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

Optimal. Leaf size=160 \[ -\frac{d^2 \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}+\frac{b^4 \log (a+b x)}{a^2 (b c-a d)^3}-\frac{\log (x) (3 a d+b c)}{a^2 c^4}+\frac{d^2 (3 b c-2 a d)}{c^3 (c+d x) (b c-a d)^2}+\frac{d^2}{2 c^2 (c+d x)^2 (b c-a d)}-\frac{1}{a c^3 x} \]

[Out]

-(1/(a*c^3*x)) + d^2/(2*c^2*(b*c - a*d)*(c + d*x)^2) + (d^2*(3*b*c - 2*a*d))/(c^3*(b*c - a*d)^2*(c + d*x)) - (
(b*c + 3*a*d)*Log[x])/(a^2*c^4) + (b^4*Log[a + b*x])/(a^2*(b*c - a*d)^3) - (d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2
*d^2)*Log[c + d*x])/(c^4*(b*c - a*d)^3)

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Rubi [A]  time = 0.159338, antiderivative size = 160, 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} \[ -\frac{d^2 \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}+\frac{b^4 \log (a+b x)}{a^2 (b c-a d)^3}-\frac{\log (x) (3 a d+b c)}{a^2 c^4}+\frac{d^2 (3 b c-2 a d)}{c^3 (c+d x) (b c-a d)^2}+\frac{d^2}{2 c^2 (c+d x)^2 (b c-a d)}-\frac{1}{a c^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)*(c + d*x)^3),x]

[Out]

-(1/(a*c^3*x)) + d^2/(2*c^2*(b*c - a*d)*(c + d*x)^2) + (d^2*(3*b*c - 2*a*d))/(c^3*(b*c - a*d)^2*(c + d*x)) - (
(b*c + 3*a*d)*Log[x])/(a^2*c^4) + (b^4*Log[a + b*x])/(a^2*(b*c - a*d)^3) - (d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2
*d^2)*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^2 (a+b x) (c+d x)^3} \, dx &=\int \left (\frac{1}{a c^3 x^2}+\frac{-b c-3 a d}{a^2 c^4 x}-\frac{b^5}{a^2 (-b c+a d)^3 (a+b x)}-\frac{d^3}{c^2 (b c-a d) (c+d x)^3}-\frac{d^3 (3 b c-2 a d)}{c^3 (b c-a d)^2 (c+d x)^2}-\frac{d^3 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac{1}{a c^3 x}+\frac{d^2}{2 c^2 (b c-a d) (c+d x)^2}+\frac{d^2 (3 b c-2 a d)}{c^3 (b c-a d)^2 (c+d x)}-\frac{(b c+3 a d) \log (x)}{a^2 c^4}+\frac{b^4 \log (a+b x)}{a^2 (b c-a d)^3}-\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.164644, size = 163, normalized size = 1.02 \[ -\frac{\left (3 a^2 d^4-8 a b c d^3+6 b^2 c^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^3}-\frac{b^4 \log (a+b x)}{a^2 (a d-b c)^3}+\frac{\log (x) (-3 a d-b c)}{a^2 c^4}+\frac{d^2 (3 b c-2 a d)}{c^3 (c+d x) (b c-a d)^2}+\frac{d^2}{2 c^2 (c+d x)^2 (b c-a d)}-\frac{1}{a c^3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a*c^3*x)) + d^2/(2*c^2*(b*c - a*d)*(c + d*x)^2) + (d^2*(3*b*c - 2*a*d))/(c^3*(b*c - a*d)^2*(c + d*x)) + (
(-(b*c) - 3*a*d)*Log[x])/(a^2*c^4) - (b^4*Log[a + b*x])/(a^2*(-(b*c) + a*d)^3) - ((6*b^2*c^2*d^2 - 8*a*b*c*d^3
+ 3*a^2*d^4)*Log[c + d*x])/(c^4*(b*c - a*d)^3)

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Maple [A]  time = 0.013, size = 216, normalized size = 1.4 \begin{align*} -{\frac{{d}^{2}}{2\,{c}^{2} \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}-2\,{\frac{{d}^{3}a}{{c}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+3\,{\frac{{d}^{2}b}{{c}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+3\,{\frac{{d}^{4}\ln \left ( dx+c \right ){a}^{2}}{{c}^{4} \left ( ad-bc \right ) ^{3}}}-8\,{\frac{{d}^{3}\ln \left ( dx+c \right ) ab}{{c}^{3} \left ( ad-bc \right ) ^{3}}}+6\,{\frac{{d}^{2}\ln \left ( dx+c \right ){b}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{3}}}-{\frac{1}{a{c}^{3}x}}-3\,{\frac{\ln \left ( x \right ) d}{a{c}^{4}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}{c}^{3}}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}{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)^3,x)

[Out]

-1/2*d^2/c^2/(a*d-b*c)/(d*x+c)^2-2*d^3/c^3/(a*d-b*c)^2/(d*x+c)*a+3*d^2/c^2/(a*d-b*c)^2/(d*x+c)*b+3*d^4/c^4/(a*
d-b*c)^3*ln(d*x+c)*a^2-8*d^3/c^3/(a*d-b*c)^3*ln(d*x+c)*a*b+6*d^2/c^2/(a*d-b*c)^3*ln(d*x+c)*b^2-1/a/c^3/x-3/a/c
^4*ln(x)*d-1/a^2/c^3*ln(x)*b-b^4/(a*d-b*c)^3/a^2*ln(b*x+a)

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Maxima [B]  time = 1.37311, size = 477, normalized size = 2.98 \begin{align*} \frac{b^{4} \log \left (b x + a\right )}{a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}} - \frac{{\left (6 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\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{2 \, b^{2} c^{4} - 4 \, a b c^{3} d + 2 \, a^{2} c^{2} d^{2} + 2 \,{\left (b^{2} c^{2} d^{2} - 5 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} +{\left (4 \, b^{2} c^{3} d - 15 \, a b c^{2} d^{2} + 9 \, a^{2} c d^{3}\right )} x}{2 \,{\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{3} + 2 \,{\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{2} +{\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x\right )}} - \frac{{\left (b c + 3 \, a d\right )} \log \left (x\right )}{a^{2} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 88.0623, size = 1238, normalized size = 7.74 \begin{align*} -\frac{2 \, a b^{3} c^{6} - 6 \, a^{2} b^{2} c^{5} d + 6 \, a^{3} b c^{4} d^{2} - 2 \, a^{4} c^{3} d^{3} + 2 \,{\left (a b^{3} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{2} +{\left (4 \, a b^{3} c^{5} d - 19 \, a^{2} b^{2} c^{4} d^{2} + 24 \, a^{3} b c^{3} d^{3} - 9 \, a^{4} c^{2} d^{4}\right )} x - 2 \,{\left (b^{4} c^{4} d^{2} x^{3} + 2 \, b^{4} c^{5} d x^{2} + b^{4} c^{6} x\right )} \log \left (b x + a\right ) + 2 \,{\left ({\left (6 \, a^{2} b^{2} c^{2} d^{4} - 8 \, a^{3} b c d^{5} + 3 \, a^{4} d^{6}\right )} x^{3} + 2 \,{\left (6 \, a^{2} b^{2} c^{3} d^{3} - 8 \, a^{3} b c^{2} d^{4} + 3 \, a^{4} c d^{5}\right )} x^{2} +{\left (6 \, a^{2} b^{2} c^{4} d^{2} - 8 \, a^{3} b c^{3} d^{3} + 3 \, a^{4} c^{2} d^{4}\right )} x\right )} \log \left (d x + c\right ) + 2 \,{\left ({\left (b^{4} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{2} d^{4} + 8 \, a^{3} b c d^{5} - 3 \, a^{4} d^{6}\right )} x^{3} + 2 \,{\left (b^{4} c^{5} d - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{2} +{\left (b^{4} c^{6} - 6 \, a^{2} b^{2} c^{4} d^{2} + 8 \, a^{3} b c^{3} d^{3} - 3 \, a^{4} c^{2} d^{4}\right )} x\right )} \log \left (x\right )}{2 \,{\left ({\left (a^{2} b^{3} c^{7} d^{2} - 3 \, a^{3} b^{2} c^{6} d^{3} + 3 \, a^{4} b c^{5} d^{4} - a^{5} c^{4} d^{5}\right )} x^{3} + 2 \,{\left (a^{2} b^{3} c^{8} d - 3 \, a^{3} b^{2} c^{7} d^{2} + 3 \, a^{4} b c^{6} d^{3} - a^{5} c^{5} d^{4}\right )} x^{2} +{\left (a^{2} b^{3} c^{9} - 3 \, a^{3} b^{2} c^{8} d + 3 \, a^{4} b c^{7} d^{2} - a^{5} c^{6} d^{3}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*b^3*c^6 - 6*a^2*b^2*c^5*d + 6*a^3*b*c^4*d^2 - 2*a^4*c^3*d^3 + 2*(a*b^3*c^4*d^2 - 6*a^2*b^2*c^3*d^3 +
8*a^3*b*c^2*d^4 - 3*a^4*c*d^5)*x^2 + (4*a*b^3*c^5*d - 19*a^2*b^2*c^4*d^2 + 24*a^3*b*c^3*d^3 - 9*a^4*c^2*d^4)*
x - 2*(b^4*c^4*d^2*x^3 + 2*b^4*c^5*d*x^2 + b^4*c^6*x)*log(b*x + a) + 2*((6*a^2*b^2*c^2*d^4 - 8*a^3*b*c*d^5 + 3
*a^4*d^6)*x^3 + 2*(6*a^2*b^2*c^3*d^3 - 8*a^3*b*c^2*d^4 + 3*a^4*c*d^5)*x^2 + (6*a^2*b^2*c^4*d^2 - 8*a^3*b*c^3*d
^3 + 3*a^4*c^2*d^4)*x)*log(d*x + c) + 2*((b^4*c^4*d^2 - 6*a^2*b^2*c^2*d^4 + 8*a^3*b*c*d^5 - 3*a^4*d^6)*x^3 + 2
*(b^4*c^5*d - 6*a^2*b^2*c^3*d^3 + 8*a^3*b*c^2*d^4 - 3*a^4*c*d^5)*x^2 + (b^4*c^6 - 6*a^2*b^2*c^4*d^2 + 8*a^3*b*
c^3*d^3 - 3*a^4*c^2*d^4)*x)*log(x))/((a^2*b^3*c^7*d^2 - 3*a^3*b^2*c^6*d^3 + 3*a^4*b*c^5*d^4 - a^5*c^4*d^5)*x^3
+ 2*(a^2*b^3*c^8*d - 3*a^3*b^2*c^7*d^2 + 3*a^4*b*c^6*d^3 - a^5*c^5*d^4)*x^2 + (a^2*b^3*c^9 - 3*a^3*b^2*c^8*d
+ 3*a^4*b*c^7*d^2 - a^5*c^6*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)/(d*x+c)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.18153, size = 452, normalized size = 2.82 \begin{align*} \frac{b^{5} \log \left ({\left | b x + a \right |}\right )}{a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}} - \frac{{\left (6 \, b^{2} c^{2} d^{3} - 8 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}} - \frac{{\left (b c + 3 \, a d\right )} \log \left ({\left | x \right |}\right )}{a^{2} c^{4}} - \frac{2 \, a b^{3} c^{6} - 6 \, a^{2} b^{2} c^{5} d + 6 \, a^{3} b c^{4} d^{2} - 2 \, a^{4} c^{3} d^{3} + 2 \,{\left (a b^{3} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{2} +{\left (4 \, a b^{3} c^{5} d - 19 \, a^{2} b^{2} c^{4} d^{2} + 24 \, a^{3} b c^{3} d^{3} - 9 \, a^{4} c^{2} d^{4}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} a^{2} c^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^5*log(abs(b*x + a))/(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3) - (6*b^2*c^2*d^3 - 8*a*b*c
*d^4 + 3*a^2*d^5)*log(abs(d*x + c))/(b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 - a^3*c^4*d^4) - (b*c + 3*a
*d)*log(abs(x))/(a^2*c^4) - 1/2*(2*a*b^3*c^6 - 6*a^2*b^2*c^5*d + 6*a^3*b*c^4*d^2 - 2*a^4*c^3*d^3 + 2*(a*b^3*c^
4*d^2 - 6*a^2*b^2*c^3*d^3 + 8*a^3*b*c^2*d^4 - 3*a^4*c*d^5)*x^2 + (4*a*b^3*c^5*d - 19*a^2*b^2*c^4*d^2 + 24*a^3*
b*c^3*d^3 - 9*a^4*c^2*d^4)*x)/((b*c - a*d)^3*(d*x + c)^2*a^2*c^4*x)