Integrand size = 20, antiderivative size = 98 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {1}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {b}{2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {b^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {b^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
1/4/(-a*d+b*c)/(d*x^2+c)^2+1/2*b/(-a*d+b*c)^2/(d*x^2+c)+1/2*b^2*ln(b*x^2+a )/(-a*d+b*c)^3-1/2*b^2*ln(d*x^2+c)/(-a*d+b*c)^3
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {(b c-a d) \left (3 b c-a d+2 b d x^2\right )+2 b^2 \left (c+d x^2\right )^2 \log \left (a+b x^2\right )-2 b^2 \left (c+d x^2\right )^2 \log \left (c+d x^2\right )}{4 (b c-a d)^3 \left (c+d x^2\right )^2} \]
((b*c - a*d)*(3*b*c - a*d + 2*b*d*x^2) + 2*b^2*(c + d*x^2)^2*Log[a + b*x^2 ] - 2*b^2*(c + d*x^2)^2*Log[c + d*x^2])/(4*(b*c - a*d)^3*(c + d*x^2)^2)
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96, 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.
\(\displaystyle \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}dx^2\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{2} \int \left (\frac {b^3}{(b c-a d)^3 \left (b x^2+a\right )}-\frac {d b^2}{(b c-a d)^3 \left (d x^2+c\right )}-\frac {d b}{(b c-a d)^2 \left (d x^2+c\right )^2}-\frac {d}{(b c-a d) \left (d x^2+c\right )^3}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {b^2 \log \left (a+b x^2\right )}{(b c-a d)^3}-\frac {b^2 \log \left (c+d x^2\right )}{(b c-a d)^3}+\frac {b}{\left (c+d x^2\right ) (b c-a d)^2}+\frac {1}{2 \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
(1/(2*(b*c - a*d)*(c + d*x^2)^2) + b/((b*c - a*d)^2*(c + d*x^2)) + (b^2*Lo g[a + b*x^2])/(b*c - a*d)^3 - (b^2*Log[c + d*x^2])/(b*c - a*d)^3)/2
3.3.55.3.1 Defintions of rubi rules used
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 = 2.71 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.13
method | result | size |
default | \(-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{3}}+\frac {d \left (-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 d \left (d \,x^{2}+c \right )^{2}}+\frac {b^{2} \ln \left (d \,x^{2}+c \right )}{d}+\frac {b \left (a d -b c \right )}{d \left (d \,x^{2}+c \right )}\right )}{2 \left (a d -b c \right )^{3}}\) | \(111\) |
risch | \(\frac {\frac {b d \,x^{2}}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}-\frac {a d -3 b c}{4 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d \,x^{2}+c \right )^{2}}-\frac {b^{2} \ln \left (-b \,x^{2}-a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{2} \ln \left (d \,x^{2}+c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}\) | \(181\) |
norman | \(\frac {\frac {-a \,d^{3}+3 b c \,d^{2}}{4 d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b d \,x^{2}}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}}{\left (d \,x^{2}+c \right )^{2}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{2} \ln \left (d \,x^{2}+c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}\) | \(187\) |
parallelrisch | \(-\frac {2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{2} d^{4}-2 \ln \left (d \,x^{2}+c \right ) x^{4} b^{2} d^{4}+4 \ln \left (b \,x^{2}+a \right ) x^{2} b^{2} c \,d^{3}-4 \ln \left (d \,x^{2}+c \right ) x^{2} b^{2} c \,d^{3}-2 x^{2} a b \,d^{4}+2 x^{2} b^{2} c \,d^{3}+2 \ln \left (b \,x^{2}+a \right ) b^{2} c^{2} d^{2}-2 \ln \left (d \,x^{2}+c \right ) b^{2} c^{2} d^{2}+a^{2} d^{4}-4 a b c \,d^{3}+3 b^{2} c^{2} d^{2}}{4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d \,x^{2}+c \right )^{2} d^{2}}\) | \(218\) |
-1/2*b^2/(a*d-b*c)^3*ln(b*x^2+a)+1/2*d/(a*d-b*c)^3*(-1/2*(a^2*d^2-2*a*b*c* d+b^2*c^2)/d/(d*x^2+c)^2+b^2/d*ln(d*x^2+c)+b/d*(a*d-b*c)/(d*x^2+c))
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (90) = 180\).
Time = 0.25 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.59 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \, {\left (b^{2} d^{2} x^{4} + 2 \, b^{2} c d x^{2} + b^{2} c^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{2} d^{2} x^{4} + 2 \, b^{2} c d x^{2} + b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{4} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x^{2}\right )}} \]
1/4*(3*b^2*c^2 - 4*a*b*c*d + a^2*d^2 + 2*(b^2*c*d - a*b*d^2)*x^2 + 2*(b^2* d^2*x^4 + 2*b^2*c*d*x^2 + b^2*c^2)*log(b*x^2 + a) - 2*(b^2*d^2*x^4 + 2*b^2 *c*d*x^2 + b^2*c^2)*log(d*x^2 + c))/(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3 *d^2 - a^3*c^2*d^3 + (b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3* d^5)*x^4 + 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*x ^2)
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (80) = 160\).
Time = 2.41 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.99 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{2} \log {\left (x^{2} + \frac {- \frac {a^{4} b^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b^{3} c d^{3}}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{4} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{5} c^{3} d}{\left (a d - b c\right )^{3}} + a b^{2} d - \frac {b^{6} c^{4}}{\left (a d - b c\right )^{3}} + b^{3} c}{2 b^{3} d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac {b^{2} \log {\left (x^{2} + \frac {\frac {a^{4} b^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b^{3} c d^{3}}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{4} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{5} c^{3} d}{\left (a d - b c\right )^{3}} + a b^{2} d + \frac {b^{6} c^{4}}{\left (a d - b c\right )^{3}} + b^{3} c}{2 b^{3} d} \right )}}{2 \left (a d - b c\right )^{3}} + \frac {- a d + 3 b c + 2 b d x^{2}}{4 a^{2} c^{2} d^{2} - 8 a b c^{3} d + 4 b^{2} c^{4} + x^{4} \cdot \left (4 a^{2} d^{4} - 8 a b c d^{3} + 4 b^{2} c^{2} d^{2}\right ) + x^{2} \cdot \left (8 a^{2} c d^{3} - 16 a b c^{2} d^{2} + 8 b^{2} c^{3} d\right )} \]
b**2*log(x**2 + (-a**4*b**2*d**4/(a*d - b*c)**3 + 4*a**3*b**3*c*d**3/(a*d - b*c)**3 - 6*a**2*b**4*c**2*d**2/(a*d - b*c)**3 + 4*a*b**5*c**3*d/(a*d - b*c)**3 + a*b**2*d - b**6*c**4/(a*d - b*c)**3 + b**3*c)/(2*b**3*d))/(2*(a* d - b*c)**3) - b**2*log(x**2 + (a**4*b**2*d**4/(a*d - b*c)**3 - 4*a**3*b** 3*c*d**3/(a*d - b*c)**3 + 6*a**2*b**4*c**2*d**2/(a*d - b*c)**3 - 4*a*b**5* c**3*d/(a*d - b*c)**3 + a*b**2*d + b**6*c**4/(a*d - b*c)**3 + b**3*c)/(2*b **3*d))/(2*(a*d - b*c)**3) + (-a*d + 3*b*c + 2*b*d*x**2)/(4*a**2*c**2*d**2 - 8*a*b*c**3*d + 4*b**2*c**4 + x**4*(4*a**2*d**4 - 8*a*b*c*d**3 + 4*b**2* c**2*d**2) + x**2*(8*a**2*c*d**3 - 16*a*b*c**2*d**2 + 8*b**2*c**3*d))
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (90) = 180\).
Time = 0.21 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.15 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{2} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {b^{2} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {2 \, b d x^{2} + 3 \, b c - a d}{4 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}} \]
1/2*b^2*log(b*x^2 + a)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/2*b^2*log(d*x^2 + c)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d ^3) + 1/4*(2*b*d*x^2 + 3*b*c - a*d)/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^4 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2)
Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.78 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac {b^{2} d \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )}} + \frac {3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} c d - a b d^{2}\right )} x^{2}}{4 \, {\left (d x^{2} + c\right )}^{2} {\left (b c - a d\right )}^{3}} \]
1/2*b^3*log(abs(b*x^2 + a))/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a ^3*b*d^3) - 1/2*b^2*d*log(abs(d*x^2 + c))/(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3 *a^2*b*c*d^3 - a^3*d^4) + 1/4*(3*b^2*c^2 - 4*a*b*c*d + a^2*d^2 + 2*(b^2*c* d - a*b*d^2)*x^2)/((d*x^2 + c)^2*(b*c - a*d)^3)
Time = 0.30 (sec) , antiderivative size = 340, normalized size of antiderivative = 3.47 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {a^2\,d^2+3\,b^2\,c^2+b^2\,c^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}+b^2\,d^2\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}-2\,a\,b\,d^2\,x^2+2\,b^2\,c\,d\,x^2-4\,a\,b\,c\,d+b^2\,c\,d\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,8{}\mathrm {i}}{-4\,a^3\,c^2\,d^3-8\,a^3\,c\,d^4\,x^2-4\,a^3\,d^5\,x^4+12\,a^2\,b\,c^3\,d^2+24\,a^2\,b\,c^2\,d^3\,x^2+12\,a^2\,b\,c\,d^4\,x^4-12\,a\,b^2\,c^4\,d-24\,a\,b^2\,c^3\,d^2\,x^2-12\,a\,b^2\,c^2\,d^3\,x^4+4\,b^3\,c^5+8\,b^3\,c^4\,d\,x^2+4\,b^3\,c^3\,d^2\,x^4} \]
(a^2*d^2 + 3*b^2*c^2 + b^2*c^2*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d *x^2 + b*c*x^2))*4i + b^2*d^2*x^4*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*4i - 2*a*b*d^2*x^2 + 2*b^2*c*d*x^2 - 4*a*b*c*d + b^2*c *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))*8i)/(4* b^3*c^5 - 4*a^3*c^2*d^3 - 4*a^3*d^5*x^4 + 12*a^2*b*c^3*d^2 - 8*a^3*c*d^4*x ^2 + 8*b^3*c^4*d*x^2 + 4*b^3*c^3*d^2*x^4 - 12*a*b^2*c^4*d + 12*a^2*b*c*d^4 *x^4 - 24*a*b^2*c^3*d^2*x^2 + 24*a^2*b*c^2*d^3*x^2 - 12*a*b^2*c^2*d^3*x^4)