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

Optimal. Leaf size=105 \[ \frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{e \left (a+b x^3\right )^4}{12 b}+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0969183, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1582, 1850} \[ \frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{e \left (a+b x^3\right )^4}{12 b}+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0036741, size = 134, normalized size = 1.28 \[ \frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+\frac{1}{2} a^2 b e x^6+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{3} a b^2 e x^9+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11}+\frac{1}{12} b^3 e x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 113, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}e{x}^{12}}{12}}+{\frac{{b}^{3}d{x}^{11}}{11}}+{\frac{{b}^{3}c{x}^{10}}{10}}+{\frac{a{b}^{2}e{x}^{9}}{3}}+{\frac{3\,a{b}^{2}d{x}^{8}}{8}}+{\frac{3\,a{b}^{2}c{x}^{7}}{7}}+{\frac{{a}^{2}be{x}^{6}}{2}}+{\frac{3\,{a}^{2}bd{x}^{5}}{5}}+{\frac{3\,{a}^{2}bc{x}^{4}}{4}}+{\frac{{a}^{3}e{x}^{3}}{3}}+{\frac{{a}^{3}d{x}^{2}}{2}}+{a}^{3}cx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.93441, size = 151, normalized size = 1.44 \begin{align*} \frac{1}{12} \, b^{3} e x^{12} + \frac{1}{11} \, b^{3} d x^{11} + \frac{1}{10} \, b^{3} c x^{10} + \frac{1}{3} \, a b^{2} e x^{9} + \frac{3}{8} \, a b^{2} d x^{8} + \frac{3}{7} \, a b^{2} c x^{7} + \frac{1}{2} \, a^{2} b e x^{6} + \frac{3}{5} \, a^{2} b d x^{5} + \frac{3}{4} \, a^{2} b c x^{4} + \frac{1}{3} \, a^{3} e x^{3} + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.28377, size = 274, normalized size = 2.61 \begin{align*} \frac{1}{12} x^{12} e b^{3} + \frac{1}{11} x^{11} d b^{3} + \frac{1}{10} x^{10} c b^{3} + \frac{1}{3} x^{9} e b^{2} a + \frac{3}{8} x^{8} d b^{2} a + \frac{3}{7} x^{7} c b^{2} a + \frac{1}{2} x^{6} e b a^{2} + \frac{3}{5} x^{5} d b a^{2} + \frac{3}{4} x^{4} c b a^{2} + \frac{1}{3} x^{3} e a^{3} + \frac{1}{2} x^{2} d a^{3} + x c a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.075057, size = 134, normalized size = 1.28 \begin{align*} a^{3} c x + \frac{a^{3} d x^{2}}{2} + \frac{a^{3} e x^{3}}{3} + \frac{3 a^{2} b c x^{4}}{4} + \frac{3 a^{2} b d x^{5}}{5} + \frac{a^{2} b e x^{6}}{2} + \frac{3 a b^{2} c x^{7}}{7} + \frac{3 a b^{2} d x^{8}}{8} + \frac{a b^{2} e x^{9}}{3} + \frac{b^{3} c x^{10}}{10} + \frac{b^{3} d x^{11}}{11} + \frac{b^{3} e x^{12}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.08264, size = 157, normalized size = 1.5 \begin{align*} \frac{1}{12} \, b^{3} x^{12} e + \frac{1}{11} \, b^{3} d x^{11} + \frac{1}{10} \, b^{3} c x^{10} + \frac{1}{3} \, a b^{2} x^{9} e + \frac{3}{8} \, a b^{2} d x^{8} + \frac{3}{7} \, a b^{2} c x^{7} + \frac{1}{2} \, a^{2} b x^{6} e + \frac{3}{5} \, a^{2} b d x^{5} + \frac{3}{4} \, a^{2} b c x^{4} + \frac{1}{3} \, a^{3} x^{3} e + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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