Optimal. Leaf size=124 \[ \frac{49}{100} \sqrt{5 x^2+2 x+3} x^3+\frac{203}{100} \sqrt{5 x^2+2 x+3} x^2-\frac{8749 \sqrt{5 x^2+2 x+3} x}{1250}-\frac{5086 \sqrt{5 x^2+2 x+3}}{3125}-\frac{8 (136602 x+12983)}{21875 \sqrt{5 x^2+2 x+3}}+\frac{89583 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1250 \sqrt{5}} \]
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Rubi [A] time = 0.160835, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1660, 1661, 640, 619, 215} \[ \frac{49}{100} \sqrt{5 x^2+2 x+3} x^3+\frac{203}{100} \sqrt{5 x^2+2 x+3} x^2-\frac{8749 \sqrt{5 x^2+2 x+3} x}{1250}-\frac{5086 \sqrt{5 x^2+2 x+3}}{3125}-\frac{8 (136602 x+12983)}{21875 \sqrt{5 x^2+2 x+3}}+\frac{89583 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1250 \sqrt{5}} \]
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 )^2 \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx &=-\frac{8 (12983+136602 x)}{21875 \sqrt{3+2 x+5 x^2}}+\frac{1}{28} \int \frac{\frac{4291112}{3125}-\frac{296716 x}{625}-\frac{194012 x^2}{125}+\frac{23716 x^3}{25}+\frac{1372 x^4}{5}}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{8 (12983+136602 x)}{21875 \sqrt{3+2 x+5 x^2}}+\frac{49}{100} x^3 \sqrt{3+2 x+5 x^2}+\frac{1}{560} \int \frac{\frac{17164448}{625}-\frac{1186864 x}{125}-\frac{837788 x^2}{25}+17052 x^3}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{8 (12983+136602 x)}{21875 \sqrt{3+2 x+5 x^2}}+\frac{203}{100} x^2 \sqrt{3+2 x+5 x^2}+\frac{49}{100} x^3 \sqrt{3+2 x+5 x^2}+\frac{\int \frac{\frac{51493344}{125}-\frac{6118392 x}{25}-\frac{2939664 x^2}{5}}{\sqrt{3+2 x+5 x^2}} \, dx}{8400}\\ &=-\frac{8 (12983+136602 x)}{21875 \sqrt{3+2 x+5 x^2}}-\frac{8749 x \sqrt{3+2 x+5 x^2}}{1250}+\frac{203}{100} x^2 \sqrt{3+2 x+5 x^2}+\frac{49}{100} x^3 \sqrt{3+2 x+5 x^2}+\frac{\int \frac{\frac{147081648}{25}-\frac{3417792 x}{5}}{\sqrt{3+2 x+5 x^2}} \, dx}{84000}\\ &=-\frac{8 (12983+136602 x)}{21875 \sqrt{3+2 x+5 x^2}}-\frac{5086 \sqrt{3+2 x+5 x^2}}{3125}-\frac{8749 x \sqrt{3+2 x+5 x^2}}{1250}+\frac{203}{100} x^2 \sqrt{3+2 x+5 x^2}+\frac{49}{100} x^3 \sqrt{3+2 x+5 x^2}+\frac{89583 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{1250}\\ &=-\frac{8 (12983+136602 x)}{21875 \sqrt{3+2 x+5 x^2}}-\frac{5086 \sqrt{3+2 x+5 x^2}}{3125}-\frac{8749 x \sqrt{3+2 x+5 x^2}}{1250}+\frac{203}{100} x^2 \sqrt{3+2 x+5 x^2}+\frac{49}{100} x^3 \sqrt{3+2 x+5 x^2}+\frac{89583 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{2500 \sqrt{70}}\\ &=-\frac{8 (12983+136602 x)}{21875 \sqrt{3+2 x+5 x^2}}-\frac{5086 \sqrt{3+2 x+5 x^2}}{3125}-\frac{8749 x \sqrt{3+2 x+5 x^2}}{1250}+\frac{203}{100} x^2 \sqrt{3+2 x+5 x^2}+\frac{49}{100} x^3 \sqrt{3+2 x+5 x^2}+\frac{89583 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{1250 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.274322, size = 65, normalized size = 0.52 \[ \frac{\frac{5 \left (42875 x^5+194775 x^4-515655 x^3-280805 x^2-1298674 x-168536\right )}{\sqrt{5 x^2+2 x+3}}+1254162 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{87500} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 132, normalized size = 1.1 \begin{align*} -{\frac{28506}{3125}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}+{\frac{49\,{x}^{5}}{20}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}+{\frac{1113\,{x}^{4}}{100}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{8023\,{x}^{2}}{500}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}+{\frac{89583\,\sqrt{5}}{6250}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }-{\frac{55640\,x+11128}{21875}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{89583\,x}{1250}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{14733\,{x}^{3}}{500}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57881, size = 154, normalized size = 1.24 \begin{align*} \frac{49 \, x^{5}}{20 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} + \frac{1113 \, x^{4}}{100 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} - \frac{14733 \, x^{3}}{500 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} - \frac{8023 \, x^{2}}{500 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} + \frac{89583}{6250} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{649337 \, x}{8750 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} - \frac{42134}{4375 \, \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 = 1.30554, size = 305, normalized size = 2.46 \begin{align*} \frac{627081 \, \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 (42875 \, x^{5} + 194775 \, x^{4} - 515655 \, x^{3} - 280805 \, x^{2} - 1298674 \, x - 168536\right )} \sqrt{5 \, x^{2} + 2 \, x + 3}}{87500 \,{\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{\left (x^{2} + 5 x + 2\right ) \left (7 x^{2} - 4 x - 1\right )^{2}}{\left (5 x^{2} + 2 x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16135, size = 96, normalized size = 0.77 \begin{align*} -\frac{89583}{6250} \, \sqrt{5} \log \left (-\sqrt{5}{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) + \frac{{\left (35 \,{\left ({\left (35 \,{\left (35 \, x + 159\right )} x - 14733\right )} x - 8023\right )} x - 1298674\right )} x - 168536}{17500 \, \sqrt{5 \, x^{2} + 2 \, x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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