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

Optimal. Leaf size=99 \[ a^2 A x+\frac{1}{4} a^2 D x^4+\frac{1}{5} b x^5 (2 a C+A b)+\frac{1}{3} a x^3 (a C+2 A b)+\frac{B \left (a+b x^2\right )^3}{6 b}+\frac{1}{3} a b D x^6+\frac{1}{7} b^2 C x^7+\frac{1}{8} b^2 D x^8 \]

[Out]

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

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Rubi [A]  time = 0.182586, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ a^2 A x+\frac{1}{4} a^2 D x^4+\frac{1}{5} b x^5 (2 a C+A b)+\frac{1}{3} a x^3 (a C+2 A b)+\frac{B \left (a+b x^2\right )^3}{6 b}+\frac{1}{3} a b D x^6+\frac{1}{7} b^2 C x^7+\frac{1}{8} b^2 D x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \left (a + b x^{2}\right )^{3}}{6 b} + \frac{C b^{2} x^{7}}{7} + \frac{D a^{2} x^{4}}{4} + \frac{D a b x^{6}}{3} + \frac{D b^{2} x^{8}}{8} + a^{2} \int A\, dx + \frac{a x^{3} \left (2 A b + C a\right )}{3} + \frac{b x^{5} \left (A b + 2 C a\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0839187, size = 88, normalized size = 0.89 \[ \frac{1}{840} \left (70 a^2 x (12 A+x (6 B+x (4 C+3 D x)))+28 a b x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+b^2 x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))\right ) \]

Antiderivative was successfully verified.

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

[Out]

(70*a^2*x*(12*A + x*(6*B + x*(4*C + 3*D*x))) + 28*a*b*x^3*(20*A + x*(15*B + 2*x*
(6*C + 5*D*x))) + b^2*x^5*(168*A + 5*x*(28*B + 3*x*(8*C + 7*D*x))))/840

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Maple [A]  time = 0.001, size = 99, normalized size = 1. \[{\frac{{b}^{2}D{x}^{8}}{8}}+{\frac{{b}^{2}C{x}^{7}}{7}}+{\frac{ \left ({b}^{2}B+2\,abD \right ){x}^{6}}{6}}+{\frac{ \left ({b}^{2}A+2\,abC \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,abB+{a}^{2}D \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,abA+{a}^{2}C \right ){x}^{3}}{3}}+{\frac{B{x}^{2}{a}^{2}}{2}}+{a}^{2}Ax \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33552, size = 132, normalized size = 1.33 \[ \frac{1}{8} \, D b^{2} x^{8} + \frac{1}{7} \, C b^{2} x^{7} + \frac{1}{6} \,{\left (2 \, D a b + B b^{2}\right )} x^{6} + \frac{1}{5} \,{\left (2 \, C a b + A b^{2}\right )} x^{5} + \frac{1}{2} \, B a^{2} x^{2} + \frac{1}{4} \,{\left (D a^{2} + 2 \, B a b\right )} x^{4} + A a^{2} x + \frac{1}{3} \,{\left (C a^{2} + 2 \, A a b\right )} x^{3} \]

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.200348, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} b^{2} D + \frac{1}{7} x^{7} b^{2} C + \frac{1}{3} x^{6} b a D + \frac{1}{6} x^{6} b^{2} B + \frac{2}{5} x^{5} b a C + \frac{1}{5} x^{5} b^{2} A + \frac{1}{4} x^{4} a^{2} D + \frac{1}{2} x^{4} b a B + \frac{1}{3} x^{3} a^{2} C + \frac{2}{3} x^{3} b a A + \frac{1}{2} x^{2} a^{2} B + x a^{2} A \]

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.071296, size = 107, normalized size = 1.08 \[ A a^{2} x + \frac{B a^{2} x^{2}}{2} + \frac{C b^{2} x^{7}}{7} + \frac{D b^{2} x^{8}}{8} + x^{6} \left (\frac{B b^{2}}{6} + \frac{D a b}{3}\right ) + x^{5} \left (\frac{A b^{2}}{5} + \frac{2 C a b}{5}\right ) + x^{4} \left (\frac{B a b}{2} + \frac{D a^{2}}{4}\right ) + x^{3} \left (\frac{2 A a b}{3} + \frac{C a^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221864, size = 138, normalized size = 1.39 \[ \frac{1}{8} \, D b^{2} x^{8} + \frac{1}{7} \, C b^{2} x^{7} + \frac{1}{3} \, D a b x^{6} + \frac{1}{6} \, B b^{2} x^{6} + \frac{2}{5} \, C a b x^{5} + \frac{1}{5} \, A b^{2} x^{5} + \frac{1}{4} \, D a^{2} x^{4} + \frac{1}{2} \, B a b x^{4} + \frac{1}{3} \, C a^{2} x^{3} + \frac{2}{3} \, A a b x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x \]

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)^2,x, algorithm="giac")

[Out]

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