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

Optimal. Leaf size=99 \[ \frac{1}{4} a^4 A x^4+\frac{1}{5} a^3 x^5 (a B+4 A b)+\frac{1}{3} a^2 b x^6 (2 a B+3 A b)+\frac{1}{8} b^3 x^8 (4 a B+A b)+\frac{2}{7} a b^2 x^7 (3 a B+2 A b)+\frac{1}{9} b^4 B x^9 \]

[Out]

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

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Rubi [A]  time = 0.177884, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{1}{4} a^4 A x^4+\frac{1}{5} a^3 x^5 (a B+4 A b)+\frac{1}{3} a^2 b x^6 (2 a B+3 A b)+\frac{1}{8} b^3 x^8 (4 a B+A b)+\frac{2}{7} a b^2 x^7 (3 a B+2 A b)+\frac{1}{9} b^4 B x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.683, size = 104, normalized size = 1.05 \[ \frac{B \left (a + b x\right )^{9}}{9 b^{5}} - \frac{a^{3} \left (a + b x\right )^{5} \left (A b - B a\right )}{5 b^{5}} + \frac{a^{2} \left (a + b x\right )^{6} \left (3 A b - 4 B a\right )}{6 b^{5}} - \frac{3 a \left (a + b x\right )^{7} \left (A b - 2 B a\right )}{7 b^{5}} + \frac{\left (a + b x\right )^{8} \left (A b - 4 B a\right )}{8 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0225089, size = 99, normalized size = 1. \[ \frac{1}{4} a^4 A x^4+\frac{1}{5} a^3 x^5 (a B+4 A b)+\frac{1}{3} a^2 b x^6 (2 a B+3 A b)+\frac{1}{8} b^3 x^8 (4 a B+A b)+\frac{2}{7} a b^2 x^7 (3 a B+2 A b)+\frac{1}{9} b^4 B x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 100, normalized size = 1. \[{\frac{{b}^{4}B{x}^{9}}{9}}+{\frac{ \left ( A{b}^{4}+4\,Ba{b}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 4\,Aa{b}^{3}+6\,B{a}^{2}{b}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,A{a}^{2}{b}^{2}+4\,B{a}^{3}b \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,A{a}^{3}b+B{a}^{4} \right ){x}^{5}}{5}}+{\frac{{a}^{4}A{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.682076, size = 134, normalized size = 1.35 \[ \frac{1}{9} \, B b^{4} x^{9} + \frac{1}{4} \, A a^{4} x^{4} + \frac{1}{8} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{8} + \frac{2}{7} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{7} + \frac{1}{3} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.280812, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} b^{4} B + \frac{1}{2} x^{8} b^{3} a B + \frac{1}{8} x^{8} b^{4} A + \frac{6}{7} x^{7} b^{2} a^{2} B + \frac{4}{7} x^{7} b^{3} a A + \frac{2}{3} x^{6} b a^{3} B + x^{6} b^{2} a^{2} A + \frac{1}{5} x^{5} a^{4} B + \frac{4}{5} x^{5} b a^{3} A + \frac{1}{4} x^{4} a^{4} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.157852, size = 105, normalized size = 1.06 \[ \frac{A a^{4} x^{4}}{4} + \frac{B b^{4} x^{9}}{9} + x^{8} \left (\frac{A b^{4}}{8} + \frac{B a b^{3}}{2}\right ) + x^{7} \left (\frac{4 A a b^{3}}{7} + \frac{6 B a^{2} b^{2}}{7}\right ) + x^{6} \left (A a^{2} b^{2} + \frac{2 B a^{3} b}{3}\right ) + x^{5} \left (\frac{4 A a^{3} b}{5} + \frac{B a^{4}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.274678, size = 135, normalized size = 1.36 \[ \frac{1}{9} \, B b^{4} x^{9} + \frac{1}{2} \, B a b^{3} x^{8} + \frac{1}{8} \, A b^{4} x^{8} + \frac{6}{7} \, B a^{2} b^{2} x^{7} + \frac{4}{7} \, A a b^{3} x^{7} + \frac{2}{3} \, B a^{3} b x^{6} + A a^{2} b^{2} x^{6} + \frac{1}{5} \, B a^{4} x^{5} + \frac{4}{5} \, A a^{3} b x^{5} + \frac{1}{4} \, A a^{4} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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