3.1.1 \(\int x^2 (a x^2+b x^3+c x^4) \, dx\)

Optimal. Leaf size=25 \[ \frac {a x^5}{5}+\frac {b x^6}{6}+\frac {c x^7}{7} \]

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {14} \begin {gather*} \frac {a x^5}{5}+\frac {b x^6}{6}+\frac {c x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a*x^2 + b*x^3 + c*x^4),x]

[Out]

(a*x^5)/5 + (b*x^6)/6 + (c*x^7)/7

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+c x^4\right ) \, dx &=\int \left (a x^4+b x^5+c x^6\right ) \, dx\\ &=\frac {a x^5}{5}+\frac {b x^6}{6}+\frac {c x^7}{7}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {a x^5}{5}+\frac {b x^6}{6}+\frac {c x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a*x^2 + b*x^3 + c*x^4),x]

[Out]

(a*x^5)/5 + (b*x^6)/6 + (c*x^7)/7

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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+c x^4\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^2*(a*x^2 + b*x^3 + c*x^4),x]

[Out]

IntegrateAlgebraic[x^2*(a*x^2 + b*x^3 + c*x^4), x]

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fricas [A]  time = 0.89, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{7} x^{7} c + \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*(c*x^4+b*x^3+a*x^2),x, algorithm="fricas")

[Out]

1/7*x^7*c + 1/6*x^6*b + 1/5*x^5*a

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giac [A]  time = 0.54, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, c x^{7} + \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*(c*x^4+b*x^3+a*x^2),x, algorithm="giac")

[Out]

1/7*c*x^7 + 1/6*b*x^6 + 1/5*a*x^5

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maple [A]  time = 0.00, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{7} c \,x^{7}+\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*(c*x^4+b*x^3+a*x^2),x)

[Out]

1/5*a*x^5+1/6*b*x^6+1/7*c*x^7

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maxima [A]  time = 0.44, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, c x^{7} + \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*(c*x^4+b*x^3+a*x^2),x, algorithm="maxima")

[Out]

1/7*c*x^7 + 1/6*b*x^6 + 1/5*a*x^5

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mupad [B]  time = 0.03, size = 19, normalized size = 0.76 \begin {gather*} \frac {x^5\,\left (30\,c\,x^2+35\,b\,x+42\,a\right )}{210} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a*x^2 + b*x^3 + c*x^4),x)

[Out]

(x^5*(42*a + 35*b*x + 30*c*x^2))/210

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sympy [A]  time = 0.06, size = 19, normalized size = 0.76 \begin {gather*} \frac {a x^{5}}{5} + \frac {b x^{6}}{6} + \frac {c x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(c*x**4+b*x**3+a*x**2),x)

[Out]

a*x**5/5 + b*x**6/6 + c*x**7/7

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