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

Optimal. Leaf size=87 \[ \frac{a^2 (a+b x)^7 (A b-a B)}{7 b^4}+\frac{(a+b x)^9 (A b-3 a B)}{9 b^4}-\frac{a (a+b x)^8 (2 A b-3 a B)}{8 b^4}+\frac{B (a+b x)^{10}}{10 b^4} \]

[Out]

(a^2*(A*b - a*B)*(a + b*x)^7)/(7*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x)^8)/(8*b^4)
+ ((A*b - 3*a*B)*(a + b*x)^9)/(9*b^4) + (B*(a + b*x)^10)/(10*b^4)

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Rubi [A]  time = 0.219425, antiderivative size = 87, 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 \[ \frac{a^2 (a+b x)^7 (A b-a B)}{7 b^4}+\frac{(a+b x)^9 (A b-3 a B)}{9 b^4}-\frac{a (a+b x)^8 (2 A b-3 a B)}{8 b^4}+\frac{B (a+b x)^{10}}{10 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(A*b - a*B)*(a + b*x)^7)/(7*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x)^8)/(8*b^4)
+ ((A*b - 3*a*B)*(a + b*x)^9)/(9*b^4) + (B*(a + b*x)^10)/(10*b^4)

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Rubi in Sympy [A]  time = 42.5533, size = 78, normalized size = 0.9 \[ \frac{B \left (a + b x\right )^{10}}{10 b^{4}} + \frac{a^{2} \left (a + b x\right )^{7} \left (A b - B a\right )}{7 b^{4}} - \frac{a \left (a + b x\right )^{8} \left (2 A b - 3 B a\right )}{8 b^{4}} + \frac{\left (a + b x\right )^{9} \left (A b - 3 B a\right )}{9 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x)**10/(10*b**4) + a**2*(a + b*x)**7*(A*b - B*a)/(7*b**4) - a*(a + b*x)
**8*(2*A*b - 3*B*a)/(8*b**4) + (a + b*x)**9*(A*b - 3*B*a)/(9*b**4)

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Mathematica [A]  time = 0.0341675, size = 143, normalized size = 1.64 \[ \frac{1}{3} a^6 A x^3+\frac{1}{4} a^5 x^4 (a B+6 A b)+\frac{3}{5} a^4 b x^5 (2 a B+5 A b)+\frac{5}{6} a^3 b^2 x^6 (3 a B+4 A b)+\frac{5}{7} a^2 b^3 x^7 (4 a B+3 A b)+\frac{1}{9} b^5 x^9 (6 a B+A b)+\frac{3}{8} a b^4 x^8 (5 a B+2 A b)+\frac{1}{10} b^6 B x^{10} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 148, normalized size = 1.7 \[{\frac{B{b}^{6}{x}^{10}}{10}}+{\frac{ \left ( A{b}^{6}+6\,Ba{b}^{5} \right ){x}^{9}}{9}}+{\frac{ \left ( 6\,Aa{b}^{5}+15\,B{a}^{2}{b}^{4} \right ){x}^{8}}{8}}+{\frac{ \left ( 15\,A{a}^{2}{b}^{4}+20\,B{a}^{3}{b}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 20\,A{a}^{3}{b}^{3}+15\,B{b}^{2}{a}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,A{b}^{2}{a}^{4}+6\,B{a}^{5}b \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,A{a}^{5}b+B{a}^{6} \right ){x}^{4}}{4}}+{\frac{A{a}^{6}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*B*b^6*x^10+1/9*(A*b^6+6*B*a*b^5)*x^9+1/8*(6*A*a*b^5+15*B*a^2*b^4)*x^8+1/7*(
15*A*a^2*b^4+20*B*a^3*b^3)*x^7+1/6*(20*A*a^3*b^3+15*B*a^4*b^2)*x^6+1/5*(15*A*a^4
*b^2+6*B*a^5*b)*x^5+1/4*(6*A*a^5*b+B*a^6)*x^4+1/3*A*a^6*x^3

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Maxima [A]  time = 0.677308, size = 198, normalized size = 2.28 \[ \frac{1}{10} \, B b^{6} x^{10} + \frac{1}{3} \, A a^{6} x^{3} + \frac{1}{9} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{9} + \frac{3}{8} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{8} + \frac{5}{7} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{7} + \frac{5}{6} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{6} + \frac{3}{5} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.257999, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} b^{6} B + \frac{2}{3} x^{9} b^{5} a B + \frac{1}{9} x^{9} b^{6} A + \frac{15}{8} x^{8} b^{4} a^{2} B + \frac{3}{4} x^{8} b^{5} a A + \frac{20}{7} x^{7} b^{3} a^{3} B + \frac{15}{7} x^{7} b^{4} a^{2} A + \frac{5}{2} x^{6} b^{2} a^{4} B + \frac{10}{3} x^{6} b^{3} a^{3} A + \frac{6}{5} x^{5} b a^{5} B + 3 x^{5} b^{2} a^{4} A + \frac{1}{4} x^{4} a^{6} B + \frac{3}{2} x^{4} b a^{5} A + \frac{1}{3} x^{3} a^{6} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*x^10*b^6*B + 2/3*x^9*b^5*a*B + 1/9*x^9*b^6*A + 15/8*x^8*b^4*a^2*B + 3/4*x^8
*b^5*a*A + 20/7*x^7*b^3*a^3*B + 15/7*x^7*b^4*a^2*A + 5/2*x^6*b^2*a^4*B + 10/3*x^
6*b^3*a^3*A + 6/5*x^5*b*a^5*B + 3*x^5*b^2*a^4*A + 1/4*x^4*a^6*B + 3/2*x^4*b*a^5*
A + 1/3*x^3*a^6*A

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Sympy [A]  time = 0.190753, size = 163, normalized size = 1.87 \[ \frac{A a^{6} x^{3}}{3} + \frac{B b^{6} x^{10}}{10} + x^{9} \left (\frac{A b^{6}}{9} + \frac{2 B a b^{5}}{3}\right ) + x^{8} \left (\frac{3 A a b^{5}}{4} + \frac{15 B a^{2} b^{4}}{8}\right ) + x^{7} \left (\frac{15 A a^{2} b^{4}}{7} + \frac{20 B a^{3} b^{3}}{7}\right ) + x^{6} \left (\frac{10 A a^{3} b^{3}}{3} + \frac{5 B a^{4} b^{2}}{2}\right ) + x^{5} \left (3 A a^{4} b^{2} + \frac{6 B a^{5} b}{5}\right ) + x^{4} \left (\frac{3 A a^{5} b}{2} + \frac{B a^{6}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**6*x**3/3 + B*b**6*x**10/10 + x**9*(A*b**6/9 + 2*B*a*b**5/3) + x**8*(3*A*a*b
**5/4 + 15*B*a**2*b**4/8) + x**7*(15*A*a**2*b**4/7 + 20*B*a**3*b**3/7) + x**6*(1
0*A*a**3*b**3/3 + 5*B*a**4*b**2/2) + x**5*(3*A*a**4*b**2 + 6*B*a**5*b/5) + x**4*
(3*A*a**5*b/2 + B*a**6/4)

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GIAC/XCAS [A]  time = 0.265579, size = 201, normalized size = 2.31 \[ \frac{1}{10} \, B b^{6} x^{10} + \frac{2}{3} \, B a b^{5} x^{9} + \frac{1}{9} \, A b^{6} x^{9} + \frac{15}{8} \, B a^{2} b^{4} x^{8} + \frac{3}{4} \, A a b^{5} x^{8} + \frac{20}{7} \, B a^{3} b^{3} x^{7} + \frac{15}{7} \, A a^{2} b^{4} x^{7} + \frac{5}{2} \, B a^{4} b^{2} x^{6} + \frac{10}{3} \, A a^{3} b^{3} x^{6} + \frac{6}{5} \, B a^{5} b x^{5} + 3 \, A a^{4} b^{2} x^{5} + \frac{1}{4} \, B a^{6} x^{4} + \frac{3}{2} \, A a^{5} b x^{4} + \frac{1}{3} \, A a^{6} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*B*b^6*x^10 + 2/3*B*a*b^5*x^9 + 1/9*A*b^6*x^9 + 15/8*B*a^2*b^4*x^8 + 3/4*A*a
*b^5*x^8 + 20/7*B*a^3*b^3*x^7 + 15/7*A*a^2*b^4*x^7 + 5/2*B*a^4*b^2*x^6 + 10/3*A*
a^3*b^3*x^6 + 6/5*B*a^5*b*x^5 + 3*A*a^4*b^2*x^5 + 1/4*B*a^6*x^4 + 3/2*A*a^5*b*x^
4 + 1/3*A*a^6*x^3