Optimal. Leaf size=101 \[ -\frac{7}{20} \sqrt{5 x^2+2 x+3} x^3-\frac{571}{300} \sqrt{5 x^2+2 x+3} x^2+\frac{59}{30} \sqrt{5 x^2+2 x+3} x+\frac{463}{125} \sqrt{5 x^2+2 x+3}-\frac{1901 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{250 \sqrt{5}} \]
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Rubi [A] time = 0.145073, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {1661, 640, 619, 215} \[ -\frac{7}{20} \sqrt{5 x^2+2 x+3} x^3-\frac{571}{300} \sqrt{5 x^2+2 x+3} x^2+\frac{59}{30} \sqrt{5 x^2+2 x+3} x+\frac{463}{125} \sqrt{5 x^2+2 x+3}-\frac{1901 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{250 \sqrt{5}} \]
Antiderivative was successfully verified.
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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 )}{\sqrt{3+2 x+5 x^2}} \, dx &=-\frac{7}{20} x^3 \sqrt{3+2 x+5 x^2}+\frac{1}{20} \int \frac{40+260 x+203 x^2-571 x^3}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{571}{300} x^2 \sqrt{3+2 x+5 x^2}-\frac{7}{20} x^3 \sqrt{3+2 x+5 x^2}+\frac{1}{300} \int \frac{600+7326 x+5900 x^2}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=\frac{59}{30} x \sqrt{3+2 x+5 x^2}-\frac{571}{300} x^2 \sqrt{3+2 x+5 x^2}-\frac{7}{20} x^3 \sqrt{3+2 x+5 x^2}+\frac{\int \frac{-11700+55560 x}{\sqrt{3+2 x+5 x^2}} \, dx}{3000}\\ &=\frac{463}{125} \sqrt{3+2 x+5 x^2}+\frac{59}{30} x \sqrt{3+2 x+5 x^2}-\frac{571}{300} x^2 \sqrt{3+2 x+5 x^2}-\frac{7}{20} x^3 \sqrt{3+2 x+5 x^2}-\frac{1901}{250} \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=\frac{463}{125} \sqrt{3+2 x+5 x^2}+\frac{59}{30} x \sqrt{3+2 x+5 x^2}-\frac{571}{300} x^2 \sqrt{3+2 x+5 x^2}-\frac{7}{20} x^3 \sqrt{3+2 x+5 x^2}-\frac{1901 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{500 \sqrt{70}}\\ &=\frac{463}{125} \sqrt{3+2 x+5 x^2}+\frac{59}{30} x \sqrt{3+2 x+5 x^2}-\frac{571}{300} x^2 \sqrt{3+2 x+5 x^2}-\frac{7}{20} x^3 \sqrt{3+2 x+5 x^2}-\frac{1901 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{250 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.0860674, size = 55, normalized size = 0.54 \[ \frac{-5 \sqrt{5 x^2+2 x+3} \left (525 x^3+2855 x^2-2950 x-5556\right )-11406 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{7500} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 79, normalized size = 0.8 \begin{align*} -{\frac{7\,{x}^{3}}{20}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{571\,{x}^{2}}{300}\sqrt{5\,{x}^{2}+2\,x+3}}+{\frac{59\,x}{30}\sqrt{5\,{x}^{2}+2\,x+3}}+{\frac{463}{125}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{1901\,\sqrt{5}}{1250}{\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 = 1.53065, size = 108, normalized size = 1.07 \begin{align*} -\frac{7}{20} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x^{3} - \frac{571}{300} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x^{2} + \frac{59}{30} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x - \frac{1901}{1250} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) + \frac{463}{125} \, \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 = 1.24932, size = 207, normalized size = 2.05 \begin{align*} -\frac{1}{1500} \,{\left (525 \, x^{3} + 2855 \, x^{2} - 2950 \, x - 5556\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{1901}{2500} \, \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{13 x}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int - \frac{7 x^{2}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{31 x^{3}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{7 x^{4}}{\sqrt{5 x^{2} + 2 x + 3}}\, dx - \int - \frac{2}{\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 = 1.19552, size = 84, normalized size = 0.83 \begin{align*} -\frac{1}{1500} \,{\left (5 \,{\left ({\left (105 \, x + 571\right )} x - 590\right )} x - 5556\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{1901}{1250} \, \sqrt{5} \log \left (-\sqrt{5}{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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