3.6 \(\int \frac{(A+B x) \left (b x+c x^2\right )}{x} \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{2} x^2 (A c+b B)+A b x+\frac{1}{3} B c x^3 \]

[Out]

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

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Rubi [A]  time = 0.0352864, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{2} x^2 (A c+b B)+A b x+\frac{1}{3} B c x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00804245, size = 28, normalized size = 1. \[ \frac{1}{2} x^2 (A c+b B)+A b x+\frac{1}{3} B c x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 25, normalized size = 0.9 \[ Abx+{\frac{ \left ( Ac+Bb \right ){x}^{2}}{2}}+{\frac{Bc{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)/x,x)

[Out]

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

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Maxima [A]  time = 0.692587, size = 32, normalized size = 1.14 \[ \frac{1}{3} \, B c x^{3} + A b x + \frac{1}{2} \,{\left (B b + A c\right )} x^{2} \]

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]

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

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Fricas [A]  time = 0.259988, size = 32, normalized size = 1.14 \[ \frac{1}{3} \, B c x^{3} + A b x + \frac{1}{2} \,{\left (B b + A c\right )} x^{2} \]

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]

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

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Sympy [A]  time = 0.087667, size = 26, normalized size = 0.93 \[ A b x + \frac{B c x^{3}}{3} + x^{2} \left (\frac{A c}{2} + \frac{B b}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.266359, size = 35, normalized size = 1.25 \[ \frac{1}{3} \, B c x^{3} + \frac{1}{2} \, B b x^{2} + \frac{1}{2} \, A c x^{2} + A b x \]

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]

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