3.1347 \(\int \frac{a+b x}{(c+d x)^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.0496636, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(c + d*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.04086, size = 26, normalized size = 0.84 \[ \frac{b \log{\left (c + d x \right )}}{d^{2}} - \frac{a d - b c}{d^{2} \left (c + d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(d*x+c)**2,x)

[Out]

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

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Mathematica [A]  time = 0.017017, size = 31, normalized size = 1. \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 39, normalized size = 1.3 \[{\frac{b\ln \left ( dx+c \right ) }{{d}^{2}}}-{\frac{a}{d \left ( dx+c \right ) }}+{\frac{bc}{{d}^{2} \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(d*x+c)^2,x)

[Out]

b*ln(d*x+c)/d^2-1/d/(d*x+c)*a+1/d^2/(d*x+c)*b*c

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Maxima [A]  time = 1.34677, size = 46, normalized size = 1.48 \[ \frac{b c - a d}{d^{3} x + c d^{2}} + \frac{b \log \left (d x + c\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224569, size = 50, normalized size = 1.61 \[ \frac{b c - a d +{\left (b d x + b c\right )} \log \left (d x + c\right )}{d^{3} x + c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.648158, size = 27, normalized size = 0.87 \[ \frac{b \log{\left (c + d x \right )}}{d^{2}} - \frac{a d - b c}{c d^{2} + d^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(d*x+c)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226467, size = 77, normalized size = 2.48 \[ -\frac{b{\left (\frac{{\rm ln}\left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{d} - \frac{c}{{\left (d x + c\right )} d}\right )}}{d} - \frac{a}{{\left (d x + c\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(d*x + c)^2,x, algorithm="giac")

[Out]

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