Optimal. Leaf size=155 \[ a^2 d^2 x+\frac{1}{9} x^9 \left (2 c e (a e+2 b d)+b^2 e^2+c^2 d^2\right )+\frac{2}{7} x^7 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+\frac{1}{5} x^5 \left (4 a b d e+a \left (a e^2+2 c d^2\right )+b^2 d^2\right )+\frac{2}{3} a d x^3 (a e+b d)+\frac{2}{11} c e x^{11} (b e+c d)+\frac{1}{13} c^2 e^2 x^{13} \]
[Out]
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Rubi [A] time = 0.281059, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ a^2 d^2 x+\frac{1}{9} x^9 \left (2 c e (a e+2 b d)+b^2 e^2+c^2 d^2\right )+\frac{2}{7} x^7 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+\frac{1}{5} x^5 \left (4 a b d e+a \left (a e^2+2 c d^2\right )+b^2 d^2\right )+\frac{2}{3} a d x^3 (a e+b d)+\frac{2}{11} c e x^{11} (b e+c d)+\frac{1}{13} c^2 e^2 x^{13} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x^2)^2*(a + b*x^2 + c*x^4)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 a d x^{3} \left (a e + b d\right )}{3} + \frac{c^{2} e^{2} x^{13}}{13} + \frac{2 c e x^{11} \left (b e + c d\right )}{11} + d^{2} \int a^{2}\, dx + x^{9} \left (\frac{2 a c e^{2}}{9} + \frac{b^{2} e^{2}}{9} + \frac{4 b c d e}{9} + \frac{c^{2} d^{2}}{9}\right ) + x^{7} \left (\frac{2 a b e^{2}}{7} + \frac{4 a c d e}{7} + \frac{2 b^{2} d e}{7} + \frac{2 b c d^{2}}{7}\right ) + x^{5} \left (\frac{a^{2} e^{2}}{5} + \frac{4 a b d e}{5} + \frac{2 a c d^{2}}{5} + \frac{b^{2} d^{2}}{5}\right ) \]
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)**2,x)
[Out]
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Mathematica [A] time = 0.0999601, size = 156, normalized size = 1.01 \[ \frac{1}{5} x^5 \left (a^2 e^2+4 a b d e+2 a c d^2+b^2 d^2\right )+a^2 d^2 x+\frac{1}{9} x^9 \left (2 a c e^2+b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{2}{7} x^7 \left (a b e^2+2 a c d e+b^2 d e+b c d^2\right )+\frac{2}{3} a d x^3 (a e+b d)+\frac{2}{11} c e x^{11} (b e+c d)+\frac{1}{13} c^2 e^2 x^{13} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x^2)^2*(a + b*x^2 + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.001, size = 155, normalized size = 1. \[{\frac{{c}^{2}{e}^{2}{x}^{13}}{13}}+{\frac{ \left ( 2\,bc{e}^{2}+2\,de{c}^{2} \right ){x}^{11}}{11}}+{\frac{ \left ({c}^{2}{d}^{2}+4\,bcde+{e}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,bc{d}^{2}+2\,de \left ( 2\,ac+{b}^{2} \right ) +2\,ab{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ({d}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,abde+{a}^{2}{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,de{a}^{2}+2\,a{d}^{2}b \right ){x}^{3}}{3}}+{a}^{2}{d}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)^2*(c*x^4+b*x^2+a)^2,x)
[Out]
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Maxima [A] time = 0.737121, size = 198, normalized size = 1.28 \[ \frac{1}{13} \, c^{2} e^{2} x^{13} + \frac{2}{11} \,{\left (c^{2} d e + b c e^{2}\right )} x^{11} + \frac{1}{9} \,{\left (c^{2} d^{2} + 4 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{9} + \frac{2}{7} \,{\left (b c d^{2} + a b e^{2} +{\left (b^{2} + 2 \, a c\right )} d e\right )} x^{7} + \frac{1}{5} \,{\left (4 \, a b d e + a^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{5} + a^{2} d^{2} x + \frac{2}{3} \,{\left (a b d^{2} + a^{2} d e\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2*(e*x^2 + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242174, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} e^{2} c^{2} + \frac{2}{11} x^{11} e d c^{2} + \frac{2}{11} x^{11} e^{2} c b + \frac{1}{9} x^{9} d^{2} c^{2} + \frac{4}{9} x^{9} e d c b + \frac{1}{9} x^{9} e^{2} b^{2} + \frac{2}{9} x^{9} e^{2} c a + \frac{2}{7} x^{7} d^{2} c b + \frac{2}{7} x^{7} e d b^{2} + \frac{4}{7} x^{7} e d c a + \frac{2}{7} x^{7} e^{2} b a + \frac{1}{5} x^{5} d^{2} b^{2} + \frac{2}{5} x^{5} d^{2} c a + \frac{4}{5} x^{5} e d b a + \frac{1}{5} x^{5} e^{2} a^{2} + \frac{2}{3} x^{3} d^{2} b a + \frac{2}{3} x^{3} e d a^{2} + x d^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2*(e*x^2 + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.190843, size = 192, normalized size = 1.24 \[ a^{2} d^{2} x + \frac{c^{2} e^{2} x^{13}}{13} + x^{11} \left (\frac{2 b c e^{2}}{11} + \frac{2 c^{2} d e}{11}\right ) + x^{9} \left (\frac{2 a c e^{2}}{9} + \frac{b^{2} e^{2}}{9} + \frac{4 b c d e}{9} + \frac{c^{2} d^{2}}{9}\right ) + x^{7} \left (\frac{2 a b e^{2}}{7} + \frac{4 a c d e}{7} + \frac{2 b^{2} d e}{7} + \frac{2 b c d^{2}}{7}\right ) + x^{5} \left (\frac{a^{2} e^{2}}{5} + \frac{4 a b d e}{5} + \frac{2 a c d^{2}}{5} + \frac{b^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{2 a^{2} d e}{3} + \frac{2 a b d^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x**2+d)**2*(c*x**4+b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.267394, size = 244, normalized size = 1.57 \[ \frac{1}{13} \, c^{2} x^{13} e^{2} + \frac{2}{11} \, c^{2} d x^{11} e + \frac{2}{11} \, b c x^{11} e^{2} + \frac{1}{9} \, c^{2} d^{2} x^{9} + \frac{4}{9} \, b c d x^{9} e + \frac{1}{9} \, b^{2} x^{9} e^{2} + \frac{2}{9} \, a c x^{9} e^{2} + \frac{2}{7} \, b c d^{2} x^{7} + \frac{2}{7} \, b^{2} d x^{7} e + \frac{4}{7} \, a c d x^{7} e + \frac{2}{7} \, a b x^{7} e^{2} + \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{2}{5} \, a c d^{2} x^{5} + \frac{4}{5} \, a b d x^{5} e + \frac{1}{5} \, a^{2} x^{5} e^{2} + \frac{2}{3} \, a b d^{2} x^{3} + \frac{2}{3} \, a^{2} d x^{3} e + a^{2} d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2*(e*x^2 + d)^2,x, algorithm="giac")
[Out]