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

Optimal. Leaf size=109 \[ \frac{1}{4} a^2 A x^4+\frac{1}{5} a^2 B x^5+\frac{1}{8} b x^8 (2 a C+A b)+\frac{1}{6} a x^6 (a C+2 A b)+\frac{1}{9} b x^9 (2 a D+b B)+\frac{1}{7} a x^7 (a D+2 b B)+\frac{1}{10} b^2 C x^{10}+\frac{1}{11} b^2 D x^{11} \]

[Out]

(a^2*A*x^4)/4 + (a^2*B*x^5)/5 + (a*(2*A*b + a*C)*x^6)/6 + (a*(2*b*B + a*D)*x^7)/7 + (b*(A*b + 2*a*C)*x^8)/8 +
(b*(b*B + 2*a*D)*x^9)/9 + (b^2*C*x^10)/10 + (b^2*D*x^11)/11

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Rubi [A]  time = 0.124172, 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} \[ \frac{1}{4} a^2 A x^4+\frac{1}{5} a^2 B x^5+\frac{1}{8} b x^8 (2 a C+A b)+\frac{1}{6} a x^6 (a C+2 A b)+\frac{1}{9} b x^9 (2 a D+b B)+\frac{1}{7} a x^7 (a D+2 b B)+\frac{1}{10} b^2 C x^{10}+\frac{1}{11} b^2 D x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*x^4)/4 + (a^2*B*x^5)/5 + (a*(2*A*b + a*C)*x^6)/6 + (a*(2*b*B + a*D)*x^7)/7 + (b*(A*b + 2*a*C)*x^8)/8 +
(b*(b*B + 2*a*D)*x^9)/9 + (b^2*C*x^10)/10 + (b^2*D*x^11)/11

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^3 \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^3+a^2 B x^4+a (2 A b+a C) x^5+a (2 b B+a D) x^6+b (A b+2 a C) x^7+b (b B+2 a D) x^8+b^2 C x^9+b^2 D x^{10}\right ) \, dx\\ &=\frac{1}{4} a^2 A x^4+\frac{1}{5} a^2 B x^5+\frac{1}{6} a (2 A b+a C) x^6+\frac{1}{7} a (2 b B+a D) x^7+\frac{1}{8} b (A b+2 a C) x^8+\frac{1}{9} b (b B+2 a D) x^9+\frac{1}{10} b^2 C x^{10}+\frac{1}{11} b^2 D x^{11}\\ \end{align*}

Mathematica [A]  time = 0.0370082, size = 98, normalized size = 0.9 \[ a^2 \left (\frac{A x^4}{4}+\frac{B x^5}{5}+\frac{1}{42} x^6 (7 C+6 D x)\right )+\frac{1}{252} a b x^6 (84 A+x (72 B+7 x (9 C+8 D x)))+\frac{b^2 x^8 \left (495 A+4 x \left (110 B+99 C x+90 D x^2\right )\right )}{3960} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*((A*x^4)/4 + (B*x^5)/5 + (x^6*(7*C + 6*D*x))/42) + (b^2*x^8*(495*A + 4*x*(110*B + 99*C*x + 90*D*x^2)))/396
0 + (a*b*x^6*(84*A + x*(72*B + 7*x*(9*C + 8*D*x))))/252

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Maple [A]  time = 0.001, size = 102, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}D{x}^{11}}{11}}+{\frac{{b}^{2}C{x}^{10}}{10}}+{\frac{ \left ({b}^{2}B+2\,abD \right ){x}^{9}}{9}}+{\frac{ \left ( A{b}^{2}+2\,abC \right ){x}^{8}}{8}}+{\frac{ \left ( 2\,Bba+{a}^{2}D \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,Aab+{a}^{2}C \right ){x}^{6}}{6}}+{\frac{{a}^{2}B{x}^{5}}{5}}+{\frac{{a}^{2}A{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^2*D*x^11+1/10*b^2*C*x^10+1/9*(B*b^2+2*D*a*b)*x^9+1/8*(A*b^2+2*C*a*b)*x^8+1/7*(2*B*a*b+D*a^2)*x^7+1/6*(2
*A*a*b+C*a^2)*x^6+1/5*a^2*B*x^5+1/4*a^2*A*x^4

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Maxima [A]  time = 1.1289, size = 136, normalized size = 1.25 \begin{align*} \frac{1}{11} \, D b^{2} x^{11} + \frac{1}{10} \, C b^{2} x^{10} + \frac{1}{9} \,{\left (2 \, D a b + B b^{2}\right )} x^{9} + \frac{1}{8} \,{\left (2 \, C a b + A b^{2}\right )} x^{8} + \frac{1}{5} \, B a^{2} x^{5} + \frac{1}{7} \,{\left (D a^{2} + 2 \, B a b\right )} x^{7} + \frac{1}{4} \, A a^{2} x^{4} + \frac{1}{6} \,{\left (C a^{2} + 2 \, A a b\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*D*b^2*x^11 + 1/10*C*b^2*x^10 + 1/9*(2*D*a*b + B*b^2)*x^9 + 1/8*(2*C*a*b + A*b^2)*x^8 + 1/5*B*a^2*x^5 + 1/
7*(D*a^2 + 2*B*a*b)*x^7 + 1/4*A*a^2*x^4 + 1/6*(C*a^2 + 2*A*a*b)*x^6

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Fricas [A]  time = 1.22682, size = 263, normalized size = 2.41 \begin{align*} \frac{1}{11} x^{11} b^{2} D + \frac{1}{10} x^{10} b^{2} C + \frac{2}{9} x^{9} b a D + \frac{1}{9} x^{9} b^{2} B + \frac{1}{4} x^{8} b a C + \frac{1}{8} x^{8} b^{2} A + \frac{1}{7} x^{7} a^{2} D + \frac{2}{7} x^{7} b a B + \frac{1}{6} x^{6} a^{2} C + \frac{1}{3} x^{6} b a A + \frac{1}{5} x^{5} a^{2} B + \frac{1}{4} x^{4} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*b^2*D + 1/10*x^10*b^2*C + 2/9*x^9*b*a*D + 1/9*x^9*b^2*B + 1/4*x^8*b*a*C + 1/8*x^8*b^2*A + 1/7*x^7*a^
2*D + 2/7*x^7*b*a*B + 1/6*x^6*a^2*C + 1/3*x^6*b*a*A + 1/5*x^5*a^2*B + 1/4*x^4*a^2*A

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Sympy [A]  time = 0.076776, size = 110, normalized size = 1.01 \begin{align*} \frac{A a^{2} x^{4}}{4} + \frac{B a^{2} x^{5}}{5} + \frac{C b^{2} x^{10}}{10} + \frac{D b^{2} x^{11}}{11} + x^{9} \left (\frac{B b^{2}}{9} + \frac{2 D a b}{9}\right ) + x^{8} \left (\frac{A b^{2}}{8} + \frac{C a b}{4}\right ) + x^{7} \left (\frac{2 B a b}{7} + \frac{D a^{2}}{7}\right ) + x^{6} \left (\frac{A a b}{3} + \frac{C a^{2}}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*x**4/4 + B*a**2*x**5/5 + C*b**2*x**10/10 + D*b**2*x**11/11 + x**9*(B*b**2/9 + 2*D*a*b/9) + x**8*(A*b**2
/8 + C*a*b/4) + x**7*(2*B*a*b/7 + D*a**2/7) + x**6*(A*a*b/3 + C*a**2/6)

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Giac [A]  time = 1.12518, size = 142, normalized size = 1.3 \begin{align*} \frac{1}{11} \, D b^{2} x^{11} + \frac{1}{10} \, C b^{2} x^{10} + \frac{2}{9} \, D a b x^{9} + \frac{1}{9} \, B b^{2} x^{9} + \frac{1}{4} \, C a b x^{8} + \frac{1}{8} \, A b^{2} x^{8} + \frac{1}{7} \, D a^{2} x^{7} + \frac{2}{7} \, B a b x^{7} + \frac{1}{6} \, C a^{2} x^{6} + \frac{1}{3} \, A a b x^{6} + \frac{1}{5} \, B a^{2} x^{5} + \frac{1}{4} \, A a^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*D*b^2*x^11 + 1/10*C*b^2*x^10 + 2/9*D*a*b*x^9 + 1/9*B*b^2*x^9 + 1/4*C*a*b*x^8 + 1/8*A*b^2*x^8 + 1/7*D*a^2*
x^7 + 2/7*B*a*b*x^7 + 1/6*C*a^2*x^6 + 1/3*A*a*b*x^6 + 1/5*B*a^2*x^5 + 1/4*A*a^2*x^4