3.9 \(\int \frac{(A+B x) \left (b x+c x^2\right )}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0367552, antiderivative size = 27, 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 \[ -\frac{A c+b B}{x}-\frac{A b}{2 x^2}+B c \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2))/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.58092, size = 22, normalized size = 0.81 \[ - \frac{A b}{2 x^{2}} + B c \log{\left (x \right )} - \frac{A c + B b}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0171201, size = 28, normalized size = 1.04 \[ \frac{-A c-b B}{x}-\frac{A b}{2 x^2}+B c \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2))/x^4,x]

[Out]

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

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Maple [A]  time = 0.008, size = 28, normalized size = 1. \[ Bc\ln \left ( x \right ) -{\frac{Ab}{2\,{x}^{2}}}-{\frac{Ac}{x}}-{\frac{Bb}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)/x^4,x)

[Out]

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

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Maxima [A]  time = 0.690632, size = 34, normalized size = 1.26 \[ B c \log \left (x\right ) - \frac{A b + 2 \,{\left (B b + A c\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.27448, size = 39, normalized size = 1.44 \[ \frac{2 \, B c x^{2} \log \left (x\right ) - A b - 2 \,{\left (B b + A c\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.53715, size = 26, normalized size = 0.96 \[ B c \log{\left (x \right )} - \frac{A b + x \left (2 A c + 2 B b\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.267946, size = 35, normalized size = 1.3 \[ B c{\rm ln}\left ({\left | x \right |}\right ) - \frac{A b + 2 \,{\left (B b + A c\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^4,x, algorithm="giac")

[Out]

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