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

Optimal. Leaf size=23 \[ \frac{2 \sqrt{b x+c x^2}}{c \sqrt{x}} \]

[Out]

(2*Sqrt[b*x + c*x^2])/(c*Sqrt[x])

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Rubi [A]  time = 0.0274766, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{2 \sqrt{b x+c x^2}}{c \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b*x + c*x^2])/(c*Sqrt[x])

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Rubi in Sympy [A]  time = 3.64854, size = 19, normalized size = 0.83 \[ \frac{2 \sqrt{b x + c x^{2}}}{c \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b*x + c*x**2)/(c*sqrt(x))

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Mathematica [A]  time = 0.0160273, size = 21, normalized size = 0.91 \[ \frac{2 \sqrt{x (b+c x)}}{c \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x*(b + c*x)])/(c*Sqrt[x])

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Maple [A]  time = 0.003, size = 25, normalized size = 1.1 \[ 2\,{\frac{ \left ( cx+b \right ) \sqrt{x}}{c\sqrt{c{x}^{2}+bx}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.71986, size = 16, normalized size = 0.7 \[ \frac{2 \, \sqrt{c x + b}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c*x + b)/c

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Fricas [A]  time = 0.219362, size = 26, normalized size = 1.13 \[ \frac{2 \, \sqrt{c x^{2} + b x}}{c \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c*x^2 + b*x)/(c*sqrt(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{\sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x)/sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.209112, size = 28, normalized size = 1.22 \[ \frac{2 \, \sqrt{c x + b}}{c} - \frac{2 \, \sqrt{b}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c*x + b)/c - 2*sqrt(b)/c