Optimal. Leaf size=51 \[ \frac{1}{10} \tan ^{-1}\left (\frac{5 (x+2)}{2 \sqrt{5 x^2+2 x-7}}\right )+\frac{1}{5} \tanh ^{-1}\left (\frac{5 (x+1)}{\sqrt{5 x^2+2 x-7}}\right ) \]
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Rubi [A] time = 0.0656829, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {986, 1029, 203, 207} \[ \frac{1}{10} \tan ^{-1}\left (\frac{5 (x+2)}{2 \sqrt{5 x^2+2 x-7}}\right )+\frac{1}{5} \tanh ^{-1}\left (\frac{5 (x+1)}{\sqrt{5 x^2+2 x-7}}\right ) \]
Antiderivative was successfully verified.
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Rule 986
Rule 1029
Rule 203
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-7+2 x+5 x^2} \left (8+12 x+5 x^2\right )} \, dx &=-\left (\frac{1}{50} \int \frac{-100-50 x}{\sqrt{-7+2 x+5 x^2} \left (8+12 x+5 x^2\right )} \, dx\right )+\frac{1}{50} \int \frac{-50-50 x}{\sqrt{-7+2 x+5 x^2} \left (8+12 x+5 x^2\right )} \, dx\\ &=400 \operatorname{Subst}\left (\int \frac{1}{160000+100 x^2} \, dx,x,\frac{200+100 x}{\sqrt{-7+2 x+5 x^2}}\right )+1600 \operatorname{Subst}\left (\int \frac{1}{-640000+100 x^2} \, dx,x,\frac{-400-400 x}{\sqrt{-7+2 x+5 x^2}}\right )\\ &=\frac{1}{10} \tan ^{-1}\left (\frac{5 (2+x)}{2 \sqrt{-7+2 x+5 x^2}}\right )+\frac{1}{5} \tanh ^{-1}\left (\frac{5 (1+x)}{\sqrt{-7+2 x+5 x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0402311, size = 81, normalized size = 1.59 \[ \left (\frac{1}{10}-\frac{i}{20}\right ) \tanh ^{-1}\left (\frac{\left (\frac{1}{100}+\frac{i}{50}\right ) ((100-40 i) x+(164-8 i))}{\sqrt{5 x^2+2 x-7}}\right )-\left (\frac{1}{20}-\frac{i}{10}\right ) \tan ^{-1}\left (\frac{\left (\frac{1}{50}+\frac{i}{100}\right ) ((-100-40 i) x-(164+8 i))}{\sqrt{5 x^2+2 x-7}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.108, size = 144, normalized size = 2.8 \begin{align*} -{\frac{1}{10}\sqrt{-4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+9} \left ( 2\,{\it Artanh} \left ( 1/5\,\sqrt{-4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+9} \right ) +\arctan \left ({\frac{10+5\,x}{-2-2\,x}\sqrt{-4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+9} \left ( 4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}-9 \right ) ^{-1}} \right ) \right ){\frac{1}{\sqrt{-{ \left ( 4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}-9 \right ) \left ( 1+{\frac{2+x}{-1-x}} \right ) ^{-2}}}}} \left ( 1+{\frac{2+x}{-1-x}} \right ) ^{-1}} \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}{{\left (5 \, x^{2} + 12 \, x + 8\right )} \sqrt{5 \, x^{2} + 2 \, x - 7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91039, size = 424, normalized size = 8.31 \begin{align*} \frac{1}{20} \, \arctan \left (\frac{27 \, x^{2} + 20 \, \sqrt{5 \, x^{2} + 2 \, x - 7}{\left (x + 2\right )} + 36 \, x}{31 \, x^{2} + 16 \, x - 56}\right ) + \frac{1}{20} \, \arctan \left (-\frac{27 \, x^{2} - 20 \, \sqrt{5 \, x^{2} + 2 \, x - 7}{\left (x + 2\right )} + 36 \, x}{31 \, x^{2} + 16 \, x - 56}\right ) + \frac{1}{20} \, \log \left (\frac{15 \, x^{2} + 5 \, \sqrt{5 \, x^{2} + 2 \, x - 7}{\left (x + 1\right )} + 26 \, x + 9}{x^{2}}\right ) - \frac{1}{20} \, \log \left (\frac{15 \, x^{2} - 5 \, \sqrt{5 \, x^{2} + 2 \, x - 7}{\left (x + 1\right )} + 26 \, x + 9}{x^{2}}\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}{\sqrt{\left (x - 1\right ) \left (5 x + 7\right )} \left (5 x^{2} + 12 x + 8\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.49367, size = 277, normalized size = 5.43 \begin{align*} -\frac{1}{10} \, \arctan \left (-\frac{5 \, \sqrt{5} x + 6 \, \sqrt{5} - 5 \, \sqrt{5 \, x^{2} + 2 \, x - 7} + 5}{2 \,{\left (\sqrt{5} + 5\right )}}\right ) - \frac{1}{10} \, \arctan \left (\frac{5 \, \sqrt{5} x + 6 \, \sqrt{5} - 5 \, \sqrt{5 \, x^{2} + 2 \, x - 7} - 5}{2 \,{\left (\sqrt{5} - 5\right )}}\right ) + \frac{1}{10} \, \log \left (5 \,{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x - 7}\right )}^{2} + 2 \,{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x - 7}\right )}{\left (6 \, \sqrt{5} + 5\right )} + 20 \, \sqrt{5} + 65\right ) - \frac{1}{10} \, \log \left (5 \,{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x - 7}\right )}^{2} + 2 \,{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x - 7}\right )}{\left (6 \, \sqrt{5} - 5\right )} - 20 \, \sqrt{5} + 65\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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