3.11 \(\int \frac{(A+B x) \left (b x+c x^2\right )}{x^6} \, dx\)

Optimal. Leaf size=33 \[ -\frac{A c+b B}{3 x^3}-\frac{A b}{4 x^4}-\frac{B c}{2 x^2} \]

[Out]

-(A*b)/(4*x^4) - (b*B + A*c)/(3*x^3) - (B*c)/(2*x^2)

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Rubi [A]  time = 0.042456, antiderivative size = 33, 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 \[ -\frac{A c+b B}{3 x^3}-\frac{A b}{4 x^4}-\frac{B c}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2))/x^6,x]

[Out]

-(A*b)/(4*x^4) - (b*B + A*c)/(3*x^3) - (B*c)/(2*x^2)

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Rubi in Sympy [A]  time = 6.92919, size = 31, normalized size = 0.94 \[ - \frac{A b}{4 x^{4}} - \frac{B c}{2 x^{2}} - \frac{\frac{A c}{3} + \frac{B b}{3}}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)/x**6,x)

[Out]

-A*b/(4*x**4) - B*c/(2*x**2) - (A*c/3 + B*b/3)/x**3

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Mathematica [A]  time = 0.0150472, size = 29, normalized size = 0.88 \[ -\frac{3 A b+4 A c x+4 b B x+6 B c x^2}{12 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2))/x^6,x]

[Out]

-(3*A*b + 4*b*B*x + 4*A*c*x + 6*B*c*x^2)/(12*x^4)

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Maple [A]  time = 0.008, size = 28, normalized size = 0.9 \[ -{\frac{Ab}{4\,{x}^{4}}}-{\frac{Ac+Bb}{3\,{x}^{3}}}-{\frac{Bc}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)/x^6,x)

[Out]

-1/4*A*b/x^4-1/3*(A*c+B*b)/x^3-1/2*B*c/x^2

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Maxima [A]  time = 0.692057, size = 36, normalized size = 1.09 \[ -\frac{6 \, B c x^{2} + 3 \, A b + 4 \,{\left (B b + A c\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

-1/12*(6*B*c*x^2 + 3*A*b + 4*(B*b + A*c)*x)/x^4

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Fricas [A]  time = 0.285036, size = 36, normalized size = 1.09 \[ -\frac{6 \, B c x^{2} + 3 \, A b + 4 \,{\left (B b + A c\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

-1/12*(6*B*c*x^2 + 3*A*b + 4*(B*b + A*c)*x)/x^4

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Sympy [A]  time = 2.00428, size = 31, normalized size = 0.94 \[ - \frac{3 A b + 6 B c x^{2} + x \left (4 A c + 4 B b\right )}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)/x**6,x)

[Out]

-(3*A*b + 6*B*c*x**2 + x*(4*A*c + 4*B*b))/(12*x**4)

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GIAC/XCAS [A]  time = 0.268773, size = 36, normalized size = 1.09 \[ -\frac{6 \, B c x^{2} + 4 \, B b x + 4 \, A c x + 3 \, A b}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^6,x, algorithm="giac")

[Out]

-1/12*(6*B*c*x^2 + 4*B*b*x + 4*A*c*x + 3*A*b)/x^4