Optimal. Leaf size=73 \[ \frac{1}{5} x^5 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9 \]
[Out]
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Rubi [A] time = 0.116278, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{1}{5} x^5 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9 \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{c e^{2} x^{9}}{9} + d^{2} \int a\, dx + \frac{d x^{3} \left (2 a e + b d\right )}{3} + \frac{e x^{7} \left (b e + 2 c d\right )}{7} + x^{5} \left (\frac{a e^{2}}{5} + \frac{2 b d e}{5} + \frac{c d^{2}}{5}\right ) \]
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)
[Out]
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Mathematica [A] time = 0.0277905, size = 73, normalized size = 1. \[ \frac{1}{5} x^5 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac{1}{7} e x^7 (b e+2 c d)+\frac{1}{9} c e^2 x^9 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]
[Out]
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Maple [A] time = 0., size = 70, normalized size = 1. \[{\frac{c{e}^{2}{x}^{9}}{9}}+{\frac{ \left ( b{e}^{2}+2\,cde \right ){x}^{7}}{7}}+{\frac{ \left ( a{e}^{2}+2\,bde+c{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,ade+b{d}^{2} \right ){x}^{3}}{3}}+a{d}^{2}x \]
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)
[Out]
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Maxima [A] time = 0.700414, size = 93, normalized size = 1.27 \[ \frac{1}{9} \, c e^{2} x^{9} + \frac{1}{7} \,{\left (2 \, c d e + b e^{2}\right )} x^{7} + \frac{1}{5} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{5} + a d^{2} x + \frac{1}{3} \,{\left (b d^{2} + 2 \, a d e\right )} x^{3} \]
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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246286, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e^{2} c + \frac{2}{7} x^{7} e d c + \frac{1}{7} x^{7} e^{2} b + \frac{1}{5} x^{5} d^{2} c + \frac{2}{5} x^{5} e d b + \frac{1}{5} x^{5} e^{2} a + \frac{1}{3} x^{3} d^{2} b + \frac{2}{3} x^{3} e d a + x d^{2} a \]
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, algorithm="fricas")
[Out]
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Sympy [A] time = 0.126845, size = 78, normalized size = 1.07 \[ a d^{2} x + \frac{c e^{2} x^{9}}{9} + x^{7} \left (\frac{b e^{2}}{7} + \frac{2 c d e}{7}\right ) + x^{5} \left (\frac{a e^{2}}{5} + \frac{2 b d e}{5} + \frac{c d^{2}}{5}\right ) + x^{3} \left (\frac{2 a d e}{3} + \frac{b d^{2}}{3}\right ) \]
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)
[Out]
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GIAC/XCAS [A] time = 0.268713, size = 103, normalized size = 1.41 \[ \frac{1}{9} \, c x^{9} e^{2} + \frac{2}{7} \, c d x^{7} e + \frac{1}{7} \, b x^{7} e^{2} + \frac{1}{5} \, c d^{2} x^{5} + \frac{2}{5} \, b d x^{5} e + \frac{1}{5} \, a x^{5} e^{2} + \frac{1}{3} \, b d^{2} x^{3} + \frac{2}{3} \, a d x^{3} e + a d^{2} x \]
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, algorithm="giac")
[Out]