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

Optimal. Leaf size=133 \[ a^2 d^3 x+a^2 d^2 e x^3+\frac{1}{11} c e x^{11} \left (2 a e^2+3 c d^2\right )+\frac{1}{9} c d x^9 \left (6 a e^2+c d^2\right )+\frac{1}{7} a e x^7 \left (a e^2+6 c d^2\right )+\frac{1}{5} a d x^5 \left (3 a e^2+2 c d^2\right )+\frac{3}{13} c^2 d e^2 x^{13}+\frac{1}{15} c^2 e^3 x^{15} \]

[Out]

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

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Rubi [A]  time = 0.222269, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ a^2 d^3 x+a^2 d^2 e x^3+\frac{1}{11} c e x^{11} \left (2 a e^2+3 c d^2\right )+\frac{1}{9} c d x^9 \left (6 a e^2+c d^2\right )+\frac{1}{7} a e x^7 \left (a e^2+6 c d^2\right )+\frac{1}{5} a d x^5 \left (3 a e^2+2 c d^2\right )+\frac{3}{13} c^2 d e^2 x^{13}+\frac{1}{15} c^2 e^3 x^{15} \]

Antiderivative was successfully verified.

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

[Out]

a^2*d^3*x + a^2*d^2*e*x^3 + (a*d*(2*c*d^2 + 3*a*e^2)*x^5)/5 + (a*e*(6*c*d^2 + a*
e^2)*x^7)/7 + (c*d*(c*d^2 + 6*a*e^2)*x^9)/9 + (c*e*(3*c*d^2 + 2*a*e^2)*x^11)/11
+ (3*c^2*d*e^2*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. \[ a^{2} d^{2} e x^{3} + \frac{a d x^{5} \left (3 a e^{2} + 2 c d^{2}\right )}{5} + \frac{a e x^{7} \left (a e^{2} + 6 c d^{2}\right )}{7} + \frac{3 c^{2} d e^{2} x^{13}}{13} + \frac{c^{2} e^{3} x^{15}}{15} + \frac{c d x^{9} \left (6 a e^{2} + c d^{2}\right )}{9} + \frac{c e x^{11} \left (2 a e^{2} + 3 c d^{2}\right )}{11} + d^{3} \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0396229, size = 133, normalized size = 1. \[ a^2 d^3 x+a^2 d^2 e x^3+\frac{1}{11} c e x^{11} \left (2 a e^2+3 c d^2\right )+\frac{1}{9} c d x^9 \left (6 a e^2+c d^2\right )+\frac{1}{7} a e x^7 \left (a e^2+6 c d^2\right )+\frac{1}{5} a d x^5 \left (3 a e^2+2 c d^2\right )+\frac{3}{13} c^2 d e^2 x^{13}+\frac{1}{15} c^2 e^3 x^{15} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 130, normalized size = 1. \[{\frac{{c}^{2}{e}^{3}{x}^{15}}{15}}+{\frac{3\,{c}^{2}d{e}^{2}{x}^{13}}{13}}+{\frac{ \left ( 2\,{e}^{3}ac+3\,{d}^{2}e{c}^{2} \right ){x}^{11}}{11}}+{\frac{ \left ( 6\,acd{e}^{2}+{c}^{2}{d}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}{e}^{3}+6\,ac{d}^{2}e \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,d{a}^{2}{e}^{2}+2\,{d}^{3}ac \right ){x}^{5}}{5}}+{a}^{2}{d}^{2}e{x}^{3}+{a}^{2}{d}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)^3*(c*x^4+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.737076, size = 174, normalized size = 1.31 \[ \frac{1}{15} \, c^{2} e^{3} x^{15} + \frac{3}{13} \, c^{2} d e^{2} x^{13} + \frac{1}{11} \,{\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{11} + \frac{1}{9} \,{\left (c^{2} d^{3} + 6 \, a c d e^{2}\right )} x^{9} + a^{2} d^{2} e x^{3} + \frac{1}{7} \,{\left (6 \, a c d^{2} e + a^{2} e^{3}\right )} x^{7} + a^{2} d^{3} x + \frac{1}{5} \,{\left (2 \, a c d^{3} + 3 \, a^{2} d e^{2}\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255478, size = 1, normalized size = 0.01 \[ \frac{1}{15} x^{15} e^{3} c^{2} + \frac{3}{13} x^{13} e^{2} d c^{2} + \frac{3}{11} x^{11} e d^{2} c^{2} + \frac{2}{11} x^{11} e^{3} c a + \frac{1}{9} x^{9} d^{3} c^{2} + \frac{2}{3} x^{9} e^{2} d c a + \frac{6}{7} x^{7} e d^{2} c a + \frac{1}{7} x^{7} e^{3} a^{2} + \frac{2}{5} x^{5} d^{3} c a + \frac{3}{5} x^{5} e^{2} d a^{2} + x^{3} e d^{2} a^{2} + x d^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + 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 + 3/11*x^11*e*d^2*c^2 + 2/11*x^11*e^3*c*
a + 1/9*x^9*d^3*c^2 + 2/3*x^9*e^2*d*c*a + 6/7*x^7*e*d^2*c*a + 1/7*x^7*e^3*a^2 +
2/5*x^5*d^3*c*a + 3/5*x^5*e^2*d*a^2 + x^3*e*d^2*a^2 + x*d^3*a^2

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Sympy [A]  time = 0.166246, size = 144, normalized size = 1.08 \[ a^{2} d^{3} x + a^{2} d^{2} e x^{3} + \frac{3 c^{2} d e^{2} x^{13}}{13} + \frac{c^{2} e^{3} x^{15}}{15} + x^{11} \left (\frac{2 a c e^{3}}{11} + \frac{3 c^{2} d^{2} e}{11}\right ) + x^{9} \left (\frac{2 a c d e^{2}}{3} + \frac{c^{2} d^{3}}{9}\right ) + x^{7} \left (\frac{a^{2} e^{3}}{7} + \frac{6 a c d^{2} e}{7}\right ) + x^{5} \left (\frac{3 a^{2} d e^{2}}{5} + \frac{2 a c d^{3}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269175, size = 173, normalized size = 1.3 \[ \frac{1}{15} \, c^{2} x^{15} e^{3} + \frac{3}{13} \, c^{2} d x^{13} e^{2} + \frac{3}{11} \, c^{2} d^{2} x^{11} e + \frac{1}{9} \, c^{2} d^{3} x^{9} + \frac{2}{11} \, a c x^{11} e^{3} + \frac{2}{3} \, a c d x^{9} e^{2} + \frac{6}{7} \, a c d^{2} x^{7} e + \frac{2}{5} \, a c d^{3} x^{5} + \frac{1}{7} \, a^{2} x^{7} e^{3} + \frac{3}{5} \, a^{2} d x^{5} e^{2} + a^{2} d^{2} x^{3} e + a^{2} d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + 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 + 3/11*c^2*d^2*x^11*e + 1/9*c^2*d^3*x^9
+ 2/11*a*c*x^11*e^3 + 2/3*a*c*d*x^9*e^2 + 6/7*a*c*d^2*x^7*e + 2/5*a*c*d^3*x^5 +
1/7*a^2*x^7*e^3 + 3/5*a^2*d*x^5*e^2 + a^2*d^2*x^3*e + a^2*d^3*x