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

Optimal. Leaf size=110 \[ \frac{3}{7} a^2 b d x^7+\frac{3}{8} a^2 b e x^8+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{c \left (a+b x^3\right )^4}{12 b}+\frac{1}{13} b^3 d x^{13}+\frac{1}{14} b^3 e x^{14} \]

[Out]

(a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (3*a^2*b*d*x^7)/7 + (3*a^2*b*e*x^8)/8 + (3*a*b^2*d*x^10)/10 + (3*a*b^2*e*x^11)
/11 + (b^3*d*x^13)/13 + (b^3*e*x^14)/14 + (c*(a + b*x^3)^4)/(12*b)

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Rubi [A]  time = 0.0787603, antiderivative size = 110, 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} \[ \frac{3}{7} a^2 b d x^7+\frac{3}{8} a^2 b e x^8+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{c \left (a+b x^3\right )^4}{12 b}+\frac{1}{13} b^3 d x^{13}+\frac{1}{14} b^3 e x^{14} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (3*a^2*b*d*x^7)/7 + (3*a^2*b*e*x^8)/8 + (3*a*b^2*d*x^10)/10 + (3*a*b^2*e*x^11)
/11 + (b^3*d*x^13)/13 + (b^3*e*x^14)/14 + (c*(a + b*x^3)^4)/(12*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 )^3 \, dx &=\frac{c \left (a+b x^3\right )^4}{12 b}+\int \left (a+b x^3\right )^3 \left (-c x^2+x^2 \left (c+d x+e x^2\right )\right ) \, dx\\ &=\frac{c \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 d x^3+a^3 e x^4+3 a^2 b d x^6+3 a^2 b e x^7+3 a b^2 d x^9+3 a b^2 e x^{10}+b^3 d x^{12}+b^3 e x^{13}\right ) \, dx\\ &=\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{3}{7} a^2 b d x^7+\frac{3}{8} a^2 b e x^8+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{13} b^3 d x^{13}+\frac{1}{14} b^3 e x^{14}+\frac{c \left (a+b x^3\right )^4}{12 b}\\ \end{align*}

Mathematica [A]  time = 0.0038531, size = 139, normalized size = 1.26 \[ \frac{1}{2} a^2 b c x^6+\frac{3}{7} a^2 b d x^7+\frac{3}{8} a^2 b e x^8+\frac{1}{3} a^3 c x^3+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{12} b^3 c x^{12}+\frac{1}{13} b^3 d x^{13}+\frac{1}{14} b^3 e x^{14} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*c*x^3)/3 + (a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (a^2*b*c*x^6)/2 + (3*a^2*b*d*x^7)/7 + (3*a^2*b*e*x^8)/8 + (a*b
^2*c*x^9)/3 + (3*a*b^2*d*x^10)/10 + (3*a*b^2*e*x^11)/11 + (b^3*c*x^12)/12 + (b^3*d*x^13)/13 + (b^3*e*x^14)/14

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Maple [A]  time = 0.001, size = 116, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}e{x}^{14}}{14}}+{\frac{{b}^{3}d{x}^{13}}{13}}+{\frac{{b}^{3}c{x}^{12}}{12}}+{\frac{3\,a{b}^{2}e{x}^{11}}{11}}+{\frac{3\,a{b}^{2}d{x}^{10}}{10}}+{\frac{a{b}^{2}c{x}^{9}}{3}}+{\frac{3\,{a}^{2}be{x}^{8}}{8}}+{\frac{3\,{a}^{2}bd{x}^{7}}{7}}+{\frac{{a}^{2}bc{x}^{6}}{2}}+{\frac{{a}^{3}e{x}^{5}}{5}}+{\frac{{a}^{3}d{x}^{4}}{4}}+{\frac{{a}^{3}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)^3,x)

[Out]

1/14*b^3*e*x^14+1/13*b^3*d*x^13+1/12*b^3*c*x^12+3/11*a*b^2*e*x^11+3/10*a*b^2*d*x^10+1/3*a*b^2*c*x^9+3/8*a^2*b*
e*x^8+3/7*a^2*b*d*x^7+1/2*a^2*b*c*x^6+1/5*a^3*e*x^5+1/4*a^3*d*x^4+1/3*a^3*c*x^3

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Maxima [A]  time = 0.941417, size = 155, normalized size = 1.41 \begin{align*} \frac{1}{14} \, b^{3} e x^{14} + \frac{1}{13} \, b^{3} d x^{13} + \frac{1}{12} \, b^{3} c x^{12} + \frac{3}{11} \, a b^{2} e x^{11} + \frac{3}{10} \, a b^{2} d x^{10} + \frac{1}{3} \, a b^{2} c x^{9} + \frac{3}{8} \, a^{2} b e x^{8} + \frac{3}{7} \, a^{2} b d x^{7} + \frac{1}{2} \, a^{2} b c x^{6} + \frac{1}{5} \, a^{3} e x^{5} + \frac{1}{4} \, a^{3} d x^{4} + \frac{1}{3} \, a^{3} 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)^3,x, algorithm="maxima")

[Out]

1/14*b^3*e*x^14 + 1/13*b^3*d*x^13 + 1/12*b^3*c*x^12 + 3/11*a*b^2*e*x^11 + 3/10*a*b^2*d*x^10 + 1/3*a*b^2*c*x^9
+ 3/8*a^2*b*e*x^8 + 3/7*a^2*b*d*x^7 + 1/2*a^2*b*c*x^6 + 1/5*a^3*e*x^5 + 1/4*a^3*d*x^4 + 1/3*a^3*c*x^3

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Fricas [A]  time = 1.30698, size = 288, normalized size = 2.62 \begin{align*} \frac{1}{14} x^{14} e b^{3} + \frac{1}{13} x^{13} d b^{3} + \frac{1}{12} x^{12} c b^{3} + \frac{3}{11} x^{11} e b^{2} a + \frac{3}{10} x^{10} d b^{2} a + \frac{1}{3} x^{9} c b^{2} a + \frac{3}{8} x^{8} e b a^{2} + \frac{3}{7} x^{7} d b a^{2} + \frac{1}{2} x^{6} c b a^{2} + \frac{1}{5} x^{5} e a^{3} + \frac{1}{4} x^{4} d a^{3} + \frac{1}{3} x^{3} c a^{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)^3,x, algorithm="fricas")

[Out]

1/14*x^14*e*b^3 + 1/13*x^13*d*b^3 + 1/12*x^12*c*b^3 + 3/11*x^11*e*b^2*a + 3/10*x^10*d*b^2*a + 1/3*x^9*c*b^2*a
+ 3/8*x^8*e*b*a^2 + 3/7*x^7*d*b*a^2 + 1/2*x^6*c*b*a^2 + 1/5*x^5*e*a^3 + 1/4*x^4*d*a^3 + 1/3*x^3*c*a^3

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Sympy [A]  time = 0.075398, size = 138, normalized size = 1.25 \begin{align*} \frac{a^{3} c x^{3}}{3} + \frac{a^{3} d x^{4}}{4} + \frac{a^{3} e x^{5}}{5} + \frac{a^{2} b c x^{6}}{2} + \frac{3 a^{2} b d x^{7}}{7} + \frac{3 a^{2} b e x^{8}}{8} + \frac{a b^{2} c x^{9}}{3} + \frac{3 a b^{2} d x^{10}}{10} + \frac{3 a b^{2} e x^{11}}{11} + \frac{b^{3} c x^{12}}{12} + \frac{b^{3} d x^{13}}{13} + \frac{b^{3} e x^{14}}{14} \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)**3,x)

[Out]

a**3*c*x**3/3 + a**3*d*x**4/4 + a**3*e*x**5/5 + a**2*b*c*x**6/2 + 3*a**2*b*d*x**7/7 + 3*a**2*b*e*x**8/8 + a*b*
*2*c*x**9/3 + 3*a*b**2*d*x**10/10 + 3*a*b**2*e*x**11/11 + b**3*c*x**12/12 + b**3*d*x**13/13 + b**3*e*x**14/14

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Giac [A]  time = 1.05876, size = 161, normalized size = 1.46 \begin{align*} \frac{1}{14} \, b^{3} x^{14} e + \frac{1}{13} \, b^{3} d x^{13} + \frac{1}{12} \, b^{3} c x^{12} + \frac{3}{11} \, a b^{2} x^{11} e + \frac{3}{10} \, a b^{2} d x^{10} + \frac{1}{3} \, a b^{2} c x^{9} + \frac{3}{8} \, a^{2} b x^{8} e + \frac{3}{7} \, a^{2} b d x^{7} + \frac{1}{2} \, a^{2} b c x^{6} + \frac{1}{5} \, a^{3} x^{5} e + \frac{1}{4} \, a^{3} d x^{4} + \frac{1}{3} \, a^{3} 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)^3,x, algorithm="giac")

[Out]

1/14*b^3*x^14*e + 1/13*b^3*d*x^13 + 1/12*b^3*c*x^12 + 3/11*a*b^2*x^11*e + 3/10*a*b^2*d*x^10 + 1/3*a*b^2*c*x^9
+ 3/8*a^2*b*x^8*e + 3/7*a^2*b*d*x^7 + 1/2*a^2*b*c*x^6 + 1/5*a^3*x^5*e + 1/4*a^3*d*x^4 + 1/3*a^3*c*x^3