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

Optimal result

Integrand size = 16, antiderivative size = 22 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2 x+a b x^2+\frac {b^2 x^3}{3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2 x+a b x^2+\frac {b^2 x^3}{3} \]

[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*} \text {integral}& = a^2 x+a b x^2+\frac {b^2 x^3}{3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2 x+a b x^2+\frac {b^2 x^3}{3} \]

[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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(a^{2} x +a b \,x^{2}+\frac {1}{3} b^{2} x^{3}\)	\(21\)
parallelrisch	\(a^{2} x +a b \,x^{2}+\frac {1}{3} b^{2} x^{3}\)	\(21\)
parts	\(a^{2} x +a b \,x^{2}+\frac {1}{3} b^{2} x^{3}\)	\(21\)
gosper	\(\frac {x \left (b^{2} x^{2}+3 a b x +3 a^{2}\right )}{3}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]

[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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2\,x+a\,b\,x^2+\frac {b^2\,x^3}{3} \]

[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