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

Optimal. Leaf size=144 \[ \frac {2 b^3 (b c-2 a d) \log (a+b x)}{a^3 (b c-a d)^3}-\frac {2 \log (x) (a d+b c)}{a^3 c^3}-\frac {b^3}{a^2 (a+b x) (b c-a d)^2}-\frac {1}{a^2 c^2 x}+\frac {2 d^3 (2 b c-a d) \log (c+d x)}{c^3 (b c-a d)^3}-\frac {d^3}{c^2 (c+d x) (b c-a d)^2} \]

[Out]

-1/a^2/c^2/x-b^3/a^2/(-a*d+b*c)^2/(b*x+a)-d^3/c^2/(-a*d+b*c)^2/(d*x+c)-2*(a*d+b*c)*ln(x)/a^3/c^3+2*b^3*(-2*a*d
+b*c)*ln(b*x+a)/a^3/(-a*d+b*c)^3+2*d^3*(-a*d+2*b*c)*ln(d*x+c)/c^3/(-a*d+b*c)^3

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \[ -\frac {b^3}{a^2 (a+b x) (b c-a d)^2}+\frac {2 b^3 (b c-2 a d) \log (a+b x)}{a^3 (b c-a d)^3}-\frac {2 \log (x) (a d+b c)}{a^3 c^3}-\frac {1}{a^2 c^2 x}-\frac {d^3}{c^2 (c+d x) (b c-a d)^2}+\frac {2 d^3 (2 b c-a d) \log (c+d x)}{c^3 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 88

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*} \int \frac {1}{x^2 (a+b x)^2 (c+d x)^2} \, dx &=\int \left (\frac {1}{a^2 c^2 x^2}-\frac {2 (b c+a d)}{a^3 c^3 x}+\frac {b^4}{a^2 (-b c+a d)^2 (a+b x)^2}+\frac {2 b^4 (-b c+2 a d)}{a^3 (-b c+a d)^3 (a+b x)}+\frac {d^4}{c^2 (b c-a d)^2 (c+d x)^2}+\frac {2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac {1}{a^2 c^2 x}-\frac {b^3}{a^2 (b c-a d)^2 (a+b x)}-\frac {d^3}{c^2 (b c-a d)^2 (c+d x)}-\frac {2 (b c+a d) \log (x)}{a^3 c^3}+\frac {2 b^3 (b c-2 a d) \log (a+b x)}{a^3 (b c-a d)^3}+\frac {2 d^3 (2 b c-a d) \log (c+d x)}{c^3 (b c-a d)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 145, normalized size = 1.01 \[ \frac {2 b^3 (2 a d-b c) \log (a+b x)}{a^3 (a d-b c)^3}-\frac {2 \log (x) (a d+b c)}{a^3 c^3}-\frac {b^3}{a^2 (a+b x) (b c-a d)^2}-\frac {1}{a^2 c^2 x}+\frac {2 d^3 (2 b c-a d) \log (c+d x)}{c^3 (b c-a d)^3}-\frac {d^3}{c^2 (c+d x) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 28.64, size = 653, normalized size = 4.53 \[ -\frac {a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + 2 \, {\left (a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{2} + {\left (2 \, a b^{4} c^{5} - 3 \, a^{2} b^{3} c^{4} d + 3 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}\right )} x - 2 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2}\right )} x^{3} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2}\right )} x^{2} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{3} + {\left (2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{2} + {\left (2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x\right )} \log \left (d x + c\right ) + 2 \, {\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{3} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{2} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x\right )} \log \relax (x)}{{\left (a^{3} b^{4} c^{6} d - 3 \, a^{4} b^{3} c^{5} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{3} - a^{6} b c^{3} d^{4}\right )} x^{3} + {\left (a^{3} b^{4} c^{7} - 2 \, a^{4} b^{3} c^{6} d + 2 \, a^{6} b c^{4} d^{3} - a^{7} c^{3} d^{4}\right )} x^{2} + {\left (a^{4} b^{3} c^{7} - 3 \, a^{5} b^{2} c^{6} d + 3 \, a^{6} b c^{5} d^{2} - a^{7} c^{4} d^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3 + 2*(a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 + 2*a^3*b
^2*c^2*d^3 - a^4*b*c*d^4)*x^2 + (2*a*b^4*c^5 - 3*a^2*b^3*c^4*d + 3*a^4*b*c^2*d^3 - 2*a^5*c*d^4)*x - 2*((b^5*c^
4*d - 2*a*b^4*c^3*d^2)*x^3 + (b^5*c^5 - a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2)*x^2 + (a*b^4*c^5 - 2*a^2*b^3*c^4*d)*x
)*log(b*x + a) - 2*((2*a^3*b^2*c*d^4 - a^4*b*d^5)*x^3 + (2*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 - a^5*d^5)*x^2 + (2*a
^4*b*c^2*d^3 - a^5*c*d^4)*x)*log(d*x + c) + 2*((b^5*c^4*d - 2*a*b^4*c^3*d^2 + 2*a^3*b^2*c*d^4 - a^4*b*d^5)*x^3
+ (b^5*c^5 - a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 - a^5*d^5)*x^2 + (a*b^4*c^5 -
2*a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x)*log(x))/((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*
d^3 - a^6*b*c^3*d^4)*x^3 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x^2 + (a^4*b^3*c^7
- 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x)

________________________________________________________________________________________

giac [B]  time = 1.10, size = 553, normalized size = 3.84 \[ -\frac {b^{7}}{{\left (a^{2} b^{6} c^{2} - 2 \, a^{3} b^{5} c d + a^{4} b^{4} d^{2}\right )} {\left (b x + a\right )}} - \frac {{\left (b^{4} c - 2 \, a b^{3} d\right )} \log \left ({\left | -\frac {b c}{b x + a} + \frac {a b c}{{\left (b x + a\right )}^{2}} + \frac {2 \, a d}{b x + a} - \frac {a^{2} d}{{\left (b x + a\right )}^{2}} - d \right |}\right )}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}} + \frac {{\left (b^{6} c^{4} - 2 \, a b^{5} c^{3} d + 4 \, a^{3} b^{3} c d^{3} - 2 \, a^{4} b^{2} d^{4}\right )} \log \left (\frac {{\left | -\frac {2 \, a b^{2} c}{b x + a} + b^{2} c - 2 \, a b d + \frac {2 \, a^{2} b d}{b x + a} - b^{2} {\left | c \right |} \right |}}{{\left | -\frac {2 \, a b^{2} c}{b x + a} + b^{2} c - 2 \, a b d + \frac {2 \, a^{2} b d}{b x + a} + b^{2} {\left | c \right |} \right |}}\right )}{{\left (a^{3} b^{3} c^{5} - 3 \, a^{4} b^{2} c^{4} d + 3 \, a^{5} b c^{3} d^{2} - a^{6} c^{2} d^{3}\right )} b^{2} {\left | c \right |}} - \frac {\frac {b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}}{a b c - a^{2} d} + \frac {b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + 2 \, a^{4} b^{2} d^{4}}{{\left (a b c - a^{2} d\right )} {\left (b x + a\right )} b}}{{\left (b c - a d\right )}^{2} a^{2} {\left (\frac {b c}{b x + a} - \frac {a b c}{{\left (b x + a\right )}^{2}} - \frac {2 \, a d}{b x + a} + \frac {a^{2} d}{{\left (b x + a\right )}^{2}} + d\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^7/((a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*(b*x + a)) - (b^4*c - 2*a*b^3*d)*log(abs(-b*c/(b*x + a) + a*
b*c/(b*x + a)^2 + 2*a*d/(b*x + a) - a^2*d/(b*x + a)^2 - d))/(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a
^6*d^3) + (b^6*c^4 - 2*a*b^5*c^3*d + 4*a^3*b^3*c*d^3 - 2*a^4*b^2*d^4)*log(abs(-2*a*b^2*c/(b*x + a) + b^2*c - 2
*a*b*d + 2*a^2*b*d/(b*x + a) - b^2*abs(c))/abs(-2*a*b^2*c/(b*x + a) + b^2*c - 2*a*b*d + 2*a^2*b*d/(b*x + a) +
b^2*abs(c)))/((a^3*b^3*c^5 - 3*a^4*b^2*c^4*d + 3*a^5*b*c^3*d^2 - a^6*c^2*d^3)*b^2*abs(c)) - ((b^4*c^3*d - 3*a*
b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - 2*a^3*b*d^4)/(a*b*c - a^2*d) + (b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 -
4*a^3*b^3*c*d^3 + 2*a^4*b^2*d^4)/((a*b*c - a^2*d)*(b*x + a)*b))/((b*c - a*d)^2*a^2*(b*c/(b*x + a) - a*b*c/(b*x
+ a)^2 - 2*a*d/(b*x + a) + a^2*d/(b*x + a)^2 + d)*c^2)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 185, normalized size = 1.28 \[ \frac {2 a \,d^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{3}}+\frac {4 b^{3} d \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} a^{2}}-\frac {2 b^{4} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} a^{3}}-\frac {4 b \,d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} c^{2}}-\frac {b^{3}}{\left (a d -b c \right )^{2} \left (b x +a \right ) a^{2}}-\frac {d^{3}}{\left (a d -b c \right )^{2} \left (d x +c \right ) c^{2}}-\frac {2 d \ln \relax (x )}{a^{2} c^{3}}-\frac {2 b \ln \relax (x )}{a^{3} c^{2}}-\frac {1}{a^{2} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.23, size = 373, normalized size = 2.59 \[ \frac {2 \, {\left (b^{4} c - 2 \, a b^{3} d\right )} \log \left (b x + a\right )}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}} + \frac {2 \, {\left (2 \, b c d^{3} - a d^{4}\right )} \log \left (d x + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac {a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d - a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} x}{{\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{3} + {\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{2} + {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x} - \frac {2 \, {\left (b c + a d\right )} \log \relax (x)}{a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.93, size = 304, normalized size = 2.11 \[ -\frac {\frac {1}{a\,c}+\frac {2\,x^2\,\left (a^2\,b\,d^3-a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}{a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (a\,d+b\,c\right )\,\left (2\,a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{a^2\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{b\,d\,x^3+\left (a\,d+b\,c\right )\,x^2+a\,c\,x}-\frac {\ln \left (a+b\,x\right )\,\left (2\,b^4\,c-4\,a\,b^3\,d\right )}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}-\frac {\ln \left (c+d\,x\right )\,\left (2\,a\,d^4-4\,b\,c\,d^3\right )}{-a^3\,c^3\,d^3+3\,a^2\,b\,c^4\,d^2-3\,a\,b^2\,c^5\,d+b^3\,c^6}-\frac {2\,\ln \relax (x)\,\left (a\,d+b\,c\right )}{a^3\,c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________