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]

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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Rubi [A]  time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 55, normalized size = 1.00 \[ \frac {1}{3} a c x^3+\frac {1}{4} a d x^4+\frac {1}{5} a e x^5+\frac {1}{6} b c x^6+\frac {1}{7} b d x^7+\frac {1}{8} b e x^8 \]

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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fricas [A]  time = 0.72, size = 43, normalized size = 0.78 \[ \frac {1}{8} x^{8} e b + \frac {1}{7} x^{7} d b + \frac {1}{6} x^{6} c b + \frac {1}{5} x^{5} e a + \frac {1}{4} x^{4} d a + \frac {1}{3} x^{3} c a \]

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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giac [A]  time = 0.20, size = 45, normalized size = 0.82 \[ \frac {1}{8} \, b x^{8} e + \frac {1}{7} \, b d x^{7} + \frac {1}{6} \, b c x^{6} + \frac {1}{5} \, a x^{5} e + \frac {1}{4} \, a d x^{4} + \frac {1}{3} \, a c x^{3} \]

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

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maple [A]  time = 0.04, size = 44, normalized size = 0.80 \[ \frac {1}{8} b e \,x^{8}+\frac {1}{7} b d \,x^{7}+\frac {1}{6} b c \,x^{6}+\frac {1}{5} a e \,x^{5}+\frac {1}{4} a d \,x^{4}+\frac {1}{3} a c \,x^{3} \]

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 = 1.33, size = 43, normalized size = 0.78 \[ \frac {1}{8} \, b e x^{8} + \frac {1}{7} \, b d x^{7} + \frac {1}{6} \, b c x^{6} + \frac {1}{5} \, a e x^{5} + \frac {1}{4} \, a d x^{4} + \frac {1}{3} \, a c x^{3} \]

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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mupad [B]  time = 0.03, size = 43, normalized size = 0.78 \[ \frac {b\,e\,x^8}{8}+\frac {b\,d\,x^7}{7}+\frac {b\,c\,x^6}{6}+\frac {a\,e\,x^5}{5}+\frac {a\,d\,x^4}{4}+\frac {a\,c\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.08, size = 49, normalized size = 0.89 \[ \frac {a c x^{3}}{3} + \frac {a d x^{4}}{4} + \frac {a e x^{5}}{5} + \frac {b c x^{6}}{6} + \frac {b d x^{7}}{7} + \frac {b e x^{8}}{8} \]

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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