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

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

[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)

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Rubi [A]  time = 0.153899, antiderivative size = 144, normalized size of antiderivative = 1., 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 = 0.17403, size = 145, normalized size = 1.01 \[ -\frac{b^3}{a^2 (a+b x) (b c-a d)^2}+\frac{2 b^3 (2 a d-b c) \log (a+b x)}{a^3 (a d-b c)^3}-\frac{2 \log (x) (a d+b c)}{a^3 c^3}-\frac{1}{a^2 c^2 x}-\frac{d^3}{c^2 (c+d x) (b c-a d)^2}+\frac{2 d^3 (2 b c-a d) \log (c+d x)}{c^3 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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)

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Maple [A]  time = 0.014, size = 185, normalized size = 1.3 \begin{align*} -{\frac{{d}^{3}}{{c}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+2\,{\frac{{d}^{4}\ln \left ( dx+c \right ) a}{{c}^{3} \left ( ad-bc \right ) ^{3}}}-4\,{\frac{{d}^{3}\ln \left ( dx+c \right ) b}{{c}^{2} \left ( ad-bc \right ) ^{3}}}-{\frac{1}{{a}^{2}{c}^{2}x}}-2\,{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{3}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}{c}^{2}}}-{\frac{{b}^{3}}{ \left ( ad-bc \right ) ^{2}{a}^{2} \left ( bx+a \right ) }}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{3}{a}^{2}}}-2\,{\frac{{b}^{4}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{3}{a}^{3}}} \end{align*}

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]

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

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Maxima [B]  time = 1.2089, size = 504, normalized size = 3.5 \begin{align*} \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 \left (x\right )}{a^{3} c^{3}} \end{align*}

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)

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Fricas [B]  time = 133.347, size = 1262, normalized size = 8.76 \begin{align*} -\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 \left (x\right )}{{\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} \end{align*}

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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Giac [B]  time = 1.26289, size = 747, normalized size = 5.19 \begin{align*} -\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}} \end{align*}

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)