3.52 \(\int \left (a+c x^2\right ) \, dx\)

Optimal. Leaf size=12 \[ a x+\frac{c x^3}{3} \]

[Out]

a*x + (c*x^3)/3

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Rubi [A]  time = 0.00791862, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ a x+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[a + c*x^2,x]

[Out]

a*x + (c*x^3)/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(c*x**2+a,x)

[Out]

c*x**3/3 + Integral(a, x)

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Mathematica [A]  time = 0.000075836, size = 12, normalized size = 1. \[ a x+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[a + c*x^2,x]

[Out]

a*x + (c*x^3)/3

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Maple [A]  time = 0.001, size = 11, normalized size = 0.9 \[ ax+{\frac{c{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(c*x^2+a,x)

[Out]

a*x+1/3*c*x^3

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Maxima [A]  time = 0.697821, size = 14, normalized size = 1.17 \[ \frac{1}{3} \, c x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3 + a*x

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Fricas [A]  time = 0.189583, size = 1, normalized size = 0.08 \[ \frac{1}{3} x^{3} c + x a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3*c + x*a

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Sympy [A]  time = 0.06707, size = 8, normalized size = 0.67 \[ a x + \frac{c x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(c*x**2+a,x)

[Out]

a*x + c*x**3/3

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GIAC/XCAS [A]  time = 0.208659, size = 14, normalized size = 1.17 \[ \frac{1}{3} \, c x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*c*x^3 + a*x