3.1.17 \(\int x (b x^2+c x^4) \, dx\)

Optimal. Leaf size=17 \[ \frac {b x^4}{4}+\frac {c x^6}{6} \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \begin {gather*} \frac {b x^4}{4}+\frac {c x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(b*x^2 + c*x^4),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {b x^4}{4}+\frac {c x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(b*x^2 + c*x^4),x]

[Out]

(b*x^4)/4 + (c*x^6)/6

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.66, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} x^{6} c + \frac {1}{4} x^{4} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*c + 1/4*x^4*b

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giac [A]  time = 0.17, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} \frac {1}{6} c \,x^{6}+\frac {1}{4} b \,x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4+1/6*c*x^6

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maxima [A]  time = 1.30, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {c\,x^6}{6}+\frac {b\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2 + c*x^4),x)

[Out]

(b*x^4)/4 + (c*x^6)/6

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sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \begin {gather*} \frac {b x^{4}}{4} + \frac {c x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x**4/4 + c*x**6/6

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