Integrand size = 23, antiderivative size = 107 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^2} \, dx=\frac {d}{\left (b^2 c^2-a^2 d^2\right ) (c+d x)}-\frac {b \log (a-b x)}{2 a (b c+a d)^2}+\frac {b \log (a+b x)}{2 a (b c-a d)^2}-\frac {2 b^2 c d \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^2} \]
d/(-a^2*d^2+b^2*c^2)/(d*x+c)-1/2*b*ln(-b*x+a)/a/(a*d+b*c)^2+1/2*b*ln(b*x+a )/a/(-a*d+b*c)^2-2*b^2*c*d*ln(d*x+c)/(-a^2*d^2+b^2*c^2)^2
Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^2} \, dx=\frac {1}{2} \left (-\frac {b \log (a-b x)}{a (b c+a d)^2}+\frac {\frac {b \log (a+b x)}{a}-\frac {2 d \left (-b^2 c^2+a^2 d^2+2 b^2 c (c+d x) \log (c+d x)\right )}{(b c+a d)^2 (c+d x)}}{(b c-a d)^2}\right ) \]
(-((b*Log[a - b*x])/(a*(b*c + a*d)^2)) + ((b*Log[a + b*x])/a - (2*d*(-(b^2 *c^2) + a^2*d^2 + 2*b^2*c*(c + d*x)*Log[c + d*x]))/((b*c + a*d)^2*(c + d*x )))/(b*c - a*d)^2)/2
Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {82, 477, 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.
\(\displaystyle \int \frac {1}{(a-b x) (a+b x) (c+d x)^2} \, dx\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \int \frac {1}{\left (a^2-b^2 x^2\right ) (c+d x)^2}dx\) |
\(\Big \downarrow \) 477 |
\(\displaystyle \frac {\int \left (\frac {a b^2}{2 (b c+a d)^2 (a-b x)}+\frac {a b^2}{2 (b c-a d)^2 (a+b x)}-\frac {2 a^2 c d^2 b^2}{\left (b^2 c^2-a^2 d^2\right )^2 (c+d x)}-\frac {a^2 d^2}{\left (b^2 c^2-a^2 d^2\right ) (c+d x)^2}\right )dx}{a^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {a^2 d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )}-\frac {2 a^2 b^2 c d \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^2}-\frac {a b \log (a-b x)}{2 (a d+b c)^2}+\frac {a b \log (a+b x)}{2 (b c-a d)^2}}{a^2}\) |
((a^2*d)/((b^2*c^2 - a^2*d^2)*(c + d*x)) - (a*b*Log[a - b*x])/(2*(b*c + a* d)^2) + (a*b*Log[a + b*x])/(2*(b*c - a*d)^2) - (2*a^2*b^2*c*d*Log[c + d*x] )/(b^2*c^2 - a^2*d^2)^2)/a^2
3.16.36.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ a^p Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 ]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & & NiceSqrtQ[-b/a] && !FractionalPowerFactorQ[Rt[-b/a, 2]]
Time = 0.93 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {d}{\left (a d +b c \right ) \left (a d -b c \right ) \left (d x +c \right )}-\frac {2 b^{2} d c \ln \left (d x +c \right )}{\left (a d +b c \right )^{2} \left (a d -b c \right )^{2}}+\frac {b \ln \left (b x +a \right )}{2 a \left (a d -b c \right )^{2}}-\frac {b \ln \left (-b x +a \right )}{2 a \left (a d +b c \right )^{2}}\) | \(107\) |
risch | \(-\frac {d}{\left (a^{2} d^{2}-b^{2} c^{2}\right ) \left (d x +c \right )}+\frac {b \ln \left (b x +a \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a}-\frac {b \ln \left (b x -a \right )}{2 \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) a}-\frac {2 b^{2} c d \ln \left (-d x -c \right )}{a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}}\) | \(149\) |
norman | \(\frac {d^{2} x}{c \left (a^{2} d^{2}-b^{2} c^{2}\right ) \left (d x +c \right )}+\frac {b \ln \left (b x +a \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a}-\frac {b \ln \left (-b x +a \right )}{2 \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) a}-\frac {2 b^{2} c d \ln \left (d x +c \right )}{a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}}\) | \(150\) |
parallelrisch | \(-\frac {\ln \left (b x -a \right ) x \,a^{2} b c \,d^{3}-2 \ln \left (b x -a \right ) x a \,b^{2} c^{2} d^{2}+\ln \left (b x -a \right ) x \,b^{3} c^{3} d -\ln \left (b x +a \right ) x \,a^{2} b c \,d^{3}-2 \ln \left (b x +a \right ) x a \,b^{2} c^{2} d^{2}-\ln \left (b x +a \right ) x \,b^{3} c^{3} d +4 \ln \left (d x +c \right ) x a \,b^{2} c^{2} d^{2}+\ln \left (b x -a \right ) a^{2} b \,c^{2} d^{2}-2 \ln \left (b x -a \right ) a \,b^{2} c^{3} d +\ln \left (b x -a \right ) b^{3} c^{4}-\ln \left (b x +a \right ) a^{2} b \,c^{2} d^{2}-2 \ln \left (b x +a \right ) a \,b^{2} c^{3} d -\ln \left (b x +a \right ) b^{3} c^{4}+4 \ln \left (d x +c \right ) a \,b^{2} c^{3} d -2 x \,a^{3} d^{4}+2 x a \,b^{2} c^{2} d^{2}}{2 \left (a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}\right ) a \left (d x +c \right ) c}\) | \(313\) |
-d/(a*d+b*c)/(a*d-b*c)/(d*x+c)-2*b^2*d*c/(a*d+b*c)^2/(a*d-b*c)^2*ln(d*x+c) +1/2/a*b/(a*d-b*c)^2*ln(b*x+a)-1/2*b*ln(-b*x+a)/a/(a*d+b*c)^2
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (103) = 206\).
Time = 0.49 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.28 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^2} \, dx=\frac {2 \, a b^{2} c^{2} d - 2 \, a^{3} d^{3} + {\left (b^{3} c^{3} + 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) - {\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x - a\right ) - 4 \, {\left (a b^{2} c d^{2} x + a b^{2} c^{2} d\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{4} c^{5} - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{5} c d^{4} + {\left (a b^{4} c^{4} d - 2 \, a^{3} b^{2} c^{2} d^{3} + a^{5} d^{5}\right )} x\right )}} \]
1/2*(2*a*b^2*c^2*d - 2*a^3*d^3 + (b^3*c^3 + 2*a*b^2*c^2*d + a^2*b*c*d^2 + (b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b*x + a) - (b^3*c^3 - 2*a*b ^2*c^2*d + a^2*b*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b* x - a) - 4*(a*b^2*c*d^2*x + a*b^2*c^2*d)*log(d*x + c))/(a*b^4*c^5 - 2*a^3* b^2*c^3*d^2 + a^5*c*d^4 + (a*b^4*c^4*d - 2*a^3*b^2*c^2*d^3 + a^5*d^5)*x)
Timed out. \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^2} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^2} \, dx=-\frac {2 \, b^{2} c d \log \left (d x + c\right )}{b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}} + \frac {b \log \left (b x + a\right )}{2 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} - \frac {b \log \left (b x - a\right )}{2 \, {\left (a b^{2} c^{2} + 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac {d}{b^{2} c^{3} - a^{2} c d^{2} + {\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x} \]
-2*b^2*c*d*log(d*x + c)/(b^4*c^4 - 2*a^2*b^2*c^2*d^2 + a^4*d^4) + 1/2*b*lo g(b*x + a)/(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2) - 1/2*b*log(b*x - a)/(a*b^2 *c^2 + 2*a^2*b*c*d + a^3*d^2) + d/(b^2*c^3 - a^2*c*d^2 + (b^2*c^2*d - a^2* d^3)*x)
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (103) = 206\).
Time = 0.29 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.66 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^2} \, dx=\frac {b^{2} c d \log \left ({\left | b^{2} - \frac {2 \, b^{2} c}{d x + c} + \frac {b^{2} c^{2}}{{\left (d x + c\right )}^{2}} - \frac {a^{2} d^{2}}{{\left (d x + c\right )}^{2}} \right |}\right )}{b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}} + \frac {d^{3}}{{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )} {\left (d x + c\right )}} - \frac {{\left (b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} \log \left (\frac {{\left | 2 \, b^{2} c d - \frac {2 \, b^{2} c^{2} d}{d x + c} + \frac {2 \, a^{2} d^{3}}{d x + c} - 2 \, d^{2} {\left | a \right |} {\left | b \right |} \right |}}{{\left | 2 \, b^{2} c d - \frac {2 \, b^{2} c^{2} d}{d x + c} + \frac {2 \, a^{2} d^{3}}{d x + c} + 2 \, d^{2} {\left | a \right |} {\left | b \right |} \right |}}\right )}{2 \, {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{4} d^{4}\right )} d^{2} {\left | a \right |} {\left | b \right |}} \]
b^2*c*d*log(abs(b^2 - 2*b^2*c/(d*x + c) + b^2*c^2/(d*x + c)^2 - a^2*d^2/(d *x + c)^2))/(b^4*c^4 - 2*a^2*b^2*c^2*d^2 + a^4*d^4) + d^3/((b^2*c^2*d^2 - a^2*d^4)*(d*x + c)) - 1/2*(b^4*c^2*d^2 + a^2*b^2*d^4)*log(abs(2*b^2*c*d - 2*b^2*c^2*d/(d*x + c) + 2*a^2*d^3/(d*x + c) - 2*d^2*abs(a)*abs(b))/abs(2*b ^2*c*d - 2*b^2*c^2*d/(d*x + c) + 2*a^2*d^3/(d*x + c) + 2*d^2*abs(a)*abs(b) ))/((b^4*c^4 - 2*a^2*b^2*c^2*d^2 + a^4*d^4)*d^2*abs(a)*abs(b))
Time = 1.71 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(a-b x) (a+b x) (c+d x)^2} \, dx=\frac {b\,\ln \left (a+b\,x\right )}{2\,\left (a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2\right )}-\frac {b\,\ln \left (a-b\,x\right )}{2\,\left (a^3\,d^2+2\,a^2\,b\,c\,d+a\,b^2\,c^2\right )}-\frac {d}{\left (a^2\,d^2-b^2\,c^2\right )\,\left (c+d\,x\right )}-\frac {2\,b^2\,c\,d\,\ln \left (c+d\,x\right )}{a^4\,d^4-2\,a^2\,b^2\,c^2\,d^2+b^4\,c^4} \]