Integrand size = 20, antiderivative size = 70 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {1}{2 (b c-a d) \left (c+d x^2\right )}+\frac {b \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac {b \log \left (c+d x^2\right )}{2 (b c-a d)^2} \] Output:
1/2/(-a*d+b*c)/(d*x^2+c)+1/2*b*ln(b*x^2+a)/(-a*d+b*c)^2-1/2*b*ln(d*x^2+c)/ (-a*d+b*c)^2
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b c-a d+b \left (c+d x^2\right ) \log \left (a+b x^2\right )-b \left (c+d x^2\right ) \log \left (c+d x^2\right )}{2 (b c-a d)^2 \left (c+d x^2\right )} \] Input:
Integrate[x/((a + b*x^2)*(c + d*x^2)^2),x]
Output:
(b*c - a*d + b*(c + d*x^2)*Log[a + b*x^2] - b*(c + d*x^2)*Log[c + d*x^2])/ (2*(b*c - a*d)^2*(c + d*x^2))
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {353, 54, 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.
\(\displaystyle \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (\frac {b^2}{(b c-a d)^2 \left (b x^2+a\right )}-\frac {d b}{(b c-a d)^2 \left (d x^2+c\right )}-\frac {d}{(b c-a d) \left (d x^2+c\right )^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{\left (c+d x^2\right ) (b c-a d)}+\frac {b \log \left (a+b x^2\right )}{(b c-a d)^2}-\frac {b \log \left (c+d x^2\right )}{(b c-a d)^2}\right )\) |
Input:
Int[x/((a + b*x^2)*(c + d*x^2)^2),x]
Output:
(1/((b*c - a*d)*(c + d*x^2)) + (b*Log[a + b*x^2])/(b*c - a*d)^2 - (b*Log[c + d*x^2])/(b*c - a*d)^2)/2
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Time = 0.57 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {b \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{2}}+\frac {d \left (-\frac {a d -b c}{d \left (x^{2} d +c \right )}-\frac {b \ln \left (x^{2} d +c \right )}{d}\right )}{2 \left (a d -b c \right )^{2}}\) | \(73\) |
risch | \(-\frac {1}{2 \left (a d -b c \right ) \left (x^{2} d +c \right )}+\frac {b \ln \left (b \,x^{2}+a \right )}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}-\frac {b \ln \left (-x^{2} d -c \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(94\) |
norman | \(\frac {d \,x^{2}}{2 c \left (a d -b c \right ) \left (x^{2} d +c \right )}+\frac {b \ln \left (b \,x^{2}+a \right )}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}-\frac {b \ln \left (x^{2} d +c \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(98\) |
parallelrisch | \(\frac {\ln \left (b \,x^{2}+a \right ) x^{2} b \,d^{2}-\ln \left (x^{2} d +c \right ) x^{2} b \,d^{2}+\ln \left (b \,x^{2}+a \right ) b c d -\ln \left (x^{2} d +c \right ) b c d -a \,d^{2}+b c d}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (x^{2} d +c \right ) d}\) | \(107\) |
Input:
int(x/(b*x^2+a)/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
1/2*b/(a*d-b*c)^2*ln(b*x^2+a)+1/2*d/(a*d-b*c)^2*(-(a*d-b*c)/d/(d*x^2+c)-b/ d*ln(d*x^2+c))
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.47 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b c - a d + {\left (b d x^{2} + b c\right )} \log \left (b x^{2} + a\right ) - {\left (b d x^{2} + b c\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )}} \] Input:
integrate(x/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")
Output:
1/2*(b*c - a*d + (b*d*x^2 + b*c)*log(b*x^2 + a) - (b*d*x^2 + b*c)*log(d*x^ 2 + c))/(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + a^ 2*d^3)*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (56) = 112\).
Time = 0.86 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.54 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=- \frac {b \log {\left (x^{2} + \frac {- \frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac {b \log {\left (x^{2} + \frac {\frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{2 \left (a d - b c\right )^{2}} - \frac {1}{2 a c d - 2 b c^{2} + x^{2} \cdot \left (2 a d^{2} - 2 b c d\right )} \] Input:
integrate(x/(b*x**2+a)/(d*x**2+c)**2,x)
Output:
-b*log(x**2 + (-a**3*b*d**3/(a*d - b*c)**2 + 3*a**2*b**2*c*d**2/(a*d - b*c )**2 - 3*a*b**3*c**2*d/(a*d - b*c)**2 + a*b*d + b**4*c**3/(a*d - b*c)**2 + b**2*c)/(2*b**2*d))/(2*(a*d - b*c)**2) + b*log(x**2 + (a**3*b*d**3/(a*d - b*c)**2 - 3*a**2*b**2*c*d**2/(a*d - b*c)**2 + 3*a*b**3*c**2*d/(a*d - b*c) **2 + a*b*d - b**4*c**3/(a*d - b*c)**2 + b**2*c)/(2*b**2*d))/(2*(a*d - b*c )**2) - 1/(2*a*c*d - 2*b*c**2 + x**2*(2*a*d**2 - 2*b*c*d))
Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.41 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {b \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {1}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}} \] Input:
integrate(x/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="maxima")
Output:
1/2*b*log(b*x^2 + a)/(b^2*c^2 - 2*a*b*c*d + a^2*d^2) - 1/2*b*log(d*x^2 + c )/(b^2*c^2 - 2*a*b*c*d + a^2*d^2) + 1/2/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x ^2)
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.21 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b d \log \left ({\left | b - \frac {b c}{d x^{2} + c} + \frac {a d}{d x^{2} + c} \right |}\right )}{2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} + \frac {d}{2 \, {\left (b c d - a d^{2}\right )} {\left (d x^{2} + c\right )}} \] Input:
integrate(x/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")
Output:
1/2*b*d*log(abs(b - b*c/(d*x^2 + c) + a*d/(d*x^2 + c)))/(b^2*c^2*d - 2*a*b *c*d^2 + a^2*d^3) + 1/2*d/((b*c*d - a*d^2)*(d*x^2 + c))
Time = 0.63 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.29 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {-a\,d+c\,\left (b+b\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}\right )+b\,d\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}}{2\,a^2\,c\,d^2+2\,a^2\,d^3\,x^2-4\,a\,b\,c^2\,d-4\,a\,b\,c\,d^2\,x^2+2\,b^2\,c^3+2\,b^2\,c^2\,d\,x^2} \] Input:
int(x/((a + b*x^2)*(c + d*x^2)^2),x)
Output:
(c*(b + b*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*2i) - a*d + b*d*x^2*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2) )*2i)/(2*b^2*c^3 + 2*a^2*c*d^2 + 2*a^2*d^3*x^2 + 2*b^2*c^2*d*x^2 - 4*a*b*c ^2*d - 4*a*b*c*d^2*x^2)
Time = 0.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.97 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {\mathrm {log}\left (b \,x^{2}+a \right ) b \,c^{2}+\mathrm {log}\left (b \,x^{2}+a \right ) b c d \,x^{2}-\mathrm {log}\left (d \,x^{2}+c \right ) b \,c^{2}-\mathrm {log}\left (d \,x^{2}+c \right ) b c d \,x^{2}+a \,d^{2} x^{2}-b c d \,x^{2}}{2 c \left (a^{2} d^{3} x^{2}-2 a b c \,d^{2} x^{2}+b^{2} c^{2} d \,x^{2}+a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:
int(x/(b*x^2+a)/(d*x^2+c)^2,x)
Output:
(log(a + b*x**2)*b*c**2 + log(a + b*x**2)*b*c*d*x**2 - log(c + d*x**2)*b*c **2 - log(c + d*x**2)*b*c*d*x**2 + a*d**2*x**2 - b*c*d*x**2)/(2*c*(a**2*c* d**2 + a**2*d**3*x**2 - 2*a*b*c**2*d - 2*a*b*c*d**2*x**2 + b**2*c**3 + b** 2*c**2*d*x**2))