\(\int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx\) [97]

Optimal result

Integrand size = 19, antiderivative size = 23 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {662} \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}} \]

[In]

Int[Sqrt[x]/Sqrt[b*x + c*x^2],x]

[Out]

(2*Sqrt[b*x + c*x^2])/(c*Sqrt[x])

Rule 662

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*} \text {integral}& = \frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x (b+c x)}}{c \sqrt {x}} \]

[In]

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

[Out]

(2*Sqrt[x*(b + c*x)])/(c*Sqrt[x])

Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {2 \sqrt {x \left (c x +b \right )}}{\sqrt {x}\, c}\)	\(18\)
risch	\(\frac {2 \left (c x +b \right ) \sqrt {x}}{\sqrt {x \left (c x +b \right )}\, c}\)	\(23\)
gosper	\(\frac {2 \left (c x +b \right ) \sqrt {x}}{c \sqrt {c \,x^{2}+b x}}\)	\(25\)

[In]

int(x^(1/2)/(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, \sqrt {c x^{2} + b x}}{c \sqrt {x}} \]

[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))

Sympy [F]

\[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\sqrt {x}}{\sqrt {x \left (b + c x\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, \sqrt {c x + b}}{c} \]

[In]

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

[Out]

2*sqrt(c*x + b)/c

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, \sqrt {c x + b}}{c} - \frac {2 \, \sqrt {b}}{c} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\sqrt {x}}{\sqrt {c\,x^2+b\,x}} \,d x \]

[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)