3.1535 \(\int \frac{1}{(a-b x) (a+b x) (c+d x)} \, dx\)

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 = 0.0582971, 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, Rules used = {72} \[ -\frac{d \log (c+d x)}{b^2 c^2-a^2 d^2}-\frac{\log (a-b x)}{2 a (a d+b c)}+\frac{\log (a+b x)}{2 a (b c-a d)} \]

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)

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0325434, 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 = 0.008, size = 72, normalized size = 1. \begin{align*}{\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 ) }{ \left ( 2\,ad+2\,bc \right ) a}} \end{align*}

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

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Maxima [A]  time = 1.14839, size = 96, normalized size = 1.3 \begin{align*} -\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 )}} \end{align*}

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 = 1.44592, size = 139, normalized size = 1.88 \begin{align*} -\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 )}} \end{align*}

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 [B]  time = 3.55717, size = 668, normalized size = 9.03 \begin{align*} \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 )} \end{align*}

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)) - 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)) - 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 [A]  time = 2.9097, size = 126, normalized size = 1.7 \begin{align*} \frac{b^{2} \log \left ({\left | b x + a \right |}\right )}{2 \,{\left (a b^{3} c - a^{2} b^{2} d\right )}} - \frac{b^{2} \log \left ({\left | b x - a \right |}\right )}{2 \,{\left (a b^{3} c + a^{2} b^{2} d\right )}} - \frac{d^{2} \log \left ({\left | d x + c \right |}\right )}{b^{2} c^{2} d - a^{2} d^{3}} \end{align*}

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