Integrand size = 16, antiderivative size = 36 \[ \int \frac {b^2 x^m}{\left (b+a x^2\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {a x^2}{b}\right )}{1+m} \] Output:
x^(1+m)*hypergeom([2, 1/2+1/2*m],[3/2+1/2*m],-a*x^2/b)/(1+m)
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {b^2 x^m}{\left (b+a x^2\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},1+\frac {1+m}{2},-\frac {a x^2}{b}\right )}{1+m} \] Input:
Integrate[(b^2*x^m)/(b + a*x^2)^2,x]
Output:
(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, 1 + (1 + m)/2, -((a*x^2)/b)])/( 1 + m)
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {27, 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.
\(\displaystyle \int \frac {b^2 x^m}{\left (a x^2+b\right )^2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b^2 \int \frac {x^m}{\left (a x^2+b\right )^2}dx\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},-\frac {a x^2}{b}\right )}{m+1}\) |
Input:
Int[(b^2*x^m)/(b + a*x^2)^2,x]
Output:
(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((a*x^2)/b)])/(1 + m)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
\[\int \frac {b^{2} x^{m}}{\left (a \,x^{2}+b \right )^{2}}d x\]
Input:
int(b^2*x^m/(a*x^2+b)^2,x)
Output:
int(b^2*x^m/(a*x^2+b)^2,x)
\[ \int \frac {b^2 x^m}{\left (b+a x^2\right )^2} \, dx=\int { \frac {b^{2} x^{m}}{{\left (a x^{2} + b\right )}^{2}} \,d x } \] Input:
integrate(b^2*x^m/(a*x^2+b)^2,x, algorithm="fricas")
Output:
integral(b^2*x^m/(a^2*x^4 + 2*a*b*x^2 + b^2), x)
Result contains complex when optimal does not.
Time = 2.78 (sec) , antiderivative size = 381, normalized size of antiderivative = 10.58 \[ \int \frac {b^2 x^m}{\left (b+a x^2\right )^2} \, dx=b^{2} \left (- \frac {a m^{2} x^{2} x^{m + 1} \Phi \left (\frac {a x^{2} e^{i \pi }}{b}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a b^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a x^{2} x^{m + 1} \Phi \left (\frac {a x^{2} e^{i \pi }}{b}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a b^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {b m^{2} x^{m + 1} \Phi \left (\frac {a x^{2} e^{i \pi }}{b}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a b^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 b m x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a b^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {b x^{m + 1} \Phi \left (\frac {a x^{2} e^{i \pi }}{b}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a b^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 b x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a b^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}\right ) \] Input:
integrate(b**2*x**m/(a*x**2+b)**2,x)
Output:
b**2*(-a*m**2*x**2*x**(m + 1)*lerchphi(a*x**2*exp_polar(I*pi)/b, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a*b**2*x**2*gamma(m/2 + 3/2) + 8*b**3*gamma(m/2 + 3/2)) + a*x**2*x**(m + 1)*lerchphi(a*x**2*exp_polar(I*pi)/b, 1, m/2 + 1/2 )*gamma(m/2 + 1/2)/(8*a*b**2*x**2*gamma(m/2 + 3/2) + 8*b**3*gamma(m/2 + 3/ 2)) - b*m**2*x**(m + 1)*lerchphi(a*x**2*exp_polar(I*pi)/b, 1, m/2 + 1/2)*g amma(m/2 + 1/2)/(8*a*b**2*x**2*gamma(m/2 + 3/2) + 8*b**3*gamma(m/2 + 3/2)) + 2*b*m*x**(m + 1)*gamma(m/2 + 1/2)/(8*a*b**2*x**2*gamma(m/2 + 3/2) + 8*b **3*gamma(m/2 + 3/2)) + b*x**(m + 1)*lerchphi(a*x**2*exp_polar(I*pi)/b, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a*b**2*x**2*gamma(m/2 + 3/2) + 8*b**3*gamm a(m/2 + 3/2)) + 2*b*x**(m + 1)*gamma(m/2 + 1/2)/(8*a*b**2*x**2*gamma(m/2 + 3/2) + 8*b**3*gamma(m/2 + 3/2)))
\[ \int \frac {b^2 x^m}{\left (b+a x^2\right )^2} \, dx=\int { \frac {b^{2} x^{m}}{{\left (a x^{2} + b\right )}^{2}} \,d x } \] Input:
integrate(b^2*x^m/(a*x^2+b)^2,x, algorithm="maxima")
Output:
b^2*integrate(x^m/(a*x^2 + b)^2, x)
\[ \int \frac {b^2 x^m}{\left (b+a x^2\right )^2} \, dx=\int { \frac {b^{2} x^{m}}{{\left (a x^{2} + b\right )}^{2}} \,d x } \] Input:
integrate(b^2*x^m/(a*x^2+b)^2,x, algorithm="giac")
Output:
integrate(b^2*x^m/(a*x^2 + b)^2, x)
Timed out. \[ \int \frac {b^2 x^m}{\left (b+a x^2\right )^2} \, dx=\int \frac {b^2\,x^m}{{\left (a\,x^2+b\right )}^2} \,d x \] Input:
int((b^2*x^m)/(b + a*x^2)^2,x)
Output:
int((b^2*x^m)/(b + a*x^2)^2, x)
\[ \int \frac {b^2 x^m}{\left (b+a x^2\right )^2} \, dx=\left (\int \frac {x^{m}}{a^{2} x^{4}+2 a b \,x^{2}+b^{2}}d x \right ) b^{2} \] Input:
int(b^2*x^m/(a*x^2+b)^2,x)
Output:
int(x**m/(a**2*x**4 + 2*a*b*x**2 + b**2),x)*b**2