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

Optimal. Leaf size=44 \[ \frac {c \log (c+d x)}{d (b c-a d)}-\frac {a \log (a+b x)}{b (b c-a d)} \]

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Rubi [A]  time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {72} \begin {gather*} \frac {c \log (c+d x)}{d (b c-a d)}-\frac {a \log (a+b x)}{b (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x)*(c + d*x)),x]

[Out]

-((a*Log[a + b*x])/(b*(b*c - a*d))) + (c*Log[c + d*x])/(d*(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 {x}{(a+b x) (c+d x)} \, dx &=\int \left (-\frac {a}{(b c-a d) (a+b x)}+\frac {c}{(b c-a d) (c+d x)}\right ) \, dx\\ &=-\frac {a \log (a+b x)}{b (b c-a d)}+\frac {c \log (c+d x)}{d (b c-a d)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.86 \begin {gather*} -\frac {a d \log (a+b x)-b c \log (c+d x)}{b^2 c d-a b d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/((a + b*x)*(c + d*x)),x]

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.20, size = 38, normalized size = 0.86 \begin {gather*} -\frac {a d \log \left (b x + a\right ) - b c \log \left (d x + c\right )}{b^{2} c d - a b d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.04, size = 46, normalized size = 1.05 \begin {gather*} -\frac {a \log \left ({\left | b x + a \right |}\right )}{b^{2} c - a b d} + \frac {c \log \left ({\left | d x + c \right |}\right )}{b c d - a d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 45, normalized size = 1.02 \begin {gather*} \frac {a \ln \left (b x +a \right )}{\left (a d -b c \right ) b}-\frac {c \ln \left (d x +c \right )}{\left (a d -b c \right ) d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.16, size = 44, normalized size = 1.00 \begin {gather*} -\frac {a \log \left (b x + a\right )}{b^{2} c - a b d} + \frac {c \log \left (d x + c\right )}{b c d - a d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.12, size = 37, normalized size = 0.84 \begin {gather*} \frac {a\,d\,\ln \left (a+b\,x\right )-b\,c\,\ln \left (c+d\,x\right )}{b\,d\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((a + b*x)*(c + d*x)),x)

[Out]

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

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sympy [B]  time = 0.83, size = 138, normalized size = 3.14 \begin {gather*} \frac {a \log {\left (x + \frac {\frac {a^{3} d^{2}}{b \left (a d - b c\right )} - \frac {2 a^{2} c d}{a d - b c} + \frac {a b c^{2}}{a d - b c} + 2 a c}{a d + b c} \right )}}{b \left (a d - b c\right )} - \frac {c \log {\left (x + \frac {- \frac {a^{2} c d}{a d - b c} + \frac {2 a b c^{2}}{a d - b c} + 2 a c - \frac {b^{2} c^{3}}{d \left (a d - b c\right )}}{a d + b c} \right )}}{d \left (a d - b c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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