3.11 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^8} \, dx\)

Optimal. Leaf size=31 \[ -\frac{A c+b B}{3 x^3}-\frac{A b}{5 x^5}-\frac{B c}{x} \]

[Out]

-(A*b)/(5*x^5) - (b*B + A*c)/(3*x^3) - (B*c)/x

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Rubi [A]  time = 0.063164, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{A c+b B}{3 x^3}-\frac{A b}{5 x^5}-\frac{B c}{x} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4))/x^8,x]

[Out]

-(A*b)/(5*x^5) - (b*B + A*c)/(3*x^3) - (B*c)/x

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Rubi in Sympy [A]  time = 8.18023, size = 27, normalized size = 0.87 \[ - \frac{A b}{5 x^{5}} - \frac{B c}{x} - \frac{\frac{A c}{3} + \frac{B b}{3}}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)/x**8,x)

[Out]

-A*b/(5*x**5) - B*c/x - (A*c/3 + B*b/3)/x**3

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Mathematica [A]  time = 0.019407, size = 33, normalized size = 1.06 \[ \frac{-A c-b B}{3 x^3}-\frac{A b}{5 x^5}-\frac{B c}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^8,x]

[Out]

-(A*b)/(5*x^5) + (-(b*B) - A*c)/(3*x^3) - (B*c)/x

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Maple [A]  time = 0.008, size = 28, normalized size = 0.9 \[ -{\frac{Ac+Bb}{3\,{x}^{3}}}-{\frac{Ab}{5\,{x}^{5}}}-{\frac{Bc}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)/x^8,x)

[Out]

-1/3*(A*c+B*b)/x^3-1/5*A*b/x^5-B*c/x

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Maxima [A]  time = 1.37732, size = 39, normalized size = 1.26 \[ -\frac{15 \, B c x^{4} + 5 \,{\left (B b + A c\right )} x^{2} + 3 \, A b}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)/x^8,x, algorithm="maxima")

[Out]

-1/15*(15*B*c*x^4 + 5*(B*b + A*c)*x^2 + 3*A*b)/x^5

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Fricas [A]  time = 0.221486, size = 39, normalized size = 1.26 \[ -\frac{15 \, B c x^{4} + 5 \,{\left (B b + A c\right )} x^{2} + 3 \, A b}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)/x^8,x, algorithm="fricas")

[Out]

-1/15*(15*B*c*x^4 + 5*(B*b + A*c)*x^2 + 3*A*b)/x^5

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Sympy [A]  time = 1.01271, size = 32, normalized size = 1.03 \[ - \frac{3 A b + 15 B c x^{4} + x^{2} \left (5 A c + 5 B b\right )}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)/x**8,x)

[Out]

-(3*A*b + 15*B*c*x**4 + x**2*(5*A*c + 5*B*b))/(15*x**5)

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GIAC/XCAS [A]  time = 0.205191, size = 42, normalized size = 1.35 \[ -\frac{15 \, B c x^{4} + 5 \, B b x^{2} + 5 \, A c x^{2} + 3 \, A b}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)/x^8,x, algorithm="giac")

[Out]

-1/15*(15*B*c*x^4 + 5*B*b*x^2 + 5*A*c*x^2 + 3*A*b)/x^5