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