Optimal. Leaf size=229 \[ \frac{14807 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{577500 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}-\frac{(14773 x+15647) \sqrt{3 x^2+5 x+2}}{57750 (2 x+3)^{7/2}}+\frac{5861 \sqrt{3 x^2+5 x+2}}{618750 \sqrt{2 x+3}}+\frac{14807 \sqrt{3 x^2+5 x+2}}{866250 (2 x+3)^{3/2}}-\frac{5861 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{412500 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.152224, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {810, 834, 843, 718, 424, 419} \[ \frac{(367 x+258) \left (3 x^2+5 x+2\right )^{3/2}}{495 (2 x+3)^{11/2}}-\frac{(14773 x+15647) \sqrt{3 x^2+5 x+2}}{57750 (2 x+3)^{7/2}}+\frac{5861 \sqrt{3 x^2+5 x+2}}{618750 \sqrt{2 x+3}}+\frac{14807 \sqrt{3 x^2+5 x+2}}{866250 (2 x+3)^{3/2}}+\frac{14807 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{577500 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{5861 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{412500 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx &=\frac{(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac{1}{330} \int \frac{(-194-303 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx\\ &=-\frac{(15647+14773 x) \sqrt{2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac{(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}+\frac{\int \frac{12185+13059 x}{(3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}} \, dx}{115500}\\ &=\frac{14807 \sqrt{2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}-\frac{(15647+14773 x) \sqrt{2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac{(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac{\int \frac{-23059-\frac{44421 x}{2}}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx}{866250}\\ &=\frac{14807 \sqrt{2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac{5861 \sqrt{2+5 x+3 x^2}}{618750 \sqrt{3+2 x}}-\frac{(15647+14773 x) \sqrt{2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac{(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}+\frac{\int \frac{-\frac{73569}{4}-\frac{123081 x}{4}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{2165625}\\ &=\frac{14807 \sqrt{2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac{5861 \sqrt{2+5 x+3 x^2}}{618750 \sqrt{3+2 x}}-\frac{(15647+14773 x) \sqrt{2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac{(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac{5861 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{825000}+\frac{14807 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{1155000}\\ &=\frac{14807 \sqrt{2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac{5861 \sqrt{2+5 x+3 x^2}}{618750 \sqrt{3+2 x}}-\frac{(15647+14773 x) \sqrt{2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac{(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac{\left (5861 \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 )}{412500 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (14807 \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 )}{577500 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{14807 \sqrt{2+5 x+3 x^2}}{866250 (3+2 x)^{3/2}}+\frac{5861 \sqrt{2+5 x+3 x^2}}{618750 \sqrt{3+2 x}}-\frac{(15647+14773 x) \sqrt{2+5 x+3 x^2}}{57750 (3+2 x)^{7/2}}+\frac{(258+367 x) \left (2+5 x+3 x^2\right )^{3/2}}{495 (3+2 x)^{11/2}}-\frac{5861 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{412500 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{14807 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{577500 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.498019, size = 227, normalized size = 0.99 \[ -\frac{2 (2 x+3)^5 \left (3394 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+82054 \left (3 x^2+5 x+2\right )+41027 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )-4 \left (3 x^2+5 x+2\right ) \left (1312864 x^5+11031040 x^4+41848650 x^3+65139670 x^2+42879355 x+9919671\right )}{17325000 (2 x+3)^{11/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 575, normalized size = 2.5 \begin{align*}{\frac{1}{86625000} \left ( 1312864\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{5}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+1056256\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{5}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+9846480\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+7921920\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+29539440\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+23765760\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+44309160\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+35648640\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+78771840\,{x}^{7}+33231870\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+26736480\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+793148800\,{x}^{6}+9969561\,\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 ) +8020944\,\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 ) +3666537560\,{x}^{5}+8534486800\,{x}^{4}+10760674300\,{x}^{3}+7488702560\,{x}^{2}+2707141300\,x+396786840 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}} \left ( 3+2\,x \right ) ^{-{\frac{11}{2}}}} \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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{13}{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{{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}{128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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