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

Optimal. Leaf size=31 \[ \frac{e \left (a+c x^2\right )^2}{4 c}+a d x+\frac{1}{3} c d x^3 \]

[Out]

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

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Rubi [A]  time = 0.0280731, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{e \left (a+c x^2\right )^2}{4 c}+a d x+\frac{1}{3} c d x^3 \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(a + c*x^2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*e*Integral(x, x) + c*d*x**3/3 + c*e*x**4/4 + d*Integral(a, x)

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Mathematica [A]  time = 0.00287985, size = 32, normalized size = 1.03 \[ a d x+\frac{1}{2} a e x^2+\frac{1}{3} c d x^3+\frac{1}{4} c e x^4 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 27, normalized size = 0.9 \[{\frac{ce{x}^{4}}{4}}+{\frac{cd{x}^{3}}{3}}+{\frac{ae{x}^{2}}{2}}+adx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+a),x)

[Out]

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

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Maxima [A]  time = 0.717783, size = 35, normalized size = 1.13 \[ \frac{1}{4} \, c e x^{4} + \frac{1}{3} \, c d x^{3} + \frac{1}{2} \, a e x^{2} + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.18282, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} e c + \frac{1}{3} x^{3} d c + \frac{1}{2} x^{2} e a + x d a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.079065, size = 29, normalized size = 0.94 \[ a d x + \frac{a e x^{2}}{2} + \frac{c d x^{3}}{3} + \frac{c e x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(c*x**2+a),x)

[Out]

a*d*x + a*e*x**2/2 + c*d*x**3/3 + c*e*x**4/4

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GIAC/XCAS [A]  time = 0.203953, size = 38, normalized size = 1.23 \[ \frac{1}{4} \, c x^{4} e + \frac{1}{3} \, c d x^{3} + \frac{1}{2} \, a x^{2} e + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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