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

Optimal. Leaf size=17 \[ \frac {a x^5}{5}+\frac {b x^6}{6} \]

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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \begin {gather*} \frac {a x^5}{5}+\frac {b x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a*x^2 + b*x^3),x]

[Out]

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

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a*x^2 + b*x^3),x]

[Out]

(a*x^5)/5 + (b*x^6)/6

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} x^{6} b + \frac {1}{5} x^{5} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*b + 1/5*x^5*a

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giac [A]  time = 0.19, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, b x^{6} + \frac {1}{5} \, a x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*x^6 + 1/5*a*x^5

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maple [A]  time = 0.04, size = 14, normalized size = 0.82 \begin {gather*} \frac {1}{6} b \,x^{6}+\frac {1}{5} a \,x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*x^5+1/6*b*x^6

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maxima [A]  time = 1.33, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, b x^{6} + \frac {1}{5} \, a x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b*x^6 + 1/5*a*x^5

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {x^5\,\left (6\,a+5\,b\,x\right )}{30} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a*x^2 + b*x^3),x)

[Out]

(x^5*(6*a + 5*b*x))/30

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sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{5}}{5} + \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**2),x)

[Out]

a*x**5/5 + b*x**6/6

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