Integrand size = 22, antiderivative size = 100 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {d}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\log (x)}{a c^2}-\frac {b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac {d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2} \] Output:
-1/2*d/c/(-a*d+b*c)/(d*x^2+c)+ln(x)/a/c^2-1/2*b^2*ln(b*x^2+a)/a/(-a*d+b*c) ^2+1/2*d*(-a*d+2*b*c)*ln(d*x^2+c)/c^2/(-a*d+b*c)^2
Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {1}{2} \left (-\frac {d}{c (b c-a d) \left (c+d x^2\right )}+\frac {2 \log (x)}{a c^2}-\frac {b^2 \log \left (a+b x^2\right )}{a (b c-a d)^2}+\frac {d (2 b c-a d) \log \left (c+d x^2\right )}{c^2 (b c-a d)^2}\right ) \] Input:
Integrate[1/(x*(a + b*x^2)*(c + d*x^2)^2),x]
Output:
(-(d/(c*(b*c - a*d)*(c + d*x^2))) + (2*Log[x])/(a*c^2) - (b^2*Log[a + b*x^ 2])/(a*(b*c - a*d)^2) + (d*(2*b*c - a*d)*Log[c + d*x^2])/(c^2*(b*c - a*d)^ 2))/2
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 93, 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.
\(\displaystyle \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^2}dx^2\) |
\(\Big \downarrow \) 93 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {b^3}{a (a d-b c)^2 \left (b x^2+a\right )}+\frac {d^2 (2 b c-a d)}{c^2 (b c-a d)^2 \left (d x^2+c\right )}+\frac {1}{a c^2 x^2}+\frac {d^2}{c (b c-a d) \left (d x^2+c\right )^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {b^2 \log \left (a+b x^2\right )}{a (b c-a d)^2}+\frac {d (2 b c-a d) \log \left (c+d x^2\right )}{c^2 (b c-a d)^2}-\frac {d}{c \left (c+d x^2\right ) (b c-a d)}+\frac {\log \left (x^2\right )}{a c^2}\right )\) |
Input:
Int[1/(x*(a + b*x^2)*(c + d*x^2)^2),x]
Output:
(-(d/(c*(b*c - a*d)*(c + d*x^2))) + Log[x^2]/(a*c^2) - (b^2*Log[a + b*x^2] )/(a*(b*c - a*d)^2) + (d*(2*b*c - a*d)*Log[c + d*x^2])/(c^2*(b*c - a*d)^2) )/2
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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] && IntegerQ [(m - 1)/2]
Time = 0.57 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a \left (a d -b c \right )^{2}}+\frac {\ln \left (x \right )}{a \,c^{2}}-\frac {d^{2} \left (-\frac {\left (a d -b c \right ) c}{d \left (x^{2} d +c \right )}+\frac {\left (a d -2 b c \right ) \ln \left (x^{2} d +c \right )}{d}\right )}{2 \left (a d -b c \right )^{2} c^{2}}\) | \(99\) |
norman | \(-\frac {d^{2} x^{2}}{2 c^{2} \left (a d -b c \right ) \left (x^{2} d +c \right )}+\frac {\ln \left (x \right )}{a \,c^{2}}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {d \left (a d -2 b c \right ) \ln \left (x^{2} d +c \right )}{2 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(125\) |
risch | \(\frac {d}{2 \left (a d -b c \right ) c \left (x^{2} d +c \right )}+\frac {\ln \left (x \right )}{a \,c^{2}}-\frac {d^{2} \ln \left (-x^{2} d -c \right ) a}{2 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {d \ln \left (-x^{2} d -c \right ) b}{c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(158\) |
parallelrisch | \(\frac {2 \ln \left (x \right ) x^{2} a^{2} d^{3}-4 \ln \left (x \right ) x^{2} a b c \,d^{2}+2 \ln \left (x \right ) x^{2} b^{2} c^{2} d -\ln \left (b \,x^{2}+a \right ) x^{2} b^{2} c^{2} d -\ln \left (x^{2} d +c \right ) x^{2} a^{2} d^{3}+2 \ln \left (x^{2} d +c \right ) x^{2} a b c \,d^{2}-a^{2} d^{3} x^{2}+a b c \,d^{2} x^{2}+2 \ln \left (x \right ) a^{2} c \,d^{2}-4 \ln \left (x \right ) a b \,c^{2} d +2 \ln \left (x \right ) b^{2} c^{3}-\ln \left (b \,x^{2}+a \right ) b^{2} c^{3}-\ln \left (x^{2} d +c \right ) a^{2} c \,d^{2}+2 \ln \left (x^{2} d +c \right ) a b \,c^{2} d}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a \left (x^{2} d +c \right ) c^{2}}\) | \(241\) |
Input:
int(1/x/(b*x^2+a)/(d*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*b^2/a/(a*d-b*c)^2*ln(b*x^2+a)+ln(x)/a/c^2-1/2*d^2/(a*d-b*c)^2/c^2*(-( a*d-b*c)*c/d/(d*x^2+c)+(a*d-2*b*c)/d*ln(d*x^2+c))
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (94) = 188\).
Time = 0.56 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.19 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {a b c^{2} d - a^{2} c d^{2} + {\left (b^{2} c^{2} d x^{2} + b^{2} c^{3}\right )} \log \left (b x^{2} + a\right ) - {\left (2 \, a b c^{2} d - a^{2} c d^{2} + {\left (2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a b^{2} c^{5} - 2 \, a^{2} b c^{4} d + a^{3} c^{3} d^{2} + {\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )} x^{2}\right )}} \] Input:
integrate(1/x/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")
Output:
-1/2*(a*b*c^2*d - a^2*c*d^2 + (b^2*c^2*d*x^2 + b^2*c^3)*log(b*x^2 + a) - ( 2*a*b*c^2*d - a^2*c*d^2 + (2*a*b*c*d^2 - a^2*d^3)*x^2)*log(d*x^2 + c) - 2* (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)*log(x))/(a*b^2*c^5 - 2*a^2*b*c^4*d + a^3*c^3*d^2 + (a*b^2*c^4*d - 2*a^ 2*b*c^3*d^2 + a^3*c^2*d^3)*x^2)
Timed out. \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/x/(b*x**2+a)/(d*x**2+c)**2,x)
Output:
Timed out
Time = 0.07 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {b^{2} \log \left (b x^{2} + a\right )}{2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac {{\left (2 \, b c d - a d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac {d}{2 \, {\left (b c^{3} - a c^{2} d + {\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a c^{2}} \] Input:
integrate(1/x/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="maxima")
Output:
-1/2*b^2*log(b*x^2 + a)/(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2) + 1/2*(2*b*c*d - a*d^2)*log(d*x^2 + c)/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2) - 1/2*d/(b* c^3 - a*c^2*d + (b*c^2*d - a*c*d^2)*x^2) + 1/2*log(x^2)/(a*c^2)
Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.85 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}} + \frac {{\left (2 \, b c d^{2} - a d^{3}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )}} - \frac {2 \, b c d^{2} x^{2} - a d^{3} x^{2} + 3 \, b c^{2} d - 2 \, a c d^{2}}{2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a c^{2}} \] Input:
integrate(1/x/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")
Output:
-1/2*b^3*log(abs(b*x^2 + a))/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2) + 1/2 *(2*b*c*d^2 - a*d^3)*log(abs(d*x^2 + c))/(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2* c^2*d^3) - 1/2*(2*b*c*d^2*x^2 - a*d^3*x^2 + 3*b*c^2*d - 2*a*c*d^2)/((b^2*c ^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*(d*x^2 + c)) + 1/2*log(x^2)/(a*c^2)
Time = 1.14 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {\ln \left (x\right )}{a\,c^2}-\frac {\ln \left (d\,x^2+c\right )\,\left (a\,d^2-2\,b\,c\,d\right )}{2\,a^2\,c^2\,d^2-4\,a\,b\,c^3\,d+2\,b^2\,c^4}-\frac {b^2\,\ln \left (b\,x^2+a\right )}{2\,a^3\,d^2-4\,a^2\,b\,c\,d+2\,a\,b^2\,c^2}+\frac {d}{2\,c\,\left (d\,x^2+c\right )\,\left (a\,d-b\,c\right )} \] Input:
int(1/(x*(a + b*x^2)*(c + d*x^2)^2),x)
Output:
log(x)/(a*c^2) - (log(c + d*x^2)*(a*d^2 - 2*b*c*d))/(2*b^2*c^4 + 2*a^2*c^2 *d^2 - 4*a*b*c^3*d) - (b^2*log(a + b*x^2))/(2*a^3*d^2 + 2*a*b^2*c^2 - 4*a^ 2*b*c*d) + d/(2*c*(c + d*x^2)*(a*d - b*c))
Time = 0.19 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.66 \[ \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {-\mathrm {log}\left (b \,x^{2}+a \right ) b^{2} c^{3}-\mathrm {log}\left (b \,x^{2}+a \right ) b^{2} c^{2} d \,x^{2}-\mathrm {log}\left (d \,x^{2}+c \right ) a^{2} c \,d^{2}-\mathrm {log}\left (d \,x^{2}+c \right ) a^{2} d^{3} x^{2}+2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a b \,c^{2} d +2 \,\mathrm {log}\left (d \,x^{2}+c \right ) a b c \,d^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) a^{2} c \,d^{2}+2 \,\mathrm {log}\left (x \right ) a^{2} d^{3} x^{2}-4 \,\mathrm {log}\left (x \right ) a b \,c^{2} d -4 \,\mathrm {log}\left (x \right ) a b c \,d^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) b^{2} c^{3}+2 \,\mathrm {log}\left (x \right ) b^{2} c^{2} d \,x^{2}-a^{2} d^{3} x^{2}+a b c \,d^{2} x^{2}}{2 a \,c^{2} \left (a^{2} d^{3} x^{2}-2 a b c \,d^{2} x^{2}+b^{2} c^{2} d \,x^{2}+a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:
int(1/x/(b*x^2+a)/(d*x^2+c)^2,x)
Output:
( - log(a + b*x**2)*b**2*c**3 - log(a + b*x**2)*b**2*c**2*d*x**2 - log(c + d*x**2)*a**2*c*d**2 - log(c + d*x**2)*a**2*d**3*x**2 + 2*log(c + d*x**2)* a*b*c**2*d + 2*log(c + d*x**2)*a*b*c*d**2*x**2 + 2*log(x)*a**2*c*d**2 + 2* log(x)*a**2*d**3*x**2 - 4*log(x)*a*b*c**2*d - 4*log(x)*a*b*c*d**2*x**2 + 2 *log(x)*b**2*c**3 + 2*log(x)*b**2*c**2*d*x**2 - a**2*d**3*x**2 + a*b*c*d** 2*x**2)/(2*a*c**2*(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))