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

Optimal. Leaf size=22 \[ a^2 x+a b x^2+\frac {b^2 x^3}{3} \]

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Rubi [A]  time = 0.00, antiderivative size = 22, 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*} a^2 x+a b x^2+\frac {b^2 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a^2 + 2*a*b*x + b^2*x^2,x]

[Out]

a^2*x + a*b*x^2 + (b^2*x^3)/3

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} a^2 x+a b x^2+\frac {b^2 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a^2 + 2*a*b*x + b^2*x^2,x]

[Out]

a^2*x + a*b*x^2 + (b^2*x^3)/3

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.35, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{3} x^{3} b^{2} + x^{2} b a + x a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*b^2 + x^2*b*a + x*a^2

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giac [A]  time = 0.18, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*x^3 + a*b*x^2 + a^2*x

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maple [A]  time = 0.04, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{3} b^{2} x^{3}+a b \,x^{2}+a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.52, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*x^3 + a*b*x^2 + a^2*x

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mupad [B]  time = 0.03, size = 20, normalized size = 0.91 \begin {gather*} a^2\,x+a\,b\,x^2+\frac {b^2\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^2 + b^2*x^2 + 2*a*b*x,x)

[Out]

a^2*x + (b^2*x^3)/3 + a*b*x^2

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sympy [A]  time = 0.06, size = 19, normalized size = 0.86 \begin {gather*} a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x + a*b*x**2 + b**2*x**3/3

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