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

Optimal. Leaf size=53 \[ -\frac{b \log (a+b x)}{a (b c-a d)}+\frac{d \log (c+d x)}{c (b c-a d)}+\frac{\log (x)}{a c} \]

[Out]

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

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Rubi [A]  time = 0.034005, antiderivative size = 53, 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 = {72} \[ -\frac{b \log (a+b x)}{a (b c-a d)}+\frac{d \log (c+d x)}{c (b c-a d)}+\frac{\log (x)}{a c} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)*(c + d*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.0223204, size = 48, normalized size = 0.91 \[ \frac{-b c \log (a+b x)+a d \log (c+d x)-a d \log (x)+b c \log (x)}{a b c^2-a^2 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 54, normalized size = 1. \begin{align*} -{\frac{d\ln \left ( dx+c \right ) }{c \left ( ad-bc \right ) }}+{\frac{\ln \left ( x \right ) }{ac}}+{\frac{b\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x+a)/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.07197, size = 72, normalized size = 1.36 \begin{align*} -\frac{b \log \left (b x + a\right )}{a b c - a^{2} d} + \frac{d \log \left (d x + c\right )}{b c^{2} - a c d} + \frac{\log \left (x\right )}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.0936, size = 109, normalized size = 2.06 \begin{align*} -\frac{b c \log \left (b x + a\right ) - a d \log \left (d x + c\right ) -{\left (b c - a d\right )} \log \left (x\right )}{a b c^{2} - a^{2} c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 27.2194, size = 583, normalized size = 11. \begin{align*} - \frac{d \log{\left (x + \frac{- \frac{2 a^{6} d^{6}}{\left (a d - b c\right )^{2}} + \frac{6 a^{5} b c d^{5}}{\left (a d - b c\right )^{2}} - \frac{8 a^{4} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 a^{4} b c d^{4}}{a d - b c} + 2 a^{4} d^{4} + \frac{6 a^{3} b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{2}} - \frac{6 a^{3} b^{2} c^{2} d^{3}}{a d - b c} - 3 a^{3} b c d^{3} - \frac{2 a^{2} b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{3} c^{3} d^{2}}{a d - b c} + 2 a^{2} b^{2} c^{2} d^{2} - 3 a b^{3} c^{3} d + 2 b^{4} c^{4}}{2 a^{3} b d^{4} - 3 a^{2} b^{2} c d^{3} - 3 a b^{3} c^{2} d^{2} + 2 b^{4} c^{3} d} \right )}}{c \left (a d - b c\right )} + \frac{b \log{\left (x + \frac{- \frac{2 a^{4} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{2}} + 2 a^{4} d^{4} + \frac{6 a^{3} b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{3} b^{2} c^{2} d^{3}}{a d - b c} - 3 a^{3} b c d^{3} - \frac{8 a^{2} b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{2}} + \frac{6 a^{2} b^{3} c^{3} d^{2}}{a d - b c} + 2 a^{2} b^{2} c^{2} d^{2} + \frac{6 a b^{5} c^{5} d}{\left (a d - b c\right )^{2}} - \frac{3 a b^{4} c^{4} d}{a d - b c} - 3 a b^{3} c^{3} d - \frac{2 b^{6} c^{6}}{\left (a d - b c\right )^{2}} + 2 b^{4} c^{4}}{2 a^{3} b d^{4} - 3 a^{2} b^{2} c d^{3} - 3 a b^{3} c^{2} d^{2} + 2 b^{4} c^{3} d} \right )}}{a \left (a d - b c\right )} + \frac{\log{\left (x \right )}}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)/(d*x+c),x)

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError