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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ a x+\frac {c x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[a + c*x^2,x]

[Out]

a*x + (c*x^3)/3

Rubi steps

\begin {align*} \int \left (a+c x^2\right ) \, dx &=a x+\frac {c x^3}{3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ a x+\frac {c x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[a + c*x^2,x]

[Out]

a*x + (c*x^3)/3

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fricas [A]  time = 0.71, size = 10, normalized size = 0.83 \[ \frac {1}{3} x^{3} c + x a \]

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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giac [A]  time = 0.35, size = 10, normalized size = 0.83 \[ \frac {1}{3} \, c x^{3} + a x \]

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

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maple [A]  time = 0.04, size = 11, normalized size = 0.92 \[ \frac {1}{3} c \,x^{3}+a x \]

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 = 1.29, size = 10, normalized size = 0.83 \[ \frac {1}{3} \, c x^{3} + a x \]

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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mupad [B]  time = 0.02, size = 10, normalized size = 0.83 \[ \frac {c\,x^3}{3}+a\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + c*x^2,x)

[Out]

a*x + (c*x^3)/3

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sympy [A]  time = 0.06, 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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