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

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

[Out]

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

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Rubi [A]  time = 0.150429, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{2 \sqrt{a+b x}}{b-c}-\frac{2 \sqrt{a+c x}}{b-c}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{b-c}+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{b-c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.1754, size = 76, normalized size = 0.78 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{b - c} + \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + c x}}{\sqrt{a}} \right )}}{b - c} + \frac{2 \sqrt{a + b x}}{b - c} - \frac{2 \sqrt{a + c x}}{b - c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0620588, size = 75, normalized size = 0.77 \[ \frac{2 \left (\sqrt{a+b x}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{a+c x}+\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )\right )}{b-c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 73, normalized size = 0.8 \[{\frac{1}{b-c} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-{\frac{1}{b-c} \left ( 2\,\sqrt{cx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(b-c)*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-1/(b-c)*(2*(c
*x+a)^(1/2)-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. \[ \int \frac{1}{\sqrt{b x + a} + \sqrt{c x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.279846, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{a} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \sqrt{a} \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \, \sqrt{b x + a} + 2 \, \sqrt{c x + a}}{b - c}, -\frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) - \sqrt{-a} \arctan \left (\frac{\sqrt{c x + a}}{\sqrt{-a}}\right ) - \sqrt{b x + a} + \sqrt{c x + a}\right )}}{b - c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out