Optimal. Leaf size=48 \[ -\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5 \]
[Out]
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Rubi [A] time = 0.0780941, 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 \[ -\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(c + d*x^2))/x^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} c}{x} + \frac{a \left (a d + 2 b c\right ) \int d\, dx}{d} + \frac{b^{2} d x^{5}}{5} + \frac{b x^{3} \left (2 a d + b c\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)/x**2,x)
[Out]
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Mathematica [A] time = 0.0273953, size = 48, normalized size = 1. \[ -\frac{a^2 c}{x}+\frac{1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac{1}{5} b^2 d x^5 \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(c + d*x^2))/x^2,x]
[Out]
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Maple [A] time = 0.006, size = 49, normalized size = 1. \[{\frac{{b}^{2}d{x}^{5}}{5}}+{\frac{2\,{x}^{3}abd}{3}}+{\frac{{x}^{3}{b}^{2}c}{3}}+x{a}^{2}d+2\,xabc-{\frac{{a}^{2}c}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)/x^2,x)
[Out]
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Maxima [A] time = 1.35448, size = 65, normalized size = 1.35 \[ \frac{1}{5} \, b^{2} d x^{5} + \frac{1}{3} \,{\left (b^{2} c + 2 \, a b d\right )} x^{3} - \frac{a^{2} c}{x} +{\left (2 \, a b c + a^{2} d\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)/x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22208, size = 72, normalized size = 1.5 \[ \frac{3 \, b^{2} d x^{6} + 5 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - 15 \, a^{2} c + 15 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}}{15 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)/x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.2204, size = 48, normalized size = 1. \[ - \frac{a^{2} c}{x} + \frac{b^{2} d x^{5}}{5} + x^{3} \left (\frac{2 a b d}{3} + \frac{b^{2} c}{3}\right ) + x \left (a^{2} d + 2 a b c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.221327, size = 65, normalized size = 1.35 \[ \frac{1}{5} \, b^{2} d x^{5} + \frac{1}{3} \, b^{2} c x^{3} + \frac{2}{3} \, a b d x^{3} + 2 \, a b c x + a^{2} d x - \frac{a^{2} c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)/x^2,x, algorithm="giac")
[Out]