3.5 \(\int (a+b x^2) \, dx\)

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

[Out]

a*x + (b*x^3)/3

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Rubi [A]  time = 0.0022155, 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., Rules used = {} \[ a x+\frac{b x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[a + b*x^2,x]

[Out]

a*x + (b*x^3)/3

Rubi steps

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

Mathematica [A]  time = 0.0000288, size = 12, normalized size = 1. \[ a x+\frac{b x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*x^2,x]

[Out]

a*x + (b*x^3)/3

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Maple [A]  time = 0., size = 11, normalized size = 0.9 \begin{align*} ax+{\frac{b{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(b*x^2+a,x)

[Out]

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

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Maxima [A]  time = 1.69238, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{3} \, b x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b*x^2+a,x, algorithm="maxima")

[Out]

1/3*b*x^3 + a*x

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Fricas [A]  time = 1.26278, size = 23, normalized size = 1.92 \begin{align*} \frac{1}{3} x^{3} b + x a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*b + x*a

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Sympy [A]  time = 0.080374, size = 8, normalized size = 0.67 \begin{align*} a x + \frac{b x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b*x**2+a,x)

[Out]

a*x + b*x**3/3

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Giac [A]  time = 2.03595, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{3} \, b x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b*x^2+a,x, algorithm="giac")

[Out]

1/3*b*x^3 + a*x