3.265 \(\int x^3 \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx\)

Optimal. Leaf size=78 \[ \frac{1}{8} x^8 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{6} d x^6 (2 a e+b d)+\frac{1}{4} a d^2 x^4+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{12} c e^2 x^{12} \]

[Out]

(a*d^2*x^4)/4 + (d*(b*d + 2*a*e)*x^6)/6 + ((c*d^2 + e*(2*b*d + a*e))*x^8)/8 + (e
*(2*c*d + b*e)*x^10)/10 + (c*e^2*x^12)/12

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Rubi [A]  time = 0.322713, antiderivative size = 78, 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 \[ \frac{1}{8} x^8 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{6} d x^6 (2 a e+b d)+\frac{1}{4} a d^2 x^4+\frac{1}{10} e x^{10} (b e+2 c d)+\frac{1}{12} c e^2 x^{12} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]

[Out]

(a*d^2*x^4)/4 + (d*(b*d + 2*a*e)*x^6)/6 + ((c*d^2 + e*(2*b*d + a*e))*x^8)/8 + (e
*(2*c*d + b*e)*x^10)/10 + (c*e^2*x^12)/12

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a d^{2} \int ^{x^{2}} x\, dx}{2} + \frac{c e^{2} x^{12}}{12} + \frac{d x^{6} \left (2 a e + b d\right )}{6} + \frac{e x^{10} \left (b e + 2 c d\right )}{10} + x^{8} \left (\frac{a e^{2}}{8} + \frac{b d e}{4} + \frac{c d^{2}}{8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x**2+d)**2*(c*x**4+b*x**2+a),x)

[Out]

a*d**2*Integral(x, (x, x**2))/2 + c*e**2*x**12/12 + d*x**6*(2*a*e + b*d)/6 + e*x
**10*(b*e + 2*c*d)/10 + x**8*(a*e**2/8 + b*d*e/4 + c*d**2/8)

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Mathematica [A]  time = 0.0401243, size = 72, normalized size = 0.92 \[ \frac{1}{120} x^4 \left (15 x^4 \left (e (a e+2 b d)+c d^2\right )+20 d x^2 (2 a e+b d)+30 a d^2+12 e x^6 (b e+2 c d)+10 c e^2 x^8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(d + e*x^2)^2*(a + b*x^2 + c*x^4),x]

[Out]

(x^4*(30*a*d^2 + 20*d*(b*d + 2*a*e)*x^2 + 15*(c*d^2 + e*(2*b*d + a*e))*x^4 + 12*
e*(2*c*d + b*e)*x^6 + 10*c*e^2*x^8))/120

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Maple [A]  time = 0.002, size = 73, normalized size = 0.9 \[{\frac{c{e}^{2}{x}^{12}}{12}}+{\frac{ \left ( b{e}^{2}+2\,cde \right ){x}^{10}}{10}}+{\frac{ \left ( a{e}^{2}+2\,bde+c{d}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 2\,ade+b{d}^{2} \right ){x}^{6}}{6}}+{\frac{a{d}^{2}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x^2+d)^2*(c*x^4+b*x^2+a),x)

[Out]

1/12*c*e^2*x^12+1/10*(b*e^2+2*c*d*e)*x^10+1/8*(a*e^2+2*b*d*e+c*d^2)*x^8+1/6*(2*a
*d*e+b*d^2)*x^6+1/4*a*d^2*x^4

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Maxima [A]  time = 0.702282, size = 97, normalized size = 1.24 \[ \frac{1}{12} \, c e^{2} x^{12} + \frac{1}{10} \,{\left (2 \, c d e + b e^{2}\right )} x^{10} + \frac{1}{8} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{8} + \frac{1}{4} \, a d^{2} x^{4} + \frac{1}{6} \,{\left (b d^{2} + 2 \, a d e\right )} x^{6} \]

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^3,x, algorithm="maxima")

[Out]

1/12*c*e^2*x^12 + 1/10*(2*c*d*e + b*e^2)*x^10 + 1/8*(c*d^2 + 2*b*d*e + a*e^2)*x^
8 + 1/4*a*d^2*x^4 + 1/6*(b*d^2 + 2*a*d*e)*x^6

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Fricas [A]  time = 0.232526, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} e^{2} c + \frac{1}{5} x^{10} e d c + \frac{1}{10} x^{10} e^{2} b + \frac{1}{8} x^{8} d^{2} c + \frac{1}{4} x^{8} e d b + \frac{1}{8} x^{8} e^{2} a + \frac{1}{6} x^{6} d^{2} b + \frac{1}{3} x^{6} e d a + \frac{1}{4} x^{4} d^{2} a \]

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^3,x, algorithm="fricas")

[Out]

1/12*x^12*e^2*c + 1/5*x^10*e*d*c + 1/10*x^10*e^2*b + 1/8*x^8*d^2*c + 1/4*x^8*e*d
*b + 1/8*x^8*e^2*a + 1/6*x^6*d^2*b + 1/3*x^6*e*d*a + 1/4*x^4*d^2*a

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Sympy [A]  time = 0.127716, size = 76, normalized size = 0.97 \[ \frac{a d^{2} x^{4}}{4} + \frac{c e^{2} x^{12}}{12} + x^{10} \left (\frac{b e^{2}}{10} + \frac{c d e}{5}\right ) + x^{8} \left (\frac{a e^{2}}{8} + \frac{b d e}{4} + \frac{c d^{2}}{8}\right ) + x^{6} \left (\frac{a d e}{3} + \frac{b d^{2}}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(e*x**2+d)**2*(c*x**4+b*x**2+a),x)

[Out]

a*d**2*x**4/4 + c*e**2*x**12/12 + x**10*(b*e**2/10 + c*d*e/5) + x**8*(a*e**2/8 +
b*d*e/4 + c*d**2/8) + x**6*(a*d*e/3 + b*d**2/6)

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GIAC/XCAS [A]  time = 0.26878, size = 107, normalized size = 1.37 \[ \frac{1}{12} \, c x^{12} e^{2} + \frac{1}{5} \, c d x^{10} e + \frac{1}{10} \, b x^{10} e^{2} + \frac{1}{8} \, c d^{2} x^{8} + \frac{1}{4} \, b d x^{8} e + \frac{1}{8} \, a x^{8} e^{2} + \frac{1}{6} \, b d^{2} x^{6} + \frac{1}{3} \, a d x^{6} e + \frac{1}{4} \, a d^{2} x^{4} \]

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^3,x, algorithm="giac")

[Out]

1/12*c*x^12*e^2 + 1/5*c*d*x^10*e + 1/10*b*x^10*e^2 + 1/8*c*d^2*x^8 + 1/4*b*d*x^8
*e + 1/8*a*x^8*e^2 + 1/6*b*d^2*x^6 + 1/3*a*d*x^6*e + 1/4*a*d^2*x^4