3.72 \(\int x (a+b x^2)^2 (A+B x+C x^2+D x^3) \, dx\)

Optimal. Leaf size=104 \[ \frac{1}{3} a^2 B x^3+\frac{1}{4} a^2 C x^4+\frac{A \left (a+b x^2\right )^3}{6 b}+\frac{1}{7} b x^7 (2 a D+b B)+\frac{1}{5} a x^5 (a D+2 b B)+\frac{1}{3} a b C x^6+\frac{1}{8} b^2 C x^8+\frac{1}{9} b^2 D x^9 \]

[Out]

(a^2*B*x^3)/3 + (a^2*C*x^4)/4 + (a*(2*b*B + a*D)*x^5)/5 + (a*b*C*x^6)/3 + (b*(b*B + 2*a*D)*x^7)/7 + (b^2*C*x^8
)/8 + (b^2*D*x^9)/9 + (A*(a + b*x^2)^3)/(6*b)

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Rubi [A]  time = 0.0738811, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1582, 1810} \[ \frac{1}{3} a^2 B x^3+\frac{1}{4} a^2 C x^4+\frac{A \left (a+b x^2\right )^3}{6 b}+\frac{1}{7} b x^7 (2 a D+b B)+\frac{1}{5} a x^5 (a D+2 b B)+\frac{1}{3} a b C x^6+\frac{1}{8} b^2 C x^8+\frac{1}{9} b^2 D x^9 \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]

[Out]

(a^2*B*x^3)/3 + (a^2*C*x^4)/4 + (a*(2*b*B + a*D)*x^5)/5 + (a*b*C*x^6)/3 + (b*(b*B + 2*a*D)*x^7)/7 + (b^2*C*x^8
)/8 + (b^2*D*x^9)/9 + (A*(a + b*x^2)^3)/(6*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 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int x \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx &=\frac{A \left (a+b x^2\right )^3}{6 b}+\int \left (a+b x^2\right )^2 \left (-A x+x \left (A+B x+C x^2+D x^3\right )\right ) \, dx\\ &=\frac{A \left (a+b x^2\right )^3}{6 b}+\int \left (a^2 B x^2+a^2 C x^3+a (2 b B+a D) x^4+2 a b C x^5+b (b B+2 a D) x^6+b^2 C x^7+b^2 D x^8\right ) \, dx\\ &=\frac{1}{3} a^2 B x^3+\frac{1}{4} a^2 C x^4+\frac{1}{5} a (2 b B+a D) x^5+\frac{1}{3} a b C x^6+\frac{1}{7} b (b B+2 a D) x^7+\frac{1}{8} b^2 C x^8+\frac{1}{9} b^2 D x^9+\frac{A \left (a+b x^2\right )^3}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.0383493, size = 92, normalized size = 0.88 \[ \frac{42 a^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))+12 a b x^4 (105 A+2 x (42 B+5 x (7 C+6 D x)))+5 b^2 x^6 (84 A+x (72 B+7 x (9 C+8 D x)))}{2520} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]

[Out]

(42*a^2*x^2*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))) + 12*a*b*x^4*(105*A + 2*x*(42*B + 5*x*(7*C + 6*D*x))) + 5*b^
2*x^6*(84*A + x*(72*B + 7*x*(9*C + 8*D*x))))/2520

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Maple [A]  time = 0.001, size = 102, normalized size = 1. \begin{align*}{\frac{{b}^{2}D{x}^{9}}{9}}+{\frac{{b}^{2}C{x}^{8}}{8}}+{\frac{ \left ({b}^{2}B+2\,abD \right ){x}^{7}}{7}}+{\frac{ \left ( A{b}^{2}+2\,abC \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,Bba+{a}^{2}D \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Aab+{a}^{2}C \right ){x}^{4}}{4}}+{\frac{{a}^{2}B{x}^{3}}{3}}+{\frac{{a}^{2}A{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x)

[Out]

1/9*b^2*D*x^9+1/8*b^2*C*x^8+1/7*(B*b^2+2*D*a*b)*x^7+1/6*(A*b^2+2*C*a*b)*x^6+1/5*(2*B*a*b+D*a^2)*x^5+1/4*(2*A*a
*b+C*a^2)*x^4+1/3*a^2*B*x^3+1/2*a^2*A*x^2

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Maxima [A]  time = 0.990046, size = 136, normalized size = 1.31 \begin{align*} \frac{1}{9} \, D b^{2} x^{9} + \frac{1}{8} \, C b^{2} x^{8} + \frac{1}{7} \,{\left (2 \, D a b + B b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (2 \, C a b + A b^{2}\right )} x^{6} + \frac{1}{3} \, B a^{2} x^{3} + \frac{1}{5} \,{\left (D a^{2} + 2 \, B a b\right )} x^{5} + \frac{1}{2} \, A a^{2} x^{2} + \frac{1}{4} \,{\left (C a^{2} + 2 \, A a b\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")

[Out]

1/9*D*b^2*x^9 + 1/8*C*b^2*x^8 + 1/7*(2*D*a*b + B*b^2)*x^7 + 1/6*(2*C*a*b + A*b^2)*x^6 + 1/3*B*a^2*x^3 + 1/5*(D
*a^2 + 2*B*a*b)*x^5 + 1/2*A*a^2*x^2 + 1/4*(C*a^2 + 2*A*a*b)*x^4

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Fricas [A]  time = 1.28963, size = 258, normalized size = 2.48 \begin{align*} \frac{1}{9} x^{9} b^{2} D + \frac{1}{8} x^{8} b^{2} C + \frac{2}{7} x^{7} b a D + \frac{1}{7} x^{7} b^{2} B + \frac{1}{3} x^{6} b a C + \frac{1}{6} x^{6} b^{2} A + \frac{1}{5} x^{5} a^{2} D + \frac{2}{5} x^{5} b a B + \frac{1}{4} x^{4} a^{2} C + \frac{1}{2} x^{4} b a A + \frac{1}{3} x^{3} a^{2} B + \frac{1}{2} x^{2} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")

[Out]

1/9*x^9*b^2*D + 1/8*x^8*b^2*C + 2/7*x^7*b*a*D + 1/7*x^7*b^2*B + 1/3*x^6*b*a*C + 1/6*x^6*b^2*A + 1/5*x^5*a^2*D
+ 2/5*x^5*b*a*B + 1/4*x^4*a^2*C + 1/2*x^4*b*a*A + 1/3*x^3*a^2*B + 1/2*x^2*a^2*A

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Sympy [A]  time = 0.078765, size = 110, normalized size = 1.06 \begin{align*} \frac{A a^{2} x^{2}}{2} + \frac{B a^{2} x^{3}}{3} + \frac{C b^{2} x^{8}}{8} + \frac{D b^{2} x^{9}}{9} + x^{7} \left (\frac{B b^{2}}{7} + \frac{2 D a b}{7}\right ) + x^{6} \left (\frac{A b^{2}}{6} + \frac{C a b}{3}\right ) + x^{5} \left (\frac{2 B a b}{5} + \frac{D a^{2}}{5}\right ) + x^{4} \left (\frac{A a b}{2} + \frac{C a^{2}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a)**2*(D*x**3+C*x**2+B*x+A),x)

[Out]

A*a**2*x**2/2 + B*a**2*x**3/3 + C*b**2*x**8/8 + D*b**2*x**9/9 + x**7*(B*b**2/7 + 2*D*a*b/7) + x**6*(A*b**2/6 +
C*a*b/3) + x**5*(2*B*a*b/5 + D*a**2/5) + x**4*(A*a*b/2 + C*a**2/4)

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Giac [A]  time = 1.16452, size = 142, normalized size = 1.37 \begin{align*} \frac{1}{9} \, D b^{2} x^{9} + \frac{1}{8} \, C b^{2} x^{8} + \frac{2}{7} \, D a b x^{7} + \frac{1}{7} \, B b^{2} x^{7} + \frac{1}{3} \, C a b x^{6} + \frac{1}{6} \, A b^{2} x^{6} + \frac{1}{5} \, D a^{2} x^{5} + \frac{2}{5} \, B a b x^{5} + \frac{1}{4} \, C a^{2} x^{4} + \frac{1}{2} \, A a b x^{4} + \frac{1}{3} \, B a^{2} x^{3} + \frac{1}{2} \, A a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")

[Out]

1/9*D*b^2*x^9 + 1/8*C*b^2*x^8 + 2/7*D*a*b*x^7 + 1/7*B*b^2*x^7 + 1/3*C*a*b*x^6 + 1/6*A*b^2*x^6 + 1/5*D*a^2*x^5
+ 2/5*B*a*b*x^5 + 1/4*C*a^2*x^4 + 1/2*A*a*b*x^4 + 1/3*B*a^2*x^3 + 1/2*A*a^2*x^2