Optimal. Leaf size=141 \[ -\frac{161 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{\sqrt{3 x^2+5 x+2} (x+21)}{3 \sqrt{2 x+3}}+\frac{121 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.0831997, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {812, 843, 718, 424, 419} \[ -\frac{\sqrt{3 x^2+5 x+2} (x+21)}{3 \sqrt{2 x+3}}-\frac{161 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{121 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 812
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx &=-\frac{(21+x) \sqrt{2+5 x+3 x^2}}{3 \sqrt{3+2 x}}-\frac{1}{6} \int \frac{-101-121 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{(21+x) \sqrt{2+5 x+3 x^2}}{3 \sqrt{3+2 x}}+\frac{121}{12} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx-\frac{161}{12} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{(21+x) \sqrt{2+5 x+3 x^2}}{3 \sqrt{3+2 x}}+\frac{\left (121 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 x^2}{3}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{6 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{\left (161 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{6 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{(21+x) \sqrt{2+5 x+3 x^2}}{3 \sqrt{3+2 x}}+\frac{121 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{161 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{6 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.364869, size = 185, normalized size = 1.31 \[ \frac{-122 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+10 \left (-9 x^3+159 x^2+284 x+116\right )+605 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{90 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 136, normalized size = 1. \begin{align*} -{\frac{1}{1080\,{x}^{3}+3420\,{x}^{2}+3420\,x+1080}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 40\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +121\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +180\,{x}^{3}+4080\,{x}^{2}+6420\,x+2520 \right ) } \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{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}{\left (x - 5\right )}}{4 \, x^{2} + 12 \, x + 9}, x\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 \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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