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

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

[Out]

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

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Rubi [A]  time = 0.199084, 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 \[ -\frac{\sqrt{a+b x}}{x (a-c)}+\frac{\sqrt{b x+c}}{x (a-c)}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a} (a-c)}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{\sqrt{c} (a-c)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.6214, size = 76, normalized size = 0.74 \[ \frac{b \operatorname{atanh}{\left (\frac{\sqrt{b x + c}}{\sqrt{c}} \right )}}{\sqrt{c} \left (a - c\right )} - \frac{\sqrt{a + b x}}{x \left (a - c\right )} + \frac{\sqrt{b x + c}}{x \left (a - c\right )} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a} \left (a - c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.18027, size = 81, normalized size = 0.79 \[ \frac{-\sqrt{a+b x}-\frac{b x \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}+\sqrt{b x+c}+\frac{b x \tanh ^{-1}\left (\frac{\sqrt{b x+c}}{\sqrt{c}}\right )}{\sqrt{c}}}{a x-c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 88, normalized size = 0.9 \[ 2\,{\frac{b}{a-c} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-2\,{\frac{b}{a-c} \left ( -1/2\,{\frac{\sqrt{bx+c}}{bx}}-1/2\,{\frac{1}{\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{bx+c}}{\sqrt{c}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2}{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.313997, size = 1, normalized size = 0.01 \[ \left [-\frac{b \sqrt{c} x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + \sqrt{a} b x \log \left (\frac{{\left (b x + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{b x + c} c}{x}\right ) + 2 \, \sqrt{b x + a} \sqrt{a} \sqrt{c} - 2 \, \sqrt{b x + c} \sqrt{a} \sqrt{c}}{2 \,{\left (a - c\right )} \sqrt{a} \sqrt{c} x}, -\frac{2 \, \sqrt{a} b x \arctan \left (\frac{c}{\sqrt{b x + c} \sqrt{-c}}\right ) + b \sqrt{-c} x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \, \sqrt{b x + a} \sqrt{a} \sqrt{-c} - 2 \, \sqrt{b x + c} \sqrt{a} \sqrt{-c}}{2 \,{\left (a - c\right )} \sqrt{a} \sqrt{-c} x}, \frac{2 \, b \sqrt{c} x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - \sqrt{-a} b x \log \left (\frac{{\left (b x + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{b x + c} c}{x}\right ) - 2 \, \sqrt{b x + a} \sqrt{-a} \sqrt{c} + 2 \, \sqrt{b x + c} \sqrt{-a} \sqrt{c}}{2 \, \sqrt{-a}{\left (a - c\right )} \sqrt{c} x}, \frac{b \sqrt{-c} x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - \sqrt{-a} b x \arctan \left (\frac{c}{\sqrt{b x + c} \sqrt{-c}}\right ) - \sqrt{b x + a} \sqrt{-a} \sqrt{-c} + \sqrt{b x + c} \sqrt{-a} \sqrt{-c}}{\sqrt{-a}{\left (a - c\right )} \sqrt{-c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(b*sqrt(c)*x*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) + sqrt(a)*b*
x*log(((b*x + 2*c)*sqrt(c) - 2*sqrt(b*x + c)*c)/x) + 2*sqrt(b*x + a)*sqrt(a)*sqr
t(c) - 2*sqrt(b*x + c)*sqrt(a)*sqrt(c))/((a - c)*sqrt(a)*sqrt(c)*x), -1/2*(2*sqr
t(a)*b*x*arctan(c/(sqrt(b*x + c)*sqrt(-c))) + b*sqrt(-c)*x*log(((b*x + 2*a)*sqrt
(a) + 2*sqrt(b*x + a)*a)/x) + 2*sqrt(b*x + a)*sqrt(a)*sqrt(-c) - 2*sqrt(b*x + c)
*sqrt(a)*sqrt(-c))/((a - c)*sqrt(a)*sqrt(-c)*x), 1/2*(2*b*sqrt(c)*x*arctan(a/(sq
rt(b*x + a)*sqrt(-a))) - sqrt(-a)*b*x*log(((b*x + 2*c)*sqrt(c) - 2*sqrt(b*x + c)
*c)/x) - 2*sqrt(b*x + a)*sqrt(-a)*sqrt(c) + 2*sqrt(b*x + c)*sqrt(-a)*sqrt(c))/(s
qrt(-a)*(a - c)*sqrt(c)*x), (b*sqrt(-c)*x*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - s
qrt(-a)*b*x*arctan(c/(sqrt(b*x + c)*sqrt(-c))) - sqrt(b*x + a)*sqrt(-a)*sqrt(-c)
+ sqrt(b*x + c)*sqrt(-a)*sqrt(-c))/(sqrt(-a)*(a - c)*sqrt(-c)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (\sqrt{a + b x} + \sqrt{b x + c}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out