3.7 \(\int \frac{\left (a+b x^2\right ) \left (A+B x^2\right )}{x^4} \, dx\)

Optimal. Leaf size=26 \[ -\frac{a B+A b}{x}-\frac{a A}{3 x^3}+b B x \]

[Out]

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

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Rubi [A]  time = 0.0497919, antiderivative size = 26, 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 \[ -\frac{a B+A b}{x}-\frac{a A}{3 x^3}+b B x \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a}{3 x^{3}} + b \int B\, dx - \frac{A b + B a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0204076, size = 27, normalized size = 1.04 \[ \frac{-a B-A b}{x}-\frac{a A}{3 x^3}+b B x \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 25, normalized size = 1. \[ bBx-{\frac{Aa}{3\,{x}^{3}}}-{\frac{Ab+Ba}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(B*x^2+A)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.34497, size = 35, normalized size = 1.35 \[ B b x - \frac{3 \,{\left (B a + A b\right )} x^{2} + A a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225187, size = 39, normalized size = 1.5 \[ \frac{3 \, B b x^{4} - 3 \,{\left (B a + A b\right )} x^{2} - A a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.44718, size = 26, normalized size = 1. \[ B b x - \frac{A a + x^{2} \left (3 A b + 3 B a\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.254484, size = 38, normalized size = 1.46 \[ B b x - \frac{3 \, B a x^{2} + 3 \, A b x^{2} + A a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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