3.1576 \(\int \frac{b+2 c x}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=16 \[ 2 \sqrt{a+b x+c x^2} \]

[Out]

2*Sqrt[a + b*x + c*x^2]

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Rubi [A]  time = 0.0100859, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ 2 \sqrt{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[a + b*x + c*x^2]

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Rubi in Sympy [A]  time = 3.8349, size = 14, normalized size = 0.88 \[ 2 \sqrt{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)/(c*x**2+b*x+a)**(1/2),x)

[Out]

2*sqrt(a + b*x + c*x**2)

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Mathematica [A]  time = 0.0159483, size = 15, normalized size = 0.94 \[ 2 \sqrt{a+x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[a + x*(b + c*x)]

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Maple [A]  time = 0.004, size = 15, normalized size = 0.9 \[ 2\,\sqrt{c{x}^{2}+bx+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)/(c*x^2+b*x+a)^(1/2),x)

[Out]

2*(c*x^2+b*x+a)^(1/2)

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Maxima [A]  time = 0.715839, size = 19, normalized size = 1.19 \[ 2 \, \sqrt{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c*x^2 + b*x + a)

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Fricas [A]  time = 0.313119, size = 19, normalized size = 1.19 \[ 2 \, \sqrt{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c*x^2 + b*x + a)

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Sympy [A]  time = 0.336394, size = 14, normalized size = 0.88 \[ 2 \sqrt{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)/(c*x**2+b*x+a)**(1/2),x)

[Out]

2*sqrt(a + b*x + c*x**2)

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GIAC/XCAS [A]  time = 0.266094, size = 19, normalized size = 1.19 \[ 2 \, \sqrt{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c*x^2 + b*x + a)