3.1518 \(\int \frac{1}{(a-b x) (a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=74 \[ -\frac{d \log (c+d x)}{b^2 c^2-a^2 d^2}-\frac{\log (a-b x)}{2 a (a d+b c)}+\frac{\log (a+b x)}{2 a (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.129186, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{d \log (c+d x)}{b^2 c^2-a^2 d^2}-\frac{\log (a-b x)}{2 a (a d+b c)}+\frac{\log (a+b x)}{2 a (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.8556, size = 54, normalized size = 0.73 \[ \frac{d \log{\left (c + d x \right )}}{a^{2} d^{2} - b^{2} c^{2}} - \frac{\log{\left (a - b x \right )}}{2 a \left (a d + b c\right )} - \frac{\log{\left (a + b x \right )}}{2 a \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0541274, size = 68, normalized size = 0.92 \[ \frac{(b c-a d) \log (a-b x)-(a d+b c) \log (a+b x)+2 a d \log (c+d x)}{2 a (a d-b c) (a d+b c)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 72, normalized size = 1. \[{\frac{d\ln \left ( dx+c \right ) }{ \left ( ad+bc \right ) \left ( ad-bc \right ) }}-{\frac{\ln \left ( bx+a \right ) }{2\,a \left ( ad-bc \right ) }}-{\frac{\ln \left ( bx-a \right ) }{2\,a \left ( ad+bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x+a)/(b*x+a)/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.34369, size = 96, normalized size = 1.3 \[ -\frac{d \log \left (d x + c\right )}{b^{2} c^{2} - a^{2} d^{2}} + \frac{\log \left (b x + a\right )}{2 \,{\left (a b c - a^{2} d\right )}} - \frac{\log \left (b x - a\right )}{2 \,{\left (a b c + a^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239294, size = 86, normalized size = 1.16 \[ -\frac{2 \, a d \log \left (d x + c\right ) -{\left (b c + a d\right )} \log \left (b x + a\right ) +{\left (b c - a d\right )} \log \left (b x - a\right )}{2 \,{\left (a b^{2} c^{2} - a^{3} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.1517, size = 668, normalized size = 9.03 \[ \frac{d \log{\left (x + \frac{\frac{12 a^{8} d^{8}}{\left (a d - b c\right )^{2} \left (a d + b c\right )^{2}} - \frac{20 a^{6} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{2} \left (a d + b c\right )^{2}} - \frac{6 a^{6} d^{6}}{\left (a d - b c\right ) \left (a d + b c\right )} + \frac{4 a^{4} b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{2} \left (a d + b c\right )^{2}} + \frac{12 a^{4} b^{2} c^{2} d^{4}}{\left (a d - b c\right ) \left (a d + b c\right )} - 6 a^{4} d^{4} + \frac{4 a^{2} b^{6} c^{6} d^{2}}{\left (a d - b c\right )^{2} \left (a d + b c\right )^{2}} - \frac{6 a^{2} b^{4} c^{4} d^{2}}{\left (a d - b c\right ) \left (a d + b c\right )} - a^{2} b^{2} c^{2} d^{2} - b^{4} c^{4}}{9 a^{2} b^{2} c d^{3} - b^{4} c^{3} d} \right )}}{\left (a d - b c\right ) \left (a d + b c\right )} - \frac{\log{\left (x + \frac{\frac{3 a^{6} d^{6}}{\left (a d + b c\right )^{2}} + \frac{3 a^{5} d^{5}}{a d + b c} - \frac{5 a^{4} b^{2} c^{2} d^{4}}{\left (a d + b c\right )^{2}} - 6 a^{4} d^{4} - \frac{6 a^{3} b^{2} c^{2} d^{3}}{a d + b c} + \frac{a^{2} b^{4} c^{4} d^{2}}{\left (a d + b c\right )^{2}} - a^{2} b^{2} c^{2} d^{2} + \frac{3 a b^{4} c^{4} d}{a d + b c} + \frac{b^{6} c^{6}}{\left (a d + b c\right )^{2}} - b^{4} c^{4}}{9 a^{2} b^{2} c d^{3} - b^{4} c^{3} d} \right )}}{2 a \left (a d + b c\right )} - \frac{\log{\left (x + \frac{\frac{3 a^{6} d^{6}}{\left (a d - b c\right )^{2}} + \frac{3 a^{5} d^{5}}{a d - b c} - \frac{5 a^{4} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{2}} - 6 a^{4} d^{4} - \frac{6 a^{3} b^{2} c^{2} d^{3}}{a d - b c} + \frac{a^{2} b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{2}} - a^{2} b^{2} c^{2} d^{2} + \frac{3 a b^{4} c^{4} d}{a d - b c} + \frac{b^{6} c^{6}}{\left (a d - b c\right )^{2}} - b^{4} c^{4}}{9 a^{2} b^{2} c d^{3} - b^{4} c^{3} d} \right )}}{2 a \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(x + (12*a**8*d**8/((a*d - b*c)**2*(a*d + b*c)**2) - 20*a**6*b**2*c**2*d**6
/((a*d - b*c)**2*(a*d + b*c)**2) - 6*a**6*d**6/((a*d - b*c)*(a*d + b*c)) + 4*a**
4*b**4*c**4*d**4/((a*d - b*c)**2*(a*d + b*c)**2) + 12*a**4*b**2*c**2*d**4/((a*d
- b*c)*(a*d + b*c)) - 6*a**4*d**4 + 4*a**2*b**6*c**6*d**2/((a*d - b*c)**2*(a*d +
b*c)**2) - 6*a**2*b**4*c**4*d**2/((a*d - b*c)*(a*d + b*c)) - a**2*b**2*c**2*d**
2 - b**4*c**4)/(9*a**2*b**2*c*d**3 - b**4*c**3*d))/((a*d - b*c)*(a*d + b*c)) - l
og(x + (3*a**6*d**6/(a*d + b*c)**2 + 3*a**5*d**5/(a*d + b*c) - 5*a**4*b**2*c**2*
d**4/(a*d + b*c)**2 - 6*a**4*d**4 - 6*a**3*b**2*c**2*d**3/(a*d + b*c) + a**2*b**
4*c**4*d**2/(a*d + b*c)**2 - a**2*b**2*c**2*d**2 + 3*a*b**4*c**4*d/(a*d + b*c) +
b**6*c**6/(a*d + b*c)**2 - b**4*c**4)/(9*a**2*b**2*c*d**3 - b**4*c**3*d))/(2*a*
(a*d + b*c)) - log(x + (3*a**6*d**6/(a*d - b*c)**2 + 3*a**5*d**5/(a*d - b*c) - 5
*a**4*b**2*c**2*d**4/(a*d - b*c)**2 - 6*a**4*d**4 - 6*a**3*b**2*c**2*d**3/(a*d -
b*c) + a**2*b**4*c**4*d**2/(a*d - b*c)**2 - a**2*b**2*c**2*d**2 + 3*a*b**4*c**4
*d/(a*d - b*c) + b**6*c**6/(a*d - b*c)**2 - b**4*c**4)/(9*a**2*b**2*c*d**3 - b**
4*c**3*d))/(2*a*(a*d - b*c))

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GIAC/XCAS [A]  time = 0.207777, size = 126, normalized size = 1.7 \[ \frac{b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{2 \,{\left (a b^{3} c - a^{2} b^{2} d\right )}} - \frac{b^{2}{\rm ln}\left ({\left | b x - a \right |}\right )}{2 \,{\left (a b^{3} c + a^{2} b^{2} d\right )}} - \frac{d^{2}{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{2} c^{2} d - a^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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