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

Optimal. Leaf size=26 \[ -\frac{a A}{2 x^2}-\frac{a B}{x}+A c \log (x)+B c x \]

[Out]

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

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Rubi [A]  time = 0.0417559, antiderivative size = 26, 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 \[ -\frac{a A}{2 x^2}-\frac{a B}{x}+A c \log (x)+B c x \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00730489, size = 26, normalized size = 1. \[ -\frac{a A}{2 x^2}-\frac{a B}{x}+A c \log (x)+B c x \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 25, normalized size = 1. \[ -{\frac{aA}{2\,{x}^{2}}}-{\frac{Ba}{x}}+Bcx+Ac\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)/x^3,x)

[Out]

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

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Maxima [A]  time = 0.684125, size = 32, normalized size = 1.23 \[ B c x + A c \log \left (x\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.291383, size = 42, normalized size = 1.62 \[ \frac{2 \, B c x^{3} + 2 \, A c x^{2} \log \left (x\right ) - 2 \, B a x - A a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.39434, size = 26, normalized size = 1. \[ A c \log{\left (x \right )} + B c x - \frac{A a + 2 B a x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271655, size = 34, normalized size = 1.31 \[ B c x + A c{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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