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

Optimal. Leaf size=42 \[ \frac{1}{2} x^2 (a B+A b)+a A x+\frac{1}{3} x^3 (A c+b B)+\frac{1}{4} B c x^4 \]

[Out]

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

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Rubi [A]  time = 0.0632424, antiderivative size = 42, 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 \[ \frac{1}{2} x^2 (a B+A b)+a A x+\frac{1}{3} x^3 (A c+b B)+\frac{1}{4} B c x^4 \]

Antiderivative was successfully verified.

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

[Out]

a*A*x + ((A*b + 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. \[ \frac{B c x^{4}}{4} + a \int A\, dx + x^{3} \left (\frac{A c}{3} + \frac{B b}{3}\right ) + \left (A b + B a\right ) \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0221979, size = 42, normalized size = 1. \[ \frac{1}{2} x^2 (a B+A b)+a A x+\frac{1}{3} x^3 (A c+b B)+\frac{1}{4} B c x^4 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 37, normalized size = 0.9 \[ aAx+{\frac{ \left ( Ab+Ba \right ){x}^{2}}{2}}+{\frac{ \left ( Ac+Bb \right ){x}^{3}}{3}}+{\frac{Bc{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a),x)

[Out]

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

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Maxima [A]  time = 0.694617, size = 49, normalized size = 1.17 \[ \frac{1}{4} \, B c x^{4} + \frac{1}{3} \,{\left (B b + A c\right )} x^{3} + A a x + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244917, size = 1, normalized size = 0.02 \[ \frac{1}{4} x^{4} c B + \frac{1}{3} x^{3} b B + \frac{1}{3} x^{3} c A + \frac{1}{2} x^{2} a B + \frac{1}{2} x^{2} b A + x a A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(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*a*B + 1/2*x^2*b*A + x*a*A

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Sympy [A]  time = 0.088659, size = 39, normalized size = 0.93 \[ A a x + \frac{B c x^{4}}{4} + x^{3} \left (\frac{A c}{3} + \frac{B b}{3}\right ) + x^{2} \left (\frac{A b}{2} + \frac{B a}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270589, size = 54, normalized size = 1.29 \[ \frac{1}{4} \, B c x^{4} + \frac{1}{3} \, B b x^{3} + \frac{1}{3} \, A c x^{3} + \frac{1}{2} \, B a x^{2} + \frac{1}{2} \, A b x^{2} + A a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(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*B*a*x^2 + 1/2*A*b*x^2 + A*a*x