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

Optimal. Leaf size=18 \[ \frac {2 a c \log (a+b x)}{b}-c x \]

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Rubi [A]  time = 0.01, antiderivative size = 18, 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 c \log (a+b x)}{b}-c x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*c - b*c*x)/(a + b*x),x]

[Out]

-(c*x) + (2*a*c*Log[a + b*x])/b

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

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} c \left (\frac {2 a \log (a+b x)}{b}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*c - b*c*x)/(a + b*x),x]

[Out]

c*(-x + (2*a*Log[a + b*x])/b)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.35, size = 20, normalized size = 1.11 \begin {gather*} -\frac {b c x - 2 \, a c \log \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*c*x - 2*a*c*log(b*x + a))/b

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giac [A]  time = 1.01, size = 19, normalized size = 1.06 \begin {gather*} -c x + \frac {2 \, a c \log \left ({\left | b x + a \right |}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*x + 2*a*c*log(abs(b*x + a))/b

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maple [A]  time = 0.00, size = 19, normalized size = 1.06 \begin {gather*} \frac {2 a c \ln \left (b x +a \right )}{b}-c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 18, normalized size = 1.00 \begin {gather*} -c x + \frac {2 \, a c \log \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c*x + 2*a*c*log(b*x + a)/b

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mupad [B]  time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} \frac {2\,a\,c\,\ln \left (a+b\,x\right )}{b}-c\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)/(a + b*x),x)

[Out]

(2*a*c*log(a + b*x))/b - c*x

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sympy [A]  time = 0.12, size = 15, normalized size = 0.83 \begin {gather*} \frac {2 a c \log {\left (a + b x \right )}}{b} - c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*c*log(a + b*x)/b - c*x

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