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

Optimal. Leaf size=133 \[ \frac{2 b x}{(a-c)^2}-\frac{2 \sqrt{a+b x} \sqrt{b x+c}}{(a-c)^2}-\frac{2 (a+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{(a-c)^2}+\frac{4 \sqrt{a} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right )}{(a-c)^2}+\frac{(a+c) \log (x)}{(a-c)^2} \]

[Out]

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

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Rubi [A]  time = 0.542579, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{2 b x}{(a-c)^2}-\frac{2 \sqrt{a+b x} \sqrt{b x+c}}{(a-c)^2}-\frac{2 (a+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{(a-c)^2}+\frac{4 \sqrt{a} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right )}{(a-c)^2}+\frac{(a+c) \log (x)}{(a-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.3208, size = 119, normalized size = 0.89 \[ \frac{4 \sqrt{a} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{b x + c}} \right )}}{\left (a - c\right )^{2}} + \frac{2 b x}{\left (a - c\right )^{2}} + \frac{\left (a + c\right ) \log{\left (x \right )}}{\left (a - c\right )^{2}} - \frac{2 \left (a + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b x + c}} \right )}}{\left (a - c\right )^{2}} - \frac{2 \sqrt{a + b x} \sqrt{b x + c}}{\left (a - c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.137924, size = 140, normalized size = 1.05 \[ \frac{-2 \sqrt{a+b x} \sqrt{b x+c}-(a+c) \log \left (2 \sqrt{a+b x} \sqrt{b x+c}+a+2 b x+c\right )+2 \sqrt{a} \sqrt{c} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{b x+c}+a b x+2 a c+b c x\right )+\left (-2 \sqrt{a} \sqrt{c}+a+c\right ) \log (x)+2 b x}{(a-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.015, size = 258, normalized size = 1.9 \[{\frac{a\ln \left ( x \right ) }{ \left ( a-c \right ) ^{2}}}+{\frac{c\ln \left ( x \right ) }{ \left ( a-c \right ) ^{2}}}+2\,{\frac{bx}{ \left ( a-c \right ) ^{2}}}+{\frac{{\it csgn} \left ( b \right ) }{ \left ( a-c \right ) ^{2}}\sqrt{bx+a}\sqrt{bx+c} \left ( 2\,{\it csgn} \left ( b \right ) \ln \left ({\frac{abx+bcx+2\,\sqrt{ac}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,ac}{x}} \right ) ac-2\,{\it csgn} \left ( b \right ) \sqrt{ac}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}-\ln \left ({\frac{{\it csgn} \left ( b \right ) }{2} \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ) } \right ) \sqrt{ac}a-\ln \left ({\frac{{\it csgn} \left ( b \right ) }{2} \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ) } \right ) \sqrt{ac}c \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.308151, size = 1, normalized size = 0.01 \[ \left [\frac{16 \, b^{2} x^{2} - 2 \,{\left (8 \, b x + 2 \,{\left (a + c\right )} \log \left (x\right ) + a + c\right )} \sqrt{b x + a} \sqrt{b x + c} - a^{2} + 6 \, a c - c^{2} + 10 \,{\left (a b + b c\right )} x - 2 \,{\left (2 \, \sqrt{b x + a} \sqrt{b x + c}{\left (a + c\right )} - a^{2} - 2 \, a c - c^{2} - 2 \,{\left (a b + b c\right )} x\right )} \log \left (-2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b x + c} - a - c\right ) + 2 \,{\left (a^{2} + 2 \, a c + c^{2} + 2 \,{\left (a b + b c\right )} x\right )} \log \left (x\right ) + 4 \,{\left (\sqrt{a c}{\left (2 \, b x + a + c\right )} - 2 \, \sqrt{a c} \sqrt{b x + a} \sqrt{b x + c}\right )} \log \left (\frac{2 \, b^{2} x^{2} - 2 \, \sqrt{a c} b x - 2 \, \sqrt{b x + a} \sqrt{b x + c}{\left (b x - \sqrt{a c}\right )} + 2 \, a c +{\left (a b + b c\right )} x}{2 \, b x^{2} - 2 \, \sqrt{b x + a} \sqrt{b x + c} x +{\left (a + c\right )} x}\right )}{2 \,{\left (a^{3} - a^{2} c - a c^{2} + c^{3} - 2 \,{\left (a^{2} - 2 \, a c + c^{2}\right )} \sqrt{b x + a} \sqrt{b x + c} + 2 \,{\left (a^{2} b - 2 \, a b c + b c^{2}\right )} x\right )}}, \frac{16 \, b^{2} x^{2} - 2 \,{\left (8 \, b x + 2 \,{\left (a + c\right )} \log \left (x\right ) + a + c\right )} \sqrt{b x + a} \sqrt{b x + c} - a^{2} + 6 \, a c - c^{2} + 10 \,{\left (a b + b c\right )} x + 8 \,{\left (\sqrt{-a c}{\left (2 \, b x + a + c\right )} - 2 \, \sqrt{-a c} \sqrt{b x + a} \sqrt{b x + c}\right )} \arctan \left (-\frac{b x - \sqrt{b x + a} \sqrt{b x + c}}{\sqrt{-a c}}\right ) - 2 \,{\left (2 \, \sqrt{b x + a} \sqrt{b x + c}{\left (a + c\right )} - a^{2} - 2 \, a c - c^{2} - 2 \,{\left (a b + b c\right )} x\right )} \log \left (-2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b x + c} - a - c\right ) + 2 \,{\left (a^{2} + 2 \, a c + c^{2} + 2 \,{\left (a b + b c\right )} x\right )} \log \left (x\right )}{2 \,{\left (a^{3} - a^{2} c - a c^{2} + c^{3} - 2 \,{\left (a^{2} - 2 \, a c + c^{2}\right )} \sqrt{b x + a} \sqrt{b x + c} + 2 \,{\left (a^{2} b - 2 \, a b c + b c^{2}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out