3.9.15 \(\int x (a+b x^2+c x^4) \, dx\) [815]

Optimal. Leaf size=25 \[ \frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14} \begin {gather*} \frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2 + c*x^4),x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {a x^2}{2}+\frac {b x^4}{4}+\frac {c x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2 + c*x^4),x]

[Out]

(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6

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Maple [A]
time = 0.01, size = 20, normalized size = 0.80

method	result	size
gosper	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)
default	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)
norman	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)
risch	\(\frac {1}{2} a \,x^{2}+\frac {1}{4} b \,x^{4}+\frac {1}{6} c \,x^{6}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4 + 1/2*a*x^2

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Fricas [A]
time = 0.36, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4 + 1/2*a*x^2

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Sympy [A]
time = 0.01, size = 19, normalized size = 0.76 \begin {gather*} \frac {a x^{2}}{2} + \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+a),x)

[Out]

a*x**2/2 + b*x**4/4 + c*x**6/6

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Giac [A]
time = 3.55, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, c x^{6} + \frac {1}{4} \, b x^{4} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*x^6 + 1/4*b*x^4 + 1/2*a*x^2

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Mupad [B]
time = 0.03, size = 19, normalized size = 0.76 \begin {gather*} \frac {c\,x^6}{6}+\frac {b\,x^4}{4}+\frac {a\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2 + c*x^4),x)

[Out]

(a*x^2)/2 + (b*x^4)/4 + (c*x^6)/6

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