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

Optimal. Leaf size=55 \[ \frac{1}{5} A b^2 x^5+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{8} B c^2 x^8 \]

[Out]

(A*b^2*x^5)/5 + (b*(b*B + 2*A*c)*x^6)/6 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^8)/
8

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Rubi [A]  time = 0.151775, antiderivative size = 55, 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 \[ \frac{1}{5} A b^2 x^5+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{8} B c^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

(A*b^2*x^5)/5 + (b*(b*B + 2*A*c)*x^6)/6 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^8)/
8

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Rubi in Sympy [A]  time = 15.2087, size = 49, normalized size = 0.89 \[ \frac{A b^{2} x^{5}}{5} + \frac{B c^{2} x^{8}}{8} + \frac{b x^{6} \left (2 A c + B b\right )}{6} + \frac{c x^{7} \left (A c + 2 B b\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**2*x**5/5 + B*c**2*x**8/8 + b*x**6*(2*A*c + B*b)/6 + c*x**7*(A*c + 2*B*b)/7

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Mathematica [A]  time = 0.0157755, size = 55, normalized size = 1. \[ \frac{1}{5} A b^2 x^5+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{6} b x^6 (2 A c+b B)+\frac{1}{8} B c^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

(A*b^2*x^5)/5 + (b*(b*B + 2*A*c)*x^6)/6 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^8)/
8

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Maple [A]  time = 0., size = 52, normalized size = 1. \[{\frac{B{c}^{2}{x}^{8}}{8}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{6}}{6}}+{\frac{A{b}^{2}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*B*c^2*x^8+1/7*(A*c^2+2*B*b*c)*x^7+1/6*(2*A*b*c+B*b^2)*x^6+1/5*A*b^2*x^5

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Maxima [A]  time = 0.690626, size = 69, normalized size = 1.25 \[ \frac{1}{8} \, B c^{2} x^{8} + \frac{1}{5} \, A b^{2} x^{5} + \frac{1}{7} \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{2} + 2 \, A b c\right )} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*B*c^2*x^8 + 1/5*A*b^2*x^5 + 1/7*(2*B*b*c + A*c^2)*x^7 + 1/6*(B*b^2 + 2*A*b*c
)*x^6

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Fricas [A]  time = 0.250004, size = 1, normalized size = 0.02 \[ \frac{1}{8} x^{8} c^{2} B + \frac{2}{7} x^{7} c b B + \frac{1}{7} x^{7} c^{2} A + \frac{1}{6} x^{6} b^{2} B + \frac{1}{3} x^{6} c b A + \frac{1}{5} x^{5} b^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*c^2*B + 2/7*x^7*c*b*B + 1/7*x^7*c^2*A + 1/6*x^6*b^2*B + 1/3*x^6*c*b*A +
1/5*x^5*b^2*A

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Sympy [A]  time = 0.121497, size = 54, normalized size = 0.98 \[ \frac{A b^{2} x^{5}}{5} + \frac{B c^{2} x^{8}}{8} + x^{7} \left (\frac{A c^{2}}{7} + \frac{2 B b c}{7}\right ) + x^{6} \left (\frac{A b c}{3} + \frac{B b^{2}}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**2*x**5/5 + B*c**2*x**8/8 + x**7*(A*c**2/7 + 2*B*b*c/7) + x**6*(A*b*c/3 + B*
b**2/6)

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GIAC/XCAS [A]  time = 0.264982, size = 72, normalized size = 1.31 \[ \frac{1}{8} \, B c^{2} x^{8} + \frac{2}{7} \, B b c x^{7} + \frac{1}{7} \, A c^{2} x^{7} + \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{3} \, A b c x^{6} + \frac{1}{5} \, A b^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*B*c^2*x^8 + 2/7*B*b*c*x^7 + 1/7*A*c^2*x^7 + 1/6*B*b^2*x^6 + 1/3*A*b*c*x^6 +
1/5*A*b^2*x^5