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

Optimal. Leaf size=60 \[ a^2 d x+\frac{1}{3} a^2 e x^3+\frac{2}{5} a c d x^5+\frac{2}{7} a c e x^7+\frac{1}{9} c^2 d x^9+\frac{1}{11} c^2 e x^{11} \]

[Out]

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

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Rubi [A]  time = 0.0646378, antiderivative size = 60, 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 \[ a^2 d x+\frac{1}{3} a^2 e x^3+\frac{2}{5} a c d x^5+\frac{2}{7} a c e x^7+\frac{1}{9} c^2 d x^9+\frac{1}{11} c^2 e x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00456328, size = 60, normalized size = 1. \[ a^2 d x+\frac{1}{3} a^2 e x^3+\frac{2}{5} a c d x^5+\frac{2}{7} a c e x^7+\frac{1}{9} c^2 d x^9+\frac{1}{11} c^2 e x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 51, normalized size = 0.9 \[{a}^{2}dx+{\frac{{a}^{2}e{x}^{3}}{3}}+{\frac{2\,acd{x}^{5}}{5}}+{\frac{2\,ace{x}^{7}}{7}}+{\frac{{c}^{2}d{x}^{9}}{9}}+{\frac{{c}^{2}e{x}^{11}}{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)*(c*x^4+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.732181, size = 68, normalized size = 1.13 \[ \frac{1}{11} \, c^{2} e x^{11} + \frac{1}{9} \, c^{2} d x^{9} + \frac{2}{7} \, a c e x^{7} + \frac{2}{5} \, a c d x^{5} + \frac{1}{3} \, a^{2} e x^{3} + a^{2} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256993, size = 1, normalized size = 0.02 \[ \frac{1}{11} x^{11} e c^{2} + \frac{1}{9} x^{9} d c^{2} + \frac{2}{7} x^{7} e c a + \frac{2}{5} x^{5} d c a + \frac{1}{3} x^{3} e a^{2} + x d a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.108154, size = 60, normalized size = 1. \[ a^{2} d x + \frac{a^{2} e x^{3}}{3} + \frac{2 a c d x^{5}}{5} + \frac{2 a c e x^{7}}{7} + \frac{c^{2} d x^{9}}{9} + \frac{c^{2} e x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268951, size = 72, normalized size = 1.2 \[ \frac{1}{11} \, c^{2} x^{11} e + \frac{1}{9} \, c^{2} d x^{9} + \frac{2}{7} \, a c x^{7} e + \frac{2}{5} \, a c d x^{5} + \frac{1}{3} \, a^{2} x^{3} e + a^{2} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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