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

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

[Out]

(a*x^5)/5 + (b*x^6)/6 + (c*x^7)/7

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Rubi [A]  time = 0.0074164, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {14} \[ \frac{a x^5}{5}+\frac{b x^6}{6}+\frac{c x^7}{7} \]

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*}

Mathematica [A]  time = 0.0022189, size = 25, normalized size = 1. \[ \frac{a x^5}{5}+\frac{b x^6}{6}+\frac{c x^7}{7} \]

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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Maple [A]  time = 0., size = 20, normalized size = 0.8 \begin{align*}{\frac{a{x}^{5}}{5}}+{\frac{b{x}^{6}}{6}}+{\frac{c{x}^{7}}{7}} \end{align*}

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 = 1.09851, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{7} \, c x^{7} + \frac{1}{6} \, b x^{6} + \frac{1}{5} \, a x^{5} \end{align*}

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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Fricas [A]  time = 1.27924, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{7} x^{7} c + \frac{1}{6} x^{6} b + \frac{1}{5} x^{5} a \end{align*}

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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Sympy [A]  time = 0.058303, size = 19, normalized size = 0.76 \begin{align*} \frac{a x^{5}}{5} + \frac{b x^{6}}{6} + \frac{c x^{7}}{7} \end{align*}

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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Giac [A]  time = 1.09862, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{7} \, c x^{7} + \frac{1}{6} \, b x^{6} + \frac{1}{5} \, a x^{5} \end{align*}

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