3.11.58 \(\int \frac {a c-b c x}{(a+b x)^2} \, dx\) [1058]

Optimal. Leaf size=27 \[ -\frac {2 a c}{b (a+b x)}-\frac {c \log (a+b x)}{b} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 27, 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 = {45} \begin {gather*} -\frac {2 a c}{b (a+b x)}-\frac {c \log (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*c - b*c*x)/(a + b*x)^2,x]

[Out]

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

Rule 45

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

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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.85 \begin {gather*} -\frac {c \left (\frac {2 a}{a+b x}+\log (a+b x)\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*c - b*c*x)/(a + b*x)^2,x]

[Out]

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

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Mathics [A]
time = 1.74, size = 29, normalized size = 1.07 \begin {gather*} \frac {c \left (-2 a-\text {Log}\left [a+b x\right ] \left (a+b x\right )\right )}{b \left (a+b x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(a + b*x)^2*(a*c - b*c*x)^1,x]')

[Out]

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

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Maple [A]
time = 0.14, size = 28, normalized size = 1.04

method	result	size
norman	\(\frac {2 c x}{b x +a}-\frac {c \ln \left (b x +a \right )}{b}\)	\(25\)
default	\(c \left (-\frac {2 a}{b \left (b x +a \right )}-\frac {\ln \left (b x +a \right )}{b}\right )\)	\(28\)
risch	\(-\frac {2 a c}{b \left (b x +a \right )}-\frac {c \ln \left (b x +a \right )}{b}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*c*x+a*c)/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 28, normalized size = 1.04 \begin {gather*} -\frac {2 \, a c}{b^{2} x + a b} - \frac {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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.29, size = 33, normalized size = 1.22 \begin {gather*} -\frac {2 \, a c + {\left (b c x + a c\right )} \log \left (b x + a\right )}{b^{2} x + a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*c*x+a*c)/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.09, size = 24, normalized size = 0.89 \begin {gather*} - \frac {2 a c}{a b + b^{2} x} - \frac {c \log {\left (a + b x \right )}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*c*x+a*c)/(b*x+a)**2,x)

[Out]

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

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Giac [A]
time = 0.00, size = 26, normalized size = 0.96 \begin {gather*} -\frac {2 c a}{b \left (x b+a\right )}-\frac {c \ln \left |x b+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)^2,x)

[Out]

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

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Mupad [B]
time = 0.04, size = 27, normalized size = 1.00 \begin {gather*} -\frac {c\,\ln \left (a+b\,x\right )}{b}-\frac {2\,a\,c}{b\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)/(a + b*x)^2,x)

[Out]

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

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