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

Optimal. Leaf size=65 \[ -\frac{\log (x) (2 A b-a B)}{a^3}+\frac{(2 A b-a B) \log (a+b x)}{a^3}-\frac{A b-a B}{a^2 (a+b x)}-\frac{A}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.117456, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{\log (x) (2 A b-a B)}{a^3}+\frac{(2 A b-a B) \log (a+b x)}{a^3}-\frac{A b-a B}{a^2 (a+b x)}-\frac{A}{a^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.2582, size = 54, normalized size = 0.83 \[ - \frac{A}{a^{2} x} - \frac{A b - B a}{a^{2} \left (a + b x\right )} - \frac{\left (2 A b - B a\right ) \log{\left (x \right )}}{a^{3}} + \frac{\left (2 A b - B a\right ) \log{\left (a + b x \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0673055, size = 56, normalized size = 0.86 \[ \frac{\frac{a (a B-A b)}{a+b x}+\log (x) (a B-2 A b)+(2 A b-a B) \log (a+b x)-\frac{a A}{x}}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 78, normalized size = 1.2 \[ -{\frac{A}{{a}^{2}x}}-2\,{\frac{A\ln \left ( x \right ) b}{{a}^{3}}}+{\frac{\ln \left ( x \right ) B}{{a}^{2}}}+2\,{\frac{\ln \left ( bx+a \right ) Ab}{{a}^{3}}}-{\frac{\ln \left ( bx+a \right ) B}{{a}^{2}}}-{\frac{Ab}{{a}^{2} \left ( bx+a \right ) }}+{\frac{B}{a \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^2/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.34597, size = 90, normalized size = 1.38 \[ -\frac{A a -{\left (B a - 2 \, A b\right )} x}{a^{2} b x^{2} + a^{3} x} - \frac{{\left (B a - 2 \, A b\right )} \log \left (b x + a\right )}{a^{3}} + \frac{{\left (B a - 2 \, A b\right )} \log \left (x\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21003, size = 144, normalized size = 2.22 \[ -\frac{A a^{2} -{\left (B a^{2} - 2 \, A a b\right )} x +{\left ({\left (B a b - 2 \, A b^{2}\right )} x^{2} +{\left (B a^{2} - 2 \, A a b\right )} x\right )} \log \left (b x + a\right ) -{\left ({\left (B a b - 2 \, A b^{2}\right )} x^{2} +{\left (B a^{2} - 2 \, A a b\right )} x\right )} \log \left (x\right )}{a^{3} b x^{2} + a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.86659, size = 128, normalized size = 1.97 \[ \frac{- A a + x \left (- 2 A b + B a\right )}{a^{3} x + a^{2} b x^{2}} + \frac{\left (- 2 A b + B a\right ) \log{\left (x + \frac{- 2 A a b + B a^{2} - a \left (- 2 A b + B a\right )}{- 4 A b^{2} + 2 B a b} \right )}}{a^{3}} - \frac{\left (- 2 A b + B a\right ) \log{\left (x + \frac{- 2 A a b + B a^{2} + a \left (- 2 A b + B a\right )}{- 4 A b^{2} + 2 B a b} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.321837, size = 116, normalized size = 1.78 \[ \frac{A b}{a^{3}{\left (\frac{a}{b x + a} - 1\right )}} + \frac{{\left (B a b - 2 \, A b^{2}\right )}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{3} b} + \frac{\frac{B a b^{2}}{b x + a} - \frac{A b^{3}}{b x + a}}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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