3.243 \(\int \frac{1}{x \left (\sqrt{a+b x}+\sqrt{c+b x}\right )} \, dx\)

Optimal. Leaf size=97 \[ \frac{2 \sqrt{a+b x}}{a-c}-\frac{2 \sqrt{b x+c}}{a-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a-c}+\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{a-c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.197203, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{2 \sqrt{a+b x}}{a-c}-\frac{2 \sqrt{b x+c}}{a-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a-c}+\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{a-c} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(Sqrt[a + b*x] + Sqrt[c + b*x])),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.0611, size = 76, normalized size = 0.78 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a - c} + \frac{2 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{b x + c}}{\sqrt{c}} \right )}}{a - c} + \frac{2 \sqrt{a + b x}}{a - c} - \frac{2 \sqrt{b x + c}}{a - c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/((b*x+a)**(1/2)+(b*x+c)**(1/2)),x)

[Out]

-2*sqrt(a)*atanh(sqrt(a + b*x)/sqrt(a))/(a - c) + 2*sqrt(c)*atanh(sqrt(b*x + c)/
sqrt(c))/(a - c) + 2*sqrt(a + b*x)/(a - c) - 2*sqrt(b*x + c)/(a - c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0769796, size = 75, normalized size = 0.77 \[ \frac{2 \left (\sqrt{a+b x}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{b x+c}+\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )\right )}{a-c} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[c + b*x])),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 73, normalized size = 0.8 \[{\frac{1}{a-c} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-{\frac{1}{a-c} \left ( 2\,\sqrt{bx+c}-2\,\sqrt{c}{\it Artanh} \left ({\frac{\sqrt{bx+c}}{\sqrt{c}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x*(sqrt(b*x + a) + sqrt(b*x + c))),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.342296, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \sqrt{c} \log \left (\frac{b x - 2 \, \sqrt{b x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \, \sqrt{b x + a} + 2 \, \sqrt{b x + c}}{a - c}, \frac{2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{b x + c}}{\sqrt{-c}}\right ) - \sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a} - 2 \, \sqrt{b x + c}}{a - c}, -\frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) + \sqrt{c} \log \left (\frac{b x - 2 \, \sqrt{b x + c} \sqrt{c} + 2 \, c}{x}\right ) - 2 \, \sqrt{b x + a} + 2 \, \sqrt{b x + c}}{a - c}, -\frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) - \sqrt{-c} \arctan \left (\frac{\sqrt{b x + c}}{\sqrt{-c}}\right ) - \sqrt{b x + a} + \sqrt{b x + c}\right )}}{a - c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x*(sqrt(b*x + a) + sqrt(b*x + c))),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (\sqrt{a + b x} + \sqrt{b x + c}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x*(sqrt(b*x + a) + sqrt(b*x + c))),x, algorithm="giac")

[Out]

Exception raised: TypeError