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

Optimal. Leaf size=223 \[ \frac{1}{7} x^7 \left (a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right )+a^2 d^3 x+\frac{1}{11} e x^{11} \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+\frac{1}{5} d x^5 \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+\frac{1}{9} x^9 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+\frac{1}{3} a d^2 x^3 (3 a e+2 b d)+\frac{1}{13} c e^2 x^{13} (2 b e+3 c d)+\frac{1}{15} c^2 e^3 x^{15} \]

[Out]

a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^3)/3 + (d*(b^2*d^2 + 6*a*b*d*e + a*(2*c*d^2
+ 3*a*e^2))*x^5)/5 + ((2*b*c*d^3 + 3*b^2*d^2*e + 6*a*c*d^2*e + 6*a*b*d*e^2 + a^
2*e^3)*x^7)/7 + ((c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b*d + 2*a*e))*x^9)/9
+ (e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(3*b*d + a*e))*x^11)/11 + (c*e^2*(3*c*d + 2*b*
e)*x^13)/13 + (c^2*e^3*x^15)/15

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Rubi [A]  time = 0.424471, antiderivative size = 223, 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 \[ \frac{1}{7} x^7 \left (a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right )+a^2 d^3 x+\frac{1}{11} e x^{11} \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+\frac{1}{5} d x^5 \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+\frac{1}{9} x^9 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+\frac{1}{3} a d^2 x^3 (3 a e+2 b d)+\frac{1}{13} c e^2 x^{13} (2 b e+3 c d)+\frac{1}{15} c^2 e^3 x^{15} \]

Antiderivative was successfully verified.

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

[Out]

a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^3)/3 + (d*(b^2*d^2 + 6*a*b*d*e + a*(2*c*d^2
+ 3*a*e^2))*x^5)/5 + ((2*b*c*d^3 + 3*b^2*d^2*e + 6*a*c*d^2*e + 6*a*b*d*e^2 + a^
2*e^3)*x^7)/7 + ((c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b*d + 2*a*e))*x^9)/9
+ (e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(3*b*d + a*e))*x^11)/11 + (c*e^2*(3*c*d + 2*b*
e)*x^13)/13 + (c^2*e^3*x^15)/15

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a d^{2} x^{3} \left (3 a e + 2 b d\right )}{3} + \frac{c^{2} e^{3} x^{15}}{15} + \frac{c e^{2} x^{13} \left (2 b e + 3 c d\right )}{13} + d^{3} \int a^{2}\, dx + \frac{d x^{5} \left (3 a^{2} e^{2} + 6 a b d e + 2 a c d^{2} + b^{2} d^{2}\right )}{5} + \frac{e x^{11} \left (2 a c e^{2} + b^{2} e^{2} + 6 b c d e + 3 c^{2} d^{2}\right )}{11} + x^{9} \left (\frac{2 a b e^{3}}{9} + \frac{2 a c d e^{2}}{3} + \frac{b^{2} d e^{2}}{3} + \frac{2 b c d^{2} e}{3} + \frac{c^{2} d^{3}}{9}\right ) + x^{7} \left (\frac{a^{2} e^{3}}{7} + \frac{6 a b d e^{2}}{7} + \frac{6 a c d^{2} e}{7} + \frac{3 b^{2} d^{2} e}{7} + \frac{2 b c d^{3}}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d**2*x**3*(3*a*e + 2*b*d)/3 + c**2*e**3*x**15/15 + c*e**2*x**13*(2*b*e + 3*c*d
)/13 + d**3*Integral(a**2, x) + d*x**5*(3*a**2*e**2 + 6*a*b*d*e + 2*a*c*d**2 + b
**2*d**2)/5 + e*x**11*(2*a*c*e**2 + b**2*e**2 + 6*b*c*d*e + 3*c**2*d**2)/11 + x*
*9*(2*a*b*e**3/9 + 2*a*c*d*e**2/3 + b**2*d*e**2/3 + 2*b*c*d**2*e/3 + c**2*d**3/9
) + x**7*(a**2*e**3/7 + 6*a*b*d*e**2/7 + 6*a*c*d**2*e/7 + 3*b**2*d**2*e/7 + 2*b*
c*d**3/7)

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Mathematica [A]  time = 0.176502, size = 223, normalized size = 1. \[ \frac{1}{7} x^7 \left (a^2 e^3+6 a b d e^2+6 a c d^2 e+3 b^2 d^2 e+2 b c d^3\right )+a^2 d^3 x+\frac{1}{11} e x^{11} \left (2 c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+\frac{1}{5} d x^5 \left (6 a b d e+a \left (3 a e^2+2 c d^2\right )+b^2 d^2\right )+\frac{1}{9} x^9 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+\frac{1}{3} a d^2 x^3 (3 a e+2 b d)+\frac{1}{13} c e^2 x^{13} (2 b e+3 c d)+\frac{1}{15} c^2 e^3 x^{15} \]

Antiderivative was successfully verified.

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

[Out]

a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^3)/3 + (d*(b^2*d^2 + 6*a*b*d*e + a*(2*c*d^2
+ 3*a*e^2))*x^5)/5 + ((2*b*c*d^3 + 3*b^2*d^2*e + 6*a*c*d^2*e + 6*a*b*d*e^2 + a^
2*e^3)*x^7)/7 + ((c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b*d + 2*a*e))*x^9)/9
+ (e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(3*b*d + a*e))*x^11)/11 + (c*e^2*(3*c*d + 2*b*
e)*x^13)/13 + (c^2*e^3*x^15)/15

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Maple [A]  time = 0.002, size = 219, normalized size = 1. \[{\frac{{c}^{2}{e}^{3}{x}^{15}}{15}}+{\frac{ \left ( 2\,{e}^{3}bc+3\,{e}^{2}d{c}^{2} \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,{d}^{2}e{c}^{2}+6\,{e}^{2}dbc+{e}^{3} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{11}}{11}}+{\frac{ \left ({c}^{2}{d}^{3}+6\,bc{d}^{2}e+3\,{e}^{2}d \left ( 2\,ac+{b}^{2} \right ) +2\,ab{e}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,bc{d}^{3}+3\,{d}^{2}e \left ( 2\,ac+{b}^{2} \right ) +6\,abd{e}^{2}+{a}^{2}{e}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ({d}^{3} \left ( 2\,ac+{b}^{2} \right ) +6\,ab{d}^{2}e+3\,d{a}^{2}{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{2}+2\,{d}^{3}ab \right ){x}^{3}}{3}}+{a}^{2}{d}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*c^2*e^3*x^15+1/13*(2*b*c*e^3+3*c^2*d*e^2)*x^13+1/11*(3*d^2*e*c^2+6*e^2*d*b*
c+e^3*(2*a*c+b^2))*x^11+1/9*(c^2*d^3+6*b*c*d^2*e+3*e^2*d*(2*a*c+b^2)+2*a*b*e^3)*
x^9+1/7*(2*b*c*d^3+3*d^2*e*(2*a*c+b^2)+6*a*b*d*e^2+a^2*e^3)*x^7+1/5*(d^3*(2*a*c+
b^2)+6*a*b*d^2*e+3*d*a^2*e^2)*x^5+1/3*(3*a^2*d^2*e+2*a*b*d^3)*x^3+a^2*d^3*x

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Maxima [A]  time = 0.714763, size = 294, normalized size = 1.32 \[ \frac{1}{15} \, c^{2} e^{3} x^{15} + \frac{1}{13} \,{\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{13} + \frac{1}{11} \,{\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{11} + \frac{1}{9} \,{\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{9} + \frac{1}{7} \,{\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{7} + a^{2} d^{3} x + \frac{1}{5} \,{\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} +{\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{5} + \frac{1}{3} \,{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2*(e*x^2 + d)^3,x, algorithm="maxima")

[Out]

1/15*c^2*e^3*x^15 + 1/13*(3*c^2*d*e^2 + 2*b*c*e^3)*x^13 + 1/11*(3*c^2*d^2*e + 6*
b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x^11 + 1/9*(c^2*d^3 + 6*b*c*d^2*e + 2*a*b*e^3 + 3
*(b^2 + 2*a*c)*d*e^2)*x^9 + 1/7*(2*b*c*d^3 + 6*a*b*d*e^2 + a^2*e^3 + 3*(b^2 + 2*
a*c)*d^2*e)*x^7 + a^2*d^3*x + 1/5*(6*a*b*d^2*e + 3*a^2*d*e^2 + (b^2 + 2*a*c)*d^3
)*x^5 + 1/3*(2*a*b*d^3 + 3*a^2*d^2*e)*x^3

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Fricas [A]  time = 0.244987, size = 1, normalized size = 0. \[ \frac{1}{15} x^{15} e^{3} c^{2} + \frac{3}{13} x^{13} e^{2} d c^{2} + \frac{2}{13} x^{13} e^{3} c b + \frac{3}{11} x^{11} e d^{2} c^{2} + \frac{6}{11} x^{11} e^{2} d c b + \frac{1}{11} x^{11} e^{3} b^{2} + \frac{2}{11} x^{11} e^{3} c a + \frac{1}{9} x^{9} d^{3} c^{2} + \frac{2}{3} x^{9} e d^{2} c b + \frac{1}{3} x^{9} e^{2} d b^{2} + \frac{2}{3} x^{9} e^{2} d c a + \frac{2}{9} x^{9} e^{3} b a + \frac{2}{7} x^{7} d^{3} c b + \frac{3}{7} x^{7} e d^{2} b^{2} + \frac{6}{7} x^{7} e d^{2} c a + \frac{6}{7} x^{7} e^{2} d b a + \frac{1}{7} x^{7} e^{3} a^{2} + \frac{1}{5} x^{5} d^{3} b^{2} + \frac{2}{5} x^{5} d^{3} c a + \frac{6}{5} x^{5} e d^{2} b a + \frac{3}{5} x^{5} e^{2} d a^{2} + \frac{2}{3} x^{3} d^{3} b a + x^{3} e d^{2} a^{2} + x d^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2*(e*x^2 + d)^3,x, algorithm="fricas")

[Out]

1/15*x^15*e^3*c^2 + 3/13*x^13*e^2*d*c^2 + 2/13*x^13*e^3*c*b + 3/11*x^11*e*d^2*c^
2 + 6/11*x^11*e^2*d*c*b + 1/11*x^11*e^3*b^2 + 2/11*x^11*e^3*c*a + 1/9*x^9*d^3*c^
2 + 2/3*x^9*e*d^2*c*b + 1/3*x^9*e^2*d*b^2 + 2/3*x^9*e^2*d*c*a + 2/9*x^9*e^3*b*a
+ 2/7*x^7*d^3*c*b + 3/7*x^7*e*d^2*b^2 + 6/7*x^7*e*d^2*c*a + 6/7*x^7*e^2*d*b*a +
1/7*x^7*e^3*a^2 + 1/5*x^5*d^3*b^2 + 2/5*x^5*d^3*c*a + 6/5*x^5*e*d^2*b*a + 3/5*x^
5*e^2*d*a^2 + 2/3*x^3*d^3*b*a + x^3*e*d^2*a^2 + x*d^3*a^2

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Sympy [A]  time = 0.238388, size = 272, normalized size = 1.22 \[ a^{2} d^{3} x + \frac{c^{2} e^{3} x^{15}}{15} + x^{13} \left (\frac{2 b c e^{3}}{13} + \frac{3 c^{2} d e^{2}}{13}\right ) + x^{11} \left (\frac{2 a c e^{3}}{11} + \frac{b^{2} e^{3}}{11} + \frac{6 b c d e^{2}}{11} + \frac{3 c^{2} d^{2} e}{11}\right ) + x^{9} \left (\frac{2 a b e^{3}}{9} + \frac{2 a c d e^{2}}{3} + \frac{b^{2} d e^{2}}{3} + \frac{2 b c d^{2} e}{3} + \frac{c^{2} d^{3}}{9}\right ) + x^{7} \left (\frac{a^{2} e^{3}}{7} + \frac{6 a b d e^{2}}{7} + \frac{6 a c d^{2} e}{7} + \frac{3 b^{2} d^{2} e}{7} + \frac{2 b c d^{3}}{7}\right ) + x^{5} \left (\frac{3 a^{2} d e^{2}}{5} + \frac{6 a b d^{2} e}{5} + \frac{2 a c d^{3}}{5} + \frac{b^{2} d^{3}}{5}\right ) + x^{3} \left (a^{2} d^{2} e + \frac{2 a b d^{3}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*d**3*x + c**2*e**3*x**15/15 + x**13*(2*b*c*e**3/13 + 3*c**2*d*e**2/13) + x*
*11*(2*a*c*e**3/11 + b**2*e**3/11 + 6*b*c*d*e**2/11 + 3*c**2*d**2*e/11) + x**9*(
2*a*b*e**3/9 + 2*a*c*d*e**2/3 + b**2*d*e**2/3 + 2*b*c*d**2*e/3 + c**2*d**3/9) +
x**7*(a**2*e**3/7 + 6*a*b*d*e**2/7 + 6*a*c*d**2*e/7 + 3*b**2*d**2*e/7 + 2*b*c*d*
*3/7) + x**5*(3*a**2*d*e**2/5 + 6*a*b*d**2*e/5 + 2*a*c*d**3/5 + b**2*d**3/5) + x
**3*(a**2*d**2*e + 2*a*b*d**3/3)

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GIAC/XCAS [A]  time = 0.268795, size = 344, normalized size = 1.54 \[ \frac{1}{15} \, c^{2} x^{15} e^{3} + \frac{3}{13} \, c^{2} d x^{13} e^{2} + \frac{2}{13} \, b c x^{13} e^{3} + \frac{3}{11} \, c^{2} d^{2} x^{11} e + \frac{6}{11} \, b c d x^{11} e^{2} + \frac{1}{9} \, c^{2} d^{3} x^{9} + \frac{1}{11} \, b^{2} x^{11} e^{3} + \frac{2}{11} \, a c x^{11} e^{3} + \frac{2}{3} \, b c d^{2} x^{9} e + \frac{1}{3} \, b^{2} d x^{9} e^{2} + \frac{2}{3} \, a c d x^{9} e^{2} + \frac{2}{7} \, b c d^{3} x^{7} + \frac{2}{9} \, a b x^{9} e^{3} + \frac{3}{7} \, b^{2} d^{2} x^{7} e + \frac{6}{7} \, a c d^{2} x^{7} e + \frac{6}{7} \, a b d x^{7} e^{2} + \frac{1}{5} \, b^{2} d^{3} x^{5} + \frac{2}{5} \, a c d^{3} x^{5} + \frac{1}{7} \, a^{2} x^{7} e^{3} + \frac{6}{5} \, a b d^{2} x^{5} e + \frac{3}{5} \, a^{2} d x^{5} e^{2} + \frac{2}{3} \, a b d^{3} x^{3} + a^{2} d^{2} x^{3} e + a^{2} d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2*(e*x^2 + d)^3,x, algorithm="giac")

[Out]

1/15*c^2*x^15*e^3 + 3/13*c^2*d*x^13*e^2 + 2/13*b*c*x^13*e^3 + 3/11*c^2*d^2*x^11*
e + 6/11*b*c*d*x^11*e^2 + 1/9*c^2*d^3*x^9 + 1/11*b^2*x^11*e^3 + 2/11*a*c*x^11*e^
3 + 2/3*b*c*d^2*x^9*e + 1/3*b^2*d*x^9*e^2 + 2/3*a*c*d*x^9*e^2 + 2/7*b*c*d^3*x^7
+ 2/9*a*b*x^9*e^3 + 3/7*b^2*d^2*x^7*e + 6/7*a*c*d^2*x^7*e + 6/7*a*b*d*x^7*e^2 +
1/5*b^2*d^3*x^5 + 2/5*a*c*d^3*x^5 + 1/7*a^2*x^7*e^3 + 6/5*a*b*d^2*x^5*e + 3/5*a^
2*d*x^5*e^2 + 2/3*a*b*d^3*x^3 + a^2*d^2*x^3*e + a^2*d^3*x