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

Optimal. Leaf size=20 \[ a x+\frac {b x^3}{3}+\frac {c x^5}{5} \]

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} a x+\frac {b x^3}{3}+\frac {c x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*x^2 + c*x^4,x]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} a x+\frac {b x^3}{3}+\frac {c x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*x^2 + c*x^4,x]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 2.47, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{5} x^{5} c + \frac {1}{3} x^{3} b + x a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*c + 1/3*x^3*b + x*a

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giac [A]  time = 0.15, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x

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maple [A]  time = 0.00, size = 17, normalized size = 0.85 \begin {gather*} \frac {1}{5} c \,x^{5}+\frac {1}{3} b \,x^{3}+a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{5} \, c x^{5} + \frac {1}{3} \, b x^{3} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x

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mupad [B]  time = 0.02, size = 16, normalized size = 0.80 \begin {gather*} \frac {c\,x^5}{5}+\frac {b\,x^3}{3}+a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*x^2 + c*x^4,x)

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

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sympy [A]  time = 0.06, size = 15, normalized size = 0.75 \begin {gather*} a x + \frac {b x^{3}}{3} + \frac {c x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x**3/3 + c*x**5/5

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