3.1.52 \(\int (a+c x^2) \, dx\) [52]

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 = {} \begin {gather*} a x+\frac {c x^3}{3} \end {gather*}

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 \begin {gather*} a x+\frac {c x^3}{3} \end {gather*}

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 = 0.02, size = 11, normalized size = 0.92

method	result	size
gosper	\(a x +\frac {1}{3} c \,x^{3}\)	\(11\)
default	\(a x +\frac {1}{3} c \,x^{3}\)	\(11\)
norman	\(a x +\frac {1}{3} c \,x^{3}\)	\(11\)
risch	\(a x +\frac {1}{3} c \,x^{3}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c*x^2+a,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{3} \, c x^{3} + a x \end {gather*}

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 = 1.57, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{3} \, c x^{3} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*x^3 + a*x

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Sympy [A]
time = 0.01, size = 8, normalized size = 0.67 \begin {gather*} a x + \frac {c x^{3}}{3} \end {gather*}

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 [A]
time = 1.35, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{3} \, c x^{3} + a x \end {gather*}

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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Mupad [B]
time = 0.02, size = 10, normalized size = 0.83 \begin {gather*} \frac {c\,x^3}{3}+a\,x \end {gather*}

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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