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

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Rubi [A]  time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \[ a^2 x+\frac {2}{3} a b x^3+\frac {b^2 x^5}{5} \]

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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fricas [A]  time = 0.69, size = 21, normalized size = 0.84 \[ \frac {1}{5} x^{5} b^{2} + \frac {2}{3} x^{3} b a + x a^{2} \]

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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giac [A]  time = 0.15, size = 21, normalized size = 0.84 \[ \frac {1}{5} \, b^{2} x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]

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

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maple [A]  time = 0.00, size = 22, normalized size = 0.88 \[ \frac {1}{5} b^{2} x^{5}+\frac {2}{3} a b \,x^{3}+a^{2} x \]

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 = 1.35, size = 21, normalized size = 0.84 \[ \frac {1}{5} \, b^{2} x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]

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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mupad [B]  time = 0.03, size = 21, normalized size = 0.84 \[ a^2\,x+\frac {2\,a\,b\,x^3}{3}+\frac {b^2\,x^5}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^2 + b^2*x^4 + 2*a*b*x^2,x)

[Out]

a^2*x + (b^2*x^5)/5 + (2*a*b*x^3)/3

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sympy [A]  time = 0.07, size = 22, normalized size = 0.88 \[ a^{2} x + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{5}}{5} \]

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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