3.266 \(\int x^2 \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx\)

Optimal. Leaf size=78 \[ \frac{1}{7} x^7 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{5} d x^5 (2 a e+b d)+\frac{1}{3} a d^2 x^3+\frac{1}{9} e x^9 (b e+2 c d)+\frac{1}{11} c e^2 x^{11} \]

[Out]

(a*d^2*x^3)/3 + (d*(b*d + 2*a*e)*x^5)/5 + ((c*d^2 + e*(2*b*d + a*e))*x^7)/7 + (e
*(2*c*d + b*e)*x^9)/9 + (c*e^2*x^11)/11

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Rubi [A]  time = 0.166276, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{1}{7} x^7 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{5} d x^5 (2 a e+b d)+\frac{1}{3} a d^2 x^3+\frac{1}{9} e x^9 (b e+2 c d)+\frac{1}{11} c e^2 x^{11} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]

[Out]

(a*d^2*x^3)/3 + (d*(b*d + 2*a*e)*x^5)/5 + ((c*d^2 + e*(2*b*d + a*e))*x^7)/7 + (e
*(2*c*d + b*e)*x^9)/9 + (c*e^2*x^11)/11

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Rubi in Sympy [A]  time = 23.8642, size = 75, normalized size = 0.96 \[ \frac{a d^{2} x^{3}}{3} + \frac{c e^{2} x^{11}}{11} + \frac{d x^{5} \left (2 a e + b d\right )}{5} + \frac{e x^{9} \left (b e + 2 c d\right )}{9} + x^{7} \left (\frac{a e^{2}}{7} + \frac{2 b d e}{7} + \frac{c d^{2}}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(e*x**2+d)**2*(c*x**4+b*x**2+a),x)

[Out]

a*d**2*x**3/3 + c*e**2*x**11/11 + d*x**5*(2*a*e + b*d)/5 + e*x**9*(b*e + 2*c*d)/
9 + x**7*(a*e**2/7 + 2*b*d*e/7 + c*d**2/7)

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Mathematica [A]  time = 0.0271464, size = 78, normalized size = 1. \[ \frac{1}{7} x^7 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{5} d x^5 (2 a e+b d)+\frac{1}{3} a d^2 x^3+\frac{1}{9} e x^9 (b e+2 c d)+\frac{1}{11} c e^2 x^{11} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]

[Out]

(a*d^2*x^3)/3 + (d*(b*d + 2*a*e)*x^5)/5 + ((c*d^2 + 2*b*d*e + a*e^2)*x^7)/7 + (e
*(2*c*d + b*e)*x^9)/9 + (c*e^2*x^11)/11

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Maple [A]  time = 0.001, size = 73, normalized size = 0.9 \[{\frac{c{e}^{2}{x}^{11}}{11}}+{\frac{ \left ( b{e}^{2}+2\,cde \right ){x}^{9}}{9}}+{\frac{ \left ( a{e}^{2}+2\,bde+c{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,ade+b{d}^{2} \right ){x}^{5}}{5}}+{\frac{a{d}^{2}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(e*x^2+d)^2*(c*x^4+b*x^2+a),x)

[Out]

1/11*c*e^2*x^11+1/9*(b*e^2+2*c*d*e)*x^9+1/7*(a*e^2+2*b*d*e+c*d^2)*x^7+1/5*(2*a*d
*e+b*d^2)*x^5+1/3*a*d^2*x^3

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Maxima [A]  time = 0.70196, size = 97, normalized size = 1.24 \[ \frac{1}{11} \, c e^{2} x^{11} + \frac{1}{9} \,{\left (2 \, c d e + b e^{2}\right )} x^{9} + \frac{1}{7} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{7} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{5} \,{\left (b d^{2} + 2 \, a d e\right )} x^{5} \]

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^2,x, algorithm="maxima")

[Out]

1/11*c*e^2*x^11 + 1/9*(2*c*d*e + b*e^2)*x^9 + 1/7*(c*d^2 + 2*b*d*e + a*e^2)*x^7
+ 1/3*a*d^2*x^3 + 1/5*(b*d^2 + 2*a*d*e)*x^5

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Fricas [A]  time = 0.228654, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} e^{2} c + \frac{2}{9} x^{9} e d c + \frac{1}{9} x^{9} e^{2} b + \frac{1}{7} x^{7} d^{2} c + \frac{2}{7} x^{7} e d b + \frac{1}{7} x^{7} e^{2} a + \frac{1}{5} x^{5} d^{2} b + \frac{2}{5} x^{5} e d a + \frac{1}{3} x^{3} d^{2} a \]

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^2,x, algorithm="fricas")

[Out]

1/11*x^11*e^2*c + 2/9*x^9*e*d*c + 1/9*x^9*e^2*b + 1/7*x^7*d^2*c + 2/7*x^7*e*d*b
+ 1/7*x^7*e^2*a + 1/5*x^5*d^2*b + 2/5*x^5*e*d*a + 1/3*x^3*d^2*a

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Sympy [A]  time = 0.130898, size = 82, normalized size = 1.05 \[ \frac{a d^{2} x^{3}}{3} + \frac{c e^{2} x^{11}}{11} + x^{9} \left (\frac{b e^{2}}{9} + \frac{2 c d e}{9}\right ) + x^{7} \left (\frac{a e^{2}}{7} + \frac{2 b d e}{7} + \frac{c d^{2}}{7}\right ) + x^{5} \left (\frac{2 a d e}{5} + \frac{b d^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(e*x**2+d)**2*(c*x**4+b*x**2+a),x)

[Out]

a*d**2*x**3/3 + c*e**2*x**11/11 + x**9*(b*e**2/9 + 2*c*d*e/9) + x**7*(a*e**2/7 +
2*b*d*e/7 + c*d**2/7) + x**5*(2*a*d*e/5 + b*d**2/5)

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GIAC/XCAS [A]  time = 0.266618, size = 107, normalized size = 1.37 \[ \frac{1}{11} \, c x^{11} e^{2} + \frac{2}{9} \, c d x^{9} e + \frac{1}{9} \, b x^{9} e^{2} + \frac{1}{7} \, c d^{2} x^{7} + \frac{2}{7} \, b d x^{7} e + \frac{1}{7} \, a x^{7} e^{2} + \frac{1}{5} \, b d^{2} x^{5} + \frac{2}{5} \, a d x^{5} e + \frac{1}{3} \, a d^{2} x^{3} \]

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^2,x, algorithm="giac")

[Out]

1/11*c*x^11*e^2 + 2/9*c*d*x^9*e + 1/9*b*x^9*e^2 + 1/7*c*d^2*x^7 + 2/7*b*d*x^7*e
+ 1/7*a*x^7*e^2 + 1/5*b*d^2*x^5 + 2/5*a*d*x^5*e + 1/3*a*d^2*x^3