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

Optimal. Leaf size=22 \[ \log (x) (A c+b B)-\frac{A b}{x}+B c x \]

[Out]

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

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Rubi [A]  time = 0.0383647, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \log (x) (A c+b B)-\frac{A b}{x}+B c x \]

Antiderivative was successfully verified.

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

[Out]

-((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. \[ - \frac{A b}{x} + c \int B\, dx + \left (A c + B b\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0139404, size = 22, normalized size = 1. \[ \log (x) (A c+b B)-\frac{A b}{x}+B c x \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 23, normalized size = 1.1 \[ Bcx+Ac\ln \left ( x \right ) +Bb\ln \left ( x \right ) -{\frac{Ab}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)/x^3,x)

[Out]

B*c*x+A*c*ln(x)+B*b*ln(x)-A*b/x

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Maxima [A]  time = 0.700996, size = 30, normalized size = 1.36 \[ B c x +{\left (B b + A c\right )} \log \left (x\right ) - \frac{A b}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*c*x + (B*b + A*c)*log(x) - A*b/x

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Fricas [A]  time = 0.27441, size = 35, normalized size = 1.59 \[ \frac{B c x^{2} +{\left (B b + A c\right )} x \log \left (x\right ) - A b}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.23159, size = 19, normalized size = 0.86 \[ - \frac{A b}{x} + B c x + \left (A c + B b\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*b/x + B*c*x + (A*c + B*b)*log(x)

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GIAC/XCAS [A]  time = 0.271979, size = 31, normalized size = 1.41 \[ B c x +{\left (B b + A c\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A b}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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