\(\int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx\) [287]

Optimal result

Integrand size = 18, antiderivative size = 123 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx=\frac {b^2}{a (b c-a d)^2 (a+b x)}+\frac {d^2}{c (b c-a d)^2 (c+d x)}+\frac {\log (x)}{a^2 c^2}-\frac {b^2 (b c-3 a d) \log (a+b x)}{a^2 (b c-a d)^3}-\frac {d^2 (3 b c-a d) \log (c+d x)}{c^2 (b c-a d)^3} \]

[Out]

b^2/a/(-a*d+b*c)^2/(b*x+a)+d^2/c/(-a*d+b*c)^2/(d*x+c)+ln(x)/a^2/c^2-b^2*(-3*a*d+b*c)*ln(b*x+a)/a^2/(-a*d+b*c)^
3-d^2*(-a*d+3*b*c)*ln(d*x+c)/c^2/(-a*d+b*c)^3

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \[ \int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx=-\frac {b^2 (b c-3 a d) \log (a+b x)}{a^2 (b c-a d)^3}+\frac {\log (x)}{a^2 c^2}+\frac {b^2}{a (a+b x) (b c-a d)^2}-\frac {d^2 (3 b c-a d) \log (c+d x)}{c^2 (b c-a d)^3}+\frac {d^2}{c (c+d x) (b c-a d)^2} \]

[In]

Int[1/(x*(a + b*x)^2*(c + d*x)^2),x]

[Out]

b^2/(a*(b*c - a*d)^2*(a + b*x)) + d^2/(c*(b*c - a*d)^2*(c + d*x)) + Log[x]/(a^2*c^2) - (b^2*(b*c - 3*a*d)*Log[
a + b*x])/(a^2*(b*c - a*d)^3) - (d^2*(3*b*c - a*d)*Log[c + d*x])/(c^2*(b*c - a*d)^3)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^2 c^2 x}-\frac {b^3}{a (-b c+a d)^2 (a+b x)^2}-\frac {b^3 (-b c+3 a d)}{a^2 (-b c+a d)^3 (a+b x)}-\frac {d^3}{c (b c-a d)^2 (c+d x)^2}-\frac {d^3 (3 b c-a d)}{c^2 (b c-a d)^3 (c+d x)}\right ) \, dx \\ & = \frac {b^2}{a (b c-a d)^2 (a+b x)}+\frac {d^2}{c (b c-a d)^2 (c+d x)}+\frac {\log (x)}{a^2 c^2}-\frac {b^2 (b c-3 a d) \log (a+b x)}{a^2 (b c-a d)^3}-\frac {d^2 (3 b c-a d) \log (c+d x)}{c^2 (b c-a d)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx=\frac {b^2}{a (b c-a d)^2 (a+b x)}+\frac {d^2}{c (b c-a d)^2 (c+d x)}+\frac {\log (x)}{a^2 c^2}+\frac {b^2 (b c-3 a d) \log (a+b x)}{a^2 (-b c+a d)^3}+\frac {d^2 (-3 b c+a d) \log (c+d x)}{c^2 (b c-a d)^3} \]

[In]

Integrate[1/(x*(a + b*x)^2*(c + d*x)^2),x]

[Out]

b^2/(a*(b*c - a*d)^2*(a + b*x)) + d^2/(c*(b*c - a*d)^2*(c + d*x)) + Log[x]/(a^2*c^2) + (b^2*(b*c - 3*a*d)*Log[
a + b*x])/(a^2*(-(b*c) + a*d)^3) + (d^2*(-3*b*c + a*d)*Log[c + d*x])/(c^2*(b*c - a*d)^3)

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {\ln \left (x \right )}{a^{2} c^{2}}+\frac {d^{2}}{c \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {d^{2} \left (a d -3 b c \right ) \ln \left (d x +c \right )}{c^{2} \left (a d -b c \right )^{3}}+\frac {b^{2}}{a \left (a d -b c \right )^{2} \left (b x +a \right )}-\frac {b^{2} \left (3 a d -b c \right ) \ln \left (b x +a \right )}{a^{2} \left (a d -b c \right )^{3}}\)	\(124\)
norman	\(\frac {\frac {\left (-a^{3} d^{3}-b^{3} c^{3}\right ) x}{c^{2} a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-a^{2} d^{2}-b^{2} c^{2}\right ) b d \,x^{2}}{c^{2} a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {\ln \left (x \right )}{a^{2} c^{2}}-\frac {b^{2} \left (3 a d -b c \right ) \ln \left (b x +a \right )}{a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d^{2} \left (a d -3 b c \right ) \ln \left (d x +c \right )}{c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(248\)
risch	\(\frac {\frac {b d \left (a d +b c \right ) x}{a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a^{2} d^{2}+b^{2} c^{2}}{a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}-\frac {d^{3} \ln \left (-d x -c \right ) a}{c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 d^{2} \ln \left (-d x -c \right ) b}{c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\ln \left (-x \right )}{a^{2} c^{2}}-\frac {3 b^{2} \ln \left (b x +a \right ) d}{a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{3} \ln \left (b x +a \right ) c}{a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(331\)
parallelrisch	\(\frac {-\ln \left (d x +c \right ) x \,a^{4} d^{4}+\ln \left (x \right ) a^{4} c \,d^{3}-\ln \left (x \right ) a \,b^{3} c^{4}+\ln \left (b x +a \right ) a \,b^{3} c^{4}-\ln \left (d x +c \right ) a^{4} c \,d^{3}-3 \ln \left (x \right ) x^{2} a^{2} b^{2} c \,d^{3}+3 \ln \left (x \right ) x^{2} a \,b^{3} c^{2} d^{2}-3 \ln \left (b x +a \right ) x^{2} a \,b^{3} c^{2} d^{2}+3 \ln \left (d x +c \right ) x^{2} a^{2} b^{2} c \,d^{3}-2 \ln \left (x \right ) x \,a^{3} b c \,d^{3}+2 \ln \left (x \right ) x a \,b^{3} c^{3} d -2 \ln \left (b x +a \right ) x a \,b^{3} c^{3} d +2 \ln \left (d x +c \right ) x \,a^{3} b c \,d^{3}+\ln \left (x \right ) x \,a^{4} d^{4}-\ln \left (x \right ) x \,b^{4} c^{4}+\ln \left (b x +a \right ) x \,b^{4} c^{4}+b^{4} c^{3} d \,x^{2}-a^{3} b \,d^{4} x^{2}+a^{2} b^{2} c \,d^{3} x^{2}-a \,b^{3} c^{2} d^{2} x^{2}+a^{3} b c \,d^{3} x -a \,b^{3} c^{3} d x +b^{4} c^{4} x -a^{4} d^{4} x +\ln \left (x \right ) x^{2} a^{3} b \,d^{4}-\ln \left (x \right ) x^{2} b^{4} c^{3} d +\ln \left (b x +a \right ) x^{2} b^{4} c^{3} d -\ln \left (d x +c \right ) x^{2} a^{3} b \,d^{4}-3 \ln \left (x \right ) a^{3} b \,c^{2} d^{2}+3 \ln \left (x \right ) a^{2} b^{2} c^{3} d -3 \ln \left (b x +a \right ) a^{2} b^{2} c^{3} d +3 \ln \left (d x +c \right ) a^{3} b \,c^{2} d^{2}-3 \ln \left (b x +a \right ) x \,a^{2} b^{2} c^{2} d^{2}+3 \ln \left (d x +c \right ) x \,a^{2} b^{2} c^{2} d^{2}}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) c^{2} a^{2}}\)	\(555\)

[In]

int(1/x/(b*x+a)^2/(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

ln(x)/a^2/c^2+d^2/c/(a*d-b*c)^2/(d*x+c)-d^2*(a*d-3*b*c)/c^2/(a*d-b*c)^3*ln(d*x+c)+b^2/a/(a*d-b*c)^2/(b*x+a)-b^
2*(3*a*d-b*c)/a^2/(a*d-b*c)^3*ln(b*x+a)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (123) = 246\).

Time = 2.69 (sec) , antiderivative size = 524, normalized size of antiderivative = 4.26 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx=\frac {a b^{3} c^{4} - a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (a b^{3} c^{3} d - a^{3} b c d^{3}\right )} x - {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2}\right )} x\right )} \log \left (b x + a\right ) - {\left (3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + {\left (3 \, a^{2} b^{2} c^{2} d^{2} + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \log \left (d x + c\right ) + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \log \left (x\right )}{a^{3} b^{3} c^{6} - 3 \, a^{4} b^{2} c^{5} d + 3 \, a^{5} b c^{4} d^{2} - a^{6} c^{3} d^{3} + {\left (a^{2} b^{4} c^{5} d - 3 \, a^{3} b^{3} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4}\right )} x^{2} + {\left (a^{2} b^{4} c^{6} - 2 \, a^{3} b^{3} c^{5} d + 2 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4}\right )} x} \]

[In]

integrate(1/x/(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

(a*b^3*c^4 - a^2*b^2*c^3*d + a^3*b*c^2*d^2 - a^4*c*d^3 + (a*b^3*c^3*d - a^3*b*c*d^3)*x - (a*b^3*c^4 - 3*a^2*b^
2*c^3*d + (b^4*c^3*d - 3*a*b^3*c^2*d^2)*x^2 + (b^4*c^4 - 2*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2)*x)*log(b*x + a) -
(3*a^3*b*c^2*d^2 - a^4*c*d^3 + (3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^2 + (3*a^2*b^2*c^2*d^2 + 2*a^3*b*c*d^3 - a^4*d^
4)*x)*log(d*x + c) + (a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2
+ 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^2 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x)*log(x))/(a^3*b^3*
c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3 + (a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 + 3*a^4*b^2*c^3*d^3
- a^5*b*c^2*d^4)*x^2 + (a^2*b^4*c^6 - 2*a^3*b^3*c^5*d + 2*a^5*b*c^3*d^3 - a^6*c^2*d^4)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx=\text {Timed out} \]

[In]

integrate(1/x/(b*x+a)**2/(d*x+c)**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (123) = 246\).

Time = 0.20 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.30 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx=-\frac {{\left (b^{3} c - 3 \, a b^{2} d\right )} \log \left (b x + a\right )}{a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}} - \frac {{\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x + c\right )}{b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}} + \frac {b^{2} c^{2} + a^{2} d^{2} + {\left (b^{2} c d + a b d^{2}\right )} x}{a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{2} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x} + \frac {\log \left (x\right )}{a^{2} c^{2}} \]

[In]

integrate(1/x/(b*x+a)^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

-(b^3*c - 3*a*b^2*d)*log(b*x + a)/(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) - (3*b*c*d^2 - a*d
^3)*log(d*x + 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^2*c^2 + a^2*d^2 + (b^2*c*d + a
*b*d^2)*x)/(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x^2 +
(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x) + log(x)/(a^2*c^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.62 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx={\left (\frac {b^{4}}{{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (3 \, b c d^{2} - a d^{3}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{5} - 3 \, a b^{3} c^{4} d + 3 \, a^{2} b^{2} c^{3} d^{2} - a^{3} b c^{2} d^{3}} - \frac {d^{3}}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} c} + \frac {\log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} b c^{2}}\right )} b \]

[In]

integrate(1/x/(b*x+a)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

(b^4/((a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*(b*x + a)) - (3*b*c*d^2 - a*d^3)*log(abs(b*c/(b*x + a) - a*d/(
b*x + a) + d))/(b^4*c^5 - 3*a*b^3*c^4*d + 3*a^2*b^2*c^3*d^2 - a^3*b*c^2*d^3) - d^3/((b*c - a*d)^3*(b*c/(b*x +
a) - a*d/(b*x + a) + d)*c) + log(abs(-a/(b*x + a) + 1))/(a^2*b*c^2))*b

Mupad [B] (verification not implemented)

Time = 1.06 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.49 \[ \int \frac {1}{x (a+b x)^2 (c+d x)^2} \, dx=\frac {\frac {a^2\,d^2+b^2\,c^2}{a\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {b\,d\,x\,\left (a\,d+b\,c\right )}{a\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^2+\left (a\,d+b\,c\right )\,x+a\,c}+\frac {\ln \left (x\right )}{a^2\,c^2}-\frac {b^2\,\ln \left (a+b\,x\right )\,\left (3\,a\,d-b\,c\right )}{a^2\,{\left (a\,d-b\,c\right )}^3}-\frac {d^2\,\ln \left (c+d\,x\right )\,\left (a\,d-3\,b\,c\right )}{c^2\,{\left (a\,d-b\,c\right )}^3} \]

[In]

int(1/(x*(a + b*x)^2*(c + d*x)^2),x)

[Out]

((a^2*d^2 + b^2*c^2)/(a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (b*d*x*(a*d + b*c))/(a*c*(a^2*d^2 + b^2*c^2 - 2*a
*b*c*d)))/(a*c + x*(a*d + b*c) + b*d*x^2) + log(x)/(a^2*c^2) - (b^2*log(a + b*x)*(3*a*d - b*c))/(a^2*(a*d - b*
c)^3) - (d^2*log(c + d*x)*(a*d - 3*b*c))/(c^2*(a*d - b*c)^3)