3.3.3 \(\int (b x+c x^2) \, dx\)

Optimal. Leaf size=17 \[ \frac {b x^2}{2}+\frac {c x^3}{3} \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \frac {b x^2}{2}+\frac {c x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[b*x + c*x^2,x]

[Out]

(b*x^2)/2 + (c*x^3)/3

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {b x^2}{2}+\frac {c x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[b*x + c*x^2,x]

[Out]

(b*x^2)/2 + (c*x^3)/3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[b*x + c*x^2,x]

[Out]

IntegrateAlgebraic[b*x + c*x^2, x]

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fricas [A]  time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} x^{3} c + \frac {1}{2} x^{2} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*c + 1/2*x^2*b

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giac [A]  time = 0.15, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*x^3 + 1/2*b*x^2

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maple [A]  time = 0.05, size = 14, normalized size = 0.82 \begin {gather*} \frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2+1/3*c*x^3

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maxima [A]  time = 1.37, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*x^3 + 1/2*b*x^2

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {x^2\,\left (3\,b+2\,c\,x\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(b*x + c*x^2,x)

[Out]

(x^2*(3*b + 2*c*x))/6

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sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \begin {gather*} \frac {b x^{2}}{2} + \frac {c x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x**2/2 + c*x**3/3

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