3.991 \(\int \frac {1}{\sqrt {a c x+b c x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {620, 206} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a c x+b c x^2}}\right )}{\sqrt {b} \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*c*x + b*c*x^2],x]

[Out]

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

Rule 206

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

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 58, normalized size = 1.45 \[ \frac {2 \sqrt {a} \sqrt {x} \sqrt {\frac {b x}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {c x (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*c*x + b*c*x^2],x]

[Out]

(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[b]*Sqrt[c*x*(a + b*x)])

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 87, normalized size = 2.18 \[ \left [\frac {\sqrt {b c} \log \left (2 \, b c x + a c + 2 \, \sqrt {b c x^{2} + a c x} \sqrt {b c}\right )}{b c}, -\frac {2 \, \sqrt {-b c} \arctan \left (\frac {\sqrt {b c x^{2} + a c x} \sqrt {-b c}}{b c x}\right )}{b c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.50, size = 50, normalized size = 1.25 \[ -\frac {\sqrt {b c} \log \left ({\left | -2 \, {\left (\sqrt {b c} x - \sqrt {b c x^{2} + a c x}\right )} b - \sqrt {b c} a \right |}\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b*c)*log(abs(-2*(sqrt(b*c)*x - sqrt(b*c*x^2 + a*c*x))*b - sqrt(b*c)*a))/(b*c)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 37, normalized size = 0.92 \[ \frac {\ln \left (\frac {b c x +\frac {1}{2} a c}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c x}\right )}{\sqrt {b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 36, normalized size = 0.90 \[ \frac {\log \left (2 \, b c x + a c + 2 \, \sqrt {b c x^{2} + a c x} \sqrt {b c}\right )}{\sqrt {b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*b*c*x + a*c + 2*sqrt(b*c*x^2 + a*c*x)*sqrt(b*c))/sqrt(b*c)

________________________________________________________________________________________

mupad [B]  time = 3.90, size = 33, normalized size = 0.82 \[ \frac {\ln \left (a\,c+2\,\sqrt {b\,c}\,\sqrt {c\,x\,\left (a+b\,x\right )}+2\,b\,c\,x\right )}{\sqrt {b\,c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*c*x + b*c*x^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a c x + b c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*c*x + b*c*x**2), x)

________________________________________________________________________________________