2}{3+m}+\frac {(b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )}{c (1+m)}\right )}{d^2} \] Input:

Integrate[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2),x]
 

Output:

(x*(e*x)^m*((-(b*B*c) + A*b*d + a*B*d)/(1 + m) + (b*B*d*x^2)/(3 + m) + ((b 
*c - a*d)*(B*c - A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2) 
/c)])/(c*(1 + m))))/d^2
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {437, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right ) (e x)^m}{c+d x^2} \, dx\)

\(\Big \downarrow \) 437

\(\displaystyle \int \left (\frac {(e x)^m \left (a A d^2-a B c d-A b c d+b B c^2\right )}{d^2 \left (c+d x^2\right )}-\frac {(e x)^m (-a B d-A b d+b B c)}{d^2}+\frac {b B (e x)^{m+2}}{d e^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(e x)^{m+1} (b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {d x^2}{c}\right )}{c d^2 e (m+1)}-\frac {(e x)^{m+1} (-a B d-A b d+b B c)}{d^2 e (m+1)}+\frac {b B (e x)^{m+3}}{d e^3 (m+3)}\)

Input:

Int[((e*x)^m*(a + b*x^2)*(A + B*x^2))/(c + d*x^2),x]
 

Output:

-(((b*B*c - A*b*d - a*B*d)*(e*x)^(1 + m))/(d^2*e*(1 + m))) + (b*B*(e*x)^(3 
 + m))/(d*e^3*(3 + m)) + ((b*c - a*d)*(B*c - A*d)*(e*x)^(1 + m)*Hypergeome 
tric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*d^2*e*(1 + m))
 

Defintions of rubi rules used

rule 437

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*( 
a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f 
, g, m}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (b \,x^{2}+a \right ) \left (x^{2} B +A \right )}{x^{2} d +c}d x\]

Input:

int((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c),x)
 

Output:

int((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c),x)
 

Fricas [F]

\[ \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{c+d x^2} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )} \left (e x\right )^{m}}{d x^{2} + c} \,d x } \] Input:

integrate((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c),x, algorithm="fricas")
 

Output:

integral((B*b*x^4 + (B*a + A*b)*x^2 + A*a)*(e*x)^m/(d*x^2 + c), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.67 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.48 \[ \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{c+d x^2} \, dx=\frac {A a e^{m} m x^{m + 1} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A a e^{m} x^{m + 1} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A b e^{m} m x^{m + 3} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 A b e^{m} x^{m + 3} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B a e^{m} m x^{m + 3} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 B a e^{m} x^{m + 3} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B b e^{m} m x^{m + 5} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {5}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {5 B b e^{m} x^{m + 5} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {5}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} \] Input:

integrate((e*x)**m*(b*x**2+a)*(B*x**2+A)/(d*x**2+c),x)
 

Output:

A*a*e**m*m*x**(m + 1)*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gam 
ma(m/2 + 1/2)/(4*c*gamma(m/2 + 3/2)) + A*a*e**m*x**(m + 1)*lerchphi(d*x**2 
*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(4*c*gamma(m/2 + 3/2)) 
+ A*b*e**m*m*x**(m + 3)*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*g 
amma(m/2 + 3/2)/(4*c*gamma(m/2 + 5/2)) + 3*A*b*e**m*x**(m + 3)*lerchphi(d* 
x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*c*gamma(m/2 + 5/ 
2)) + B*a*e**m*m*x**(m + 3)*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/ 
2)*gamma(m/2 + 3/2)/(4*c*gamma(m/2 + 5/2)) + 3*B*a*e**m*x**(m + 3)*lerchph 
i(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(4*c*gamma(m/2 
+ 5/2)) + B*b*e**m*m*x**(m + 5)*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 
+ 5/2)*gamma(m/2 + 5/2)/(4*c*gamma(m/2 + 7/2)) + 5*B*b*e**m*x**(m + 5)*ler 
chphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 5/2)*gamma(m/2 + 5/2)/(4*c*gamma( 
m/2 + 7/2))
 

Maxima [F]

\[ \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{c+d x^2} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )} \left (e x\right )^{m}}{d x^{2} + c} \,d x } \] Input:

integrate((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c),x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)*(b*x^2 + a)*(e*x)^m/(d*x^2 + c), x)
 

Giac [F]

\[ \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{c+d x^2} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )} \left (e x\right )^{m}}{d x^{2} + c} \,d x } \] Input:

integrate((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)*(b*x^2 + a)*(e*x)^m/(d*x^2 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{c+d x^2} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,\left (b\,x^2+a\right )}{d\,x^2+c} \,d x \] Input:

int(((A + B*x^2)*(e*x)^m*(a + b*x^2))/(c + d*x^2),x)
 

Output:

int(((A + B*x^2)*(e*x)^m*(a + b*x^2))/(c + d*x^2), x)
 

Reduce [F]

\[ \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{c+d x^2} \, dx=\frac {e^{m} \left (2 x^{m} a b d m x +6 x^{m} a b d x -x^{m} b^{2} c m x -3 x^{m} b^{2} c x +x^{m} b^{2} d m \,x^{3}+x^{m} b^{2} d \,x^{3}+\left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) a^{2} d^{2} m^{2}+4 \left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) a^{2} d^{2} m +3 \left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) a^{2} d^{2}-2 \left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) a b c d \,m^{2}-8 \left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) a b c d m -6 \left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) a b c d +\left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) b^{2} c^{2} m^{2}+4 \left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) b^{2} c^{2} m +3 \left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) b^{2} c^{2}\right )}{d^{2} \left (m^{2}+4 m +3\right )} \] Input:

int((e*x)^m*(b*x^2+a)*(B*x^2+A)/(d*x^2+c),x)
 

Output:

(e**m*(2*x**m*a*b*d*m*x + 6*x**m*a*b*d*x - x**m*b**2*c*m*x - 3*x**m*b**2*c 
*x + x**m*b**2*d*m*x**3 + x**m*b**2*d*x**3 + int(x**m/(c + d*x**2),x)*a**2 
*d**2*m**2 + 4*int(x**m/(c + d*x**2),x)*a**2*d**2*m + 3*int(x**m/(c + d*x* 
*2),x)*a**2*d**2 - 2*int(x**m/(c + d*x**2),x)*a*b*c*d*m**2 - 8*int(x**m/(c 
 + d*x**2),x)*a*b*c*d*m - 6*int(x**m/(c + d*x**2),x)*a*b*c*d + int(x**m/(c 
 + d*x**2),x)*b**2*c**2*m**2 + 4*int(x**m/(c + d*x**2),x)*b**2*c**2*m + 3* 
int(x**m/(c + d*x**2),x)*b**2*c**2))/(d**2*(m**2 + 4*m + 3))