Optimal. Leaf size=82 \[ -\frac{7}{50} \sqrt{5 x^2+2 x+3} x-\frac{261}{250} \sqrt{5 x^2+2 x+3}-\frac{2 (2449 x+2321)}{875 \sqrt{5 x^2+2 x+3}}+\frac{149 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{25 \sqrt{5}} \]
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Rubi [A] time = 0.0865919, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1660, 1661, 640, 619, 215} \[ -\frac{7}{50} \sqrt{5 x^2+2 x+3} x-\frac{261}{250} \sqrt{5 x^2+2 x+3}-\frac{2 (2449 x+2321)}{875 \sqrt{5 x^2+2 x+3}}+\frac{149 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{25 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 1661
Rule 640
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{\left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx &=-\frac{2 (2321+2449 x)}{875 \sqrt{3+2 x+5 x^2}}+\frac{1}{28} \int \frac{\frac{15736}{125}-\frac{3948 x}{25}-\frac{196 x^2}{5}}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{2 (2321+2449 x)}{875 \sqrt{3+2 x+5 x^2}}-\frac{7}{50} x \sqrt{3+2 x+5 x^2}+\frac{1}{280} \int \frac{\frac{34412}{25}-\frac{7308 x}{5}}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{2 (2321+2449 x)}{875 \sqrt{3+2 x+5 x^2}}-\frac{261}{250} \sqrt{3+2 x+5 x^2}-\frac{7}{50} x \sqrt{3+2 x+5 x^2}+\frac{149}{25} \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{2 (2321+2449 x)}{875 \sqrt{3+2 x+5 x^2}}-\frac{261}{250} \sqrt{3+2 x+5 x^2}-\frac{7}{50} x \sqrt{3+2 x+5 x^2}+\frac{149 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{50 \sqrt{70}}\\ &=-\frac{2 (2321+2449 x)}{875 \sqrt{3+2 x+5 x^2}}-\frac{261}{250} \sqrt{3+2 x+5 x^2}-\frac{7}{50} x \sqrt{3+2 x+5 x^2}+\frac{149 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{25 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.155307, size = 55, normalized size = 0.67 \[ \frac{149 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{25 \sqrt{5}}-\frac{245 x^3+1925 x^2+2837 x+2953}{350 \sqrt{5 x^2+2 x+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 98, normalized size = 1.2 \begin{align*} -{\frac{7\,{x}^{3}}{10}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{11\,{x}^{2}}{2}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{149\,x}{25}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{1001}{125}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{7510\,x+1502}{3500}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}+{\frac{149\,\sqrt{5}}{125}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5036, size = 108, normalized size = 1.32 \begin{align*} -\frac{7 \, x^{3}}{10 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} - \frac{11 \, x^{2}}{2 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} + \frac{149}{125} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{2837 \, x}{350 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} - \frac{2953}{350 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38676, size = 254, normalized size = 3.1 \begin{align*} \frac{1043 \, \sqrt{5}{\left (5 \, x^{2} + 2 \, x + 3\right )} \log \left (-\sqrt{5} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) - 5 \,{\left (245 \, x^{3} + 1925 \, x^{2} + 2837 \, x + 2953\right )} \sqrt{5 \, x^{2} + 2 \, x + 3}}{1750 \,{\left (5 \, x^{2} + 2 \, x + 3\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{13 x}{5 x^{2} \sqrt{5 x^{2} + 2 x + 3} + 2 x \sqrt{5 x^{2} + 2 x + 3} + 3 \sqrt{5 x^{2} + 2 x + 3}}\, dx - \int - \frac{7 x^{2}}{5 x^{2} \sqrt{5 x^{2} + 2 x + 3} + 2 x \sqrt{5 x^{2} + 2 x + 3} + 3 \sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{31 x^{3}}{5 x^{2} \sqrt{5 x^{2} + 2 x + 3} + 2 x \sqrt{5 x^{2} + 2 x + 3} + 3 \sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{7 x^{4}}{5 x^{2} \sqrt{5 x^{2} + 2 x + 3} + 2 x \sqrt{5 x^{2} + 2 x + 3} + 3 \sqrt{5 x^{2} + 2 x + 3}}\, dx - \int - \frac{2}{5 x^{2} \sqrt{5 x^{2} + 2 x + 3} + 2 x \sqrt{5 x^{2} + 2 x + 3} + 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 [A] time = 1.18432, size = 84, normalized size = 1.02 \begin{align*} -\frac{149}{125} \, \sqrt{5} \log \left (-\sqrt{5}{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) - \frac{{\left (35 \,{\left (7 \, x + 55\right )} x + 2837\right )} x + 2953}{350 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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