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

Optimal. Leaf size=30 \[ \frac {a^2 x^2}{2}+\frac {1}{2} a b x^4+\frac {b^2 x^6}{6} \]

[Out]

1/2*a^2*x^2+1/2*a*b*x^4+1/6*b^2*x^6

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {14} \[ \frac {a^2 x^2}{2}+\frac {1}{2} a b x^4+\frac {b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[x*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(a^2*x^2)/2 + (a*b*x^4)/2 + (b^2*x^6)/6

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int x \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 x+2 a b x^3+b^2 x^5\right ) \, dx\\ &=\frac {a^2 x^2}{2}+\frac {1}{2} a b x^4+\frac {b^2 x^6}{6}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.53 \[ \frac {\left (a+b x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x^2)^3/(6*b)

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fricas [A]  time = 0.66, size = 24, normalized size = 0.80 \[ \frac {1}{6} x^{6} b^{2} + \frac {1}{2} x^{4} b a + \frac {1}{2} x^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*b^2 + 1/2*x^4*b*a + 1/2*x^2*a^2

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giac [A]  time = 0.17, size = 24, normalized size = 0.80 \[ \frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 1/2*a*b*x^4 + 1/2*a^2*x^2

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maple [A]  time = 0.00, size = 25, normalized size = 0.83 \[ \frac {1}{6} b^{2} x^{6}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*x^2+1/2*a*b*x^4+1/6*b^2*x^6

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maxima [A]  time = 1.29, size = 24, normalized size = 0.80 \[ \frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 1/2*a*b*x^4 + 1/2*a^2*x^2

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mupad [B]  time = 0.03, size = 24, normalized size = 0.80 \[ \frac {a^2\,x^2}{2}+\frac {a\,b\,x^4}{2}+\frac {b^2\,x^6}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a^2 + b^2*x^4 + 2*a*b*x^2),x)

[Out]

(a^2*x^2)/2 + (b^2*x^6)/6 + (a*b*x^4)/2

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sympy [A]  time = 0.07, size = 24, normalized size = 0.80 \[ \frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6

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