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

Optimal. Leaf size=149 \[ \frac{1}{6} a^2 x^6 (a C+3 A b)+\frac{1}{4} a^3 A x^4+\frac{1}{7} a^2 x^7 (a D+3 b B)+\frac{1}{5} a^3 B x^5+\frac{1}{10} b^2 x^{10} (3 a C+A b)+\frac{3}{8} a b x^8 (a C+A b)+\frac{1}{11} b^2 x^{11} (3 a D+b B)+\frac{1}{3} a b x^9 (a D+b B)+\frac{1}{12} b^3 C x^{12}+\frac{1}{13} b^3 D x^{13} \]

[Out]

(a^3*A*x^4)/4 + (a^3*B*x^5)/5 + (a^2*(3*A*b + a*C)*x^6)/6 + (a^2*(3*b*B + a*D)*x^7)/7 + (3*a*b*(A*b + a*C)*x^8
)/8 + (a*b*(b*B + a*D)*x^9)/3 + (b^2*(A*b + 3*a*C)*x^10)/10 + (b^2*(b*B + 3*a*D)*x^11)/11 + (b^3*C*x^12)/12 +
(b^3*D*x^13)/13

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Rubi [A]  time = 0.186161, antiderivative size = 149, 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} \[ \frac{1}{6} a^2 x^6 (a C+3 A b)+\frac{1}{4} a^3 A x^4+\frac{1}{7} a^2 x^7 (a D+3 b B)+\frac{1}{5} a^3 B x^5+\frac{1}{10} b^2 x^{10} (3 a C+A b)+\frac{3}{8} a b x^8 (a C+A b)+\frac{1}{11} b^2 x^{11} (3 a D+b B)+\frac{1}{3} a b x^9 (a D+b B)+\frac{1}{12} b^3 C x^{12}+\frac{1}{13} b^3 D x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*A*x^4)/4 + (a^3*B*x^5)/5 + (a^2*(3*A*b + a*C)*x^6)/6 + (a^2*(3*b*B + a*D)*x^7)/7 + (3*a*b*(A*b + a*C)*x^8
)/8 + (a*b*(b*B + a*D)*x^9)/3 + (b^2*(A*b + 3*a*C)*x^10)/10 + (b^2*(b*B + 3*a*D)*x^11)/11 + (b^3*C*x^12)/12 +
(b^3*D*x^13)/13

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 )^3 \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (a^3 A x^3+a^3 B x^4+a^2 (3 A b+a C) x^5+a^2 (3 b B+a D) x^6+3 a b (A b+a C) x^7+3 a b (b B+a D) x^8+b^2 (A b+3 a C) x^9+b^2 (b B+3 a D) x^{10}+b^3 C x^{11}+b^3 D x^{12}\right ) \, dx\\ &=\frac{1}{4} a^3 A x^4+\frac{1}{5} a^3 B x^5+\frac{1}{6} a^2 (3 A b+a C) x^6+\frac{1}{7} a^2 (3 b B+a D) x^7+\frac{3}{8} a b (A b+a C) x^8+\frac{1}{3} a b (b B+a D) x^9+\frac{1}{10} b^2 (A b+3 a C) x^{10}+\frac{1}{11} b^2 (b B+3 a D) x^{11}+\frac{1}{12} b^3 C x^{12}+\frac{1}{13} b^3 D x^{13}\\ \end{align*}

Mathematica [A]  time = 0.0271811, size = 149, normalized size = 1. \[ \frac{1}{6} a^2 x^6 (a C+3 A b)+\frac{1}{4} a^3 A x^4+\frac{1}{7} a^2 x^7 (a D+3 b B)+\frac{1}{5} a^3 B x^5+\frac{1}{10} b^2 x^{10} (3 a C+A b)+\frac{3}{8} a b x^8 (a C+A b)+\frac{1}{11} b^2 x^{11} (3 a D+b B)+\frac{1}{3} a b x^9 (a D+b B)+\frac{1}{12} b^3 C x^{12}+\frac{1}{13} b^3 D x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*A*x^4)/4 + (a^3*B*x^5)/5 + (a^2*(3*A*b + a*C)*x^6)/6 + (a^2*(3*b*B + a*D)*x^7)/7 + (3*a*b*(A*b + a*C)*x^8
)/8 + (a*b*(b*B + a*D)*x^9)/3 + (b^2*(A*b + 3*a*C)*x^10)/10 + (b^2*(b*B + 3*a*D)*x^11)/11 + (b^3*C*x^12)/12 +
(b^3*D*x^13)/13

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Maple [A]  time = 0.003, size = 150, normalized size = 1. \begin{align*}{\frac{{b}^{3}D{x}^{13}}{13}}+{\frac{{b}^{3}C{x}^{12}}{12}}+{\frac{ \left ({b}^{3}B+3\,a{b}^{2}D \right ){x}^{11}}{11}}+{\frac{ \left ( A{b}^{3}+3\,a{b}^{2}C \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,a{b}^{2}B+3\,{a}^{2}bD \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bC \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,{a}^{2}bB+{a}^{3}D \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,A{a}^{2}b+{a}^{3}C \right ){x}^{6}}{6}}+{\frac{{a}^{3}B{x}^{5}}{5}}+{\frac{{a}^{3}A{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b^3*D*x^13+1/12*b^3*C*x^12+1/11*(B*b^3+3*D*a*b^2)*x^11+1/10*(A*b^3+3*C*a*b^2)*x^10+1/9*(3*B*a*b^2+3*D*a^2
*b)*x^9+1/8*(3*A*a*b^2+3*C*a^2*b)*x^8+1/7*(3*B*a^2*b+D*a^3)*x^7+1/6*(3*A*a^2*b+C*a^3)*x^6+1/5*a^3*B*x^5+1/4*a^
3*A*x^4

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Maxima [A]  time = 1.01083, size = 196, normalized size = 1.32 \begin{align*} \frac{1}{13} \, D b^{3} x^{13} + \frac{1}{12} \, C b^{3} x^{12} + \frac{1}{11} \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{11} + \frac{1}{10} \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{10} + \frac{1}{3} \,{\left (D a^{2} b + B a b^{2}\right )} x^{9} + \frac{1}{5} \, B a^{3} x^{5} + \frac{3}{8} \,{\left (C a^{2} b + A a b^{2}\right )} x^{8} + \frac{1}{4} \, A a^{3} x^{4} + \frac{1}{7} \,{\left (D a^{3} + 3 \, B a^{2} b\right )} x^{7} + \frac{1}{6} \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*D*b^3*x^13 + 1/12*C*b^3*x^12 + 1/11*(3*D*a*b^2 + B*b^3)*x^11 + 1/10*(3*C*a*b^2 + A*b^3)*x^10 + 1/3*(D*a^2
*b + B*a*b^2)*x^9 + 1/5*B*a^3*x^5 + 3/8*(C*a^2*b + A*a*b^2)*x^8 + 1/4*A*a^3*x^4 + 1/7*(D*a^3 + 3*B*a^2*b)*x^7
+ 1/6*(C*a^3 + 3*A*a^2*b)*x^6

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Fricas [A]  time = 1.37057, size = 382, normalized size = 2.56 \begin{align*} \frac{1}{13} x^{13} b^{3} D + \frac{1}{12} x^{12} b^{3} C + \frac{3}{11} x^{11} b^{2} a D + \frac{1}{11} x^{11} b^{3} B + \frac{3}{10} x^{10} b^{2} a C + \frac{1}{10} x^{10} b^{3} A + \frac{1}{3} x^{9} b a^{2} D + \frac{1}{3} x^{9} b^{2} a B + \frac{3}{8} x^{8} b a^{2} C + \frac{3}{8} x^{8} b^{2} a A + \frac{1}{7} x^{7} a^{3} D + \frac{3}{7} x^{7} b a^{2} B + \frac{1}{6} x^{6} a^{3} C + \frac{1}{2} x^{6} b a^{2} A + \frac{1}{5} x^{5} a^{3} B + \frac{1}{4} x^{4} a^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*b^3*D + 1/12*x^12*b^3*C + 3/11*x^11*b^2*a*D + 1/11*x^11*b^3*B + 3/10*x^10*b^2*a*C + 1/10*x^10*b^3*A
+ 1/3*x^9*b*a^2*D + 1/3*x^9*b^2*a*B + 3/8*x^8*b*a^2*C + 3/8*x^8*b^2*a*A + 1/7*x^7*a^3*D + 3/7*x^7*b*a^2*B + 1/
6*x^6*a^3*C + 1/2*x^6*b*a^2*A + 1/5*x^5*a^3*B + 1/4*x^4*a^3*A

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Sympy [A]  time = 0.091382, size = 163, normalized size = 1.09 \begin{align*} \frac{A a^{3} x^{4}}{4} + \frac{B a^{3} x^{5}}{5} + \frac{C b^{3} x^{12}}{12} + \frac{D b^{3} x^{13}}{13} + x^{11} \left (\frac{B b^{3}}{11} + \frac{3 D a b^{2}}{11}\right ) + x^{10} \left (\frac{A b^{3}}{10} + \frac{3 C a b^{2}}{10}\right ) + x^{9} \left (\frac{B a b^{2}}{3} + \frac{D a^{2} b}{3}\right ) + x^{8} \left (\frac{3 A a b^{2}}{8} + \frac{3 C a^{2} b}{8}\right ) + x^{7} \left (\frac{3 B a^{2} b}{7} + \frac{D a^{3}}{7}\right ) + x^{6} \left (\frac{A a^{2} b}{2} + \frac{C a^{3}}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**3*x**4/4 + B*a**3*x**5/5 + C*b**3*x**12/12 + D*b**3*x**13/13 + x**11*(B*b**3/11 + 3*D*a*b**2/11) + x**10*
(A*b**3/10 + 3*C*a*b**2/10) + x**9*(B*a*b**2/3 + D*a**2*b/3) + x**8*(3*A*a*b**2/8 + 3*C*a**2*b/8) + x**7*(3*B*
a**2*b/7 + D*a**3/7) + x**6*(A*a**2*b/2 + C*a**3/6)

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Giac [A]  time = 1.1554, size = 207, normalized size = 1.39 \begin{align*} \frac{1}{13} \, D b^{3} x^{13} + \frac{1}{12} \, C b^{3} x^{12} + \frac{3}{11} \, D a b^{2} x^{11} + \frac{1}{11} \, B b^{3} x^{11} + \frac{3}{10} \, C a b^{2} x^{10} + \frac{1}{10} \, A b^{3} x^{10} + \frac{1}{3} \, D a^{2} b x^{9} + \frac{1}{3} \, B a b^{2} x^{9} + \frac{3}{8} \, C a^{2} b x^{8} + \frac{3}{8} \, A a b^{2} x^{8} + \frac{1}{7} \, D a^{3} x^{7} + \frac{3}{7} \, B a^{2} b x^{7} + \frac{1}{6} \, C a^{3} x^{6} + \frac{1}{2} \, A a^{2} b x^{6} + \frac{1}{5} \, B a^{3} x^{5} + \frac{1}{4} \, A a^{3} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*D*b^3*x^13 + 1/12*C*b^3*x^12 + 3/11*D*a*b^2*x^11 + 1/11*B*b^3*x^11 + 3/10*C*a*b^2*x^10 + 1/10*A*b^3*x^10
+ 1/3*D*a^2*b*x^9 + 1/3*B*a*b^2*x^9 + 3/8*C*a^2*b*x^8 + 3/8*A*a*b^2*x^8 + 1/7*D*a^3*x^7 + 3/7*B*a^2*b*x^7 + 1/
6*C*a^3*x^6 + 1/2*A*a^2*b*x^6 + 1/5*B*a^3*x^5 + 1/4*A*a^3*x^4