3.63 \(\int \frac{a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=20 \[ d x+\frac{e x^2}{2}+\frac{f x^3}{3} \]

[Out]

d*x + (e*x^2)/2 + (f*x^3)/3

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Rubi [A]  time = 0.0291188, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 63, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.016 \[ d x+\frac{e x^2}{2}+\frac{f x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4),x]

[Out]

d*x + (e*x^2)/2 + (f*x^3)/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d+a*e*x+(a*f+b*d)*x**2+b*e*x**3+(b*f+c*d)*x**4+c*e*x**5+c*f*x**6)/(c*x**4+b*x**2+a),x)

[Out]

e*Integral(x, x) + f*x**3/3 + Integral(d, x)

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Mathematica [A]  time = 0.00337774, size = 20, normalized size = 1. \[ d x+\frac{e x^2}{2}+\frac{f x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4),x]

[Out]

d*x + (e*x^2)/2 + (f*x^3)/3

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Maple [A]  time = 0.002, size = 17, normalized size = 0.9 \[ dx+{\frac{e{x}^{2}}{2}}+{\frac{f{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6)/(c*x^4+b*x^2+a),x)

[Out]

d*x+1/2*e*x^2+1/3*f*x^3

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Maxima [A]  time = 0.701537, size = 22, normalized size = 1.1 \[ \frac{1}{3} \, f x^{3} + \frac{1}{2} \, e x^{2} + d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*f*x^6 + c*e*x^5 + b*e*x^3 + (c*d + b*f)*x^4 + a*e*x + (b*d + a*f)*x^2 + a*d)/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

1/3*f*x^3 + 1/2*e*x^2 + d*x

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Fricas [A]  time = 0.261822, size = 22, normalized size = 1.1 \[ \frac{1}{3} \, f x^{3} + \frac{1}{2} \, e x^{2} + d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*f*x^6 + c*e*x^5 + b*e*x^3 + (c*d + b*f)*x^4 + a*e*x + (b*d + a*f)*x^2 + a*d)/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

1/3*f*x^3 + 1/2*e*x^2 + d*x

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Sympy [A]  time = 0.144591, size = 15, normalized size = 0.75 \[ d x + \frac{e x^{2}}{2} + \frac{f x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d+a*e*x+(a*f+b*d)*x**2+b*e*x**3+(b*f+c*d)*x**4+c*e*x**5+c*f*x**6)/(c*x**4+b*x**2+a),x)

[Out]

d*x + e*x**2/2 + f*x**3/3

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GIAC/XCAS [A]  time = 0.317661, size = 23, normalized size = 1.15 \[ \frac{1}{3} \, f x^{3} + \frac{1}{2} \, x^{2} e + d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*f*x^6 + c*e*x^5 + b*e*x^3 + (c*d + b*f)*x^4 + a*e*x + (b*d + a*f)*x^2 + a*d)/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

1/3*f*x^3 + 1/2*x^2*e + d*x