3.1.1 \(\int x^2 (a x+b x^3) \, dx\) [1]

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

[Out]

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

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \begin {gather*} \frac {a x^4}{4}+\frac {b x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a*x + b*x^3),x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a*x + b*x^3),x]

[Out]

(a*x^4)/4 + (b*x^6)/6

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Maple [A]
time = 0.08, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {1}{4} a \,x^{4}+\frac {1}{6} b \,x^{6}\)	\(14\)
norman	\(\frac {1}{4} a \,x^{4}+\frac {1}{6} b \,x^{6}\)	\(14\)
risch	\(\frac {1}{4} a \,x^{4}+\frac {1}{6} b \,x^{6}\)	\(14\)
gosper	\(\frac {x^{4} \left (2 b \,x^{2}+3 a \right )}{12}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, b x^{6} + \frac {1}{4} \, a x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*x^6 + 1/4*a*x^4

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Fricas [A]
time = 1.95, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, b x^{6} + \frac {1}{4} \, a x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*x^6 + 1/4*a*x^4

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Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{4}}{4} + \frac {b x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**4/4 + b*x**6/6

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Giac [A]
time = 1.50, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, b x^{6} + \frac {1}{4} \, a x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*x^6 + 1/4*a*x^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a*x + b*x^3),x)

[Out]

(a*x^4)/4 + (b*x^6)/6

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