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

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

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Rubi [A]  time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.91 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x (a+b x) (c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.55, size = 50, normalized size = 0.94 \begin {gather*} -\frac {b c \log \left (b x + a\right ) - a d \log \left (d x + c\right ) - {\left (b c - a d\right )} \log \relax (x)}{a b c^{2} - a^{2} c d} \end {gather*}

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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giac [A]  time = 0.74, size = 66, normalized size = 1.25 \begin {gather*} -\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a b^{2} c - a^{2} b d} + \frac {d^{2} \log \left ({\left | d x + c \right |}\right )}{b c^{2} d - a c d^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 54, normalized size = 1.02 \begin {gather*} \frac {b \ln \left (b x +a \right )}{\left (a d -b c \right ) a}-\frac {d \ln \left (d x +c \right )}{\left (a d -b c \right ) c}+\frac {\ln \relax (x )}{a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.93, size = 53, normalized size = 1.00 \begin {gather*} -\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 \relax (x)}{a c} \end {gather*}

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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mupad [B]  time = 0.49, size = 52, normalized size = 0.98 \begin {gather*} \frac {\ln \relax (x)}{a\,c}+\frac {b\,\ln \left (a+b\,x\right )}{a^2\,d-a\,b\,c}+\frac {d\,\ln \left (c+d\,x\right )}{b\,c^2-a\,c\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)*(c + d*x)),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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