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

Optimal. Leaf size=135 \[ -\frac{2 \sqrt{a+b x} \sqrt{a+c x}}{(b-c)^2}+\frac{2 a \log (x)}{(b-c)^2}+\frac{4 a \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{(b-c)^2}-\frac{2 a (b+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right )}{\sqrt{b} \sqrt{c} (b-c)^2}+\frac{x (b+c)}{(b-c)^2} \]

[Out]

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

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Rubi [A]  time = 0.453467, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ -\frac{2 \sqrt{a+b x} \sqrt{a+c x}}{(b-c)^2}+\frac{2 a \log (x)}{(b-c)^2}+\frac{4 a \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a+c x}}\right )}{(b-c)^2}-\frac{2 a (b+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{b} \sqrt{a+c x}}\right )}{\sqrt{b} \sqrt{c} (b-c)^2}+\frac{x (b+c)}{(b-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a \log{\left (x \right )}}{\left (b - c\right )^{2}} + \frac{4 a \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a + c x}} \right )}}{\left (b - c\right )^{2}} - \frac{2 a \left (b + c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{b} \sqrt{a + c x}} \right )}}{\sqrt{b} \sqrt{c} \left (b - c\right )^{2}} - \frac{2 \sqrt{a + b x} \sqrt{a + c x}}{\left (b - c\right )^{2}} + \frac{\left (b + c\right ) \int b\, dx}{b \left (b - c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.17939, size = 128, normalized size = 0.95 \[ \frac{-2 \sqrt{a+b x} \sqrt{a+c x}+2 a \log \left (2 \sqrt{a+b x} \sqrt{a+c x}+2 a+b x+c x\right )-\frac{a (b+c) \log \left (2 \sqrt{b} \sqrt{c} \sqrt{a+b x} \sqrt{a+c x}+a b+a c+2 b c x\right )}{\sqrt{b} \sqrt{c}}+b x+c x}{(b-c)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.018, size = 266, normalized size = 2. \[{\frac{bx}{ \left ( b-c \right ) ^{2}}}+{\frac{cx}{ \left ( b-c \right ) ^{2}}}+2\,{\frac{a\ln \left ( x \right ) }{ \left ( b-c \right ) ^{2}}}-{\frac{{\it csgn} \left ( a \right ) }{ \left ( b-c \right ) ^{2}}\sqrt{bx+a}\sqrt{cx+a} \left ({\it csgn} \left ( a \right ) \ln \left ({\frac{1}{2} \left ( 2\,bcx+2\,\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}\sqrt{bc}+ab+ac \right ){\frac{1}{\sqrt{bc}}}} \right ) ab+{\it csgn} \left ( a \right ) \ln \left ({\frac{1}{2} \left ( 2\,bcx+2\,\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}\sqrt{bc}+ab+ac \right ){\frac{1}{\sqrt{bc}}}} \right ) ac+2\,{\it csgn} \left ( a \right ) \sqrt{bc}\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}-2\,\ln \left ({\frac{a \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{bc{x}^{2}+abx+acx+{a}^{2}}+bx+cx+2\,a \right ) }{x}} \right ) \sqrt{bc}a \right ){\frac{1}{\sqrt{bc{x}^{2}+abx+acx+{a}^{2}}}}{\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.318313, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(4*sqrt(b*c)*((b + c)*x + a*log(x))*sqrt(b*x + a)*sqrt(c*x + a) - (2*a^2*b + 2*
a^2*c - 2*(a*b + a*c)*sqrt(b*x + a)*sqrt(c*x + a) + (a*b^2 + 2*a*b*c + a*c^2)*x)
*log((2*a*b*c*x - 2*(b*c*x + sqrt(b*c)*a)*sqrt(b*x + a)*sqrt(c*x + a) + (2*b*c*x
^2 + 2*a^2 + (a*b + a*c)*x)*sqrt(b*c))/((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a
) + 2*a)) - 2*(2*sqrt(b*c)*sqrt(b*x + a)*sqrt(c*x + a)*a - (2*a^2 + (a*b + a*c)*
x)*sqrt(b*c))*log(-((b + c)*x - 2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)/x) - ((b^2
+ 6*b*c + c^2)*x^2 + 4*(a*b + a*c)*x + 2*(2*a^2 + (a*b + a*c)*x)*log(x))*sqrt(b*
c))/(2*(b^2 - 2*b*c + c^2)*sqrt(b*c)*sqrt(b*x + a)*sqrt(c*x + a) - (2*a*b^2 - 4*
a*b*c + 2*a*c^2 + (b^3 - b^2*c - b*c^2 + c^3)*x)*sqrt(b*c)), (4*sqrt(-b*c)*((b +
c)*x + a*log(x))*sqrt(b*x + a)*sqrt(c*x + a) + 2*(2*a^2*b + 2*a^2*c - 2*(a*b +
a*c)*sqrt(b*x + a)*sqrt(c*x + a) + (a*b^2 + 2*a*b*c + a*c^2)*x)*arctan((sqrt(-b*
c)*sqrt(b*x + a)*sqrt(c*x + a) - sqrt(-b*c)*a)/(b*c*x)) - 2*(2*sqrt(-b*c)*sqrt(b
*x + a)*sqrt(c*x + a)*a - (2*a^2 + (a*b + a*c)*x)*sqrt(-b*c))*log(-((b + c)*x -
2*sqrt(b*x + a)*sqrt(c*x + a) + 2*a)/x) - ((b^2 + 6*b*c + c^2)*x^2 + 4*(a*b + a*
c)*x + 2*(2*a^2 + (a*b + a*c)*x)*log(x))*sqrt(-b*c))/(2*(b^2 - 2*b*c + c^2)*sqrt
(-b*c)*sqrt(b*x + a)*sqrt(c*x + a) - (2*a*b^2 - 4*a*b*c + 2*a*c^2 + (b^3 - b^2*c
- b*c^2 + c^3)*x)*sqrt(-b*c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out