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

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]

a^2*d^2*x + (2*a*d*(b*d + a*e)*x^3)/3 + ((b^2*d^2 + 4*a*b*d*e + a*(2*c*d^2 + a*e
^2))*x^5)/5 + (2*(b*c*d^2 + b^2*d*e + 2*a*c*d*e + a*b*e^2)*x^7)/7 + ((c^2*d^2 +
b^2*e^2 + 2*c*e*(2*b*d + a*e))*x^9)/9 + (2*c*e*(c*d + b*e)*x^11)/11 + (c^2*e^2*x
^13)/13

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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]

a^2*d^2*x + (2*a*d*(b*d + a*e)*x^3)/3 + ((b^2*d^2 + 4*a*b*d*e + a*(2*c*d^2 + a*e
^2))*x^5)/5 + (2*(b*c*d^2 + b^2*d*e + 2*a*c*d*e + a*b*e^2)*x^7)/7 + ((c^2*d^2 +
b^2*e^2 + 2*c*e*(2*b*d + a*e))*x^9)/9 + (2*c*e*(c*d + b*e)*x^11)/11 + (c^2*e^2*x
^13)/13

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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]

2*a*d*x**3*(a*e + b*d)/3 + c**2*e**2*x**13/13 + 2*c*e*x**11*(b*e + c*d)/11 + d**
2*Integral(a**2, x) + x**9*(2*a*c*e**2/9 + b**2*e**2/9 + 4*b*c*d*e/9 + c**2*d**2
/9) + x**7*(2*a*b*e**2/7 + 4*a*c*d*e/7 + 2*b**2*d*e/7 + 2*b*c*d**2/7) + x**5*(a*
*2*e**2/5 + 4*a*b*d*e/5 + 2*a*c*d**2/5 + b**2*d**2/5)

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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]

a^2*d^2*x + (2*a*d*(b*d + a*e)*x^3)/3 + ((b^2*d^2 + 2*a*c*d^2 + 4*a*b*d*e + a^2*
e^2)*x^5)/5 + (2*(b*c*d^2 + b^2*d*e + 2*a*c*d*e + a*b*e^2)*x^7)/7 + ((c^2*d^2 +
4*b*c*d*e + b^2*e^2 + 2*a*c*e^2)*x^9)/9 + (2*c*e*(c*d + b*e)*x^11)/11 + (c^2*e^2
*x^13)/13

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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]

1/13*c^2*e^2*x^13+1/11*(2*b*c*e^2+2*c^2*d*e)*x^11+1/9*(c^2*d^2+4*b*c*d*e+e^2*(2*
a*c+b^2))*x^9+1/7*(2*b*c*d^2+2*d*e*(2*a*c+b^2)+2*a*b*e^2)*x^7+1/5*(d^2*(2*a*c+b^
2)+4*a*b*d*e+a^2*e^2)*x^5+1/3*(2*a^2*d*e+2*a*b*d^2)*x^3+a^2*d^2*x

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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]

1/13*c^2*e^2*x^13 + 2/11*(c^2*d*e + b*c*e^2)*x^11 + 1/9*(c^2*d^2 + 4*b*c*d*e + (
b^2 + 2*a*c)*e^2)*x^9 + 2/7*(b*c*d^2 + a*b*e^2 + (b^2 + 2*a*c)*d*e)*x^7 + 1/5*(4
*a*b*d*e + a^2*e^2 + (b^2 + 2*a*c)*d^2)*x^5 + a^2*d^2*x + 2/3*(a*b*d^2 + a^2*d*e
)*x^3

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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]

1/13*x^13*e^2*c^2 + 2/11*x^11*e*d*c^2 + 2/11*x^11*e^2*c*b + 1/9*x^9*d^2*c^2 + 4/
9*x^9*e*d*c*b + 1/9*x^9*e^2*b^2 + 2/9*x^9*e^2*c*a + 2/7*x^7*d^2*c*b + 2/7*x^7*e*
d*b^2 + 4/7*x^7*e*d*c*a + 2/7*x^7*e^2*b*a + 1/5*x^5*d^2*b^2 + 2/5*x^5*d^2*c*a +
4/5*x^5*e*d*b*a + 1/5*x^5*e^2*a^2 + 2/3*x^3*d^2*b*a + 2/3*x^3*e*d*a^2 + x*d^2*a^
2

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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]

a**2*d**2*x + c**2*e**2*x**13/13 + x**11*(2*b*c*e**2/11 + 2*c**2*d*e/11) + x**9*
(2*a*c*e**2/9 + b**2*e**2/9 + 4*b*c*d*e/9 + c**2*d**2/9) + x**7*(2*a*b*e**2/7 +
4*a*c*d*e/7 + 2*b**2*d*e/7 + 2*b*c*d**2/7) + x**5*(a**2*e**2/5 + 4*a*b*d*e/5 + 2
*a*c*d**2/5 + b**2*d**2/5) + x**3*(2*a**2*d*e/3 + 2*a*b*d**2/3)

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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]

1/13*c^2*x^13*e^2 + 2/11*c^2*d*x^11*e + 2/11*b*c*x^11*e^2 + 1/9*c^2*d^2*x^9 + 4/
9*b*c*d*x^9*e + 1/9*b^2*x^9*e^2 + 2/9*a*c*x^9*e^2 + 2/7*b*c*d^2*x^7 + 2/7*b^2*d*
x^7*e + 4/7*a*c*d*x^7*e + 2/7*a*b*x^7*e^2 + 1/5*b^2*d^2*x^5 + 2/5*a*c*d^2*x^5 +
4/5*a*b*d*x^5*e + 1/5*a^2*x^5*e^2 + 2/3*a*b*d^2*x^3 + 2/3*a^2*d*x^3*e + a^2*d^2*
x