Integrand size = 22, antiderivative size = 77 \[ \int \frac {(e x)^m \left (A+B x^2\right )}{c+d x^2} \, dx=\frac {B (e x)^{1+m}}{d e (1+m)}-\frac {(B c-A d) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )}{c d e (1+m)} \] Output:
B*(e*x)^(1+m)/d/e/(1+m)-(-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/e/(1+m)
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.73 \[ \int \frac {(e x)^m \left (A+B x^2\right )}{c+d x^2} \, dx=\frac {x (e x)^m \left (B c+(-B c+A d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )\right )}{c d (1+m)} \] Input:
Integrate[((e*x)^m*(A + B*x^2))/(c + d*x^2),x]
Output:
(x*(e*x)^m*(B*c + (-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2 , -((d*x^2)/c)]))/(c*d*(1 + m))
Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {363, 278}
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.
\(\displaystyle \int \frac {\left (A+B x^2\right ) (e x)^m}{c+d x^2} \, dx\) |
\(\Big \downarrow \) 363 |
\(\displaystyle \frac {B (e x)^{m+1}}{d e (m+1)}-\frac {(B c-A d) \int \frac {(e x)^m}{d x^2+c}dx}{d}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {B (e x)^{m+1}}{d e (m+1)}-\frac {(e x)^{m+1} (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {d x^2}{c}\right )}{c d e (m+1)}\) |
Input:
Int[((e*x)^m*(A + B*x^2))/(c + d*x^2),x]
Output:
(B*(e*x)^(1 + m))/(d*e*(1 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometri c2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(c*d*e*(1 + m))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
\[\int \frac {\left (e x \right )^{m} \left (x^{2} B +A \right )}{x^{2} d +c}d x\]
Input:
int((e*x)^m*(B*x^2+A)/(d*x^2+c),x)
Output:
int((e*x)^m*(B*x^2+A)/(d*x^2+c),x)
\[ \int \frac {(e x)^m \left (A+B x^2\right )}{c+d x^2} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{d x^{2} + c} \,d x } \] Input:
integrate((e*x)^m*(B*x^2+A)/(d*x^2+c),x, algorithm="fricas")
Output:
integral((B*x^2 + A)*(e*x)^m/(d*x^2 + c), x)
Result contains complex when optimal does not.
Time = 2.02 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.61 \[ \int \frac {(e x)^m \left (A+B x^2\right )}{c+d x^2} \, dx=\frac {A e^{m} m x^{m + 1} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A e^{m} x^{m + 1} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {B e^{m} m x^{m + 3} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 B e^{m} x^{m + 3} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 c \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \] Input:
integrate((e*x)**m*(B*x**2+A)/(d*x**2+c),x)
Output:
A*e**m*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*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)) + B* 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*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))
\[ \int \frac {(e x)^m \left (A+B x^2\right )}{c+d x^2} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{d x^{2} + c} \,d x } \] Input:
integrate((e*x)^m*(B*x^2+A)/(d*x^2+c),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c), x)
\[ \int \frac {(e x)^m \left (A+B x^2\right )}{c+d x^2} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{d x^{2} + c} \,d x } \] Input:
integrate((e*x)^m*(B*x^2+A)/(d*x^2+c),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*(e*x)^m/(d*x^2 + c), x)
Timed out. \[ \int \frac {(e x)^m \left (A+B x^2\right )}{c+d x^2} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m}{d\,x^2+c} \,d x \] Input:
int(((A + B*x^2)*(e*x)^m)/(c + d*x^2),x)
Output:
int(((A + B*x^2)*(e*x)^m)/(c + d*x^2), x)
\[ \int \frac {(e x)^m \left (A+B x^2\right )}{c+d x^2} \, dx=\frac {e^{m} \left (x^{m} b x +\left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) a d m +\left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) a d -\left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) b c m -\left (\int \frac {x^{m}}{d \,x^{2}+c}d x \right ) b c \right )}{d \left (m +1\right )} \] Input:
int((e*x)^m*(B*x^2+A)/(d*x^2+c),x)
Output:
(e**m*(x**m*b*x + int(x**m/(c + d*x**2),x)*a*d*m + int(x**m/(c + d*x**2),x )*a*d - int(x**m/(c + d*x**2),x)*b*c*m - int(x**m/(c + d*x**2),x)*b*c))/(d *(m + 1))