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

Optimal. Leaf size=37 \[ \frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{4} A c x^4+\frac{1}{5} B c x^5 \]

[Out]

(a*A*x^2)/2 + (a*B*x^3)/3 + (A*c*x^4)/4 + (B*c*x^5)/5

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Rubi [A]  time = 0.0580558, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{4} A c x^4+\frac{1}{5} B c x^5 \]

Antiderivative was successfully verified.

[In]  Int[x*(A + B*x)*(a + c*x^2),x]

[Out]

(a*A*x^2)/2 + (a*B*x^3)/3 + (A*c*x^4)/4 + (B*c*x^5)/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a \int x\, dx + \frac{A c x^{4}}{4} + \frac{B a x^{3}}{3} + \frac{B c x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*Integral(x, x) + A*c*x**4/4 + B*a*x**3/3 + B*c*x**5/5

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Mathematica [A]  time = 0.00298, size = 37, normalized size = 1. \[ \frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{4} A c x^4+\frac{1}{5} B c x^5 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^2)/2 + (a*B*x^3)/3 + (A*c*x^4)/4 + (B*c*x^5)/5

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Maple [A]  time = 0., size = 30, normalized size = 0.8 \[{\frac{aA{x}^{2}}{2}}+{\frac{aB{x}^{3}}{3}}+{\frac{Ac{x}^{4}}{4}}+{\frac{Bc{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)*(c*x^2+a),x)

[Out]

1/2*a*A*x^2+1/3*a*B*x^3+1/4*A*c*x^4+1/5*B*c*x^5

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Maxima [A]  time = 0.684263, size = 39, normalized size = 1.05 \[ \frac{1}{5} \, B c x^{5} + \frac{1}{4} \, A c x^{4} + \frac{1}{3} \, B a x^{3} + \frac{1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*c*x^5 + 1/4*A*c*x^4 + 1/3*B*a*x^3 + 1/2*A*a*x^2

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Fricas [A]  time = 0.255056, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} c B + \frac{1}{4} x^{4} c A + \frac{1}{3} x^{3} a B + \frac{1}{2} x^{2} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*x^5*c*B + 1/4*x^4*c*A + 1/3*x^3*a*B + 1/2*x^2*a*A

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Sympy [A]  time = 0.077932, size = 32, normalized size = 0.86 \[ \frac{A a x^{2}}{2} + \frac{A c x^{4}}{4} + \frac{B a x^{3}}{3} + \frac{B c x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**2/2 + A*c*x**4/4 + B*a*x**3/3 + B*c*x**5/5

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GIAC/XCAS [A]  time = 0.266729, size = 39, normalized size = 1.05 \[ \frac{1}{5} \, B c x^{5} + \frac{1}{4} \, A c x^{4} + \frac{1}{3} \, B a x^{3} + \frac{1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*c*x^5 + 1/4*A*c*x^4 + 1/3*B*a*x^3 + 1/2*A*a*x^2