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

Optimal. Leaf size=109 \[ \frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{7} b x^7 (2 a C+A b)+\frac{1}{5} a x^5 (a C+2 A b)+\frac{1}{8} b x^8 (2 a D+b B)+\frac{1}{6} a x^6 (a D+2 b B)+\frac{1}{9} b^2 C x^9+\frac{1}{10} b^2 D x^{10} \]

[Out]

(a^2*A*x^3)/3 + (a^2*B*x^4)/4 + (a*(2*A*b + a*C)*x^5)/5 + (a*(2*b*B + a*D)*x^6)/6 + (b*(A*b + 2*a*C)*x^7)/7 +
(b*(b*B + 2*a*D)*x^8)/8 + (b^2*C*x^9)/9 + (b^2*D*x^10)/10

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Rubi [A]  time = 0.111717, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1802} \[ \frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{7} b x^7 (2 a C+A b)+\frac{1}{5} a x^5 (a C+2 A b)+\frac{1}{8} b x^8 (2 a D+b B)+\frac{1}{6} a x^6 (a D+2 b B)+\frac{1}{9} b^2 C x^9+\frac{1}{10} b^2 D x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^3)/3 + (a^2*B*x^4)/4 + (a*(2*A*b + a*C)*x^5)/5 + (a*(2*b*B + a*D)*x^6)/6 + (b*(A*b + 2*a*C)*x^7)/7 +
(b*(b*B + 2*a*D)*x^8)/8 + (b^2*C*x^9)/9 + (b^2*D*x^10)/10

Rule 1802

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

Rubi steps

\begin{align*} \int x^2 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (a^2 A x^2+a^2 B x^3+a (2 A b+a C) x^4+a (2 b B+a D) x^5+b (A b+2 a C) x^6+b (b B+2 a D) x^7+b^2 C x^8+b^2 D x^9\right ) \, dx\\ &=\frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{1}{5} a (2 A b+a C) x^5+\frac{1}{6} a (2 b B+a D) x^6+\frac{1}{7} b (A b+2 a C) x^7+\frac{1}{8} b (b B+2 a D) x^8+\frac{1}{9} b^2 C x^9+\frac{1}{10} b^2 D x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0477629, size = 92, normalized size = 0.84 \[ \frac{42 a^2 x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+6 a b x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))+b^2 x^7 (360 A+7 x (45 B+4 x (10 C+9 D x)))}{2520} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(42*a^2*x^3*(20*A + x*(15*B + 2*x*(6*C + 5*D*x))) + 6*a*b*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x))) + b^2*x
^7*(360*A + 7*x*(45*B + 4*x*(10*C + 9*D*x))))/2520

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Maple [A]  time = 0.001, size = 102, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}D{x}^{10}}{10}}+{\frac{{b}^{2}C{x}^{9}}{9}}+{\frac{ \left ({b}^{2}B+2\,abD \right ){x}^{8}}{8}}+{\frac{ \left ( A{b}^{2}+2\,abC \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,Bba+{a}^{2}D \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,Aab+{a}^{2}C \right ){x}^{5}}{5}}+{\frac{{a}^{2}B{x}^{4}}{4}}+{\frac{{a}^{2}A{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^2*D*x^10+1/9*b^2*C*x^9+1/8*(B*b^2+2*D*a*b)*x^8+1/7*(A*b^2+2*C*a*b)*x^7+1/6*(2*B*a*b+D*a^2)*x^6+1/5*(2*A
*a*b+C*a^2)*x^5+1/4*a^2*B*x^4+1/3*a^2*A*x^3

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Maxima [A]  time = 0.97115, size = 136, normalized size = 1.25 \begin{align*} \frac{1}{10} \, D b^{2} x^{10} + \frac{1}{9} \, C b^{2} x^{9} + \frac{1}{8} \,{\left (2 \, D a b + B b^{2}\right )} x^{8} + \frac{1}{7} \,{\left (2 \, C a b + A b^{2}\right )} x^{7} + \frac{1}{4} \, B a^{2} x^{4} + \frac{1}{6} \,{\left (D a^{2} + 2 \, B a b\right )} x^{6} + \frac{1}{3} \, A a^{2} x^{3} + \frac{1}{5} \,{\left (C a^{2} + 2 \, A a b\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*D*b^2*x^10 + 1/9*C*b^2*x^9 + 1/8*(2*D*a*b + B*b^2)*x^8 + 1/7*(2*C*a*b + A*b^2)*x^7 + 1/4*B*a^2*x^4 + 1/6*
(D*a^2 + 2*B*a*b)*x^6 + 1/3*A*a^2*x^3 + 1/5*(C*a^2 + 2*A*a*b)*x^5

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Fricas [A]  time = 1.25343, size = 261, normalized size = 2.39 \begin{align*} \frac{1}{10} x^{10} b^{2} D + \frac{1}{9} x^{9} b^{2} C + \frac{1}{4} x^{8} b a D + \frac{1}{8} x^{8} b^{2} B + \frac{2}{7} x^{7} b a C + \frac{1}{7} x^{7} b^{2} A + \frac{1}{6} x^{6} a^{2} D + \frac{1}{3} x^{6} b a B + \frac{1}{5} x^{5} a^{2} C + \frac{2}{5} x^{5} b a A + \frac{1}{4} x^{4} a^{2} B + \frac{1}{3} x^{3} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*b^2*D + 1/9*x^9*b^2*C + 1/4*x^8*b*a*D + 1/8*x^8*b^2*B + 2/7*x^7*b*a*C + 1/7*x^7*b^2*A + 1/6*x^6*a^2*
D + 1/3*x^6*b*a*B + 1/5*x^5*a^2*C + 2/5*x^5*b*a*A + 1/4*x^4*a^2*B + 1/3*x^3*a^2*A

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Sympy [A]  time = 0.077921, size = 110, normalized size = 1.01 \begin{align*} \frac{A a^{2} x^{3}}{3} + \frac{B a^{2} x^{4}}{4} + \frac{C b^{2} x^{9}}{9} + \frac{D b^{2} x^{10}}{10} + x^{8} \left (\frac{B b^{2}}{8} + \frac{D a b}{4}\right ) + x^{7} \left (\frac{A b^{2}}{7} + \frac{2 C a b}{7}\right ) + x^{6} \left (\frac{B a b}{3} + \frac{D a^{2}}{6}\right ) + x^{5} \left (\frac{2 A a b}{5} + \frac{C a^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*x**3/3 + B*a**2*x**4/4 + C*b**2*x**9/9 + D*b**2*x**10/10 + x**8*(B*b**2/8 + D*a*b/4) + x**7*(A*b**2/7 +
2*C*a*b/7) + x**6*(B*a*b/3 + D*a**2/6) + x**5*(2*A*a*b/5 + C*a**2/5)

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Giac [A]  time = 1.13061, size = 142, normalized size = 1.3 \begin{align*} \frac{1}{10} \, D b^{2} x^{10} + \frac{1}{9} \, C b^{2} x^{9} + \frac{1}{4} \, D a b x^{8} + \frac{1}{8} \, B b^{2} x^{8} + \frac{2}{7} \, C a b x^{7} + \frac{1}{7} \, A b^{2} x^{7} + \frac{1}{6} \, D a^{2} x^{6} + \frac{1}{3} \, B a b x^{6} + \frac{1}{5} \, C a^{2} x^{5} + \frac{2}{5} \, A a b x^{5} + \frac{1}{4} \, B a^{2} x^{4} + \frac{1}{3} \, A a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*D*b^2*x^10 + 1/9*C*b^2*x^9 + 1/4*D*a*b*x^8 + 1/8*B*b^2*x^8 + 2/7*C*a*b*x^7 + 1/7*A*b^2*x^7 + 1/6*D*a^2*x^
6 + 1/3*B*a*b*x^6 + 1/5*C*a^2*x^5 + 2/5*A*a*b*x^5 + 1/4*B*a^2*x^4 + 1/3*A*a^2*x^3