3.13.42 \(\int \frac {c+d x}{(a+b x)^2} \, dx\) [1242]

Optimal. Leaf size=32 \[ -\frac {b c-a d}{b^2 (a+b x)}+\frac {d \log (a+b x)}{b^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \begin {gather*} \frac {d \log (a+b x)}{b^2}-\frac {b c-a d}{b^2 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a + b*x)^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.97 \begin {gather*} \frac {-b c+a d}{b^2 (a+b x)}+\frac {d \log (a+b x)}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a + b*x)^2,x]

[Out]

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

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Mathics [A]
time = 1.86, size = 32, normalized size = 1.00 \begin {gather*} \frac {a d-b c+d \text {Log}\left [a+b x\right ] \left (a+b x\right )}{b^2 \left (a+b x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.13, size = 33, normalized size = 1.03

method	result	size
norman	\(\frac {a d -b c}{b^{2} \left (b x +a \right )}+\frac {d \ln \left (b x +a \right )}{b^{2}}\)	\(32\)
default	\(-\frac {-a d +b c}{b^{2} \left (b x +a \right )}+\frac {d \ln \left (b x +a \right )}{b^{2}}\)	\(33\)
risch	\(\frac {a d}{b^{2} \left (b x +a \right )}-\frac {c}{b \left (b x +a \right )}+\frac {d \ln \left (b x +a \right )}{b^{2}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 35, normalized size = 1.09 \begin {gather*} -\frac {b c - a d}{b^{3} x + a b^{2}} + \frac {d \log \left (b x + a\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.29, size = 39, normalized size = 1.22 \begin {gather*} -\frac {b c - a d - {\left (b d x + a d\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.11, size = 27, normalized size = 0.84 \begin {gather*} \frac {a d - b c}{a b^{2} + b^{3} x} + \frac {d \log {\left (a + b x \right )}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 33, normalized size = 1.03 \begin {gather*} \frac {d a-c b}{b b \left (x b+a\right )}+\frac {d \ln \left |x b+a\right |}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.17, size = 31, normalized size = 0.97 \begin {gather*} \frac {a\,d-b\,c}{b^2\,\left (a+b\,x\right )}+\frac {d\,\ln \left (a+b\,x\right )}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(a + b*x)^2,x)

[Out]

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

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