3.5.100 \(\int (\frac {x^m}{(1-a^2 x^2)^2}+\frac {a x^{1+m}}{(1-a^2 x^2)^2}) \, dx\) [500]

3.5.100.1 Optimal result

Integrand size = 36, antiderivative size = 70 \[ \int \left (\frac {x^m}{\left (1-a^2 x^2\right )^2}+\frac {a x^{1+m}}{\left (1-a^2 x^2\right )^2}\right ) \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{2+m} \]

output

x^(1+m)*hypergeom([2, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/(1+m)+a*x^(2+m)*hype 
rgeom([2, 1+1/2*m],[2+1/2*m],a^2*x^2)/(2+m)
 

3.5.100.2 Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96 \[ \int \left (\frac {x^m}{\left (1-a^2 x^2\right )^2}+\frac {a x^{1+m}}{\left (1-a^2 x^2\right )^2}\right ) \, dx=x^{1+m} \left (\frac {a x \operatorname {Hypergeometric2F1}\left (2,1+\frac {m}{2},2+\frac {m}{2},a^2 x^2\right )}{2+m}+\frac {\operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{1+m}\right ) \]

input

Integrate[x^m/(1 - a^2*x^2)^2 + (a*x^(1 + m))/(1 - a^2*x^2)^2,x]
 

output

x^(1 + m)*((a*x*Hypergeometric2F1[2, 1 + m/2, 2 + m/2, a^2*x^2])/(2 + m) + 
 Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, a^2*x^2]/(1 + m))
 

3.5.100.3 Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {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.

\(\displaystyle \int \left (\frac {a x^{m+1}}{\left (1-a^2 x^2\right )^2}+\frac {x^m}{\left (1-a^2 x^2\right )^2}\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{m+2}\)

input

Int[x^m/(1 - a^2*x^2)^2 + (a*x^(1 + m))/(1 - a^2*x^2)^2,x]
 

output

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + 
(a*x^(2 + m)*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)
 

3.5.100.3.1 Defintions of rubi rules used

rule 2009

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

3.5.100.4 Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5.

Time = 4.83 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.53

method	result	size
meijerg	\(-\frac {\left (-a^{2}\right )^{-\frac {m}{2}} \left (\frac {x^{m} \left (-a^{2}\right )^{\frac {m}{2}} \left (-2-m \right )}{\left (2+m \right ) \left (-a^{2} x^{2}+1\right )}+\frac {x^{m} \left (-a^{2}\right )^{\frac {m}{2}} m \Phi \left (a^{2} x^{2}, 1, \frac {m}{2}\right )}{2}\right )}{2 a}+\frac {\left (-a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (-\frac {2 x^{1+m} \left (-a^{2}\right )^{\frac {1}{2}+\frac {m}{2}} \left (-1-m \right )}{\left (1+m \right ) \left (-2 a^{2} x^{2}+2\right )}+\frac {2 x^{1+m} \left (-a^{2}\right )^{\frac {1}{2}+\frac {m}{2}} \left (-\frac {m^{2}}{4}+\frac {1}{4}\right ) \Phi \left (a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{1+m}\right )}{2}\)	\(177\)

input

int(x^m/(-a^2*x^2+1)^2+a*x^(1+m)/(-a^2*x^2+1)^2,x,method=_RETURNVERBOSE)
 

output

-1/2/a*(-a^2)^(-1/2*m)*(1/(2+m)*x^m*(-a^2)^(1/2*m)*(-2-m)/(-a^2*x^2+1)+1/2 
*x^m*(-a^2)^(1/2*m)*m*LerchPhi(a^2*x^2,1,1/2*m))+1/2*(-a^2)^(-1/2-1/2*m)*( 
-2/(1+m)*x^(1+m)*(-a^2)^(1/2+1/2*m)*(-1-m)/(-2*a^2*x^2+2)+2/(1+m)*x^(1+m)* 
(-a^2)^(1/2+1/2*m)*(-1/4*m^2+1/4)*LerchPhi(a^2*x^2,1,1/2+1/2*m))
 

3.5.100.5 Fricas [F]

\[ \int \left (\frac {x^m}{\left (1-a^2 x^2\right )^2}+\frac {a x^{1+m}}{\left (1-a^2 x^2\right )^2}\right ) \, dx=\int { \frac {a x^{m + 1}}{{\left (a^{2} x^{2} - 1\right )}^{2}} + \frac {x^{m}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \]

input

integrate(x^m/(-a^2*x^2+1)^2+a*x^(1+m)/(-a^2*x^2+1)^2,x, algorithm="fricas 
")
 

output

integral((a*x^(m + 1) + x^m)/(a^4*x^4 - 2*a^2*x^2 + 1), x)
 

3.5.100.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.93 (sec) , antiderivative size = 673, normalized size of antiderivative = 9.61 \[ \int \left (\frac {x^m}{\left (1-a^2 x^2\right )^2}+\frac {a x^{1+m}}{\left (1-a^2 x^2\right )^2}\right ) \, dx=- \frac {a^{2} m^{2} x^{2} x^{m + 1} \Phi \left (a^{2} x^{2} e^{2 i \pi }, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) - 8 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a^{2} x^{2} x^{m + 1} \Phi \left (a^{2} x^{2} e^{2 i \pi }, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) - 8 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + a \left (- \frac {a^{2} m^{2} x^{2} x^{m + 2} \Phi \left (a^{2} x^{2} e^{2 i \pi }, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + 2\right ) - 8 \Gamma \left (\frac {m}{2} + 2\right )} - \frac {2 a^{2} m x^{2} x^{m + 2} \Phi \left (a^{2} x^{2} e^{2 i \pi }, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + 2\right ) - 8 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {m^{2} x^{m + 2} \Phi \left (a^{2} x^{2} e^{2 i \pi }, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + 2\right ) - 8 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {2 m x^{m + 2} \Phi \left (a^{2} x^{2} e^{2 i \pi }, 1, \frac {m}{2} + 1\right ) \Gamma \left (\frac {m}{2} + 1\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + 2\right ) - 8 \Gamma \left (\frac {m}{2} + 2\right )} - \frac {2 m x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + 2\right ) - 8 \Gamma \left (\frac {m}{2} + 2\right )} - \frac {4 x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + 2\right ) - 8 \Gamma \left (\frac {m}{2} + 2\right )}\right ) + \frac {m^{2} x^{m + 1} \Phi \left (a^{2} x^{2} e^{2 i \pi }, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) - 8 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {2 m x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) - 8 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {x^{m + 1} \Phi \left (a^{2} x^{2} e^{2 i \pi }, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) - 8 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {2 x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a^{2} x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) - 8 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \]

input

integrate(x**m/(-a**2*x**2+1)**2+a*x**(1+m)/(-a**2*x**2+1)**2,x)
 

output

-a**2*m**2*x**2*x**(m + 1)*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 
1/2)*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3/2) - 8*gamma(m/2 + 3/2)) 
+ a**2*x**2*x**(m + 1)*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1/2) 
*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3/2) - 8*gamma(m/2 + 3/2)) + a* 
(-a**2*m**2*x**2*x**(m + 2)*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 
 1)*gamma(m/2 + 1)/(8*a**2*x**2*gamma(m/2 + 2) - 8*gamma(m/2 + 2)) - 2*a** 
2*m*x**2*x**(m + 2)*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamm 
a(m/2 + 1)/(8*a**2*x**2*gamma(m/2 + 2) - 8*gamma(m/2 + 2)) + m**2*x**(m + 
2)*lerchphi(a**2*x**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(8*a** 
2*x**2*gamma(m/2 + 2) - 8*gamma(m/2 + 2)) + 2*m*x**(m + 2)*lerchphi(a**2*x 
**2*exp_polar(2*I*pi), 1, m/2 + 1)*gamma(m/2 + 1)/(8*a**2*x**2*gamma(m/2 + 
 2) - 8*gamma(m/2 + 2)) - 2*m*x**(m + 2)*gamma(m/2 + 1)/(8*a**2*x**2*gamma 
(m/2 + 2) - 8*gamma(m/2 + 2)) - 4*x**(m + 2)*gamma(m/2 + 1)/(8*a**2*x**2*g 
amma(m/2 + 2) - 8*gamma(m/2 + 2))) + m**2*x**(m + 1)*lerchphi(a**2*x**2*ex 
p_polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3 
/2) - 8*gamma(m/2 + 3/2)) - 2*m*x**(m + 1)*gamma(m/2 + 1/2)/(8*a**2*x**2*g 
amma(m/2 + 3/2) - 8*gamma(m/2 + 3/2)) - x**(m + 1)*lerchphi(a**2*x**2*exp_ 
polar(2*I*pi), 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma(m/2 + 3/2 
) - 8*gamma(m/2 + 3/2)) - 2*x**(m + 1)*gamma(m/2 + 1/2)/(8*a**2*x**2*gamma 
(m/2 + 3/2) - 8*gamma(m/2 + 3/2))
 

3.5.100.7 Maxima [F]

\[ \int \left (\frac {x^m}{\left (1-a^2 x^2\right )^2}+\frac {a x^{1+m}}{\left (1-a^2 x^2\right )^2}\right ) \, dx=\int { \frac {a x^{m + 1}}{{\left (a^{2} x^{2} - 1\right )}^{2}} + \frac {x^{m}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \]

input

integrate(x^m/(-a^2*x^2+1)^2+a*x^(1+m)/(-a^2*x^2+1)^2,x, algorithm="maxima 
")
 

output

integrate(a*x^(m + 1)/(a^2*x^2 - 1)^2 + x^m/(a^2*x^2 - 1)^2, x)
 

3.5.100.8 Giac [F]

\[ \int \left (\frac {x^m}{\left (1-a^2 x^2\right )^2}+\frac {a x^{1+m}}{\left (1-a^2 x^2\right )^2}\right ) \, dx=\int { \frac {a x^{m + 1}}{{\left (a^{2} x^{2} - 1\right )}^{2}} + \frac {x^{m}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \]

input

integrate(x^m/(-a^2*x^2+1)^2+a*x^(1+m)/(-a^2*x^2+1)^2,x, algorithm="giac")
 

output

integrate(a*x^(m + 1)/(a^2*x^2 - 1)^2 + x^m/(a^2*x^2 - 1)^2, x)
 

3.5.100.9 Mupad [F(-1)]

Timed out. \[ \int \left (\frac {x^m}{\left (1-a^2 x^2\right )^2}+\frac {a x^{1+m}}{\left (1-a^2 x^2\right )^2}\right ) \, dx=\int \frac {x^m}{{\left (a^2\,x^2-1\right )}^2}+\frac {a\,x^{m+1}}{{\left (a^2\,x^2-1\right )}^2} \,d x \]

input

int(x^m/(a^2*x^2 - 1)^2 + (a*x^(m + 1))/(a^2*x^2 - 1)^2,x)
 

output

int(x^m/(a^2*x^2 - 1)^2 + (a*x^(m + 1))/(a^2*x^2 - 1)^2, x)