3.3.37 \(\int x^3 (a^2+2 a b x^2+b^2 x^4) \, dx\)

Optimal. Leaf size=30 \[ \frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8} \]

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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {14} \begin {gather*} \frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8

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^3 \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 x^3+2 a b x^5+b^2 x^7\right ) \, dx\\ &=\frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

IntegrateAlgebraic[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4), x]

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fricas [A]  time = 0.68, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} x^{8} b^{2} + \frac {1}{3} x^{6} b a + \frac {1}{4} x^{4} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")

[Out]

1/8*x^8*b^2 + 1/3*x^6*b*a + 1/4*x^4*a^2

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giac [A]  time = 0.15, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} + \frac {1}{3} \, a b x^{6} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")

[Out]

1/8*b^2*x^8 + 1/3*a*b*x^6 + 1/4*a^2*x^4

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maple [A]  time = 0.00, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{8} b^{2} x^{8}+\frac {1}{3} a b \,x^{6}+\frac {1}{4} a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

1/4*a^2*x^4+1/3*a*b*x^6+1/8*b^2*x^8

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maxima [A]  time = 1.36, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} + \frac {1}{3} \, a b x^{6} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")

[Out]

1/8*b^2*x^8 + 1/3*a*b*x^6 + 1/4*a^2*x^4

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mupad [B]  time = 0.04, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^4}{4}+\frac {a\,b\,x^6}{3}+\frac {b^2\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a^2 + b^2*x^4 + 2*a*b*x^2),x)

[Out]

(a^2*x^4)/4 + (b^2*x^8)/8 + (a*b*x^6)/3

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sympy [A]  time = 0.07, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^{2} x^{4}}{4} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

a**2*x**4/4 + a*b*x**6/3 + b**2*x**8/8

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