3.252 \(\int x^2 (A+B x) \left (a+c x^2\right ) \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{3} a A x^3+\frac{1}{4} a B x^4+\frac{1}{5} A c x^5+\frac{1}{6} B c x^6 \]

[Out]

(a*A*x^3)/3 + (a*B*x^4)/4 + (A*c*x^5)/5 + (B*c*x^6)/6

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Rubi [A]  time = 0.0779751, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{1}{3} a A x^3+\frac{1}{4} a B x^4+\frac{1}{5} A c x^5+\frac{1}{6} B c x^6 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^3)/3 + (a*B*x^4)/4 + (A*c*x^5)/5 + (B*c*x^6)/6

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Rubi in Sympy [A]  time = 6.15792, size = 32, normalized size = 0.86 \[ \frac{A a x^{3}}{3} + \frac{A c x^{5}}{5} + \frac{B a x^{4}}{4} + \frac{B c x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**3/3 + A*c*x**5/5 + B*a*x**4/4 + B*c*x**6/6

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Mathematica [A]  time = 0.00320719, size = 37, normalized size = 1. \[ \frac{1}{3} a A x^3+\frac{1}{4} a B x^4+\frac{1}{5} A c x^5+\frac{1}{6} B c x^6 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^3)/3 + (a*B*x^4)/4 + (A*c*x^5)/5 + (B*c*x^6)/6

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Maple [A]  time = 0.001, size = 30, normalized size = 0.8 \[{\frac{aA{x}^{3}}{3}}+{\frac{aB{x}^{4}}{4}}+{\frac{Ac{x}^{5}}{5}}+{\frac{Bc{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x+A)*(c*x^2+a),x)

[Out]

1/3*a*A*x^3+1/4*a*B*x^4+1/5*A*c*x^5+1/6*B*c*x^6

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Maxima [A]  time = 0.681205, size = 39, normalized size = 1.05 \[ \frac{1}{6} \, B c x^{6} + \frac{1}{5} \, A c x^{5} + \frac{1}{4} \, B a x^{4} + \frac{1}{3} \, A a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*c*x^6 + 1/5*A*c*x^5 + 1/4*B*a*x^4 + 1/3*A*a*x^3

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Fricas [A]  time = 0.261882, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} c B + \frac{1}{5} x^{5} c A + \frac{1}{4} x^{4} a B + \frac{1}{3} x^{3} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^6*c*B + 1/5*x^5*c*A + 1/4*x^4*a*B + 1/3*x^3*a*A

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Sympy [A]  time = 0.078918, size = 32, normalized size = 0.86 \[ \frac{A a x^{3}}{3} + \frac{A c x^{5}}{5} + \frac{B a x^{4}}{4} + \frac{B c x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**3/3 + A*c*x**5/5 + B*a*x**4/4 + B*c*x**6/6

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GIAC/XCAS [A]  time = 0.269497, size = 39, normalized size = 1.05 \[ \frac{1}{6} \, B c x^{6} + \frac{1}{5} \, A c x^{5} + \frac{1}{4} \, B a x^{4} + \frac{1}{3} \, A a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*c*x^6 + 1/5*A*c*x^5 + 1/4*B*a*x^4 + 1/3*A*a*x^3