3.1110 \(\int \frac{A+B x}{(a+b x)^2} \, dx\)

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

[Out]

-((A*b - a*B)/(b^2*(a + b*x))) + (B*Log[a + b*x])/b^2

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Rubi [A]  time = 0.0508162, antiderivative size = 32, 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 \[ \frac{B \log (a+b x)}{b^2}-\frac{A b-a B}{b^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(a + b*x)^2,x]

[Out]

-((A*b - a*B)/(b^2*(a + b*x))) + (B*Log[a + b*x])/b^2

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Rubi in Sympy [A]  time = 7.95116, size = 26, normalized size = 0.81 \[ \frac{B \log{\left (a + b x \right )}}{b^{2}} - \frac{A b - B a}{b^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**2,x)

[Out]

B*log(a + b*x)/b**2 - (A*b - B*a)/(b**2*(a + b*x))

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Mathematica [A]  time = 0.0194492, size = 31, normalized size = 0.97 \[ \frac{a B-A b}{b^2 (a+b x)}+\frac{B \log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(a + b*x)^2,x]

[Out]

(-(A*b) + a*B)/(b^2*(a + b*x)) + (B*Log[a + b*x])/b^2

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Maple [A]  time = 0.003, size = 39, normalized size = 1.2 \[{\frac{B\ln \left ( bx+a \right ) }{{b}^{2}}}-{\frac{A}{b \left ( bx+a \right ) }}+{\frac{Ba}{ \left ( bx+a \right ){b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^2,x)

[Out]

B*ln(b*x+a)/b^2-1/(b*x+a)/b*A+1/(b*x+a)/b^2*B*a

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Maxima [A]  time = 1.34647, size = 46, normalized size = 1.44 \[ \frac{B a - A b}{b^{3} x + a b^{2}} + \frac{B \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*a - A*b)/(b^3*x + a*b^2) + B*log(b*x + a)/b^2

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Fricas [A]  time = 0.216087, size = 50, normalized size = 1.56 \[ \frac{B a - A b +{\left (B b x + B a\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*a - A*b + (B*b*x + B*a)*log(b*x + a))/(b^3*x + a*b^2)

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Sympy [A]  time = 1.39364, size = 27, normalized size = 0.84 \[ \frac{B \log{\left (a + b x \right )}}{b^{2}} + \frac{- A b + B a}{a b^{2} + b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)**2,x)

[Out]

B*log(a + b*x)/b**2 + (-A*b + B*a)/(a*b**2 + b**3*x)

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GIAC/XCAS [A]  time = 0.224075, size = 77, normalized size = 2.41 \[ -\frac{B{\left (\frac{{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x + a\right )} b}\right )}}{b} - \frac{A}{{\left (b x + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-B*(ln(abs(b*x + a)/((b*x + a)^2*abs(b)))/b - a/((b*x + a)*b))/b - A/((b*x + a)*
b)