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

Optimal. Leaf size=87 \[ -\frac{b^2 \log (a+b x)}{a (b c-a d)^2}+\frac{d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2}-\frac{d}{c (c+d x) (b c-a d)}+\frac{\log (x)}{a c^2} \]

[Out]

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

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Rubi [A]  time = 0.155195, antiderivative size = 87, 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 \[ -\frac{b^2 \log (a+b x)}{a (b c-a d)^2}+\frac{d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2}-\frac{d}{c (c+d x) (b c-a d)}+\frac{\log (x)}{a c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.4287, size = 71, normalized size = 0.82 \[ \frac{d}{c \left (c + d x\right ) \left (a d - b c\right )} - \frac{d \left (a d - 2 b c\right ) \log{\left (c + d x \right )}}{c^{2} \left (a d - b c\right )^{2}} - \frac{b^{2} \log{\left (a + b x \right )}}{a \left (a d - b c\right )^{2}} + \frac{\log{\left (x \right )}}{a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x+a)/(d*x+c)**2,x)

[Out]

d/(c*(c + d*x)*(a*d - b*c)) - d*(a*d - 2*b*c)*log(c + d*x)/(c**2*(a*d - b*c)**2)
- b**2*log(a + b*x)/(a*(a*d - b*c)**2) + log(x)/(a*c**2)

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Mathematica [A]  time = 0.181197, size = 83, normalized size = 0.95 \[ \frac{\frac{a d ((c+d x) (2 b c-a d) \log (c+d x)+c (a d-b c))-b^2 c^2 (c+d x) \log (a+b x)}{(c+d x) (b c-a d)^2}+\log (x)}{a c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 105, normalized size = 1.2 \[{\frac{d}{c \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{{d}^{2}\ln \left ( dx+c \right ) a}{{c}^{2} \left ( ad-bc \right ) ^{2}}}+2\,{\frac{d\ln \left ( dx+c \right ) b}{c \left ( ad-bc \right ) ^{2}}}+{\frac{\ln \left ( x \right ) }{a{c}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x+a)/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.34851, size = 173, normalized size = 1.99 \[ -\frac{b^{2} \log \left (b x + a\right )}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}} - \frac{d}{b c^{3} - a c^{2} d +{\left (b c^{2} d - a c d^{2}\right )} x} + \frac{\log \left (x\right )}{a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.9864, size = 279, normalized size = 3.21 \[ -\frac{a b c^{2} d - a^{2} c d^{2} +{\left (b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (b x + a\right ) -{\left (2 \, a b c^{2} d - a^{2} c d^{2} +{\left (2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \log \left (d x + c\right ) -{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \log \left (x\right )}{a b^{2} c^{5} - 2 \, a^{2} b c^{4} d + a^{3} c^{3} d^{2} +{\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x+a)/(d*x+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.327213, size = 381, normalized size = 4.38 \[ -\frac{1}{2} \, d{\left (\frac{{\left (2 \, b c - a d\right )}{\rm ln}\left ({\left | -b + \frac{2 \, b c}{d x + c} - \frac{b c^{2}}{{\left (d x + c\right )}^{2}} - \frac{a d}{d x + c} + \frac{a c d}{{\left (d x + c\right )}^{2}} \right |}\right )}{b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}} + \frac{2 \, d^{2}}{{\left (b c^{2} d^{2} - a c d^{3}\right )}{\left (d x + c\right )}} + \frac{{\left (2 \, b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}{\rm ln}\left (\frac{{\left | -2 \, b c d + \frac{2 \, b c^{2} d}{d x + c} + a d^{2} - \frac{2 \, a c d^{2}}{d x + c} - d^{2}{\left | a \right |} \right |}}{{\left | -2 \, b c d + \frac{2 \, b c^{2} d}{d x + c} + a d^{2} - \frac{2 \, a c d^{2}}{d x + c} + d^{2}{\left | a \right |} \right |}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} d^{2}{\left | a \right |}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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