3.10 \(\int \frac{\left (a+b x^2\right ) \left (A+B x^2\right )}{x^7} \, dx\)

Optimal. Leaf size=33 \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{b B}{2 x^2} \]

[Out]

-(a*A)/(6*x^6) - (A*b + a*B)/(4*x^4) - (b*B)/(2*x^2)

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Rubi [A]  time = 0.0698478, 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 \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{b B}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)*(A + B*x^2))/x^7,x]

[Out]

-(a*A)/(6*x^6) - (A*b + a*B)/(4*x^4) - (b*B)/(2*x^2)

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Rubi in Sympy [A]  time = 9.8017, size = 31, normalized size = 0.94 \[ - \frac{A a}{6 x^{6}} - \frac{B b}{2 x^{2}} - \frac{\frac{A b}{4} + \frac{B a}{4}}{x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)*(B*x**2+A)/x**7,x)

[Out]

-A*a/(6*x**6) - B*b/(2*x**2) - (A*b/4 + B*a/4)/x**4

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Mathematica [A]  time = 0.0155557, size = 35, normalized size = 1.06 \[ \frac{-a B-A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{b B}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)*(A + B*x^2))/x^7,x]

[Out]

-(a*A)/(6*x^6) + (-(A*b) - a*B)/(4*x^4) - (b*B)/(2*x^2)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.9 \[ -{\frac{Aa}{6\,{x}^{6}}}-{\frac{Ab+Ba}{4\,{x}^{4}}}-{\frac{Bb}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(B*x^2+A)/x^7,x)

[Out]

-1/6*a*A/x^6-1/4*(A*b+B*a)/x^4-1/2*b*B/x^2

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Maxima [A]  time = 1.33117, size = 39, normalized size = 1.18 \[ -\frac{6 \, B b x^{4} + 3 \,{\left (B a + A b\right )} x^{2} + 2 \, A a}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.2279, size = 39, normalized size = 1.18 \[ -\frac{6 \, B b x^{4} + 3 \,{\left (B a + A b\right )} x^{2} + 2 \, A a}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.61765, size = 32, normalized size = 0.97 \[ - \frac{2 A a + 6 B b x^{4} + x^{2} \left (3 A b + 3 B a\right )}{12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)*(B*x**2+A)/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.282054, size = 42, normalized size = 1.27 \[ -\frac{6 \, B b x^{4} + 3 \, B a x^{2} + 3 \, A b x^{2} + 2 \, A a}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)/x^7,x, algorithm="giac")

[Out]

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