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

Optimal. Leaf size=123 \[ \frac{\sqrt{a+b x} (a+c x)^{3/2}}{a x^2 (b-c)^2}-\frac{a}{x^2 (b-c)^2}+\frac{\sqrt{a+b x} \sqrt{a+c x}}{2 a x (b-c)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{2 a}-\frac{b+c}{x (b-c)^2} \]

[Out]

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

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Rubi [A]  time = 0.452643, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\sqrt{a+b x} (a+c x)^{3/2}}{a x^2 (b-c)^2}-\frac{a}{x^2 (b-c)^2}+\frac{\sqrt{a+b x} \sqrt{a+c x}}{2 a x (b-c)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{2 a}-\frac{b+c}{x (b-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.1656, size = 94, normalized size = 0.76 \[ - \frac{a}{x^{2} \left (b - c\right )^{2}} - \frac{b + c}{x \left (b - c\right )^{2}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a + c x}} \right )}}{2 a} + \frac{\sqrt{a + b x} \sqrt{a + c x}}{2 a x \left (b - c\right )} + \frac{\sqrt{a + b x} \left (a + c x\right )^{\frac{3}{2}}}{a x^{2} \left (b - c\right )^{2}} \]

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))**2,x)

[Out]

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

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Mathematica [A]  time = 0.212935, size = 130, normalized size = 1.06 \[ -\frac{a}{x^2 (b-c)^2}+\sqrt{a+b x} \sqrt{a+c x} \left (\frac{b+c}{2 a x (b-c)^2}+\frac{1}{x^2 (b-c)^2}\right )-\frac{\log \left (2 \sqrt{a+b x} \sqrt{a+c x}+2 a+b x+c x\right )}{4 a}+\frac{\log (x)}{4 a}+\frac{-b-c}{x (b-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a/((b - c)^2*x^2)) + (-b - c)/((b - c)^2*x) + (1/((b - c)^2*x^2) + (b + c)/(2*
a*(b - c)^2*x))*Sqrt[a + b*x]*Sqrt[a + c*x] + Log[x]/(4*a) - Log[2*a + b*x + c*x
+ 2*Sqrt[a + b*x]*Sqrt[a + c*x]]/(4*a)

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Maple [C]  time = 0.017, size = 313, normalized size = 2.5 \[ -{\frac{b}{x \left ( b-c \right ) ^{2}}}-{\frac{c}{x \left ( b-c \right ) ^{2}}}-{\frac{a}{ \left ( b-c \right ) ^{2}{x}^{2}}}+{\frac{{\it csgn} \left ( a \right ) }{4\, \left ( b-c \right ) ^{2}a{x}^{2}}\sqrt{bx+a}\sqrt{cx+a} \left ( -\ln \left ({\frac{a}{x} \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) } \right ){x}^{2}{b}^{2}+2\,\ln \left ({\frac{a \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) }{x}} \right ){x}^{2}bc-\ln \left ({\frac{a}{x} \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) } \right ){x}^{2}{c}^{2}+2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}xb+2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}xc+4\,{\it csgn} \left ( a \right ) a\sqrt{bc{x}^{2}+abx+acx+{a}^{2}} \right ){\frac{1}{\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/x/(b-c)^2*b-1/x/(b-c)^2*c-a/(b-c)^2/x^2+1/4/(b-c)^2*(b*x+a)^(1/2)*(c*x+a)^(1/
2)/a*(-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^2*b^2+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^2*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^2*c^2+2*csgn(a)*(b*c*
x^2+a*b*x+a*c*x+a^2)^(1/2)*x*b+2*csgn(a)*(b*c*x^2+a*b*x+a*c*x+a^2)^(1/2)*x*c+4*c
sgn(a)*a*(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^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.28425, size = 753, normalized size = 6.12 \[ \frac{{\left (b^{4} - 24 \, b^{3} c - 50 \, b^{2} c^{2} - 24 \, b c^{3} + c^{4}\right )} x^{4} - 256 \, a^{4} - 8 \,{\left (5 \, a b^{3} + 39 \, a b^{2} c + 39 \, a b c^{2} + 5 \, a c^{3}\right )} x^{3} - 8 \,{\left (37 \, a^{2} b^{2} + 98 \, a^{2} b c + 37 \, a^{2} c^{2}\right )} x^{2} + 4 \,{\left ({\left (b^{3} + 11 \, b^{2} c + 11 \, b c^{2} + c^{3}\right )} x^{3} + 64 \, a^{3} + 2 \,{\left (17 \, a b^{2} + 42 \, a b c + 17 \, a c^{2}\right )} x^{2} + 96 \,{\left (a^{2} b + a^{2} c\right )} x\right )} \sqrt{b x + a} \sqrt{c x + a} - 512 \,{\left (a^{3} b + a^{3} c\right )} x + 4 \,{\left ({\left (b^{4} + 4 \, b^{3} c - 10 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} x^{4} + 8 \,{\left (a b^{3} - a b^{2} c - a b c^{2} + a c^{3}\right )} x^{3} + 8 \,{\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} x^{2} - 4 \,{\left ({\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} x^{3} + 2 \,{\left (a b^{2} - 2 \, a b c + a c^{2}\right )} x^{2}\right )} \sqrt{b x + a} \sqrt{c x + a}\right )} \log \left (-\frac{{\left (b + c\right )} x - 2 \, \sqrt{b x + a} \sqrt{c x + a} + 2 \, a}{x}\right )}{16 \,{\left ({\left (a b^{4} + 4 \, a b^{3} c - 10 \, a b^{2} c^{2} + 4 \, a b c^{3} + a c^{4}\right )} x^{4} + 8 \,{\left (a^{2} b^{3} - a^{2} b^{2} c - a^{2} b c^{2} + a^{2} c^{3}\right )} x^{3} + 8 \,{\left (a^{3} b^{2} - 2 \, a^{3} b c + a^{3} c^{2}\right )} x^{2} - 4 \,{\left ({\left (a b^{3} - a b^{2} c - a b c^{2} + a c^{3}\right )} x^{3} + 2 \,{\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} x^{2}\right )} \sqrt{b x + a} \sqrt{c x + a}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*((b^4 - 24*b^3*c - 50*b^2*c^2 - 24*b*c^3 + c^4)*x^4 - 256*a^4 - 8*(5*a*b^3
+ 39*a*b^2*c + 39*a*b*c^2 + 5*a*c^3)*x^3 - 8*(37*a^2*b^2 + 98*a^2*b*c + 37*a^2*c
^2)*x^2 + 4*((b^3 + 11*b^2*c + 11*b*c^2 + c^3)*x^3 + 64*a^3 + 2*(17*a*b^2 + 42*a
*b*c + 17*a*c^2)*x^2 + 96*(a^2*b + a^2*c)*x)*sqrt(b*x + a)*sqrt(c*x + a) - 512*(
a^3*b + a^3*c)*x + 4*((b^4 + 4*b^3*c - 10*b^2*c^2 + 4*b*c^3 + c^4)*x^4 + 8*(a*b^
3 - a*b^2*c - a*b*c^2 + a*c^3)*x^3 + 8*(a^2*b^2 - 2*a^2*b*c + a^2*c^2)*x^2 - 4*(
(b^3 - b^2*c - b*c^2 + c^3)*x^3 + 2*(a*b^2 - 2*a*b*c + a*c^2)*x^2)*sqrt(b*x + a)
*sqrt(c*x + a))*log(-((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)/x))/((a*b
^4 + 4*a*b^3*c - 10*a*b^2*c^2 + 4*a*b*c^3 + a*c^4)*x^4 + 8*(a^2*b^3 - a^2*b^2*c
- a^2*b*c^2 + a^2*c^3)*x^3 + 8*(a^3*b^2 - 2*a^3*b*c + a^3*c^2)*x^2 - 4*((a*b^3 -
a*b^2*c - a*b*c^2 + a*c^3)*x^3 + 2*(a^2*b^2 - 2*a^2*b*c + a^2*c^2)*x^2)*sqrt(b*
x + a)*sqrt(c*x + a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out