3.1107 \(\int (A+B x) \left (b x+c x^2\right ) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{3} x^3 (A c+b B)+\frac{1}{2} A b x^2+\frac{1}{4} B c x^4 \]

[Out]

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

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Rubi [A]  time = 0.059585, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{3} x^3 (A c+b B)+\frac{1}{2} A b x^2+\frac{1}{4} B c x^4 \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(b*x + c*x^2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A b \int x\, dx + \frac{B c x^{4}}{4} + x^{3} \left (\frac{A c}{3} + \frac{B b}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00767991, size = 29, normalized size = 0.88 \[ \frac{1}{12} x^2 (A (6 b+4 c x)+B x (4 b+3 c x)) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(B*x*(4*b + 3*c*x) + A*(6*b + 4*c*x)))/12

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Maple [A]  time = 0.002, size = 28, normalized size = 0.9 \[{\frac{Ab{x}^{2}}{2}}+{\frac{ \left ( Ac+Bb \right ){x}^{3}}{3}}+{\frac{Bc{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x),x)

[Out]

1/2*A*b*x^2+1/3*(A*c+B*b)*x^3+1/4*B*c*x^4

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Maxima [A]  time = 0.693784, size = 36, normalized size = 1.09 \[ \frac{1}{4} \, B c x^{4} + \frac{1}{2} \, A b x^{2} + \frac{1}{3} \,{\left (B b + A c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*B*c*x^4 + 1/2*A*b*x^2 + 1/3*(B*b + A*c)*x^3

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Fricas [A]  time = 0.286645, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} c B + \frac{1}{3} x^{3} b B + \frac{1}{3} x^{3} c A + \frac{1}{2} x^{2} b A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.08411, size = 29, normalized size = 0.88 \[ \frac{A b x^{2}}{2} + \frac{B c x^{4}}{4} + x^{3} \left (\frac{A c}{3} + \frac{B b}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b*x**2/2 + B*c*x**4/4 + x**3*(A*c/3 + B*b/3)

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GIAC/XCAS [A]  time = 0.278768, size = 39, normalized size = 1.18 \[ \frac{1}{4} \, B c x^{4} + \frac{1}{3} \, B b x^{3} + \frac{1}{3} \, A c x^{3} + \frac{1}{2} \, A b x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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