3.289 \(\int \frac{1}{x^3 (a+b x)^2 (c+d x)^2} \, dx\)

Optimal. Leaf size=178 \[ \frac{\log (x) \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{a^4 c^4}+\frac{b^4}{a^3 (a+b x) (b c-a d)^2}-\frac{b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}+\frac{2 (a d+b c)}{a^3 c^3 x}-\frac{1}{2 a^2 c^2 x^2}+\frac{d^4}{c^3 (c+d x) (b c-a d)^2}-\frac{d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3} \]

[Out]

-1/(2*a^2*c^2*x^2) + (2*(b*c + a*d))/(a^3*c^3*x) + b^4/(a^3*(b*c - a*d)^2*(a + b*x)) + d^4/(c^3*(b*c - a*d)^2*
(c + d*x)) + ((3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*Log[x])/(a^4*c^4) - (b^4*(3*b*c - 5*a*d)*Log[a + b*x])/(a^4*
(b*c - a*d)^3) - (d^4*(5*b*c - 3*a*d)*Log[c + d*x])/(c^4*(b*c - a*d)^3)

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Rubi [A]  time = 0.20293, antiderivative size = 178, 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} \[ \frac{\log (x) \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{a^4 c^4}+\frac{b^4}{a^3 (a+b x) (b c-a d)^2}-\frac{b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}+\frac{2 (a d+b c)}{a^3 c^3 x}-\frac{1}{2 a^2 c^2 x^2}+\frac{d^4}{c^3 (c+d x) (b c-a d)^2}-\frac{d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*a^2*c^2*x^2) + (2*(b*c + a*d))/(a^3*c^3*x) + b^4/(a^3*(b*c - a*d)^2*(a + b*x)) + d^4/(c^3*(b*c - a*d)^2*
(c + d*x)) + ((3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*Log[x])/(a^4*c^4) - (b^4*(3*b*c - 5*a*d)*Log[a + b*x])/(a^4*
(b*c - a*d)^3) - (d^4*(5*b*c - 3*a*d)*Log[c + d*x])/(c^4*(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^3 (a+b x)^2 (c+d x)^2} \, dx &=\int \left (\frac{1}{a^2 c^2 x^3}-\frac{2 (b c+a d)}{a^3 c^3 x^2}+\frac{3 b^2 c^2+4 a b c d+3 a^2 d^2}{a^4 c^4 x}-\frac{b^5}{a^3 (-b c+a d)^2 (a+b x)^2}-\frac{b^5 (-3 b c+5 a d)}{a^4 (-b c+a d)^3 (a+b x)}-\frac{d^5}{c^3 (b c-a d)^2 (c+d x)^2}-\frac{d^5 (5 b c-3 a d)}{c^4 (b c-a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac{1}{2 a^2 c^2 x^2}+\frac{2 (b c+a d)}{a^3 c^3 x}+\frac{b^4}{a^3 (b c-a d)^2 (a+b x)}+\frac{d^4}{c^3 (b c-a d)^2 (c+d x)}+\frac{\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \log (x)}{a^4 c^4}-\frac{b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (b c-a d)^3}-\frac{d^4 (5 b c-3 a d) \log (c+d x)}{c^4 (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.179741, size = 176, normalized size = 0.99 \[ \frac{\log (x) \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{a^4 c^4}+\frac{b^4}{a^3 (a+b x) (b c-a d)^2}+\frac{b^4 (3 b c-5 a d) \log (a+b x)}{a^4 (a d-b c)^3}+\frac{2 (a d+b c)}{a^3 c^3 x}-\frac{1}{2 a^2 c^2 x^2}+\frac{d^4}{c^3 (c+d x) (b c-a d)^2}+\frac{d^4 (3 a d-5 b c) \log (c+d x)}{c^4 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*a^2*c^2*x^2) + (2*(b*c + a*d))/(a^3*c^3*x) + b^4/(a^3*(b*c - a*d)^2*(a + b*x)) + d^4/(c^3*(b*c - a*d)^2*
(c + d*x)) + ((3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*Log[x])/(a^4*c^4) + (b^4*(3*b*c - 5*a*d)*Log[a + b*x])/(a^4*
(-(b*c) + a*d)^3) + (d^4*(-5*b*c + 3*a*d)*Log[c + d*x])/(c^4*(b*c - a*d)^3)

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Maple [A]  time = 0.015, size = 223, normalized size = 1.3 \begin{align*}{\frac{{d}^{4}}{{c}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-3\,{\frac{{d}^{5}\ln \left ( dx+c \right ) a}{{c}^{4} \left ( ad-bc \right ) ^{3}}}+5\,{\frac{{d}^{4}\ln \left ( dx+c \right ) b}{{c}^{3} \left ( ad-bc \right ) ^{3}}}-{\frac{1}{2\,{a}^{2}{c}^{2}{x}^{2}}}+2\,{\frac{d}{{a}^{2}{c}^{3}x}}+2\,{\frac{b}{{a}^{3}{c}^{2}x}}+3\,{\frac{\ln \left ( x \right ){d}^{2}}{{a}^{2}{c}^{4}}}+4\,{\frac{b\ln \left ( x \right ) d}{{a}^{3}{c}^{3}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{4}{c}^{2}}}+{\frac{{b}^{4}}{ \left ( ad-bc \right ) ^{2}{a}^{3} \left ( bx+a \right ) }}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{3}{a}^{3}}}+3\,{\frac{{b}^{5}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{3}{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x+a)^2/(d*x+c)^2,x)

[Out]

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

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Maxima [B]  time = 1.33327, size = 637, normalized size = 3.58 \begin{align*} -\frac{{\left (3 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x + a\right )}{a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3}} - \frac{{\left (5 \, b c d^{4} - 3 \, a d^{5}\right )} \log \left (d x + c\right )}{b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}} - \frac{a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} - 2 \,{\left (3 \, b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2} - 2 \, a^{2} b^{2} c d^{3} + 3 \, a^{3} b d^{4}\right )} x^{3} -{\left (6 \, b^{4} c^{4} - a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - a^{3} b c d^{3} + 6 \, a^{4} d^{4}\right )} x^{2} - 3 \,{\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}{2 \,{\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{4} +{\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{3} +{\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} d^{2}\right )} x^{2}\right )}} + \frac{{\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (x\right )}{a^{4} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

-(3*b^5*c - 5*a*b^4*d)*log(b*x + a)/(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3) - (5*b*c*d^4 - 3
*a*d^5)*log(d*x + c)/(b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3) - 1/2*(a^2*b^2*c^4 - 2*a^3*b*c^
3*d + a^4*c^2*d^2 - 2*(3*b^4*c^3*d - 2*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^3 - (6*b^4*c^4 - a*b^3
*c^3*d - 6*a^2*b^2*c^2*d^2 - a^3*b*c*d^3 + 6*a^4*d^4)*x^2 - 3*(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)/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^4 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4
*d^2 + a^6*c^3*d^3)*x^3 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x^2) + (3*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^
2)*log(x)/(a^4*c^4)

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Fricas [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^3/(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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Giac [B]  time = 1.21713, size = 620, normalized size = 3.48 \begin{align*} \frac{b^{9}}{{\left (a^{3} b^{7} c^{2} - 2 \, a^{4} b^{6} c d + a^{5} b^{5} d^{2}\right )}{\left (b x + a\right )}} - \frac{{\left (5 \, b^{2} c d^{4} - 3 \, a b d^{5}\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{7} - 3 \, a b^{3} c^{6} d + 3 \, a^{2} b^{2} c^{5} d^{2} - a^{3} b c^{4} d^{3}} + \frac{{\left (3 \, b^{3} c^{2} + 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{4} b c^{4}} + \frac{5 \, b^{5} c^{5} d - 11 \, a b^{4} c^{4} d^{2} + 3 \, a^{2} b^{3} c^{3} d^{3} + 7 \, a^{3} b^{2} c^{2} d^{4} - 6 \, a^{4} b c d^{5} + \frac{5 \, b^{7} c^{6} - 22 \, a b^{6} c^{5} d + 28 \, a^{2} b^{5} c^{4} d^{2} - 2 \, a^{3} b^{4} c^{3} d^{3} - 17 \, a^{4} b^{3} c^{2} d^{4} + 12 \, a^{5} b^{2} c d^{5}}{{\left (b x + a\right )} b} - \frac{2 \,{\left (3 \, a b^{8} c^{6} - 10 \, a^{2} b^{7} c^{5} d + 10 \, a^{3} b^{6} c^{4} d^{2} - 5 \, a^{5} b^{4} c^{2} d^{4} + 3 \, a^{6} b^{3} c d^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{2 \,{\left (b c - a d\right )}^{3} a^{4}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )} c^{4}{\left (\frac{a}{b x + a} - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x+a)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

b^9/((a^3*b^7*c^2 - 2*a^4*b^6*c*d + a^5*b^5*d^2)*(b*x + a)) - (5*b^2*c*d^4 - 3*a*b*d^5)*log(abs(b*c/(b*x + a)
- a*d/(b*x + a) + d))/(b^4*c^7 - 3*a*b^3*c^6*d + 3*a^2*b^2*c^5*d^2 - a^3*b*c^4*d^3) + (3*b^3*c^2 + 4*a*b^2*c*d
+ 3*a^2*b*d^2)*log(abs(-a/(b*x + a) + 1))/(a^4*b*c^4) + 1/2*(5*b^5*c^5*d - 11*a*b^4*c^4*d^2 + 3*a^2*b^3*c^3*d
^3 + 7*a^3*b^2*c^2*d^4 - 6*a^4*b*c*d^5 + (5*b^7*c^6 - 22*a*b^6*c^5*d + 28*a^2*b^5*c^4*d^2 - 2*a^3*b^4*c^3*d^3
- 17*a^4*b^3*c^2*d^4 + 12*a^5*b^2*c*d^5)/((b*x + a)*b) - 2*(3*a*b^8*c^6 - 10*a^2*b^7*c^5*d + 10*a^3*b^6*c^4*d^
2 - 5*a^5*b^4*c^2*d^4 + 3*a^6*b^3*c*d^5)/((b*x + a)^2*b^2))/((b*c - a*d)^3*a^4*(b*c/(b*x + a) - a*d/(b*x + a)
+ d)*c^4*(a/(b*x + a) - 1)^2)