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

Optimal. Leaf size=17 \[ \frac {a x^2}{2}+\frac {b x^4}{4} \]

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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 {a x^2}{2}+\frac {b x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a*x + b*x^3,x]

[Out]

(a*x^2)/2 + (b*x^4)/4

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {a x^2}{2}+\frac {b x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a*x + b*x^3,x]

[Out]

(a*x^2)/2 + (b*x^4)/4

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[a*x + b*x^3,x]

[Out]

IntegrateAlgebraic[a*x + b*x^3, x]

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fricas [A]  time = 0.35, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} x^{4} b + \frac {1}{2} x^{2} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*b + 1/2*x^2*a

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giac [A]  time = 0.15, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4 + 1/2*a*x^2

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maple [A]  time = 0.04, size = 14, normalized size = 0.82 \begin {gather*} \frac {1}{4} b \,x^{4}+\frac {1}{2} a \,x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(b*x^3+a*x,x)

[Out]

1/2*a*x^2+1/4*b*x^4

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maxima [A]  time = 1.35, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4 + 1/2*a*x^2

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {b\,x^4}{4}+\frac {a\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a*x + b*x^3,x)

[Out]

(a*x^2)/2 + (b*x^4)/4

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sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{2}}{2} + \frac {b x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b*x**3+a*x,x)

[Out]

a*x**2/2 + b*x**4/4

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