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

Optimal. Leaf size=48 \[ -\frac{a^2 A}{3 x^3}+b x (2 a B+A b)-\frac{a (a B+2 A b)}{x}+\frac{1}{3} b^2 B x^3 \]

[Out]

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

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Rubi [A]  time = 0.0822763, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a^2 A}{3 x^3}+b x (2 a B+A b)-\frac{a (a B+2 A b)}{x}+\frac{1}{3} b^2 B x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{3 x^{3}} + \frac{B b^{2} x^{3}}{3} - \frac{a \left (2 A b + B a\right )}{x} + \frac{b \left (A b + 2 B a\right ) \int A\, dx}{A} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0346548, size = 50, normalized size = 1.04 \[ \frac{a^2 (-B)-2 a A b}{x}-\frac{a^2 A}{3 x^3}+b x (2 a B+A b)+\frac{1}{3} b^2 B x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 46, normalized size = 1. \[{\frac{{b}^{2}B{x}^{3}}{3}}+Ax{b}^{2}+2\,Bxab-{\frac{A{a}^{2}}{3\,{x}^{3}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34977, size = 68, normalized size = 1.42 \[ \frac{1}{3} \, B b^{2} x^{3} +{\left (2 \, B a b + A b^{2}\right )} x - \frac{A a^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.243247, size = 70, normalized size = 1.46 \[ \frac{B b^{2} x^{6} + 3 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - A a^{2} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.65045, size = 49, normalized size = 1.02 \[ \frac{B b^{2} x^{3}}{3} + x \left (A b^{2} + 2 B a b\right ) - \frac{A a^{2} + x^{2} \left (6 A a b + 3 B a^{2}\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227837, size = 68, normalized size = 1.42 \[ \frac{1}{3} \, B b^{2} x^{3} + 2 \, B a b x + A b^{2} x - \frac{3 \, B a^{2} x^{2} + 6 \, A a b x^{2} + A a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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