\(\int \frac {x^m (c+d x^2)}{(a+b x^2)^2} \, dx\) [1603]

Optimal result

Integrand size = 20, antiderivative size = 93 \[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {(b c-a d) x^{1+m}}{2 a b \left (a+b x^2\right )}+\frac {(a d (1+m)+b (c-c m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{2 a^2 b (1+m)} \] Output:

1/2*(-a*d+b*c)*x^(1+m)/a/b/(b*x^2+a)+1/2*(a*d*(1+m)+b*(-c*m+c))*x^(1+m)*hy 
pergeom([1, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/a^2/b/(1+m)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {x^{1+m} \left (a d \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+(b c-a d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )\right )}{a^2 b (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {362, 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.

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

\(\Big \downarrow \) 362

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 278

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])
 

rule 362

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e 
*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1))   I 
nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N 
eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || 
  !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 14.19 (sec) , antiderivative size = 906, normalized size of antiderivative = 9.74 \[ \int \frac {x^m \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

c*(-a*m**2*x**(m + 1)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gam 
ma(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*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*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*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)) - b*m**2*x**2*x**(m + 1)*lerchphi(b*x**2*ex 
p_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*x**2*x**(m + 1)*lerchphi(b*x**2*exp_p 
olar(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))) + d*(-a*m**2*x**(m + 3)*lerchphi(b*x**2*exp 
_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 
8*a**2*b*x**2*gamma(m/2 + 5/2)) - 4*a*m*x**(m + 3)*lerchphi(b*x**2*exp_pol 
ar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/2 + 5/2) + 8*a* 
*2*b*x**2*gamma(m/2 + 5/2)) + 2*a*m*x**(m + 3)*gamma(m/2 + 3/2)/(8*a**3*ga 
mma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - 3*a*x**(m + 3)*lerchphi 
(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*a**3*gamma(m/ 
2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) + 6*a*x**(m + 3)*gamma(m/2 + 3/ 
2)/(8*a**3*gamma(m/2 + 5/2) + 8*a**2*b*x**2*gamma(m/2 + 5/2)) - b*m**2*x** 
2*x**(m + 3)*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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