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

Optimal. Leaf size=25 \[ a^2 x+\frac{2}{3} a b x^3+\frac{b^2 x^5}{5} \]

[Out]

a^2*x + (2*a*b*x^3)/3 + (b^2*x^5)/5

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Rubi [A]  time = 0.0049772, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ a^2 x+\frac{2}{3} a b x^3+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[a^2 + 2*a*b*x^2 + b^2*x^4,x]

[Out]

a^2*x + (2*a*b*x^3)/3 + (b^2*x^5)/5

Rubi steps

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

Mathematica [A]  time = 0.0000442, size = 25, normalized size = 1. \[ a^2 x+\frac{2}{3} a b x^3+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[a^2 + 2*a*b*x^2 + b^2*x^4,x]

[Out]

a^2*x + (2*a*b*x^3)/3 + (b^2*x^5)/5

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Maple [A]  time = 0.039, size = 22, normalized size = 0.9 \begin{align*}{a}^{2}x+{\frac{2\,ab{x}^{3}}{3}}+{\frac{{b}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x+2/3*a*b*x^3+1/5*b^2*x^5

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Maxima [A]  time = 0.990951, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{2}{3} \, a b x^{3} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5 + 2/3*a*b*x^3 + a^2*x

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Fricas [A]  time = 1.28327, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{5} x^{5} b^{2} + \frac{2}{3} x^{3} b a + x a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*b^2 + 2/3*x^3*b*a + x*a^2

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Sympy [A]  time = 0.061637, size = 22, normalized size = 0.88 \begin{align*} a^{2} x + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x + 2*a*b*x**3/3 + b**2*x**5/5

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Giac [A]  time = 1.13614, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{2}{3} \, a b x^{3} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5 + 2/3*a*b*x^3 + a^2*x