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

Optimal. Leaf size=77 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{e \left (a+b x^3\right )^3}{9 b}+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8 \]

[Out]

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

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Rubi [A]  time = 0.0610721, antiderivative size = 77, 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} \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{e \left (a+b x^3\right )^3}{9 b}+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0027036, size = 92, normalized size = 1.19 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{3} a b e x^6+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8+\frac{1}{9} b^2 e x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 77, normalized size = 1. \begin{align*}{\frac{{b}^{2}e{x}^{9}}{9}}+{\frac{{b}^{2}d{x}^{8}}{8}}+{\frac{{b}^{2}c{x}^{7}}{7}}+{\frac{abe{x}^{6}}{3}}+{\frac{2\,abd{x}^{5}}{5}}+{\frac{abc{x}^{4}}{2}}+{\frac{{a}^{2}e{x}^{3}}{3}}+{\frac{{a}^{2}d{x}^{2}}{2}}+{a}^{2}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)^2,x)

[Out]

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

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Maxima [A]  time = 0.939354, size = 103, normalized size = 1.34 \begin{align*} \frac{1}{9} \, b^{2} e x^{9} + \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{7} \, b^{2} c x^{7} + \frac{1}{3} \, a b e x^{6} + \frac{2}{5} \, a b d x^{5} + \frac{1}{2} \, a b c x^{4} + \frac{1}{3} \, a^{2} e x^{3} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.36238, size = 185, normalized size = 2.4 \begin{align*} \frac{1}{9} x^{9} e b^{2} + \frac{1}{8} x^{8} d b^{2} + \frac{1}{7} x^{7} c b^{2} + \frac{1}{3} x^{6} e b a + \frac{2}{5} x^{5} d b a + \frac{1}{2} x^{4} c b a + \frac{1}{3} x^{3} e a^{2} + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.097755, size = 88, normalized size = 1.14 \begin{align*} a^{2} c x + \frac{a^{2} d x^{2}}{2} + \frac{a^{2} e x^{3}}{3} + \frac{a b c x^{4}}{2} + \frac{2 a b d x^{5}}{5} + \frac{a b e x^{6}}{3} + \frac{b^{2} c x^{7}}{7} + \frac{b^{2} d x^{8}}{8} + \frac{b^{2} e x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06257, size = 107, normalized size = 1.39 \begin{align*} \frac{1}{9} \, b^{2} x^{9} e + \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{7} \, b^{2} c x^{7} + \frac{1}{3} \, a b x^{6} e + \frac{2}{5} \, a b d x^{5} + \frac{1}{2} \, a b c x^{4} + \frac{1}{3} \, a^{2} x^{3} e + \frac{1}{2} \, a^{2} d x^{2} + a^{2} 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)^2,x, algorithm="giac")

[Out]

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