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