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

Optimal. Leaf size=171 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2} (b-c)}-\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{4 a^{3/2} (b-c)}-\frac{\sqrt{a+b x}}{2 x^2 (b-c)}+\frac{\sqrt{a+c x}}{2 x^2 (b-c)}-\frac{b \sqrt{a+b x}}{4 a x (b-c)}+\frac{c \sqrt{a+c x}}{4 a x (b-c)} \]

[Out]

-Sqrt[a + b*x]/(2*(b - c)*x^2) - (b*Sqrt[a + b*x])/(4*a*(b - c)*x) + Sqrt[a + c*
x]/(2*(b - c)*x^2) + (c*Sqrt[a + c*x])/(4*a*(b - c)*x) + (b^2*ArcTanh[Sqrt[a + b
*x]/Sqrt[a]])/(4*a^(3/2)*(b - c)) - (c^2*ArcTanh[Sqrt[a + c*x]/Sqrt[a]])/(4*a^(3
/2)*(b - c))

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Rubi [A]  time = 0.247181, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2} (b-c)}-\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{4 a^{3/2} (b-c)}-\frac{\sqrt{a+b x}}{2 x^2 (b-c)}+\frac{\sqrt{a+c x}}{2 x^2 (b-c)}-\frac{b \sqrt{a+b x}}{4 a x (b-c)}+\frac{c \sqrt{a+c x}}{4 a x (b-c)} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*x]/(2*(b - c)*x^2) - (b*Sqrt[a + b*x])/(4*a*(b - c)*x) + Sqrt[a + c*
x]/(2*(b - c)*x^2) + (c*Sqrt[a + c*x])/(4*a*(b - c)*x) + (b^2*ArcTanh[Sqrt[a + b
*x]/Sqrt[a]])/(4*a^(3/2)*(b - c)) - (c^2*ArcTanh[Sqrt[a + c*x]/Sqrt[a]])/(4*a^(3
/2)*(b - c))

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Rubi in Sympy [A]  time = 24.6208, size = 128, normalized size = 0.75 \[ - \frac{\sqrt{a + b x}}{2 x^{2} \left (b - c\right )} + \frac{\sqrt{a + c x}}{2 x^{2} \left (b - c\right )} - \frac{b \sqrt{a + b x}}{4 a x \left (b - c\right )} + \frac{c \sqrt{a + c x}}{4 a x \left (b - c\right )} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \left (b - c\right )} - \frac{c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} \left (b - c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)/(2*x**2*(b - c)) + sqrt(a + c*x)/(2*x**2*(b - c)) - b*sqrt(a + b*
x)/(4*a*x*(b - c)) + c*sqrt(a + c*x)/(4*a*x*(b - c)) + b**2*atanh(sqrt(a + b*x)/
sqrt(a))/(4*a**(3/2)*(b - c)) - c**2*atanh(sqrt(a + c*x)/sqrt(a))/(4*a**(3/2)*(b
- c))

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Mathematica [A]  time = 0.183354, size = 123, normalized size = 0.72 \[ \frac{b^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\sqrt{a} \left (-2 a \sqrt{a+b x}-b x \sqrt{a+b x}+2 a \sqrt{a+c x}+c x \sqrt{a+c x}\right )-c^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{4 a^{3/2} x^2 (b-c)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*(-2*a*Sqrt[a + b*x] - b*x*Sqrt[a + b*x] + 2*a*Sqrt[a + c*x] + c*x*Sqrt[
a + c*x]) + b^2*x^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] - c^2*x^2*ArcTanh[Sqrt[a + c*
x]/Sqrt[a]])/(4*a^(3/2)*(b - c)*x^2)

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Maple [A]  time = 0.015, size = 120, normalized size = 0.7 \[ 2\,{\frac{{b}^{2}}{b-c} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( bx+a \right ) ^{3/2}}{a}}-1/8\,\sqrt{bx+a} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-2\,{\frac{{c}^{2}}{b-c} \left ({\frac{1}{{c}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( cx+a \right ) ^{3/2}}{a}}-1/8\,\sqrt{cx+a} \right ) }+1/8\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(b-c)*b^2*((-1/8/a*(b*x+a)^(3/2)-1/8*(b*x+a)^(1/2))/x^2/b^2+1/8/a^(3/2)*arctan
h((b*x+a)^(1/2)/a^(1/2)))-2/(b-c)*c^2*((-1/8/a*(c*x+a)^(3/2)-1/8*(c*x+a)^(1/2))/
c^2/x^2+1/8/a^(3/2)*arctanh((c*x+a)^(1/2)/a^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.310516, size = 1, normalized size = 0.01 \[ \left [-\frac{b^{2} x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + c^{2} x^{2} \log \left (\frac{{\left (c x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x + a} a}{x}\right ) + 2 \,{\left (b x + 2 \, a\right )} \sqrt{b x + a} \sqrt{a} - 2 \,{\left (c x + 2 \, a\right )} \sqrt{c x + a} \sqrt{a}}{8 \,{\left (a b - a c\right )} \sqrt{a} x^{2}}, -\frac{b^{2} x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) - c^{2} x^{2} \arctan \left (\frac{a}{\sqrt{c x + a} \sqrt{-a}}\right ) +{\left (b x + 2 \, a\right )} \sqrt{b x + a} \sqrt{-a} -{\left (c x + 2 \, a\right )} \sqrt{c x + a} \sqrt{-a}}{4 \,{\left (a b - a c\right )} \sqrt{-a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(b^2*x^2*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) + c^2*x^2*log(((
c*x + 2*a)*sqrt(a) + 2*sqrt(c*x + a)*a)/x) + 2*(b*x + 2*a)*sqrt(b*x + a)*sqrt(a)
- 2*(c*x + 2*a)*sqrt(c*x + a)*sqrt(a))/((a*b - a*c)*sqrt(a)*x^2), -1/4*(b^2*x^2
*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - c^2*x^2*arctan(a/(sqrt(c*x + a)*sqrt(-a)))
+ (b*x + 2*a)*sqrt(b*x + a)*sqrt(-a) - (c*x + 2*a)*sqrt(c*x + a)*sqrt(-a))/((a*
b - a*c)*sqrt(-a)*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out