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

Optimal. Leaf size=75 \[ \frac{1}{6} A b^3 x^6+\frac{1}{7} b^2 x^7 (3 A c+b B)+\frac{1}{9} c^2 x^9 (A c+3 b B)+\frac{3}{8} b c x^8 (A c+b B)+\frac{1}{10} B c^3 x^{10} \]

[Out]

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

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Rubi [A]  time = 0.192809, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{6} A b^3 x^6+\frac{1}{7} b^2 x^7 (3 A c+b B)+\frac{1}{9} c^2 x^9 (A c+3 b B)+\frac{3}{8} b c x^8 (A c+b B)+\frac{1}{10} B c^3 x^{10} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.1277, size = 70, normalized size = 0.93 \[ \frac{A b^{3} x^{6}}{6} + \frac{B c^{3} x^{10}}{10} + \frac{b^{2} x^{7} \left (3 A c + B b\right )}{7} + \frac{3 b c x^{8} \left (A c + B b\right )}{8} + \frac{c^{2} x^{9} \left (A c + 3 B b\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**3*x**6/6 + B*c**3*x**10/10 + b**2*x**7*(3*A*c + B*b)/7 + 3*b*c*x**8*(A*c +
B*b)/8 + c**2*x**9*(A*c + 3*B*b)/9

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Mathematica [A]  time = 0.0198418, size = 75, normalized size = 1. \[ \frac{1}{6} A b^3 x^6+\frac{1}{7} b^2 x^7 (3 A c+b B)+\frac{1}{9} c^2 x^9 (A c+3 b B)+\frac{3}{8} b c x^8 (A c+b B)+\frac{1}{10} B c^3 x^{10} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 76, normalized size = 1. \[{\frac{B{c}^{3}{x}^{10}}{10}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,A{b}^{2}c+B{b}^{3} \right ){x}^{7}}{7}}+{\frac{A{b}^{3}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.698748, size = 99, normalized size = 1.32 \[ \frac{1}{10} \, B c^{3} x^{10} + \frac{1}{6} \, A b^{3} x^{6} + \frac{1}{9} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + \frac{3}{8} \,{\left (B b^{2} c + A b c^{2}\right )} x^{8} + \frac{1}{7} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.253067, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} c^{3} B + \frac{1}{3} x^{9} c^{2} b B + \frac{1}{9} x^{9} c^{3} A + \frac{3}{8} x^{8} c b^{2} B + \frac{3}{8} x^{8} c^{2} b A + \frac{1}{7} x^{7} b^{3} B + \frac{3}{7} x^{7} c b^{2} A + \frac{1}{6} x^{6} b^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*x^10*c^3*B + 1/3*x^9*c^2*b*B + 1/9*x^9*c^3*A + 3/8*x^8*c*b^2*B + 3/8*x^8*c^
2*b*A + 1/7*x^7*b^3*B + 3/7*x^7*c*b^2*A + 1/6*x^6*b^3*A

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Sympy [A]  time = 0.145133, size = 82, normalized size = 1.09 \[ \frac{A b^{3} x^{6}}{6} + \frac{B c^{3} x^{10}}{10} + x^{9} \left (\frac{A c^{3}}{9} + \frac{B b c^{2}}{3}\right ) + x^{8} \left (\frac{3 A b c^{2}}{8} + \frac{3 B b^{2} c}{8}\right ) + x^{7} \left (\frac{3 A b^{2} c}{7} + \frac{B b^{3}}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**3*x**6/6 + B*c**3*x**10/10 + x**9*(A*c**3/9 + B*b*c**2/3) + x**8*(3*A*b*c**
2/8 + 3*B*b**2*c/8) + x**7*(3*A*b**2*c/7 + B*b**3/7)

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GIAC/XCAS [A]  time = 0.265904, size = 104, normalized size = 1.39 \[ \frac{1}{10} \, B c^{3} x^{10} + \frac{1}{3} \, B b c^{2} x^{9} + \frac{1}{9} \, A c^{3} x^{9} + \frac{3}{8} \, B b^{2} c x^{8} + \frac{3}{8} \, A b c^{2} x^{8} + \frac{1}{7} \, B b^{3} x^{7} + \frac{3}{7} \, A b^{2} c x^{7} + \frac{1}{6} \, A b^{3} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*B*c^3*x^10 + 1/3*B*b*c^2*x^9 + 1/9*A*c^3*x^9 + 3/8*B*b^2*c*x^8 + 3/8*A*b*c^
2*x^8 + 1/7*B*b^3*x^7 + 3/7*A*b^2*c*x^7 + 1/6*A*b^3*x^6