3.3.8 \(\int x (a+b x^3) \, dx\) [208]

Optimal. Leaf size=17 \[ \frac {a x^2}{2}+\frac {b x^5}{5} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {14} \begin {gather*} \frac {a x^2}{2}+\frac {b x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^3),x]

[Out]

(a*x^2)/2 + (b*x^5)/5

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

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {a x^2}{2}+\frac {b x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^3),x]

[Out]

(a*x^2)/2 + (b*x^5)/5

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Maple [A]
time = 0.01, size = 14, normalized size = 0.82

method	result	size
gosper	\(\frac {1}{2} a \,x^{2}+\frac {1}{5} b \,x^{5}\)	\(14\)
default	\(\frac {1}{2} a \,x^{2}+\frac {1}{5} b \,x^{5}\)	\(14\)
norman	\(\frac {1}{2} a \,x^{2}+\frac {1}{5} b \,x^{5}\)	\(14\)
risch	\(\frac {1}{2} a \,x^{2}+\frac {1}{5} b \,x^{5}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/2*a*x^2

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Fricas [A]
time = 0.33, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/2*a*x^2

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Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{2}}{2} + \frac {b x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*x**5/5

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Giac [A]
time = 2.52, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/2*a*x^2

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {b\,x^5}{5}+\frac {a\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^3),x)

[Out]

(a*x^2)/2 + (b*x^5)/5

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