Optimal. Leaf size=48 \[ -\frac{a^2 c}{3 x^3}+b x (2 a d+b c)-\frac{a (a d+2 b c)}{x}+\frac{1}{3} b^2 d x^3 \]
[Out]
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Rubi [A] time = 0.0861551, 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 \[ -\frac{a^2 c}{3 x^3}+b x (2 a d+b c)-\frac{a (a d+2 b c)}{x}+\frac{1}{3} b^2 d x^3 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(c + d*x^2))/x^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} c}{3 x^{3}} - \frac{a \left (a d + 2 b c\right )}{x} + \frac{b^{2} d x^{3}}{3} + \frac{b \left (2 a d + b c\right ) \int c\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)/x**4,x)
[Out]
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Mathematica [A] time = 0.0326024, size = 50, normalized size = 1.04 \[ \frac{a^2 (-d)-2 a b c}{x}-\frac{a^2 c}{3 x^3}+b x (2 a d+b c)+\frac{1}{3} b^2 d x^3 \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(c + d*x^2))/x^4,x]
[Out]
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Maple [A] time = 0.008, size = 46, normalized size = 1. \[{\frac{{b}^{2}d{x}^{3}}{3}}+2\,xabd+x{b}^{2}c-{\frac{{a}^{2}c}{3\,{x}^{3}}}-{\frac{a \left ( ad+2\,bc \right ) }{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)/x^4,x)
[Out]
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Maxima [A] time = 1.34971, size = 68, normalized size = 1.42 \[ \frac{1}{3} \, b^{2} d x^{3} +{\left (b^{2} c + 2 \, a b d\right )} x - \frac{a^{2} c + 3 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225974, size = 70, normalized size = 1.46 \[ \frac{b^{2} d x^{6} + 3 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - a^{2} c - 3 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)/x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.65417, size = 49, normalized size = 1.02 \[ \frac{b^{2} d x^{3}}{3} + x \left (2 a b d + b^{2} c\right ) - \frac{a^{2} c + x^{2} \left (3 a^{2} d + 6 a b c\right )}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.219653, size = 68, normalized size = 1.42 \[ \frac{1}{3} \, b^{2} d x^{3} + b^{2} c x + 2 \, a b d x - \frac{6 \, a b c x^{2} + 3 \, a^{2} d x^{2} + a^{2} c}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)/x^4,x, algorithm="giac")
[Out]