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

Optimal. Leaf size=110 \[ \frac{b^3 \log (a+b x)}{a^2 (b c-a d)^2}-\frac{\log (x) (2 a d+b c)}{a^2 c^3}+\frac{d^2}{c^2 (c+d x) (b c-a d)}-\frac{d^2 (3 b c-2 a d) \log (c+d x)}{c^3 (b c-a d)^2}-\frac{1}{a c^2 x} \]

[Out]

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

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Rubi [A]  time = 0.103345, antiderivative size = 110, 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} \[ \frac{b^3 \log (a+b x)}{a^2 (b c-a d)^2}-\frac{\log (x) (2 a d+b c)}{a^2 c^3}+\frac{d^2}{c^2 (c+d x) (b c-a d)}-\frac{d^2 (3 b c-2 a d) \log (c+d x)}{c^3 (b c-a d)^2}-\frac{1}{a c^2 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)*(c + d*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0968905, size = 111, normalized size = 1.01 \[ \frac{b^3 \log (a+b x)}{a^2 (a d-b c)^2}+\frac{\log (x) (-2 a d-b c)}{a^2 c^3}+\frac{d^2}{c^2 (c+d x) (b c-a d)}+\frac{\left (2 a d^3-3 b c d^2\right ) \log (c+d x)}{c^3 (b c-a d)^2}-\frac{1}{a c^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 133, normalized size = 1.2 \begin{align*} -{\frac{{d}^{2}}{{c}^{2} \left ( ad-bc \right ) \left ( dx+c \right ) }}+2\,{\frac{{d}^{3}\ln \left ( dx+c \right ) a}{{c}^{3} \left ( ad-bc \right ) ^{2}}}-3\,{\frac{{d}^{2}\ln \left ( dx+c \right ) b}{{c}^{2} \left ( ad-bc \right ) ^{2}}}-{\frac{1}{a{c}^{2}x}}-2\,{\frac{\ln \left ( x \right ) d}{a{c}^{3}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}{c}^{2}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05985, size = 239, normalized size = 2.17 \begin{align*} \frac{b^{3} \log \left (b x + a\right )}{a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}} - \frac{{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} \log \left (d x + c\right )}{b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}} - \frac{b c^{2} - a c d +{\left (b c d - 2 \, a d^{2}\right )} x}{{\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )} x^{2} +{\left (a b c^{4} - a^{2} c^{3} d\right )} x} - \frac{{\left (b c + 2 \, a d\right )} \log \left (x\right )}{a^{2} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 34.0512, size = 574, normalized size = 5.22 \begin{align*} -\frac{a b^{2} c^{4} - 2 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2} +{\left (a b^{2} c^{3} d - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x -{\left (b^{3} c^{3} d x^{2} + b^{3} c^{4} x\right )} \log \left (b x + a\right ) +{\left ({\left (3 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4}\right )} x^{2} +{\left (3 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3}\right )} x\right )} \log \left (d x + c\right ) +{\left ({\left (b^{3} c^{3} d - 3 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4}\right )} x^{2} +{\left (b^{3} c^{4} - 3 \, a^{2} b c^{2} d^{2} + 2 \, a^{3} c d^{3}\right )} x\right )} \log \left (x\right )}{{\left (a^{2} b^{2} c^{5} d - 2 \, a^{3} b c^{4} d^{2} + a^{4} c^{3} d^{3}\right )} x^{2} +{\left (a^{2} b^{2} c^{6} - 2 \, a^{3} b c^{5} d + a^{4} c^{4} d^{2}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.20851, size = 203, normalized size = 1.85 \begin{align*} \frac{b^{3} d \log \left ({\left | b - \frac{b c}{d x + c} + \frac{a d}{d x + c} \right |}\right )}{a^{2} b^{2} c^{2} d - 2 \, a^{3} b c d^{2} + a^{4} d^{3}} + \frac{d^{5}}{{\left (b c^{3} d^{3} - a c^{2} d^{4}\right )}{\left (d x + c\right )}} + \frac{d}{a c^{3}{\left (\frac{c}{d x + c} - 1\right )}} - \frac{{\left (b c d + 2 \, a d^{2}\right )} \log \left ({\left | -\frac{c}{d x + c} + 1 \right |}\right )}{a^{2} c^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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