3.1752 \(\int \frac{a c+(b c+a d) x+b d x^2}{a+b x} \, dx\)

Optimal. Leaf size=12 \[ c x+\frac{d x^2}{2} \]

[Out]

c*x + (d*x^2)/2

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Rubi [A]  time = 0.0191222, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ c x+\frac{d x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x),x]

[Out]

c*x + (d*x^2)/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d \int x\, dx + \int c\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)/(b*x+a),x)

[Out]

d*Integral(x, x) + Integral(c, x)

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Mathematica [A]  time = 0.00153336, size = 12, normalized size = 1. \[ c x+\frac{d x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x),x]

[Out]

c*x + (d*x^2)/2

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Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \[ cx+{\frac{d{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*c+(a*d+b*c)*x+x^2*b*d)/(b*x+a),x)

[Out]

c*x+1/2*d*x^2

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Maxima [A]  time = 0.730066, size = 14, normalized size = 1.17 \[ \frac{1}{2} \, d x^{2} + c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d*x^2 + c*x

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Fricas [A]  time = 0.194038, size = 14, normalized size = 1.17 \[ \frac{1}{2} \, d x^{2} + c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)/(b*x + a),x, algorithm="fricas")

[Out]

1/2*d*x^2 + c*x

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Sympy [A]  time = 0.111742, size = 8, normalized size = 0.67 \[ c x + \frac{d x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*c+(a*d+b*c)*x+b*d*x**2)/(b*x+a),x)

[Out]

c*x + d*x**2/2

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GIAC/XCAS [A]  time = 0.208054, size = 14, normalized size = 1.17 \[ \frac{1}{2} \, d x^{2} + c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)/(b*x + a),x, algorithm="giac")

[Out]

1/2*d*x^2 + c*x