Optimal. Leaf size=202 \[ \frac{4153 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{3564 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{33} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}+\frac{1}{99} \sqrt{2 x+3} (127 x+119) \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (3987 x+1246) \sqrt{3 x^2+5 x+2}}{8910}-\frac{15283 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{17820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.129675, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ -\frac{2}{33} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}+\frac{1}{99} \sqrt{2 x+3} (127 x+119) \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (3987 x+1246) \sqrt{3 x^2+5 x+2}}{8910}+\frac{4153 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3564 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{15283 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{17820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int (5-x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}+\frac{2}{33} \int \frac{\left (287+\frac{381 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx\\ &=\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{\int \frac{\left (\frac{9807}{2}+\frac{9303 x}{2}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{2079}\\ &=-\frac{\sqrt{3+2 x} (1246+3987 x) \sqrt{2+5 x+3 x^2}}{8910}+\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}+\frac{\int \frac{-131691-\frac{320943 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{187110}\\ &=-\frac{\sqrt{3+2 x} (1246+3987 x) \sqrt{2+5 x+3 x^2}}{8910}+\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{15283 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{35640}+\frac{4153 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{7128}\\ &=-\frac{\sqrt{3+2 x} (1246+3987 x) \sqrt{2+5 x+3 x^2}}{8910}+\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{\left (15283 \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 )}{17820 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (4153 \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 )}{3564 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{\sqrt{3+2 x} (1246+3987 x) \sqrt{2+5 x+3 x^2}}{8910}+\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{15283 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{17820 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{4153 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{3564 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.352769, size = 208, normalized size = 1.03 \[ -\frac{-2824 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (87480 x^7-48600 x^6-2001510 x^5-6002964 x^4-8112483 x^3-5740860 x^2-2059597 x-293686\right ) \sqrt{2 x+3}+15283 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{53460 (2 x+3) \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 156, normalized size = 0.8 \begin{align*}{\frac{1}{3207600\,{x}^{3}+10157400\,{x}^{2}+10157400\,x+3207600}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -1749600\,{x}^{7}+972000\,{x}^{6}+40030200\,{x}^{5}+5482\,\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 ) +15283\,\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 ) +120059280\,{x}^{4}+162249660\,{x}^{3}+115734180\,{x}^{2}+42720240\,x+6485040 \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{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} \sqrt{2 \, x + 3}{\left (x - 5\right )}\,{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 (-{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, 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 - 10 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 23 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 10 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 3 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, 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 -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} \sqrt{2 \, x + 3}{\left (x - 5\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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