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

Optimal. Leaf size=96 \[ a^2 d x+\frac{1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{3} a x^3 (a e+2 b d)+\frac{1}{9} c x^9 (2 b e+c d)+\frac{1}{11} c^2 e x^{11} \]

[Out]

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

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Rubi [A]  time = 0.137177, antiderivative size = 96, 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 \[ a^2 d x+\frac{1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{3} a x^3 (a e+2 b d)+\frac{1}{9} c x^9 (2 b e+c d)+\frac{1}{11} c^2 e x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} \int d\, dx + \frac{a x^{3} \left (a e + 2 b d\right )}{3} + \frac{c^{2} e x^{11}}{11} + \frac{c x^{9} \left (2 b e + c d\right )}{9} + x^{7} \left (\frac{2 a c e}{7} + \frac{b^{2} e}{7} + \frac{2 b c d}{7}\right ) + x^{5} \left (\frac{2 a b e}{5} + \frac{2 a c d}{5} + \frac{b^{2} d}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0428684, size = 96, normalized size = 1. \[ a^2 d x+\frac{1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac{1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac{1}{3} a x^3 (a e+2 b d)+\frac{1}{9} c x^9 (2 b e+c d)+\frac{1}{11} c^2 e x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 91, normalized size = 1. \[{\frac{{c}^{2}e{x}^{11}}{11}}+{\frac{ \left ( 2\,bce+{c}^{2}d \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,bcd+e \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( d \left ( 2\,ac+{b}^{2} \right ) +2\,abe \right ){x}^{5}}{5}}+{\frac{ \left ( e{a}^{2}+2\,dab \right ){x}^{3}}{3}}+{a}^{2}dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.743316, size = 122, normalized size = 1.27 \[ \frac{1}{11} \, c^{2} e x^{11} + \frac{1}{9} \,{\left (c^{2} d + 2 \, b c e\right )} x^{9} + \frac{1}{7} \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{7} + \frac{1}{5} \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + a^{2} d x + \frac{1}{3} \,{\left (2 \, a b d + a^{2} e\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2*(e*x^2 + d),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.238333, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} e c^{2} + \frac{1}{9} x^{9} d c^{2} + \frac{2}{9} x^{9} e c b + \frac{2}{7} x^{7} d c b + \frac{1}{7} x^{7} e b^{2} + \frac{2}{7} x^{7} e c a + \frac{1}{5} x^{5} d b^{2} + \frac{2}{5} x^{5} d c a + \frac{2}{5} x^{5} e b a + \frac{2}{3} x^{3} d b a + \frac{1}{3} x^{3} e a^{2} + x d a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2*(e*x^2 + d),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.152216, size = 107, normalized size = 1.11 \[ a^{2} d x + \frac{c^{2} e x^{11}}{11} + x^{9} \left (\frac{2 b c e}{9} + \frac{c^{2} d}{9}\right ) + x^{7} \left (\frac{2 a c e}{7} + \frac{b^{2} e}{7} + \frac{2 b c d}{7}\right ) + x^{5} \left (\frac{2 a b e}{5} + \frac{2 a c d}{5} + \frac{b^{2} d}{5}\right ) + x^{3} \left (\frac{a^{2} e}{3} + \frac{2 a b d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269385, size = 143, normalized size = 1.49 \[ \frac{1}{11} \, c^{2} x^{11} e + \frac{1}{9} \, c^{2} d x^{9} + \frac{2}{9} \, b c x^{9} e + \frac{2}{7} \, b c d x^{7} + \frac{1}{7} \, b^{2} x^{7} e + \frac{2}{7} \, a c x^{7} e + \frac{1}{5} \, b^{2} d x^{5} + \frac{2}{5} \, a c d x^{5} + \frac{2}{5} \, a b x^{5} e + \frac{2}{3} \, a b d x^{3} + \frac{1}{3} \, a^{2} x^{3} e + a^{2} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2*(e*x^2 + d),x, algorithm="giac")

[Out]

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