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

Optimal. Leaf size=33 \[ \frac{1}{5} x^5 (A c+b B)+\frac{1}{4} A b x^4+\frac{1}{6} B c x^6 \]

[Out]

(A*b*x^4)/4 + ((b*B + A*c)*x^5)/5 + (B*c*x^6)/6

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Rubi [A]  time = 0.0890231, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{5} x^5 (A c+b B)+\frac{1}{4} A b x^4+\frac{1}{6} B c x^6 \]

Antiderivative was successfully verified.

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

[Out]

(A*b*x^4)/4 + ((b*B + A*c)*x^5)/5 + (B*c*x^6)/6

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Rubi in Sympy [A]  time = 7.37068, size = 29, normalized size = 0.88 \[ \frac{A b x^{4}}{4} + \frac{B c x^{6}}{6} + x^{5} \left (\frac{A c}{5} + \frac{B b}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b*x**4/4 + B*c*x**6/6 + x**5*(A*c/5 + B*b/5)

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Mathematica [A]  time = 0.00805429, size = 33, normalized size = 1. \[ \frac{1}{5} x^5 (A c+b B)+\frac{1}{4} A b x^4+\frac{1}{6} B c x^6 \]

Antiderivative was successfully verified.

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

[Out]

(A*b*x^4)/4 + ((b*B + A*c)*x^5)/5 + (B*c*x^6)/6

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Maple [A]  time = 0., size = 28, normalized size = 0.9 \[{\frac{Ab{x}^{4}}{4}}+{\frac{ \left ( Ac+Bb \right ){x}^{5}}{5}}+{\frac{Bc{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x+A)*(c*x^2+b*x),x)

[Out]

1/4*A*b*x^4+1/5*(A*c+B*b)*x^5+1/6*B*c*x^6

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Maxima [A]  time = 0.69692, size = 36, normalized size = 1.09 \[ \frac{1}{6} \, B c x^{6} + \frac{1}{4} \, A b x^{4} + \frac{1}{5} \,{\left (B b + A c\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*c*x^6 + 1/4*A*b*x^4 + 1/5*(B*b + A*c)*x^5

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Fricas [A]  time = 0.249103, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} c B + \frac{1}{5} x^{5} b B + \frac{1}{5} x^{5} c A + \frac{1}{4} x^{4} b A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^6*c*B + 1/5*x^5*b*B + 1/5*x^5*c*A + 1/4*x^4*b*A

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Sympy [A]  time = 0.092578, size = 29, normalized size = 0.88 \[ \frac{A b x^{4}}{4} + \frac{B c x^{6}}{6} + x^{5} \left (\frac{A c}{5} + \frac{B b}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b*x**4/4 + B*c*x**6/6 + x**5*(A*c/5 + B*b/5)

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GIAC/XCAS [A]  time = 0.271306, size = 39, normalized size = 1.18 \[ \frac{1}{6} \, B c x^{6} + \frac{1}{5} \, B b x^{5} + \frac{1}{5} \, A c x^{5} + \frac{1}{4} \, A b x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*c*x^6 + 1/5*B*b*x^5 + 1/5*A*c*x^5 + 1/4*A*b*x^4