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

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

[Out]

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

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Rubi [A]  time = 0.368424, 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 \[ \frac{1}{3} a^3 A x^3+\frac{1}{4} a^3 B x^4+\frac{1}{5} a^2 x^5 (a C+3 A b)+\frac{1}{6} a^2 x^6 (a D+3 b B)+\frac{1}{9} b^2 x^9 (3 a C+A b)+\frac{3}{7} a b x^7 (a C+A b)+\frac{1}{10} b^2 x^{10} (3 a D+b B)+\frac{3}{8} a b x^8 (a D+b B)+\frac{1}{11} b^3 C x^{11}+\frac{1}{12} b^3 D x^{12} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.0379, size = 141, normalized size = 0.95 \[ \frac{A a^{3} x^{3}}{3} + \frac{B a^{3} x^{4}}{4} + \frac{C b^{3} x^{11}}{11} + \frac{D b^{3} x^{12}}{12} + \frac{a^{2} x^{6} \left (3 B b + D a\right )}{6} + \frac{a^{2} x^{5} \left (3 A b + C a\right )}{5} + \frac{3 a b x^{8} \left (B b + D a\right )}{8} + \frac{3 a b x^{7} \left (A b + C a\right )}{7} + \frac{b^{2} x^{10} \left (B b + 3 D a\right )}{10} + \frac{b^{2} x^{9} \left (A b + 3 C a\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.155142, size = 125, normalized size = 0.84 \[ \frac{462 a^3 x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+99 a^2 b x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))+33 a b^2 x^7 (360 A+7 x (45 B+4 x (10 C+9 D x)))+14 b^3 x^9 \left (220 A+3 x \left (66 B+60 C x+55 D x^2\right )\right )}{27720} \]

Antiderivative was successfully verified.

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

[Out]

(14*b^3*x^9*(220*A + 3*x*(66*B + 60*C*x + 55*D*x^2)) + 462*a^3*x^3*(20*A + x*(15
*B + 2*x*(6*C + 5*D*x))) + 99*a^2*b*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x)))
+ 33*a*b^2*x^7*(360*A + 7*x*(45*B + 4*x*(10*C + 9*D*x))))/27720

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Maple [A]  time = 0.002, size = 150, normalized size = 1. \[{\frac{{b}^{3}D{x}^{12}}{12}}+{\frac{{b}^{3}C{x}^{11}}{11}}+{\frac{ \left ({b}^{3}B+3\,a{b}^{2}D \right ){x}^{10}}{10}}+{\frac{ \left ( A{b}^{3}+3\,a{b}^{2}C \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,a{b}^{2}B+3\,{a}^{2}bD \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bC \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}bB+{a}^{3}D \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{a}^{2}b+{a}^{3}C \right ){x}^{5}}{5}}+{\frac{{a}^{3}B{x}^{4}}{4}}+{\frac{{a}^{3}A{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35328, size = 196, normalized size = 1.32 \[ \frac{1}{12} \, D b^{3} x^{12} + \frac{1}{11} \, C b^{3} x^{11} + \frac{1}{10} \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{10} + \frac{1}{9} \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{9} + \frac{3}{8} \,{\left (D a^{2} b + B a b^{2}\right )} x^{8} + \frac{1}{4} \, B a^{3} x^{4} + \frac{3}{7} \,{\left (C a^{2} b + A a b^{2}\right )} x^{7} + \frac{1}{3} \, A a^{3} x^{3} + \frac{1}{6} \,{\left (D a^{3} + 3 \, B a^{2} b\right )} x^{6} + \frac{1}{5} \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.207658, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} b^{3} D + \frac{1}{11} x^{11} b^{3} C + \frac{3}{10} x^{10} b^{2} a D + \frac{1}{10} x^{10} b^{3} B + \frac{1}{3} x^{9} b^{2} a C + \frac{1}{9} x^{9} b^{3} A + \frac{3}{8} x^{8} b a^{2} D + \frac{3}{8} x^{8} b^{2} a B + \frac{3}{7} x^{7} b a^{2} C + \frac{3}{7} x^{7} b^{2} a A + \frac{1}{6} x^{6} a^{3} D + \frac{1}{2} x^{6} b a^{2} B + \frac{1}{5} x^{5} a^{3} C + \frac{3}{5} x^{5} b a^{2} A + \frac{1}{4} x^{4} a^{3} B + \frac{1}{3} x^{3} a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.086901, size = 165, normalized size = 1.11 \[ \frac{A a^{3} x^{3}}{3} + \frac{B a^{3} x^{4}}{4} + \frac{C b^{3} x^{11}}{11} + \frac{D b^{3} x^{12}}{12} + x^{10} \left (\frac{B b^{3}}{10} + \frac{3 D a b^{2}}{10}\right ) + x^{9} \left (\frac{A b^{3}}{9} + \frac{C a b^{2}}{3}\right ) + x^{8} \left (\frac{3 B a b^{2}}{8} + \frac{3 D a^{2} b}{8}\right ) + x^{7} \left (\frac{3 A a b^{2}}{7} + \frac{3 C a^{2} b}{7}\right ) + x^{6} \left (\frac{B a^{2} b}{2} + \frac{D a^{3}}{6}\right ) + x^{5} \left (\frac{3 A a^{2} b}{5} + \frac{C a^{3}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222928, size = 207, normalized size = 1.39 \[ \frac{1}{12} \, D b^{3} x^{12} + \frac{1}{11} \, C b^{3} x^{11} + \frac{3}{10} \, D a b^{2} x^{10} + \frac{1}{10} \, B b^{3} x^{10} + \frac{1}{3} \, C a b^{2} x^{9} + \frac{1}{9} \, A b^{3} x^{9} + \frac{3}{8} \, D a^{2} b x^{8} + \frac{3}{8} \, B a b^{2} x^{8} + \frac{3}{7} \, C a^{2} b x^{7} + \frac{3}{7} \, A a b^{2} x^{7} + \frac{1}{6} \, D a^{3} x^{6} + \frac{1}{2} \, B a^{2} b x^{6} + \frac{1}{5} \, C a^{3} x^{5} + \frac{3}{5} \, A a^{2} b x^{5} + \frac{1}{4} \, B a^{3} x^{4} + \frac{1}{3} \, A a^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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