3.10 \(\int \frac{(A+B x) \left (b x+c x^2\right )}{x^5} \, dx\)

Optimal. Leaf size=31 \[ -\frac{A c+b B}{2 x^2}-\frac{A b}{3 x^3}-\frac{B c}{x} \]

[Out]

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

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Rubi [A]  time = 0.0413309, antiderivative size = 31, 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 \[ -\frac{A c+b B}{2 x^2}-\frac{A b}{3 x^3}-\frac{B c}{x} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2))/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.88172, size = 27, normalized size = 0.87 \[ - \frac{A b}{3 x^{3}} - \frac{B c}{x} - \frac{\frac{A c}{2} + \frac{B b}{2}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)/x**5,x)

[Out]

-A*b/(3*x**3) - B*c/x - (A*c/2 + B*b/2)/x**2

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Mathematica [A]  time = 0.0159601, size = 28, normalized size = 0.9 \[ -\frac{A (2 b+3 c x)+3 B x (b+2 c x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2))/x^5,x]

[Out]

-(3*B*x*(b + 2*c*x) + A*(2*b + 3*c*x))/(6*x^3)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.9 \[ -{\frac{Ab}{3\,{x}^{3}}}-{\frac{Ac+Bb}{2\,{x}^{2}}}-{\frac{Bc}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)/x^5,x)

[Out]

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

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Maxima [A]  time = 0.694254, size = 36, normalized size = 1.16 \[ -\frac{6 \, B c x^{2} + 2 \, A b + 3 \,{\left (B b + A c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^5,x, algorithm="maxima")

[Out]

-1/6*(6*B*c*x^2 + 2*A*b + 3*(B*b + A*c)*x)/x^3

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Fricas [A]  time = 0.274299, size = 36, normalized size = 1.16 \[ -\frac{6 \, B c x^{2} + 2 \, A b + 3 \,{\left (B b + A c\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^5,x, algorithm="fricas")

[Out]

-1/6*(6*B*c*x^2 + 2*A*b + 3*(B*b + A*c)*x)/x^3

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Sympy [A]  time = 1.77138, size = 31, normalized size = 1. \[ - \frac{2 A b + 6 B c x^{2} + x \left (3 A c + 3 B b\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)/x**5,x)

[Out]

-(2*A*b + 6*B*c*x**2 + x*(3*A*c + 3*B*b))/(6*x**3)

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GIAC/XCAS [A]  time = 0.266927, size = 36, normalized size = 1.16 \[ -\frac{6 \, B c x^{2} + 3 \, B b x + 3 \, A c x + 2 \, A b}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^5,x, algorithm="giac")

[Out]

-1/6*(6*B*c*x^2 + 3*B*b*x + 3*A*c*x + 2*A*b)/x^3