Integrand size = 15, antiderivative size = 44 \[ \int \frac {(e x)^m}{\left (a+b x^2\right )^2} \, dx=\frac {(e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^2 e (1+m)} \] Output:
(e*x)^(1+m)*hypergeom([2, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a^2/e/(1+m)
Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {(e x)^m}{\left (a+b x^2\right )^2} \, dx=\frac {x (e x)^m \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},1+\frac {1+m}{2},-\frac {b x^2}{a}\right )}{a^2 (1+m)} \] Input:
Integrate[(e*x)^m/(a + b*x^2)^2,x]
Output:
(x*(e*x)^m*Hypergeometric2F1[2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/( a^2*(1 + m))
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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.
\(\displaystyle \int \frac {(e x)^m}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{a^2 e (m+1)}\) |
Input:
Int[(e*x)^m/(a + b*x^2)^2,x]
Output:
((e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/( a^2*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])
\[\int \frac {\left (e x \right )^{m}}{\left (b \,x^{2}+a \right )^{2}}d x\]
Input:
int((e*x)^m/(b*x^2+a)^2,x)
Output:
int((e*x)^m/(b*x^2+a)^2,x)
\[ \int \frac {(e x)^m}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x)^m/(b*x^2+a)^2,x, algorithm="fricas")
Output:
integral((e*x)^m/(b^2*x^4 + 2*a*b*x^2 + a^2), x)
Result contains complex when optimal does not.
Time = 2.90 (sec) , antiderivative size = 398, normalized size of antiderivative = 9.05 \[ \int \frac {(e x)^m}{\left (a+b x^2\right )^2} \, dx=- \frac {a e^{m} m^{2} x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 a e^{m} m x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a e^{m} x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 a e^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {b e^{m} m^{2} x^{2} x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {b e^{m} x^{2} x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \] Input:
integrate((e*x)**m/(b*x**2+a)**2,x)
Output:
-a*e**m*m**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*g amma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + 2*a*e**m*m*x**(m + 1)*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a** 2*b*x**2*gamma(m/2 + 3/2)) + a*e**m*x**(m + 1)*lerchphi(b*x**2*exp_polar(I *pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b *x**2*gamma(m/2 + 3/2)) + 2*a*e**m*x**(m + 1)*gamma(m/2 + 1/2)/(8*a**3*gam ma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) - b*e**m*m**2*x**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a **3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + b*e**m*x**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8 *a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2))
\[ \int \frac {(e x)^m}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x)^m/(b*x^2+a)^2,x, algorithm="maxima")
Output:
integrate((e*x)^m/(b*x^2 + a)^2, x)
\[ \int \frac {(e x)^m}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x)^m/(b*x^2+a)^2,x, algorithm="giac")
Output:
integrate((e*x)^m/(b*x^2 + a)^2, x)
Timed out. \[ \int \frac {(e x)^m}{\left (a+b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^m}{{\left (b\,x^2+a\right )}^2} \,d x \] Input:
int((e*x)^m/(a + b*x^2)^2,x)
Output:
int((e*x)^m/(a + b*x^2)^2, x)
\[ \int \frac {(e x)^m}{\left (a+b x^2\right )^2} \, dx=e^{m} \left (\int \frac {x^{m}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) \] Input:
int((e*x)^m/(b*x^2+a)^2,x)
Output:
e**m*int(x**m/(a**2 + 2*a*b*x**2 + b**2*x**4),x)