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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.92, size = 19, normalized size = 0.83 \[ \frac {2 \, \sqrt {c x^{2} + b x}}{c \sqrt {x}} \]

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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giac [A]  time = 0.19, size = 21, normalized size = 0.91 \[ \frac {2 \, \sqrt {c x + b}}{c} - \frac {2 \, \sqrt {b}}{c} \]

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

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maple [A]  time = 0.04, size = 25, normalized size = 1.09 \[ \frac {2 \left (c x +b \right ) \sqrt {x}}{\sqrt {c \,x^{2}+b x}\, c} \]

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 = 1.41, size = 12, normalized size = 0.52 \[ \frac {2 \, \sqrt {c x + b}}{c} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {x}}{\sqrt {c\,x^2+b\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(b*x + c*x^2)^(1/2),x)

[Out]

int(x^(1/2)/(b*x + c*x^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x}}{\sqrt {x \left (b + c x\right )}}\, dx \]

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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