Optimal. Leaf size=205 \[ -\frac{4145485 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{49896 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{(x+73) \left (3 x^2+5 x+2\right )^{5/2}}{11 \sqrt{2 x+3}}+\frac{5 \sqrt{2 x+3} (3031 x+218) \left (3 x^2+5 x+2\right )^{3/2}}{1386}-\frac{(21871-471213 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{24948}+\frac{451331 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{7128 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.129384, antiderivative size = 205, 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 = {812, 814, 843, 718, 424, 419} \[ -\frac{(x+73) \left (3 x^2+5 x+2\right )^{5/2}}{11 \sqrt{2 x+3}}+\frac{5 \sqrt{2 x+3} (3031 x+218) \left (3 x^2+5 x+2\right )^{3/2}}{1386}-\frac{(21871-471213 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{24948}-\frac{4145485 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{49896 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{451331 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{7128 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 812
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{3/2}} \, dx &=-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}-\frac{5}{22} \int \frac{(-361-433 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx\\ &=\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}+\frac{5 \int \frac{(43918+52357 x) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{2772}\\ &=-\frac{(21871-471213 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{24948}+\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}-\frac{\int \frac{-2666233-3159317 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{49896}\\ &=-\frac{(21871-471213 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{24948}+\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}+\frac{451331 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{14256}-\frac{4145485 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{99792}\\ &=-\frac{(21871-471213 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{24948}+\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}+\frac{\left (451331 \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 )}{7128 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{\left (4145485 \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 )}{49896 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{(21871-471213 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{24948}+\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}+\frac{451331 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{7128 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{4145485 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{49896 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.442033, size = 205, normalized size = 1. \[ \frac{2 \left (-336013 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{3/2} \sqrt{\frac{3 x+2}{2 x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )-183708 x^7+401436 x^6+3305934 x^5+7163046 x^4+6935769 x^3+6834513 x^2+6998740 x+2657740\right )+3159317 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{149688 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 156, normalized size = 0.8 \begin{align*} -{\frac{1}{8981280\,{x}^{3}+28440720\,{x}^{2}+28440720\,x+8981280}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 3674160\,{x}^{7}-8028720\,{x}^{6}+986168\,\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 ) +3159317\,\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 ) -66118680\,{x}^{5}-143260920\,{x}^{4}-138715380\,{x}^{3}+52868760\,{x}^{2}+175956900\,x+73217880 \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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{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{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}{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{20 \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int \frac{9 x^{5} \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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{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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