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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]