3.15.55 \(\int (a^2+2 a b x+b^2 x^2) \, dx\) [1455]

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

[Out]

a^2*x+a*b*x^2+1/3*b^2*x^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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Maple [A]
time = 0.03, size = 13, normalized size = 0.59

method	result	size
default	\(\frac {\left (b x +a \right )^{3}}{3 b}\)	\(13\)
norman	\(a^{2} x +a b \,x^{2}+\frac {1}{3} b^{2} x^{3}\)	\(21\)
risch	\(a^{2} x +a b \,x^{2}+\frac {1}{3} b^{2} x^{3}\)	\(21\)
gosper	\(\frac {x \left (b^{2} x^{2}+3 a b x +3 a^{2}\right )}{3}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(b^2*x^2+2*a*b*x+a^2,x,method=_RETURNVERBOSE)

[Out]

1/3*(b*x+a)^3/b

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Maxima [A]
time = 0.27, 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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Fricas [A]
time = 1.84, 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="fricas")

[Out]

1/3*b^2*x^3 + a*b*x^2 + a^2*x

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Sympy [A]
time = 0.01, 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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Giac [A]
time = 1.80, 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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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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