3.1.3 \(\int (a x^2+b x^3+c x^4) \, dx\)

Optimal. Leaf size=25 \[ \frac {a x^3}{3}+\frac {b x^4}{4}+\frac {c x^5}{5} \]

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Rubi [A]  time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \frac {a x^3}{3}+\frac {b x^4}{4}+\frac {c x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a*x^2 + b*x^3 + c*x^4,x]

[Out]

(a*x^3)/3 + (b*x^4)/4 + (c*x^5)/5

Rubi steps

\begin {align*} \int \left (a x^2+b x^3+c x^4\right ) \, dx &=\frac {a x^3}{3}+\frac {b x^4}{4}+\frac {c x^5}{5}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {a x^3}{3}+\frac {b x^4}{4}+\frac {c x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a*x^2 + b*x^3 + c*x^4,x]

[Out]

(a*x^3)/3 + (b*x^4)/4 + (c*x^5)/5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a x^2+b x^3+c x^4\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[a*x^2 + b*x^3 + c*x^4,x]

[Out]

IntegrateAlgebraic[a*x^2 + b*x^3 + c*x^4, x]

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fricas [A]  time = 0.84, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{5} x^{5} c + \frac {1}{4} x^{4} b + \frac {1}{3} x^{3} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x^4+b*x^3+a*x^2,x, algorithm="fricas")

[Out]

1/5*x^5*c + 1/4*x^4*b + 1/3*x^3*a

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giac [A]  time = 0.54, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, c x^{5} + \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x^4+b*x^3+a*x^2,x, algorithm="giac")

[Out]

1/5*c*x^5 + 1/4*b*x^4 + 1/3*a*x^3

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maple [A]  time = 0.00, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{5} c \,x^{5}+\frac {1}{4} b \,x^{4}+\frac {1}{3} a \,x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c*x^4+b*x^3+a*x^2,x)

[Out]

1/3*a*x^3+1/4*b*x^4+1/5*c*x^5

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maxima [A]  time = 0.43, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, c x^{5} + \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x^4+b*x^3+a*x^2,x, algorithm="maxima")

[Out]

1/5*c*x^5 + 1/4*b*x^4 + 1/3*a*x^3

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mupad [B]  time = 0.03, size = 19, normalized size = 0.76 \begin {gather*} \frac {x^3\,\left (12\,c\,x^2+15\,b\,x+20\,a\right )}{60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a*x^2 + b*x^3 + c*x^4,x)

[Out]

(x^3*(20*a + 15*b*x + 12*c*x^2))/60

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sympy [A]  time = 0.06, size = 19, normalized size = 0.76 \begin {gather*} \frac {a x^{3}}{3} + \frac {b x^{4}}{4} + \frac {c x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x**4+b*x**3+a*x**2,x)

[Out]

a*x**3/3 + b*x**4/4 + c*x**5/5

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