3.849 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )}{x^3} \, dx\)

Optimal. Leaf size=36 \[ -\frac{a B+A b}{x}-\frac{a A}{2 x^2}+\log (x) (A c+b B)+B c x \]

[Out]

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

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Rubi [A]  time = 0.0653984, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{a B+A b}{x}-\frac{a A}{2 x^2}+\log (x) (A c+b B)+B c x \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2))/x^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a}{2 x^{2}} + c \int B\, dx + \left (A c + B b\right ) \log{\left (x \right )} - \frac{A b + B a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0362528, size = 37, normalized size = 1.03 \[ \frac{-a B-A b}{x}-\frac{a A}{2 x^2}+\log (x) (A c+b B)+B c x \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2))/x^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 37, normalized size = 1. \[ Bcx+Ac\ln \left ( x \right ) +Bb\ln \left ( x \right ) -{\frac{aA}{2\,{x}^{2}}}-{\frac{Ab}{x}}-{\frac{Ba}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)/x^3,x)

[Out]

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

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Maxima [A]  time = 0.688743, size = 46, normalized size = 1.28 \[ B c x +{\left (B b + A c\right )} \log \left (x\right ) - \frac{A a + 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.269242, size = 55, normalized size = 1.53 \[ \frac{2 \, B c x^{3} + 2 \,{\left (B b + A c\right )} x^{2} \log \left (x\right ) - A a - 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.78463, size = 34, normalized size = 0.94 \[ B c x + \left (A c + B b\right ) \log{\left (x \right )} - \frac{A a + x \left (2 A b + 2 B a\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.272758, size = 47, normalized size = 1.31 \[ B c x +{\left (B b + A c\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a + 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(B*x + A)/x^3,x, algorithm="giac")

[Out]

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