Optimal. Leaf size=141 \[ \frac{2 \sqrt{a+b x} \sqrt{b x+c}}{x (a-c)^2}+\frac{2 b \log (x)}{(a-c)^2}+\frac{2 b (a+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right )}{\sqrt{a} \sqrt{c} (a-c)^2}-\frac{4 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{(a-c)^2}-\frac{a+c}{x (a-c)^2} \]
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Rubi [A] time = 0.215897, antiderivative size = 141, 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, Rules used = {6689, 97, 157, 63, 217, 206, 93, 208} \[ \frac{2 \sqrt{a+b x} \sqrt{b x+c}}{x (a-c)^2}+\frac{2 b \log (x)}{(a-c)^2}+\frac{2 b (a+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right )}{\sqrt{a} \sqrt{c} (a-c)^2}-\frac{4 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{b x+c}}\right )}{(a-c)^2}-\frac{a+c}{x (a-c)^2} \]
Antiderivative was successfully verified.
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Rule 6689
Rule 97
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (\sqrt{a+b x}+\sqrt{c+b x}\right )^2} \, dx &=\frac{\int \left (\frac{a \left (1+\frac{c}{a}\right )}{x^2}+\frac{2 b}{x}-\frac{2 \sqrt{a+b x} \sqrt{c+b x}}{x^2}\right ) \, dx}{(a-c)^2}\\ &=-\frac{a+c}{(a-c)^2 x}+\frac{2 b \log (x)}{(a-c)^2}-\frac{2 \int \frac{\sqrt{a+b x} \sqrt{c+b x}}{x^2} \, dx}{(a-c)^2}\\ &=-\frac{a+c}{(a-c)^2 x}+\frac{2 \sqrt{a+b x} \sqrt{c+b x}}{(a-c)^2 x}+\frac{2 b \log (x)}{(a-c)^2}-\frac{2 \int \frac{\frac{1}{2} b (a+c)+b^2 x}{x \sqrt{a+b x} \sqrt{c+b x}} \, dx}{(a-c)^2}\\ &=-\frac{a+c}{(a-c)^2 x}+\frac{2 \sqrt{a+b x} \sqrt{c+b x}}{(a-c)^2 x}+\frac{2 b \log (x)}{(a-c)^2}-\frac{\left (2 b^2\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+b x}} \, dx}{(a-c)^2}-\frac{(b (a+c)) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+b x}} \, dx}{(a-c)^2}\\ &=-\frac{a+c}{(a-c)^2 x}+\frac{2 \sqrt{a+b x} \sqrt{c+b x}}{(a-c)^2 x}+\frac{2 b \log (x)}{(a-c)^2}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+c+x^2}} \, dx,x,\sqrt{a+b x}\right )}{(a-c)^2}-\frac{(2 b (a+c)) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+b x}}\right )}{(a-c)^2}\\ &=-\frac{a+c}{(a-c)^2 x}+\frac{2 \sqrt{a+b x} \sqrt{c+b x}}{(a-c)^2 x}+\frac{2 b (a+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+b x}}\right )}{\sqrt{a} (a-c)^2 \sqrt{c}}+\frac{2 b \log (x)}{(a-c)^2}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+b x}}\right )}{(a-c)^2}\\ &=-\frac{a+c}{(a-c)^2 x}+\frac{2 \sqrt{a+b x} \sqrt{c+b x}}{(a-c)^2 x}-\frac{4 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{c+b x}}\right )}{(a-c)^2}+\frac{2 b (a+c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+b x}}\right )}{\sqrt{a} (a-c)^2 \sqrt{c}}+\frac{2 b \log (x)}{(a-c)^2}\\ \end{align*}
Mathematica [A] time = 0.980618, size = 205, normalized size = 1.45 \[ \frac{\frac{a \left (-\sqrt{b x+c}\right )+2 c \sqrt{a+b x}+2 b x \sqrt{a+b x}-c \sqrt{b x+c}+2 b x \log (x) \sqrt{b x+c}}{x}-4 \sqrt{b} \sqrt{b (c-a)} \sqrt{-\frac{b x+c}{a-c}} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{a+b x}}{\sqrt{b (c-a)}}\right )+\frac{2 b (a+c) \sqrt{b x+c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{b x+c}}\right )}{\sqrt{a} \sqrt{c}}}{(a-c)^2 \sqrt{b x+c}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 274, normalized size = 1.9 \begin{align*} -{\frac{a}{x \left ( a-c \right ) ^{2}}}-{\frac{c}{x \left ( a-c \right ) ^{2}}}+2\,{\frac{b\ln \left ( x \right ) }{ \left ( a-c \right ) ^{2}}}+{\frac{{\it csgn} \left ( b \right ) }{x \left ( a-c \right ) ^{2}}\sqrt{bx+a}\sqrt{bx+c} \left ({\it csgn} \left ( b \right ) \ln \left ({\frac{1}{x} \left ( abx+bcx+2\,\sqrt{ac}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,ac \right ) } \right ) xab+{\it csgn} \left ( b \right ) \ln \left ({\frac{1}{x} \left ( abx+bcx+2\,\sqrt{ac}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,ac \right ) } \right ) xbc-2\,\ln \left ( 1/2\, \left ( 2\,{\it csgn} \left ( b \right ) \sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}+2\,bx+a+c \right ){\it csgn} \left ( b \right ) \right ) xb\sqrt{ac}+2\,{\it csgn} \left ( b \right ) \sqrt{ac}\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+abx+bcx+ac}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2}{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15903, size = 873, normalized size = 6.19 \begin{align*} \left [\frac{2 \, a b c x \log \left (-2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b x + c} - a - c\right ) + 2 \, a b c x \log \left (x\right ) + 2 \, a b c x +{\left (a b + b c\right )} \sqrt{a c} x \log \left (\frac{2 \, a^{2} c + 2 \, a c^{2} + 2 \,{\left (2 \, a c + \sqrt{a c}{\left (a + c\right )}\right )} \sqrt{b x + a} \sqrt{b x + c} +{\left (a^{2} b + 2 \, a b c + b c^{2}\right )} x + 2 \,{\left (2 \, a c +{\left (a b + b c\right )} x\right )} \sqrt{a c}}{x}\right ) + 2 \, \sqrt{b x + a} \sqrt{b x + c} a c - a^{2} c - a c^{2}}{{\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3}\right )} x}, \frac{2 \, a b c x \log \left (-2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b x + c} - a - c\right ) + 2 \, a b c x \log \left (x\right ) + 2 \, a b c x - 2 \,{\left (a b + b c\right )} \sqrt{-a c} x \arctan \left (-\frac{\sqrt{-a c} b x - \sqrt{-a c} \sqrt{b x + a} \sqrt{b x + c}}{a c}\right ) + 2 \, \sqrt{b x + a} \sqrt{b x + c} a c - a^{2} c - a c^{2}}{{\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (\sqrt{a + b x} + \sqrt{b x + c}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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