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

Optimal. Leaf size=50 \[ a d x+\frac{1}{2} a e x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4+\frac{1}{5} c d x^5+\frac{1}{6} c e x^6 \]

[Out]

a*d*x + (a*e*x^2)/2 + (b*d*x^3)/3 + (b*e*x^4)/4 + (c*d*x^5)/5 + (c*e*x^6)/6

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Rubi [A]  time = 0.0874187, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a d x+\frac{1}{2} a e x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4+\frac{1}{5} c d x^5+\frac{1}{6} c e x^6 \]

Antiderivative was successfully verified.

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

[Out]

a*d*x + (a*e*x^2)/2 + (b*d*x^3)/3 + (b*e*x^4)/4 + (c*d*x^5)/5 + (c*e*x^6)/6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a e \int x\, dx + \frac{b d x^{3}}{3} + \frac{b e x^{4}}{4} + \frac{c d x^{5}}{5} + \frac{c e x^{6}}{6} + d \int a\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*e*Integral(x, x) + b*d*x**3/3 + b*e*x**4/4 + c*d*x**5/5 + c*e*x**6/6 + d*Integ
ral(a, x)

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Mathematica [A]  time = 0.00528132, size = 50, normalized size = 1. \[ a d x+\frac{1}{2} a e x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4+\frac{1}{5} c d x^5+\frac{1}{6} c e x^6 \]

Antiderivative was successfully verified.

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

[Out]

a*d*x + (a*e*x^2)/2 + (b*d*x^3)/3 + (b*e*x^4)/4 + (c*d*x^5)/5 + (c*e*x^6)/6

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Maple [A]  time = 0.001, size = 41, normalized size = 0.8 \[ adx+{\frac{ae{x}^{2}}{2}}+{\frac{bd{x}^{3}}{3}}+{\frac{be{x}^{4}}{4}}+{\frac{cd{x}^{5}}{5}}+{\frac{ce{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^4+b*x^2+a),x)

[Out]

a*d*x+1/2*a*e*x^2+1/3*b*d*x^3+1/4*b*e*x^4+1/5*c*d*x^5+1/6*c*e*x^6

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Maxima [A]  time = 0.695756, size = 54, normalized size = 1.08 \[ \frac{1}{6} \, c e x^{6} + \frac{1}{5} \, c d x^{5} + \frac{1}{4} \, b e x^{4} + \frac{1}{3} \, b d x^{3} + \frac{1}{2} \, a e x^{2} + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*c*e*x^6 + 1/5*c*d*x^5 + 1/4*b*e*x^4 + 1/3*b*d*x^3 + 1/2*a*e*x^2 + a*d*x

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Fricas [A]  time = 0.25548, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} e c + \frac{1}{5} x^{5} d c + \frac{1}{4} x^{4} e b + \frac{1}{3} x^{3} d b + \frac{1}{2} x^{2} e a + x d a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^6*e*c + 1/5*x^5*d*c + 1/4*x^4*e*b + 1/3*x^3*d*b + 1/2*x^2*e*a + x*d*a

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Sympy [A]  time = 0.089687, size = 46, normalized size = 0.92 \[ a d x + \frac{a e x^{2}}{2} + \frac{b d x^{3}}{3} + \frac{b e x^{4}}{4} + \frac{c d x^{5}}{5} + \frac{c e x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d*x + a*e*x**2/2 + b*d*x**3/3 + b*e*x**4/4 + c*d*x**5/5 + c*e*x**6/6

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GIAC/XCAS [A]  time = 0.277799, size = 58, normalized size = 1.16 \[ \frac{1}{6} \, c x^{6} e + \frac{1}{5} \, c d x^{5} + \frac{1}{4} \, b x^{4} e + \frac{1}{3} \, b d x^{3} + \frac{1}{2} \, a x^{2} e + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*c*x^6*e + 1/5*c*d*x^5 + 1/4*b*x^4*e + 1/3*b*d*x^3 + 1/2*a*x^2*e + a*d*x