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

Optimal. Leaf size=82 \[ \frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{c \left (a+b x^3\right )^3}{9 b}+\frac{2}{7} a b d x^7+\frac{1}{4} a b e x^8+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]

[Out]

(a^2*d*x^4)/4 + (a^2*e*x^5)/5 + (2*a*b*d*x^7)/7 + (a*b*e*x^8)/4 + (b^2*d*x^10)/10 + (b^2*e*x^11)/11 + (c*(a +
b*x^3)^3)/(9*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0589412, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1582, 1850} \[ \frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{c \left (a+b x^3\right )^3}{9 b}+\frac{2}{7} a b d x^7+\frac{1}{4} a b e x^8+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*d*x^4)/4 + (a^2*e*x^5)/5 + (2*a*b*d*x^7)/7 + (a*b*e*x^8)/4 + (b^2*d*x^10)/10 + (b^2*e*x^11)/11 + (c*(a +
b*x^3)^3)/(9*b)

Rule 1582

Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(Coeff[Px, x, n - 1]*(a + b*x^n)^(p + 1))/(b*n*(p +
1)), x] + Int[(Px - Coeff[Px, x, n - 1]*x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && I
GtQ[p, 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] &&  !MatchQ[P
x, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ[{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Co
eff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx &=\frac{c \left (a+b x^3\right )^3}{9 b}+\int \left (a+b x^3\right )^2 \left (-c x^2+x^2 \left (c+d x+e x^2\right )\right ) \, dx\\ &=\frac{c \left (a+b x^3\right )^3}{9 b}+\int \left (a^2 d x^3+a^2 e x^4+2 a b d x^6+2 a b e x^7+b^2 d x^9+b^2 e x^{10}\right ) \, dx\\ &=\frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{2}{7} a b d x^7+\frac{1}{4} a b e x^8+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}+\frac{c \left (a+b x^3\right )^3}{9 b}\\ \end{align*}

Mathematica [A]  time = 0.0035515, size = 97, normalized size = 1.18 \[ \frac{1}{3} a^2 c x^3+\frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{1}{3} a b c x^6+\frac{2}{7} a b d x^7+\frac{1}{4} a b e x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*c*x^3)/3 + (a^2*d*x^4)/4 + (a^2*e*x^5)/5 + (a*b*c*x^6)/3 + (2*a*b*d*x^7)/7 + (a*b*e*x^8)/4 + (b^2*c*x^9)/
9 + (b^2*d*x^10)/10 + (b^2*e*x^11)/11

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 80, normalized size = 1. \begin{align*}{\frac{{b}^{2}e{x}^{11}}{11}}+{\frac{{b}^{2}d{x}^{10}}{10}}+{\frac{{b}^{2}c{x}^{9}}{9}}+{\frac{abe{x}^{8}}{4}}+{\frac{2\,abd{x}^{7}}{7}}+{\frac{abc{x}^{6}}{3}}+{\frac{{a}^{2}e{x}^{5}}{5}}+{\frac{{a}^{2}d{x}^{4}}{4}}+{\frac{{a}^{2}c{x}^{3}}{3}} \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)^2,x)

[Out]

1/11*b^2*e*x^11+1/10*b^2*d*x^10+1/9*b^2*c*x^9+1/4*a*b*e*x^8+2/7*a*b*d*x^7+1/3*a*b*c*x^6+1/5*a^2*e*x^5+1/4*a^2*
d*x^4+1/3*a^2*c*x^3

________________________________________________________________________________________

Maxima [A]  time = 0.938983, size = 107, normalized size = 1.3 \begin{align*} \frac{1}{11} \, b^{2} e x^{11} + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b e x^{8} + \frac{2}{7} \, a b d x^{7} + \frac{1}{3} \, a b c x^{6} + \frac{1}{5} \, a^{2} e x^{5} + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{3} \, a^{2} 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)^2,x, algorithm="maxima")

[Out]

1/11*b^2*e*x^11 + 1/10*b^2*d*x^10 + 1/9*b^2*c*x^9 + 1/4*a*b*e*x^8 + 2/7*a*b*d*x^7 + 1/3*a*b*c*x^6 + 1/5*a^2*e*
x^5 + 1/4*a^2*d*x^4 + 1/3*a^2*c*x^3

________________________________________________________________________________________

Fricas [A]  time = 1.22065, size = 198, normalized size = 2.41 \begin{align*} \frac{1}{11} x^{11} e b^{2} + \frac{1}{10} x^{10} d b^{2} + \frac{1}{9} x^{9} c b^{2} + \frac{1}{4} x^{8} e b a + \frac{2}{7} x^{7} d b a + \frac{1}{3} x^{6} c b a + \frac{1}{5} x^{5} e a^{2} + \frac{1}{4} x^{4} d a^{2} + \frac{1}{3} x^{3} c a^{2} \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)^2,x, algorithm="fricas")

[Out]

1/11*x^11*e*b^2 + 1/10*x^10*d*b^2 + 1/9*x^9*c*b^2 + 1/4*x^8*e*b*a + 2/7*x^7*d*b*a + 1/3*x^6*c*b*a + 1/5*x^5*e*
a^2 + 1/4*x^4*d*a^2 + 1/3*x^3*c*a^2

________________________________________________________________________________________

Sympy [A]  time = 0.07362, size = 92, normalized size = 1.12 \begin{align*} \frac{a^{2} c x^{3}}{3} + \frac{a^{2} d x^{4}}{4} + \frac{a^{2} e x^{5}}{5} + \frac{a b c x^{6}}{3} + \frac{2 a b d x^{7}}{7} + \frac{a b e x^{8}}{4} + \frac{b^{2} c x^{9}}{9} + \frac{b^{2} d x^{10}}{10} + \frac{b^{2} e x^{11}}{11} \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)**2,x)

[Out]

a**2*c*x**3/3 + a**2*d*x**4/4 + a**2*e*x**5/5 + a*b*c*x**6/3 + 2*a*b*d*x**7/7 + a*b*e*x**8/4 + b**2*c*x**9/9 +
b**2*d*x**10/10 + b**2*e*x**11/11

________________________________________________________________________________________

Giac [A]  time = 1.08003, size = 111, normalized size = 1.35 \begin{align*} \frac{1}{11} \, b^{2} x^{11} e + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b x^{8} e + \frac{2}{7} \, a b d x^{7} + \frac{1}{3} \, a b c x^{6} + \frac{1}{5} \, a^{2} x^{5} e + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{3} \, a^{2} 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)^2,x, algorithm="giac")

[Out]

1/11*b^2*x^11*e + 1/10*b^2*d*x^10 + 1/9*b^2*c*x^9 + 1/4*a*b*x^8*e + 2/7*a*b*d*x^7 + 1/3*a*b*c*x^6 + 1/5*a^2*x^
5*e + 1/4*a^2*d*x^4 + 1/3*a^2*c*x^3