3.1340 \(\int \frac{1}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=36 \[ \frac{\log (a+b x)}{b c-a d}-\frac{\log (c+d x)}{b c-a d} \]

[Out]

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

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Rubi [A]  time = 0.0076648, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {36, 31} \[ \frac{\log (a+b x)}{b c-a d}-\frac{\log (c+d x)}{b c-a d} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(c + d*x)),x]

[Out]

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0107835, size = 26, normalized size = 0.72 \[ \frac{\log (a+b x)-\log (c+d x)}{b c-a d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)*(c + d*x)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 37, normalized size = 1. \begin{align*}{\frac{\ln \left ( dx+c \right ) }{ad-bc}}-{\frac{\ln \left ( bx+a \right ) }{ad-bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.980413, size = 49, normalized size = 1.36 \begin{align*} \frac{\log \left (b x + a\right )}{b c - a d} - \frac{\log \left (d x + c\right )}{b c - a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.71206, size = 58, normalized size = 1.61 \begin{align*} \frac{\log \left (b x + a\right ) - \log \left (d x + c\right )}{b c - a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.306326, size = 128, normalized size = 3.56 \begin{align*} \frac{\log{\left (x + \frac{- \frac{a^{2} d^{2}}{a d - b c} + \frac{2 a b c d}{a d - b c} + a d - \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} - \frac{\log{\left (x + \frac{\frac{a^{2} d^{2}}{a d - b c} - \frac{2 a b c d}{a d - b c} + a d + \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + (-a**2*d**2/(a*d - b*c) + 2*a*b*c*d/(a*d - b*c) + a*d - b**2*c**2/(a*d - b*c) + b*c)/(2*b*d))/(a*d - b
*c) - log(x + (a**2*d**2/(a*d - b*c) - 2*a*b*c*d/(a*d - b*c) + a*d + b**2*c**2/(a*d - b*c) + b*c)/(2*b*d))/(a*
d - b*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/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError