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

Optimal. Leaf size=32 \[ a d x+\frac{1}{3} a e x^3+\frac{1}{5} c d x^5+\frac{1}{7} c e x^7 \]

[Out]

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

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Rubi [A]  time = 0.0328197, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ a d x+\frac{1}{3} a e x^3+\frac{1}{5} c d x^5+\frac{1}{7} c e x^7 \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2)*(a + c*x^4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00253939, size = 32, normalized size = 1. \[ a d x+\frac{1}{3} a e x^3+\frac{1}{5} c d x^5+\frac{1}{7} c e x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 27, normalized size = 0.8 \[ adx+{\frac{ae{x}^{3}}{3}}+{\frac{cd{x}^{5}}{5}}+{\frac{ce{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)*(c*x^4+a),x)

[Out]

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

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Maxima [A]  time = 0.727102, size = 35, normalized size = 1.09 \[ \frac{1}{7} \, c e x^{7} + \frac{1}{5} \, c d x^{5} + \frac{1}{3} \, a e x^{3} + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.261315, size = 1, normalized size = 0.03 \[ \frac{1}{7} x^{7} e c + \frac{1}{5} x^{5} d c + \frac{1}{3} x^{3} e a + x d a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.074758, size = 29, normalized size = 0.91 \[ a d x + \frac{a e x^{3}}{3} + \frac{c d x^{5}}{5} + \frac{c e x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d*x + a*e*x**3/3 + c*d*x**5/5 + c*e*x**7/7

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GIAC/XCAS [A]  time = 0.268922, size = 38, normalized size = 1.19 \[ \frac{1}{7} \, c x^{7} e + \frac{1}{5} \, c d x^{5} + \frac{1}{3} \, a x^{3} e + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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