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

Optimal. Leaf size=195 \[ \frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac{b^4}{a^2 (a+b x) (b c-a d)^3}+\frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{1}{a^2 c^3 x}-\frac{2 d^3 (2 b c-a d)}{c^3 (c+d x) (b c-a d)^3}-\frac{d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]

[Out]

-(1/(a^2*c^3*x)) - b^4/(a^2*(b*c - a*d)^3*(a + b*x)) - d^3/(2*c^2*(b*c - a*d)^2*(c + d*x)^2) - (2*d^3*(2*b*c -
a*d))/(c^3*(b*c - a*d)^3*(c + d*x)) - ((2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a + b*x])
/(a^3*(b*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x])/(c^4*(b*c - a*d)^4)

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Rubi [A]  time = 0.231376, antiderivative size = 195, 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} \[ \frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac{b^4}{a^2 (a+b x) (b c-a d)^3}+\frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{1}{a^2 c^3 x}-\frac{2 d^3 (2 b c-a d)}{c^3 (c+d x) (b c-a d)^3}-\frac{d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a^2*c^3*x)) - b^4/(a^2*(b*c - a*d)^3*(a + b*x)) - d^3/(2*c^2*(b*c - a*d)^2*(c + d*x)^2) - (2*d^3*(2*b*c -
a*d))/(c^3*(b*c - a*d)^3*(c + d*x)) - ((2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a + b*x])
/(a^3*(b*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x])/(c^4*(b*c - a*d)^4)

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)^3} \, dx &=\int \left (\frac{1}{a^2 c^3 x^2}+\frac{-2 b c-3 a d}{a^3 c^4 x}-\frac{b^5}{a^2 (-b c+a d)^3 (a+b x)^2}-\frac{b^5 (-2 b c+5 a d)}{a^3 (-b c+a d)^4 (a+b x)}+\frac{d^4}{c^2 (b c-a d)^2 (c+d x)^3}+\frac{2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)^2}+\frac{d^4 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac{1}{a^2 c^3 x}-\frac{b^4}{a^2 (b c-a d)^3 (a+b x)}-\frac{d^3}{2 c^2 (b c-a d)^2 (c+d x)^2}-\frac{2 d^3 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)}-\frac{(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.263727, size = 193, normalized size = 0.99 \[ \frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}+\frac{b^4}{a^2 (a+b x) (a d-b c)^3}+\frac{b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{1}{a^2 c^3 x}+\frac{2 d^3 (a d-2 b c)}{c^3 (c+d x) (b c-a d)^3}-\frac{d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a^2*c^3*x)) + b^4/(a^2*(-(b*c) + a*d)^3*(a + b*x)) - d^3/(2*c^2*(b*c - a*d)^2*(c + d*x)^2) + (2*d^3*(-2*b
*c + a*d))/(c^3*(b*c - a*d)^3*(c + d*x)) - ((2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a + b
*x])/(a^3*(b*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x])/(c^4*(b*c - a*d)^4)

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Maple [A]  time = 0.017, size = 266, normalized size = 1.4 \begin{align*} -{\frac{{d}^{3}}{2\,{c}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}+3\,{\frac{{d}^{5}\ln \left ( dx+c \right ){a}^{2}}{{c}^{4} \left ( ad-bc \right ) ^{4}}}-10\,{\frac{{d}^{4}\ln \left ( dx+c \right ) ab}{{c}^{3} \left ( ad-bc \right ) ^{4}}}+10\,{\frac{{d}^{3}\ln \left ( dx+c \right ){b}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{4}}}-2\,{\frac{{d}^{4}a}{{c}^{3} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+4\,{\frac{{d}^{3}b}{{c}^{2} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-{\frac{1}{{a}^{2}{c}^{3}x}}-3\,{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{4}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}{c}^{3}}}+{\frac{{b}^{4}}{ \left ( ad-bc \right ) ^{3}{a}^{2} \left ( bx+a \right ) }}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{4}{a}^{2}}}+2\,{\frac{{b}^{5}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{4}{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)^3,x)

[Out]

-1/2*d^3/c^2/(a*d-b*c)^2/(d*x+c)^2+3*d^5/c^4/(a*d-b*c)^4*ln(d*x+c)*a^2-10*d^4/c^3/(a*d-b*c)^4*ln(d*x+c)*a*b+10
*d^3/c^2/(a*d-b*c)^4*ln(d*x+c)*b^2-2*d^4/c^3/(a*d-b*c)^3/(d*x+c)*a+4*d^3/c^2/(a*d-b*c)^3/(d*x+c)*b-1/a^2/c^3/x
-3/a^2/c^4*ln(x)*d-2/a^3/c^3*ln(x)*b+b^4/(a*d-b*c)^3/a^2/(b*x+a)-5*b^4/(a*d-b*c)^4/a^2*ln(b*x+a)*d+2*b^5/(a*d-
b*c)^4/a^3*ln(b*x+a)*c

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Maxima [B]  time = 3.22616, size = 863, normalized size = 4.43 \begin{align*} \frac{{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x + a\right )}{a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}} + \frac{{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x + c\right )}{b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}} - \frac{2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{3} +{\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{2} +{\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x}{2 \,{\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{4} +{\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{3} +{\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{2} +{\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x\right )}} - \frac{{\left (2 \, b c + 3 \, a d\right )} \log \left (x\right )}{a^{3} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^5*c - 5*a*b^4*d)*log(b*x + a)/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^
4) + (10*b^2*c^2*d^3 - 10*a*b*c*d^4 + 3*a^2*d^5)*log(d*x + c)/(b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2 - 4
*a^3*b*c^5*d^3 + a^4*c^4*d^4) - 1/2*(2*a*b^3*c^5 - 6*a^2*b^2*c^4*d + 6*a^3*b*c^3*d^2 - 2*a^4*c^2*d^3 + 2*(2*b^
4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 7*a^2*b^2*c*d^4 - 3*a^3*b*d^5)*x^3 + (8*b^4*c^4*d - 10*a*b^3*c^3*d^2 + 15*a^2*b^
2*c^2*d^3 + 5*a^3*b*c*d^4 - 6*a^4*d^5)*x^2 + (4*b^4*c^5 - 2*a*b^3*c^4*d - 6*a^2*b^2*c^3*d^2 + 19*a^3*b*c^2*d^3
- 9*a^4*c*d^4)*x)/((a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^4 + (2*a^2*b^4
*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^3 + (a^2*b^4*c^8 - a^3*b^3*c^7
*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2 + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d
^2 - a^6*c^5*d^3)*x) - (2*b*c + 3*a*d)*log(x)/(a^3*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^2/(b*x+a)^2/(d*x+c)^3,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**2/(b*x+a)**2/(d*x+c)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.20558, size = 698, normalized size = 3.58 \begin{align*} -\frac{b^{9}}{{\left (a^{2} b^{8} c^{3} - 3 \, a^{3} b^{7} c^{2} d + 3 \, a^{4} b^{6} c d^{2} - a^{5} b^{5} d^{3}\right )}{\left (b x + a\right )}} + \frac{{\left (10 \, b^{3} c^{2} d^{3} - 10 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{8} - 4 \, a b^{4} c^{7} d + 6 \, a^{2} b^{3} c^{6} d^{2} - 4 \, a^{3} b^{2} c^{5} d^{3} + a^{4} b c^{4} d^{4}} - \frac{{\left (2 \, b^{2} c + 3 \, a b d\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{3} b c^{4}} + \frac{2 \, b^{5} c^{5} d^{2} - 8 \, a b^{4} c^{4} d^{3} + 12 \, a^{2} b^{3} c^{3} d^{4} - 17 \, a^{3} b^{2} c^{2} d^{5} + 6 \, a^{4} b c d^{6} + \frac{4 \, b^{7} c^{6} d - 20 \, a b^{6} c^{5} d^{2} + 40 \, a^{2} b^{5} c^{4} d^{3} - 50 \, a^{3} b^{4} c^{3} d^{4} + 43 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6}}{{\left (b x + a\right )} b} + \frac{2 \,{\left (b^{9} c^{7} - 6 \, a b^{8} c^{6} d + 15 \, a^{2} b^{7} c^{5} d^{2} - 20 \, a^{3} b^{6} c^{4} d^{3} + 20 \, a^{4} b^{5} c^{3} d^{4} - 13 \, a^{5} b^{4} c^{2} d^{5} + 3 \, a^{6} b^{3} c d^{6}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{2 \,{\left (b c - a d\right )}^{4} a^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2} c^{4}{\left (\frac{a}{b x + a} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^9/((a^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*(b*x + a)) + (10*b^3*c^2*d^3 - 10*a*b^2*
c*d^4 + 3*a^2*b*d^5)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^8 - 4*a*b^4*c^7*d + 6*a^2*b^3*c^6*d^2
- 4*a^3*b^2*c^5*d^3 + a^4*b*c^4*d^4) - (2*b^2*c + 3*a*b*d)*log(abs(-a/(b*x + a) + 1))/(a^3*b*c^4) + 1/2*(2*b^5
*c^5*d^2 - 8*a*b^4*c^4*d^3 + 12*a^2*b^3*c^3*d^4 - 17*a^3*b^2*c^2*d^5 + 6*a^4*b*c*d^6 + (4*b^7*c^6*d - 20*a*b^6
*c^5*d^2 + 40*a^2*b^5*c^4*d^3 - 50*a^3*b^4*c^3*d^4 + 43*a^4*b^3*c^2*d^5 - 12*a^5*b^2*c*d^6)/((b*x + a)*b) + 2*
(b^9*c^7 - 6*a*b^8*c^6*d + 15*a^2*b^7*c^5*d^2 - 20*a^3*b^6*c^4*d^3 + 20*a^4*b^5*c^3*d^4 - 13*a^5*b^4*c^2*d^5 +
3*a^6*b^3*c*d^6)/((b*x + a)^2*b^2))/((b*c - a*d)^4*a^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*c^4*(a/(b*x + a)
- 1))