3.13 \(\int \frac{(A+B x) \left (b x+c x^2\right )}{x^8} \, dx\)

Optimal. Leaf size=33 \[ -\frac{A c+b B}{5 x^5}-\frac{A b}{6 x^6}-\frac{B c}{4 x^4} \]

[Out]

-(A*b)/(6*x^6) - (b*B + A*c)/(5*x^5) - (B*c)/(4*x^4)

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Rubi [A]  time = 0.0413783, 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 \[ -\frac{A c+b B}{5 x^5}-\frac{A b}{6 x^6}-\frac{B c}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2))/x^8,x]

[Out]

-(A*b)/(6*x^6) - (b*B + A*c)/(5*x^5) - (B*c)/(4*x^4)

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Rubi in Sympy [A]  time = 6.97477, size = 31, normalized size = 0.94 \[ - \frac{A b}{6 x^{6}} - \frac{B c}{4 x^{4}} - \frac{\frac{A c}{5} + \frac{B b}{5}}{x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)/x**8,x)

[Out]

-A*b/(6*x**6) - B*c/(4*x**4) - (A*c/5 + B*b/5)/x**5

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Mathematica [A]  time = 0.0154398, size = 31, normalized size = 0.94 \[ -\frac{2 A (5 b+6 c x)+3 B x (4 b+5 c x)}{60 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2))/x^8,x]

[Out]

-(3*B*x*(4*b + 5*c*x) + 2*A*(5*b + 6*c*x))/(60*x^6)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.9 \[ -{\frac{Ab}{6\,{x}^{6}}}-{\frac{Bc}{4\,{x}^{4}}}-{\frac{Ac+Bb}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)/x^8,x)

[Out]

-1/6*A*b/x^6-1/4*B*c/x^4-1/5*(A*c+B*b)/x^5

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Maxima [A]  time = 0.690789, size = 36, normalized size = 1.09 \[ -\frac{15 \, B c x^{2} + 10 \, A b + 12 \,{\left (B b + A c\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^8,x, algorithm="maxima")

[Out]

-1/60*(15*B*c*x^2 + 10*A*b + 12*(B*b + A*c)*x)/x^6

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Fricas [A]  time = 0.265339, size = 36, normalized size = 1.09 \[ -\frac{15 \, B c x^{2} + 10 \, A b + 12 \,{\left (B b + A c\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^8,x, algorithm="fricas")

[Out]

-1/60*(15*B*c*x^2 + 10*A*b + 12*(B*b + A*c)*x)/x^6

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Sympy [A]  time = 2.61464, size = 31, normalized size = 0.94 \[ - \frac{10 A b + 15 B c x^{2} + x \left (12 A c + 12 B b\right )}{60 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)/x**8,x)

[Out]

-(10*A*b + 15*B*c*x**2 + x*(12*A*c + 12*B*b))/(60*x**6)

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GIAC/XCAS [A]  time = 0.26745, size = 36, normalized size = 1.09 \[ -\frac{15 \, B c x^{2} + 12 \, B b x + 12 \, A c x + 10 \, A b}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/x^8,x, algorithm="giac")

[Out]

-1/60*(15*B*c*x^2 + 12*B*b*x + 12*A*c*x + 10*A*b)/x^6