3.10.63 \(\int \frac {a+b x}{a c-b c x} \, dx\)

Optimal. Leaf size=23 \[ -\frac {2 a \log (a-b x)}{b c}-\frac {x}{c} \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \begin {gather*} -\frac {2 a \log (a-b x)}{b c}-\frac {x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(a*c - b*c*x),x]

[Out]

-(x/c) - (2*a*Log[a - b*x])/(b*c)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2 a \log (a-b x)}{b c}-\frac {x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(a*c - b*c*x),x]

[Out]

-(x/c) - (2*a*Log[a - b*x])/(b*c)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.31, size = 23, normalized size = 1.00 \begin {gather*} -\frac {b x + 2 \, a \log \left (b x - a\right )}{b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x + 2*a*log(b*x - a))/(b*c)

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giac [A]  time = 0.92, size = 25, normalized size = 1.09 \begin {gather*} -\frac {x}{c} - \frac {2 \, a \log \left ({\left | b x - a \right |}\right )}{b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/c - 2*a*log(abs(b*x - a))/(b*c)

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maple [A]  time = 0.00, size = 25, normalized size = 1.09 \begin {gather*} -\frac {2 a \ln \left (b x -a \right )}{b c}-\frac {x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/c-2/c*a/b*ln(b*x-a)

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maxima [A]  time = 1.31, size = 24, normalized size = 1.04 \begin {gather*} -\frac {x}{c} - \frac {2 \, a \log \left (b x - a\right )}{b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/c - 2*a*log(b*x - a)/(b*c)

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mupad [B]  time = 0.05, size = 23, normalized size = 1.00 \begin {gather*} -\frac {b\,x+2\,a\,\ln \left (b\,x-a\right )}{b\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(a*c - b*c*x),x)

[Out]

-(b*x + 2*a*log(b*x - a))/(b*c)

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sympy [A]  time = 0.14, size = 17, normalized size = 0.74 \begin {gather*} - \frac {2 a \log {\left (- a + b x \right )}}{b c} - \frac {x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*log(-a + b*x)/(b*c) - x/c

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