3.313 \(\int x^2 (c+d x+e x^2) (a+b x^3) \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{6} b c x^6+\frac{1}{7} b d x^7+\frac{1}{8} b e x^8 \]

[Out]

(a*c*x^3)/3 + (a*d*x^4)/4 + (a*e*x^5)/5 + (b*c*x^6)/6 + (b*d*x^7)/7 + (b*e*x^8)/8

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Rubi [A]  time = 0.0554037, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {1628} \[ \frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{6} b c x^6+\frac{1}{7} b d x^7+\frac{1}{8} b e x^8 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(c + d*x + e*x^2)*(a + b*x^3),x]

[Out]

(a*c*x^3)/3 + (a*d*x^4)/4 + (a*e*x^5)/5 + (b*c*x^6)/6 + (b*d*x^7)/7 + (b*e*x^8)/8

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right ) \, dx &=\int \left (a c x^2+a d x^3+a e x^4+b c x^5+b d x^6+b e x^7\right ) \, dx\\ &=\frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{6} b c x^6+\frac{1}{7} b d x^7+\frac{1}{8} b e x^8\\ \end{align*}

Mathematica [A]  time = 0.0032462, size = 55, normalized size = 1. \[ \frac{1}{3} a c x^3+\frac{1}{4} a d x^4+\frac{1}{5} a e x^5+\frac{1}{6} b c x^6+\frac{1}{7} b d x^7+\frac{1}{8} b e x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c*x^3)/3 + (a*d*x^4)/4 + (a*e*x^5)/5 + (b*c*x^6)/6 + (b*d*x^7)/7 + (b*e*x^8)/8

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Maple [A]  time = 0.002, size = 44, normalized size = 0.8 \begin{align*}{\frac{ac{x}^{3}}{3}}+{\frac{ad{x}^{4}}{4}}+{\frac{ae{x}^{5}}{5}}+{\frac{bc{x}^{6}}{6}}+{\frac{bd{x}^{7}}{7}}+{\frac{be{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*c*x^3+1/4*a*d*x^4+1/5*a*e*x^5+1/6*b*c*x^6+1/7*b*d*x^7+1/8*b*e*x^8

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Maxima [A]  time = 0.928077, size = 58, normalized size = 1.05 \begin{align*} \frac{1}{8} \, b e x^{8} + \frac{1}{7} \, b d x^{7} + \frac{1}{6} \, b c x^{6} + \frac{1}{5} \, a e x^{5} + \frac{1}{4} \, a d x^{4} + \frac{1}{3} \, a c x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b*e*x^8 + 1/7*b*d*x^7 + 1/6*b*c*x^6 + 1/5*a*e*x^5 + 1/4*a*d*x^4 + 1/3*a*c*x^3

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Fricas [A]  time = 1.29515, size = 112, normalized size = 2.04 \begin{align*} \frac{1}{8} x^{8} e b + \frac{1}{7} x^{7} d b + \frac{1}{6} x^{6} c b + \frac{1}{5} x^{5} e a + \frac{1}{4} x^{4} d a + \frac{1}{3} x^{3} c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*e*b + 1/7*x^7*d*b + 1/6*x^6*c*b + 1/5*x^5*e*a + 1/4*x^4*d*a + 1/3*x^3*c*a

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Sympy [A]  time = 0.063593, size = 49, normalized size = 0.89 \begin{align*} \frac{a c x^{3}}{3} + \frac{a d x^{4}}{4} + \frac{a e x^{5}}{5} + \frac{b c x^{6}}{6} + \frac{b d x^{7}}{7} + \frac{b e x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x**3/3 + a*d*x**4/4 + a*e*x**5/5 + b*c*x**6/6 + b*d*x**7/7 + b*e*x**8/8

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Giac [A]  time = 1.06384, size = 61, normalized size = 1.11 \begin{align*} \frac{1}{8} \, b x^{8} e + \frac{1}{7} \, b d x^{7} + \frac{1}{6} \, b c x^{6} + \frac{1}{5} \, a x^{5} e + \frac{1}{4} \, a d x^{4} + \frac{1}{3} \, a c x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b*x^8*e + 1/7*b*d*x^7 + 1/6*b*c*x^6 + 1/5*a*x^5*e + 1/4*a*d*x^4 + 1/3*a*c*x^3