\(\int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\) [98]

Optimal result

Integrand size = 19, antiderivative size = 32 \[ \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {674, 213} \[ \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \]

[In]

Int[1/(Sqrt[x]*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/Sqrt[b]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right ) \\ & = -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {x} \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {x (b+c x)}} \]

[In]

Integrate[1/(Sqrt[x]*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[x]*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(Sqrt[b]*Sqrt[x*(b + c*x)])

Maple [A] (verified)

Time = 2.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {2 \sqrt {x \left (c x +b \right )}\, \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )}{\sqrt {x}\, \sqrt {c x +b}\, \sqrt {b}}\)	\(37\)

[In]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22 \[ \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx=\left [\frac {\log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right )}{\sqrt {b}}, \frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right )}{b}\right ] \]

[In]

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

[Out]

[log(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2)/sqrt(b), 2*sqrt(-b)*arctan(sqrt(-b)*sqrt(x)/s
qrt(c*x^2 + b*x))/b]

Sympy [F]

\[ \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{\sqrt {x} \sqrt {x \left (b + c x\right )}}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(x)*sqrt(x*(b + c*x))), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} \sqrt {x}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {2 \, \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right )}{\sqrt {-b}} \]

[In]

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

[Out]

2*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) - 2*arctan(sqrt(b)/sqrt(-b))/sqrt(-b)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{\sqrt {x}\,\sqrt {c\,x^2+b\,x}} \,d x \]

[In]

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

[Out]

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