Optimal. Leaf size=207 \[ -\frac{32513 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{840 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{(113 x+124) \left (3 x^2+5 x+2\right )^{5/2}}{63 (2 x+3)^{9/2}}+\frac{(10729 x+14311) \left (3 x^2+5 x+2\right )^{3/2}}{3150 (2 x+3)^{5/2}}-\frac{(7877 x+27213) \sqrt{3 x^2+5 x+2}}{2100 \sqrt{2 x+3}}+\frac{17699 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{600 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.128924, antiderivative size = 207, 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, 812, 843, 718, 424, 419} \[ \frac{(113 x+124) \left (3 x^2+5 x+2\right )^{5/2}}{63 (2 x+3)^{9/2}}+\frac{(10729 x+14311) \left (3 x^2+5 x+2\right )^{3/2}}{3150 (2 x+3)^{5/2}}-\frac{(7877 x+27213) \sqrt{3 x^2+5 x+2}}{2100 \sqrt{2 x+3}}-\frac{32513 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{840 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{17699 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{600 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 812
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)^{11/2}} \, dx &=\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}-\frac{1}{126} \int \frac{(548+603 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{7/2}} \, dx\\ &=\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac{\int \frac{(-60147-70893 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx}{6300}\\ &=-\frac{(27213+7877 x) \sqrt{2+5 x+3 x^2}}{2100 \sqrt{3+2 x}}+\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}-\frac{\int \frac{-941013-1115037 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{37800}\\ &=-\frac{(27213+7877 x) \sqrt{2+5 x+3 x^2}}{2100 \sqrt{3+2 x}}+\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac{17699 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{1200}-\frac{32513 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{1680}\\ &=-\frac{(27213+7877 x) \sqrt{2+5 x+3 x^2}}{2100 \sqrt{3+2 x}}+\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac{\left (17699 \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 )}{600 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{\left (32513 \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 )}{840 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{(27213+7877 x) \sqrt{2+5 x+3 x^2}}{2100 \sqrt{3+2 x}}+\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac{17699 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{600 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{32513 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{840 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.440175, size = 227, normalized size = 1.1 \[ -\frac{8 \left (3 x^2+5 x+2\right ) \left (18900 x^5+1579088 x^4+8510198 x^3+17853828 x^2+16893773 x+6047963\right )-4 (2 x+3)^4 \left (-26354 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+247786 \left (3 x^2+5 x+2\right )+123893 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )}{50400 (2 x+3)^{9/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 487, normalized size = 2.4 \begin{align*} -{\frac{1}{126000} \left ( 618752\,\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}+1982288\,\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}+3712512\,\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}+11893728\,\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}+8353152\,\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}+26760888\,\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}+1134000\,{x}^{7}+8353152\,\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}+26760888\,\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}+96635280\,{x}^{6}+3132432\,\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 ) +10035333\,\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 ) +669276680\,{x}^{5}+1985413000\,{x}^{4}+3139417100\,{x}^{3}+2766408200\,{x}^{2}+1280547220\,x+241918520 \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{5}{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 (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}}{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{5}{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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