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

Optimal. Leaf size=138 \[ -\frac{2 a}{x (b-c)^2}+\frac{2 \sqrt{a+b x} \sqrt{a+c x}}{x (b-c)^2}+\frac{2 (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{(b-c)^2}-\frac{4 \sqrt{b} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right )}{(b-c)^2}+\frac{(b+c) \log (x)}{(b-c)^2} \]

[Out]

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

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Rubi [A]  time = 0.336803, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 a}{x (b-c)^2}+\frac{2 \sqrt{a+b x} \sqrt{a+c x}}{x (b-c)^2}+\frac{2 (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{(b-c)^2}-\frac{4 \sqrt{b} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right )}{(b-c)^2}+\frac{(b+c) \log (x)}{(b-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.632, size = 121, normalized size = 0.88 \[ - \frac{2 a}{x \left (b - c\right )^{2}} - \frac{4 \sqrt{b} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{a + c x}}{\sqrt{c} \sqrt{a + b x}} \right )}}{\left (b - c\right )^{2}} + \frac{\left (b + c\right ) \log{\left (x \right )}}{\left (b - c\right )^{2}} + \frac{2 \left (b + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a + c x}} \right )}}{\left (b - c\right )^{2}} + \frac{2 \sqrt{a + b x} \sqrt{a + c x}}{x \left (b - c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a/(x*(b - c)**2) - 4*sqrt(b)*sqrt(c)*atanh(sqrt(b)*sqrt(a + c*x)/(sqrt(c)*sqr
t(a + b*x)))/(b - c)**2 + (b + c)*log(x)/(b - c)**2 + 2*(b + c)*atanh(sqrt(a + b
*x)/sqrt(a + c*x))/(b - c)**2 + 2*sqrt(a + b*x)*sqrt(a + c*x)/(x*(b - c)**2)

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Mathematica [A]  time = 0.0839888, size = 127, normalized size = 0.92 \[ \frac{2 \sqrt{a+b x} \sqrt{a+c x}+x (b+c) \log \left (2 \sqrt{a+b x} \sqrt{a+c x}+2 a+b x+c x\right )-2 \sqrt{b} \sqrt{c} x \log \left (2 \sqrt{b} \sqrt{c} \sqrt{a+b x} \sqrt{a+c x}+a b+a c+2 b c x\right )-2 a}{x (b-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.017, size = 272, normalized size = 2. \[{\frac{b\ln \left ( x \right ) }{ \left ( b-c \right ) ^{2}}}+{\frac{c\ln \left ( x \right ) }{ \left ( b-c \right ) ^{2}}}-2\,{\frac{a}{ \left ( b-c \right ) ^{2}x}}-{\frac{{\it csgn} \left ( a \right ) }{ \left ( b-c \right ) ^{2}x}\sqrt{bx+a}\sqrt{cx+a} \left ( 2\,{\it csgn} \left ( a \right ) \ln \left ( 1/2\,{\frac{2\,bcx+2\,\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}\sqrt{bc}+ab+ac}{\sqrt{bc}}} \right ) xbc-\ln \left ({\frac{a}{x} \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) } \right ) xb\sqrt{bc}-\ln \left ({\frac{a}{x} \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) } \right ) xc\sqrt{bc}-2\,{\it csgn} \left ( a \right ) \sqrt{bc}\sqrt{bc{x}^{2}+abx+acx+{a}^{2}} \right ){\frac{1}{\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}}}{\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.302274, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((b^2 - 6*b*c + c^2)*x^2 - 2*(2*(b + c)*x*log(x) - (b + c)*x - 8*a)*sqrt(b
*x + a)*sqrt(c*x + a) - 16*a^2 - 10*(a*b + a*c)*x + 2*((b^2 + 2*b*c + c^2)*x^2 +
2*(a*b + a*c)*x)*log(x) - 4*(2*sqrt(b*c)*sqrt(b*x + a)*sqrt(c*x + a)*x - ((b +
c)*x^2 + 2*a*x)*sqrt(b*c))*log((2*b*c*x^2 + 2*sqrt(b*c)*a*x - 2*sqrt(b*x + a)*sq
rt(c*x + a)*(sqrt(b*c)*x + a) + 2*a^2 + (a*b + a*c)*x)/((b + c)*x - 2*sqrt(b*x +
a)*sqrt(c*x + a) + 2*a)) + 2*(2*sqrt(b*x + a)*sqrt(c*x + a)*(b + c)*x - (b^2 +
2*b*c + c^2)*x^2 - 2*(a*b + a*c)*x)*log(-((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x +
a) + 2*a)/x))/(2*(b^2 - 2*b*c + c^2)*sqrt(b*x + a)*sqrt(c*x + a)*x - (b^3 - b^2
*c - b*c^2 + c^3)*x^2 - 2*(a*b^2 - 2*a*b*c + a*c^2)*x), -1/2*((b^2 - 6*b*c + c^2
)*x^2 - 2*(2*(b + c)*x*log(x) - (b + c)*x - 8*a)*sqrt(b*x + a)*sqrt(c*x + a) - 1
6*a^2 - 10*(a*b + a*c)*x + 8*(2*sqrt(-b*c)*sqrt(b*x + a)*sqrt(c*x + a)*x - ((b +
c)*x^2 + 2*a*x)*sqrt(-b*c))*arctan((sqrt(b*x + a)*sqrt(c*x + a) - a)/(sqrt(-b*c
)*x)) + 2*((b^2 + 2*b*c + c^2)*x^2 + 2*(a*b + a*c)*x)*log(x) + 2*(2*sqrt(b*x + a
)*sqrt(c*x + a)*(b + c)*x - (b^2 + 2*b*c + c^2)*x^2 - 2*(a*b + a*c)*x)*log(-((b
+ c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)/x))/(2*(b^2 - 2*b*c + c^2)*sqrt(b*
x + a)*sqrt(c*x + a)*x - (b^3 - b^2*c - b*c^2 + c^3)*x^2 - 2*(a*b^2 - 2*a*b*c +
a*c^2)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out