3.1150 \(\int \frac{A+B x}{\left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=62 \[ \frac{\log (x) (b B-2 A c)}{b^3}-\frac{(b B-2 A c) \log (b+c x)}{b^3}+\frac{b B-A c}{b^2 (b+c x)}-\frac{A}{b^2 x} \]

[Out]

-(A/(b^2*x)) + (b*B - A*c)/(b^2*(b + c*x)) + ((b*B - 2*A*c)*Log[x])/b^3 - ((b*B
- 2*A*c)*Log[b + c*x])/b^3

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Rubi [A]  time = 0.118902, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\log (x) (b B-2 A c)}{b^3}-\frac{(b B-2 A c) \log (b+c x)}{b^3}+\frac{b B-A c}{b^2 (b+c x)}-\frac{A}{b^2 x} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(b*x + c*x^2)^2,x]

[Out]

-(A/(b^2*x)) + (b*B - A*c)/(b^2*(b + c*x)) + ((b*B - 2*A*c)*Log[x])/b^3 - ((b*B
- 2*A*c)*Log[b + c*x])/b^3

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Rubi in Sympy [A]  time = 15.9132, size = 54, normalized size = 0.87 \[ - \frac{A}{b^{2} x} - \frac{A c - B b}{b^{2} \left (b + c x\right )} - \frac{\left (2 A c - B b\right ) \log{\left (x \right )}}{b^{3}} + \frac{\left (2 A c - B b\right ) \log{\left (b + c x \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0828449, size = 56, normalized size = 0.9 \[ \frac{\frac{b (b B-A c)}{b+c x}+\log (x) (b B-2 A c)+(2 A c-b B) \log (b+c x)-\frac{A b}{x}}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(-((A*b)/x) + (b*(b*B - A*c))/(b + c*x) + (b*B - 2*A*c)*Log[x] + (-(b*B) + 2*A*c
)*Log[b + c*x])/b^3

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Maple [A]  time = 0.016, size = 78, normalized size = 1.3 \[ -{\frac{A}{{b}^{2}x}}-2\,{\frac{Ac\ln \left ( x \right ) }{{b}^{3}}}+{\frac{\ln \left ( x \right ) B}{{b}^{2}}}-{\frac{Ac}{{b}^{2} \left ( cx+b \right ) }}+{\frac{B}{b \left ( cx+b \right ) }}+2\,{\frac{\ln \left ( cx+b \right ) Ac}{{b}^{3}}}-{\frac{\ln \left ( cx+b \right ) B}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+b*x)^2,x)

[Out]

-A/b^2/x-2/b^3*ln(x)*A*c+1/b^2*ln(x)*B-1/b^2/(c*x+b)*A*c+1/b/(c*x+b)*B+2/b^3*ln(
c*x+b)*A*c-1/b^2*ln(c*x+b)*B

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Maxima [A]  time = 0.691733, size = 90, normalized size = 1.45 \[ -\frac{A b -{\left (B b - 2 \, A c\right )} x}{b^{2} c x^{2} + b^{3} x} - \frac{{\left (B b - 2 \, A c\right )} \log \left (c x + b\right )}{b^{3}} + \frac{{\left (B b - 2 \, A c\right )} \log \left (x\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.279354, size = 144, normalized size = 2.32 \[ -\frac{A b^{2} -{\left (B b^{2} - 2 \, A b c\right )} x +{\left ({\left (B b c - 2 \, A c^{2}\right )} x^{2} +{\left (B b^{2} - 2 \, A b c\right )} x\right )} \log \left (c x + b\right ) -{\left ({\left (B b c - 2 \, A c^{2}\right )} x^{2} +{\left (B b^{2} - 2 \, A b c\right )} x\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A*b^2 - (B*b^2 - 2*A*b*c)*x + ((B*b*c - 2*A*c^2)*x^2 + (B*b^2 - 2*A*b*c)*x)*lo
g(c*x + b) - ((B*b*c - 2*A*c^2)*x^2 + (B*b^2 - 2*A*b*c)*x)*log(x))/(b^3*c*x^2 +
b^4*x)

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Sympy [A]  time = 2.47937, size = 128, normalized size = 2.06 \[ \frac{- A b + x \left (- 2 A c + B b\right )}{b^{3} x + b^{2} c x^{2}} + \frac{\left (- 2 A c + B b\right ) \log{\left (x + \frac{- 2 A b c + B b^{2} - b \left (- 2 A c + B b\right )}{- 4 A c^{2} + 2 B b c} \right )}}{b^{3}} - \frac{\left (- 2 A c + B b\right ) \log{\left (x + \frac{- 2 A b c + B b^{2} + b \left (- 2 A c + B b\right )}{- 4 A c^{2} + 2 B b c} \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(c*x**2+b*x)**2,x)

[Out]

(-A*b + x*(-2*A*c + B*b))/(b**3*x + b**2*c*x**2) + (-2*A*c + B*b)*log(x + (-2*A*
b*c + B*b**2 - b*(-2*A*c + B*b))/(-4*A*c**2 + 2*B*b*c))/b**3 - (-2*A*c + B*b)*lo
g(x + (-2*A*b*c + B*b**2 + b*(-2*A*c + B*b))/(-4*A*c**2 + 2*B*b*c))/b**3

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GIAC/XCAS [A]  time = 0.280418, size = 96, normalized size = 1.55 \[ \frac{{\left (B b - 2 \, A c\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{B b x - 2 \, A c x - A b}{{\left (c x^{2} + b x\right )} b^{2}} - \frac{{\left (B b c - 2 \, A c^{2}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b - 2*A*c)*ln(abs(x))/b^3 + (B*b*x - 2*A*c*x - A*b)/((c*x^2 + b*x)*b^2) - (B*
b*c - 2*A*c^2)*ln(abs(c*x + b))/(b^3*c)