3.56 \(\int \frac{(a+b x) (A+B x)}{x^3} \, dx\)

Optimal. Leaf size=27 \[ -\frac{a B+A b}{x}-\frac{a A}{2 x^2}+b B \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0389608, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{a B+A b}{x}-\frac{a A}{2 x^2}+b B \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(A + B*x))/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.03037, size = 22, normalized size = 0.81 \[ - \frac{A a}{2 x^{2}} + B b \log{\left (x \right )} - \frac{A b + B a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0171661, size = 28, normalized size = 1.04 \[ \frac{-a B-A b}{x}-\frac{a A}{2 x^2}+b B \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(A + B*x))/x^3,x]

[Out]

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

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Maple [A]  time = 0.007, size = 28, normalized size = 1. \[ bB\ln \left ( x \right ) -{\frac{Aa}{2\,{x}^{2}}}-{\frac{Ab}{x}}-{\frac{Ba}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.3489, size = 34, normalized size = 1.26 \[ B b \log \left (x\right ) - \frac{A a + 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.201038, size = 39, normalized size = 1.44 \[ \frac{2 \, B b x^{2} \log \left (x\right ) - A a - 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.53712, size = 26, normalized size = 0.96 \[ B b \log{\left (x \right )} - \frac{A a + x \left (2 A b + 2 B a\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.278564, size = 35, normalized size = 1.3 \[ B b{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a + 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x^3,x, algorithm="giac")

[Out]

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