Optimal. Leaf size=164 \[ -\frac{3}{14} \sqrt{\frac{4091-1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )+\frac{3}{14} \sqrt{\frac{4091+1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )-\frac{\sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{7 \sqrt{5}} \]
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Rubi [A] time = 0.230132, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {1076, 619, 215, 1032, 724, 206} \[ -\frac{3}{14} \sqrt{\frac{4091-1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )+\frac{3}{14} \sqrt{\frac{4091+1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )-\frac{\sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{7 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1076
Rule 619
Rule 215
Rule 1032
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx &=-\left (\frac{1}{7} \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx\right )-\frac{1}{7} \int \frac{-15-39 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{14 \sqrt{70}}+\frac{1}{77} \left (3 \left (143-61 \sqrt{11}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx+\frac{1}{77} \left (3 \left (143+61 \sqrt{11}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{\sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{7 \sqrt{5}}-\frac{1}{77} \left (6 \left (143-61 \sqrt{11}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2352+112 \left (4-2 \sqrt{11}\right )+20 \left (4-2 \sqrt{11}\right )^2-x^2} \, dx,x,\frac{-84-2 \left (4-2 \sqrt{11}\right )-\left (28+10 \left (4-2 \sqrt{11}\right )\right ) x}{\sqrt{3+2 x+5 x^2}}\right )-\frac{1}{77} \left (6 \left (143+61 \sqrt{11}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2352+112 \left (4+2 \sqrt{11}\right )+20 \left (4+2 \sqrt{11}\right )^2-x^2} \, dx,x,\frac{-84-2 \left (4+2 \sqrt{11}\right )-\left (28+10 \left (4+2 \sqrt{11}\right )\right ) x}{\sqrt{3+2 x+5 x^2}}\right )\\ &=-\frac{\sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{7 \sqrt{5}}-\frac{3}{14} \sqrt{\frac{4091-1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{23-\sqrt{11}+\left (17-5 \sqrt{11}\right ) x}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{3+2 x+5 x^2}}\right )+\frac{3}{14} \sqrt{\frac{4091+1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{23+\sqrt{11}+\left (17+5 \sqrt{11}\right ) x}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{3+2 x+5 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.484516, size = 157, normalized size = 0.96 \[ -\frac{3 \left (\sqrt{4091-1055 \sqrt{11}} \tanh ^{-1}\left (\frac{-5 \sqrt{11} x+17 x-\sqrt{11}+23}{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )-\sqrt{4091+1055 \sqrt{11}} \tanh ^{-1}\left (\frac{5 \sqrt{11} x+17 x+\sqrt{11}+23}{\sqrt{250+34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )\right )}{14 \sqrt{2794}}-\frac{\sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{7 \sqrt{5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 204, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{5}}{35}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }+{\frac{ \left ( 183+39\,\sqrt{11} \right ) \sqrt{11}}{154\,\sqrt{250+34\,\sqrt{11}}}{\it Artanh} \left ({\frac{49}{2\,\sqrt{250+34\,\sqrt{11}}} \left ({\frac{500}{49}}+{\frac{68\,\sqrt{11}}{49}}+ \left ({\frac{34}{7}}+{\frac{10\,\sqrt{11}}{7}} \right ) \left ( x-{\frac{2}{7}}-{\frac{\sqrt{11}}{7}} \right ) \right ){\frac{1}{\sqrt{245\, \left ( x-2/7-1/7\,\sqrt{11} \right ) ^{2}+49\, \left ({\frac{34}{7}}+{\frac{10\,\sqrt{11}}{7}} \right ) \left ( x-2/7-1/7\,\sqrt{11} \right ) +250+34\,\sqrt{11}}}}} \right ) }+{\frac{ \left ( -183+39\,\sqrt{11} \right ) \sqrt{11}}{154\,\sqrt{250-34\,\sqrt{11}}}{\it Artanh} \left ({\frac{49}{2\,\sqrt{250-34\,\sqrt{11}}} \left ({\frac{500}{49}}-{\frac{68\,\sqrt{11}}{49}}+ \left ({\frac{34}{7}}-{\frac{10\,\sqrt{11}}{7}} \right ) \left ( x-{\frac{2}{7}}+{\frac{\sqrt{11}}{7}} \right ) \right ){\frac{1}{\sqrt{245\, \left ( x-2/7+1/7\,\sqrt{11} \right ) ^{2}+49\, \left ({\frac{34}{7}}-{\frac{10\,\sqrt{11}}{7}} \right ) \left ( x-2/7+1/7\,\sqrt{11} \right ) +250-34\,\sqrt{11}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.65272, size = 628, normalized size = 3.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54786, size = 1126, normalized size = 6.87 \begin{align*} -\frac{3}{78232} \, \sqrt{2794} \sqrt{1055 \, \sqrt{11} + 4091} \log \left (\frac{3 \,{\left (\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{1055 \, \sqrt{11} + 4091}{\left (172 \, \sqrt{11} - 715\right )} + 185801 \, \sqrt{11}{\left (x + 3\right )} + 557403 \, x - 929005\right )}}{x}\right ) + \frac{3}{78232} \, \sqrt{2794} \sqrt{1055 \, \sqrt{11} + 4091} \log \left (-\frac{3 \,{\left (\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{1055 \, \sqrt{11} + 4091}{\left (172 \, \sqrt{11} - 715\right )} - 185801 \, \sqrt{11}{\left (x + 3\right )} - 557403 \, x + 929005\right )}}{x}\right ) - \frac{1}{78232} \, \sqrt{2794} \sqrt{-9495 \, \sqrt{11} + 36819} \log \left (-\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (172 \, \sqrt{11} + 715\right )} \sqrt{-9495 \, \sqrt{11} + 36819} + 557403 \, \sqrt{11}{\left (x + 3\right )} - 1672209 \, x + 2787015}{x}\right ) + \frac{1}{78232} \, \sqrt{2794} \sqrt{-9495 \, \sqrt{11} + 36819} \log \left (\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (172 \, \sqrt{11} + 715\right )} \sqrt{-9495 \, \sqrt{11} + 36819} - 557403 \, \sqrt{11}{\left (x + 3\right )} + 1672209 \, x - 2787015}{x}\right ) + \frac{1}{70} \, \sqrt{5} \log \left (\sqrt{5} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\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{5 x}{7 x^{2} \sqrt{5 x^{2} + 2 x + 3} - 4 x \sqrt{5 x^{2} + 2 x + 3} - \sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{x^{2}}{7 x^{2} \sqrt{5 x^{2} + 2 x + 3} - 4 x \sqrt{5 x^{2} + 2 x + 3} - \sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{2}{7 x^{2} \sqrt{5 x^{2} + 2 x + 3} - 4 x \sqrt{5 x^{2} + 2 x + 3} - \sqrt{5 x^{2} + 2 x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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