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 = 0.0068375, 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, Rules used = {648} \[ \frac{2 \sqrt{b x+c x^2}}{c \sqrt{x}} \]

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])

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx &=\frac{2 \sqrt{b x+c x^2}}{c \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.008983, 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 = 0.045, size = 25, normalized size = 1.1 \begin{align*} 2\,{\frac{ \left ( cx+b \right ) \sqrt{x}}{c\sqrt{c{x}^{2}+bx}}} \end{align*}

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 = 1.18947, size = 16, normalized size = 0.7 \begin{align*} \frac{2 \, \sqrt{c x + b}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(c*x + b)/c

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Fricas [A]  time = 1.99411, size = 45, normalized size = 1.96 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}}{c \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(c*x^2+b*x)^(1/2),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. \begin{align*} \int \frac{\sqrt{x}}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}

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 [A]  time = 1.26256, size = 28, normalized size = 1.22 \begin{align*} \frac{2 \, \sqrt{c x + b}}{c} - \frac{2 \, \sqrt{b}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(c*x + b)/c - 2*sqrt(b)/c