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

Optimal. Leaf size=33 \[ \frac{1}{4} x^4 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{5} B c x^5 \]

[Out]

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

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Rubi [A]  time = 0.0720951, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{4} x^4 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{5} B c x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.19502, size = 29, normalized size = 0.88 \[ \frac{A b x^{3}}{3} + \frac{B c x^{5}}{5} + x^{4} \left (\frac{A c}{4} + \frac{B b}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b*x**3/3 + B*c*x**5/5 + x**4*(A*c/4 + B*b/4)

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Mathematica [A]  time = 0.00761144, size = 33, normalized size = 1. \[ \frac{1}{4} x^4 (A c+b B)+\frac{1}{3} A b x^3+\frac{1}{5} B c x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 28, normalized size = 0.9 \[{\frac{Ab{x}^{3}}{3}}+{\frac{ \left ( Ac+Bb \right ){x}^{4}}{4}}+{\frac{Bc{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)*(c*x^2+b*x),x)

[Out]

1/3*A*b*x^3+1/4*(A*c+B*b)*x^4+1/5*B*c*x^5

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Maxima [A]  time = 0.695291, size = 36, normalized size = 1.09 \[ \frac{1}{5} \, B c x^{5} + \frac{1}{3} \, A b x^{3} + \frac{1}{4} \,{\left (B b + A c\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*c*x^5 + 1/3*A*b*x^3 + 1/4*(B*b + A*c)*x^4

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Fricas [A]  time = 0.250824, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} c B + \frac{1}{4} x^{4} b B + \frac{1}{4} x^{4} c A + \frac{1}{3} x^{3} b A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.084366, size = 29, normalized size = 0.88 \[ \frac{A b x^{3}}{3} + \frac{B c x^{5}}{5} + x^{4} \left (\frac{A c}{4} + \frac{B b}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b*x**3/3 + B*c*x**5/5 + x**4*(A*c/4 + B*b/4)

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GIAC/XCAS [A]  time = 0.266552, size = 39, normalized size = 1.18 \[ \frac{1}{5} \, B c x^{5} + \frac{1}{4} \, B b x^{4} + \frac{1}{4} \, A c x^{4} + \frac{1}{3} \, A b x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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