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

Optimal. Leaf size=103 \[ -\frac{\sqrt{a+b x}}{x (b-c)}+\frac{\sqrt{a+c x}}{x (b-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)} \]

[Out]

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

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Rubi [A]  time = 0.193023, antiderivative size = 103, 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 \[ -\frac{\sqrt{a+b x}}{x (b-c)}+\frac{\sqrt{a+c x}}{x (b-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a} (b-c)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.4099, size = 76, normalized size = 0.74 \[ - \frac{\sqrt{a + b x}}{x \left (b - c\right )} + \frac{\sqrt{a + c x}}{x \left (b - c\right )} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (b - c\right )} + \frac{c \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (b - c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.120457, size = 81, normalized size = 0.79 \[ \frac{-\sqrt{a+b x}-\frac{b x \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}+\sqrt{a+c x}+\frac{c x \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{\sqrt{a}}}{b x-c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 88, normalized size = 0.9 \[ 2\,{\frac{b}{b-c} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-2\,{\frac{c}{b-c} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.290659, size = 1, normalized size = 0.01 \[ \left [-\frac{b x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + c x \log \left (\frac{{\left (c x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x + a} a}{x}\right ) + 2 \, \sqrt{b x + a} \sqrt{a} - 2 \, \sqrt{c x + a} \sqrt{a}}{2 \, \sqrt{a}{\left (b - c\right )} x}, \frac{b x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - c x \arctan \left (\frac{a}{\sqrt{c x + a} \sqrt{-a}}\right ) - \sqrt{b x + a} \sqrt{-a} + \sqrt{c x + a} \sqrt{-a}}{\sqrt{-a}{\left (b - c\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(b*x*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) + c*x*log(((c*x + 2*
a)*sqrt(a) - 2*sqrt(c*x + a)*a)/x) + 2*sqrt(b*x + a)*sqrt(a) - 2*sqrt(c*x + a)*s
qrt(a))/(sqrt(a)*(b - c)*x), (b*x*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - c*x*arcta
n(a/(sqrt(c*x + a)*sqrt(-a))) - sqrt(b*x + a)*sqrt(-a) + sqrt(c*x + a)*sqrt(-a))
/(sqrt(-a)*(b - c)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out