Optimal. Leaf size=74 \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8 \]
[Out]
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Rubi [A] time = 0.194187, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8 \]
Antiderivative was successfully verified.
[In] Int[((d + e*x^2)^2*(a + b*x^2 + c*x^4))/x,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a d^{2} \log{\left (x^{2} \right )}}{2} + \frac{c e^{2} x^{8}}{8} + \frac{e x^{6} \left (b e + 2 c d\right )}{6} + \left (a e + \frac{b d}{2}\right ) \int ^{x^{2}} d\, dx + \left (\frac{a e^{2}}{2} + b d e + \frac{c d^{2}}{2}\right ) \int ^{x^{2}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)**2*(c*x**4+b*x**2+a)/x,x)
[Out]
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Mathematica [A] time = 0.0387653, size = 74, normalized size = 1. \[ \frac{1}{4} x^4 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8 \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x^2)^2*(a + b*x^2 + c*x^4))/x,x]
[Out]
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Maple [A] time = 0.004, size = 77, normalized size = 1. \[{\frac{c{e}^{2}{x}^{8}}{8}}+{\frac{{x}^{6}b{e}^{2}}{6}}+{\frac{{x}^{6}cde}{3}}+{\frac{{x}^{4}a{e}^{2}}{4}}+{\frac{{x}^{4}bde}{2}}+{\frac{{x}^{4}c{d}^{2}}{4}}+{x}^{2}ade+{\frac{{x}^{2}b{d}^{2}}{2}}+a{d}^{2}\ln \left ( x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)^2*(c*x^4+b*x^2+a)/x,x)
[Out]
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Maxima [A] time = 0.700413, size = 99, normalized size = 1.34 \[ \frac{1}{8} \, c e^{2} x^{8} + \frac{1}{6} \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + \frac{1}{4} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + \frac{1}{2} \, a d^{2} \log \left (x^{2}\right ) + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)*(e*x^2 + d)^2/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.252087, size = 95, normalized size = 1.28 \[ \frac{1}{8} \, c e^{2} x^{8} + \frac{1}{6} \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + \frac{1}{4} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + a d^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)*(e*x^2 + d)^2/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.26214, size = 73, normalized size = 0.99 \[ a d^{2} \log{\left (x \right )} + \frac{c e^{2} x^{8}}{8} + x^{6} \left (\frac{b e^{2}}{6} + \frac{c d e}{3}\right ) + x^{4} \left (\frac{a e^{2}}{4} + \frac{b d e}{2} + \frac{c d^{2}}{4}\right ) + x^{2} \left (a d e + \frac{b d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)**2*(c*x**4+b*x**2+a)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.269057, size = 107, normalized size = 1.45 \[ \frac{1}{8} \, c x^{8} e^{2} + \frac{1}{3} \, c d x^{6} e + \frac{1}{6} \, b x^{6} e^{2} + \frac{1}{4} \, c d^{2} x^{4} + \frac{1}{2} \, b d x^{4} e + \frac{1}{4} \, a x^{4} e^{2} + \frac{1}{2} \, b d^{2} x^{2} + a d x^{2} e + \frac{1}{2} \, a d^{2}{\rm ln}\left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)*(e*x^2 + d)^2/x,x, algorithm="giac")
[Out]