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

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

[Out]

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

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Rubi [A]  time = 0.0504188, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{a B+A b}{3 x^3}-\frac{a A}{5 x^5}-\frac{b B}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.62054, size = 27, normalized size = 0.87 \[ - \frac{A a}{5 x^{5}} - \frac{B b}{x} - \frac{\frac{A b}{3} + \frac{B a}{3}}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0207695, size = 33, normalized size = 1.06 \[ \frac{-a B-A b}{3 x^3}-\frac{a A}{5 x^5}-\frac{b B}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 28, normalized size = 0.9 \[ -{\frac{Ab+Ba}{3\,{x}^{3}}}-{\frac{Aa}{5\,{x}^{5}}}-{\frac{Bb}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(B*x^2+A)/x^6,x)

[Out]

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

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Maxima [A]  time = 1.38374, size = 39, normalized size = 1.26 \[ -\frac{15 \, B b x^{4} + 5 \,{\left (B a + A b\right )} x^{2} + 3 \, A a}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227657, size = 39, normalized size = 1.26 \[ -\frac{15 \, B b x^{4} + 5 \,{\left (B a + A b\right )} x^{2} + 3 \, A a}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.09165, size = 32, normalized size = 1.03 \[ - \frac{3 A a + 15 B b x^{4} + x^{2} \left (5 A b + 5 B a\right )}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269024, size = 42, normalized size = 1.35 \[ -\frac{15 \, B b x^{4} + 5 \, B a x^{2} + 5 \, A b x^{2} + 3 \, A a}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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