Optimal. Leaf size=202 \[ \frac{7 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{1500 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac{(211 x+189) \sqrt{3 x^2+5 x+2}}{2250 (2 x+3)^{5/2}}-\frac{23 \sqrt{3 x^2+5 x+2}}{11250 \sqrt{2 x+3}}+\frac{23 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{7500 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.126921, 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 = {810, 834, 843, 718, 424, 419} \[ \frac{(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac{(211 x+189) \sqrt{3 x^2+5 x+2}}{2250 (2 x+3)^{5/2}}-\frac{23 \sqrt{3 x^2+5 x+2}}{11250 \sqrt{2 x+3}}+\frac{7 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1500 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{23 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{7500 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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)^{11/2}} \, dx &=\frac{(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}-\frac{1}{210} \int \frac{(28-21 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx\\ &=-\frac{(189+211 x) \sqrt{2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac{(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac{\int \frac{301+147 x}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx}{31500}\\ &=-\frac{23 \sqrt{2+5 x+3 x^2}}{11250 \sqrt{3+2 x}}-\frac{(189+211 x) \sqrt{2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac{(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}-\frac{\int \frac{-546-\frac{483 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{78750}\\ &=-\frac{23 \sqrt{2+5 x+3 x^2}}{11250 \sqrt{3+2 x}}-\frac{(189+211 x) \sqrt{2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac{(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac{23 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{15000}+\frac{7 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{3000}\\ &=-\frac{23 \sqrt{2+5 x+3 x^2}}{11250 \sqrt{3+2 x}}-\frac{(189+211 x) \sqrt{2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac{(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac{\left (23 \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 )}{7500 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (7 \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 )}{1500 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{23 \sqrt{2+5 x+3 x^2}}{11250 \sqrt{3+2 x}}-\frac{(189+211 x) \sqrt{2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac{(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac{23 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{7500 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{7 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{1500 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.450862, size = 192, normalized size = 0.95 \[ \frac{-44 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{11/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+204180 x^5+822160 x^4+1297210 x^3+998860 x^2+373610 x+23 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{11/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+53980}{22500 (2 x+3)^{9/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 482, normalized size = 2.4 \begin{align*}{\frac{1}{225000} \left ( 928\,\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}-368\,\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}+5568\,\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}-2208\,\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}+12528\,\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}-4968\,\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}+12528\,\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}-4968\,\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}-22080\,{x}^{6}+4698\,\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 ) -1863\,\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 ) +1872520\,{x}^{5}+7688000\,{x}^{4}+12088900\,{x}^{3}+9181300\,{x}^{2}+3351080\,x+465280 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}} \left ( 3+2\,x \right ) ^{-{\frac{9}{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{11}{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}}{64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729}, 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{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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