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

Optimal. Leaf size=106 \[ \frac{1}{9} e^2 x^9 \left (a e^2+6 c d^2\right )+\frac{4}{7} d e x^7 \left (a e^2+c d^2\right )+\frac{1}{5} d^2 x^5 \left (6 a e^2+c d^2\right )+a d^4 x+\frac{4}{3} a d^3 e x^3+\frac{4}{11} c d e^3 x^{11}+\frac{1}{13} c e^4 x^{13} \]

[Out]

a*d^4*x + (4*a*d^3*e*x^3)/3 + (d^2*(c*d^2 + 6*a*e^2)*x^5)/5 + (4*d*e*(c*d^2 + a*
e^2)*x^7)/7 + (e^2*(6*c*d^2 + a*e^2)*x^9)/9 + (4*c*d*e^3*x^11)/11 + (c*e^4*x^13)
/13

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Rubi [A]  time = 0.162781, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{1}{9} e^2 x^9 \left (a e^2+6 c d^2\right )+\frac{4}{7} d e x^7 \left (a e^2+c d^2\right )+\frac{1}{5} d^2 x^5 \left (6 a e^2+c d^2\right )+a d^4 x+\frac{4}{3} a d^3 e x^3+\frac{4}{11} c d e^3 x^{11}+\frac{1}{13} c e^4 x^{13} \]

Antiderivative was successfully verified.

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

[Out]

a*d^4*x + (4*a*d^3*e*x^3)/3 + (d^2*(c*d^2 + 6*a*e^2)*x^5)/5 + (4*d*e*(c*d^2 + a*
e^2)*x^7)/7 + (e^2*(6*c*d^2 + a*e^2)*x^9)/9 + (4*c*d*e^3*x^11)/11 + (c*e^4*x^13)
/13

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 a d^{3} e x^{3}}{3} + \frac{4 c d e^{3} x^{11}}{11} + \frac{c e^{4} x^{13}}{13} + d^{4} \int a\, dx + \frac{d^{2} x^{5} \left (6 a e^{2} + c d^{2}\right )}{5} + \frac{4 d e x^{7} \left (a e^{2} + c d^{2}\right )}{7} + \frac{e^{2} x^{9} \left (a e^{2} + 6 c d^{2}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a*d**3*e*x**3/3 + 4*c*d*e**3*x**11/11 + c*e**4*x**13/13 + d**4*Integral(a, x)
+ d**2*x**5*(6*a*e**2 + c*d**2)/5 + 4*d*e*x**7*(a*e**2 + c*d**2)/7 + e**2*x**9*(
a*e**2 + 6*c*d**2)/9

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Mathematica [A]  time = 0.0310268, size = 106, normalized size = 1. \[ \frac{1}{9} e^2 x^9 \left (a e^2+6 c d^2\right )+\frac{4}{7} d e x^7 \left (a e^2+c d^2\right )+\frac{1}{5} d^2 x^5 \left (6 a e^2+c d^2\right )+a d^4 x+\frac{4}{3} a d^3 e x^3+\frac{4}{11} c d e^3 x^{11}+\frac{1}{13} c e^4 x^{13} \]

Antiderivative was successfully verified.

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

[Out]

a*d^4*x + (4*a*d^3*e*x^3)/3 + (d^2*(c*d^2 + 6*a*e^2)*x^5)/5 + (4*d*e*(c*d^2 + a*
e^2)*x^7)/7 + (e^2*(6*c*d^2 + a*e^2)*x^9)/9 + (4*c*d*e^3*x^11)/11 + (c*e^4*x^13)
/13

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Maple [A]  time = 0.002, size = 97, normalized size = 0.9 \[{\frac{c{e}^{4}{x}^{13}}{13}}+{\frac{4\,cd{e}^{3}{x}^{11}}{11}}+{\frac{ \left ( a{e}^{4}+6\,{d}^{2}{e}^{2}c \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,d{e}^{3}a+4\,{d}^{3}ec \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}a+{d}^{4}c \right ){x}^{5}}{5}}+{\frac{4\,a{d}^{3}e{x}^{3}}{3}}+a{d}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)^4*(c*x^4+a),x)

[Out]

1/13*c*e^4*x^13+4/11*c*d*e^3*x^11+1/9*(a*e^4+6*c*d^2*e^2)*x^9+1/7*(4*a*d*e^3+4*c
*d^3*e)*x^7+1/5*(6*a*d^2*e^2+c*d^4)*x^5+4/3*a*d^3*e*x^3+a*d^4*x

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Maxima [A]  time = 0.699931, size = 127, normalized size = 1.2 \[ \frac{1}{13} \, c e^{4} x^{13} + \frac{4}{11} \, c d e^{3} x^{11} + \frac{1}{9} \,{\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{9} + \frac{4}{3} \, a d^{3} e x^{3} + \frac{4}{7} \,{\left (c d^{3} e + a d e^{3}\right )} x^{7} + a d^{4} x + \frac{1}{5} \,{\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*c*e^4*x^13 + 4/11*c*d*e^3*x^11 + 1/9*(6*c*d^2*e^2 + a*e^4)*x^9 + 4/3*a*d^3*
e*x^3 + 4/7*(c*d^3*e + a*d*e^3)*x^7 + a*d^4*x + 1/5*(c*d^4 + 6*a*d^2*e^2)*x^5

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Fricas [A]  time = 0.274835, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} e^{4} c + \frac{4}{11} x^{11} e^{3} d c + \frac{2}{3} x^{9} e^{2} d^{2} c + \frac{1}{9} x^{9} e^{4} a + \frac{4}{7} x^{7} e d^{3} c + \frac{4}{7} x^{7} e^{3} d a + \frac{1}{5} x^{5} d^{4} c + \frac{6}{5} x^{5} e^{2} d^{2} a + \frac{4}{3} x^{3} e d^{3} a + x d^{4} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*x^13*e^4*c + 4/11*x^11*e^3*d*c + 2/3*x^9*e^2*d^2*c + 1/9*x^9*e^4*a + 4/7*x^
7*e*d^3*c + 4/7*x^7*e^3*d*a + 1/5*x^5*d^4*c + 6/5*x^5*e^2*d^2*a + 4/3*x^3*e*d^3*
a + x*d^4*a

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Sympy [A]  time = 0.13508, size = 110, normalized size = 1.04 \[ a d^{4} x + \frac{4 a d^{3} e x^{3}}{3} + \frac{4 c d e^{3} x^{11}}{11} + \frac{c e^{4} x^{13}}{13} + x^{9} \left (\frac{a e^{4}}{9} + \frac{2 c d^{2} e^{2}}{3}\right ) + x^{7} \left (\frac{4 a d e^{3}}{7} + \frac{4 c d^{3} e}{7}\right ) + x^{5} \left (\frac{6 a d^{2} e^{2}}{5} + \frac{c d^{4}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d**4*x + 4*a*d**3*e*x**3/3 + 4*c*d*e**3*x**11/11 + c*e**4*x**13/13 + x**9*(a*e
**4/9 + 2*c*d**2*e**2/3) + x**7*(4*a*d*e**3/7 + 4*c*d**3*e/7) + x**5*(6*a*d**2*e
**2/5 + c*d**4/5)

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GIAC/XCAS [A]  time = 0.267683, size = 127, normalized size = 1.2 \[ \frac{1}{13} \, c x^{13} e^{4} + \frac{4}{11} \, c d x^{11} e^{3} + \frac{2}{3} \, c d^{2} x^{9} e^{2} + \frac{4}{7} \, c d^{3} x^{7} e + \frac{1}{9} \, a x^{9} e^{4} + \frac{1}{5} \, c d^{4} x^{5} + \frac{4}{7} \, a d x^{7} e^{3} + \frac{6}{5} \, a d^{2} x^{5} e^{2} + \frac{4}{3} \, a d^{3} x^{3} e + a d^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*c*x^13*e^4 + 4/11*c*d*x^11*e^3 + 2/3*c*d^2*x^9*e^2 + 4/7*c*d^3*x^7*e + 1/9*
a*x^9*e^4 + 1/5*c*d^4*x^5 + 4/7*a*d*x^7*e^3 + 6/5*a*d^2*x^5*e^2 + 4/3*a*d^3*x^3*
e + a*d^4*x