Optimal. Leaf size=202 \[ \frac{3212 \sqrt{3} \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}}-\frac{2 (47 x+37)}{5 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{23464 \sqrt{3 x^2+5 x+2}}{125 \sqrt{2 x+3}}+\frac{4 (2409 x+2054)}{25 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}-\frac{11732 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{125 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.131407, 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 = {822, 834, 843, 718, 424, 419} \[ -\frac{2 (47 x+37)}{5 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{23464 \sqrt{3 x^2+5 x+2}}{125 \sqrt{2 x+3}}+\frac{4 (2409 x+2054)}{25 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}+\frac{3212 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}}-\frac{11732 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{125 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (37+47 x)}{5 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{15} \int \frac{918+705 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (2054+2409 x)}{25 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}+\frac{4}{75} \int \frac{6441+7227 x}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (2054+2409 x)}{25 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}+\frac{23464 \sqrt{2+5 x+3 x^2}}{125 \sqrt{3+2 x}}-\frac{8}{375} \int \frac{10764+\frac{26397 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (2054+2409 x)}{25 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}+\frac{23464 \sqrt{2+5 x+3 x^2}}{125 \sqrt{3+2 x}}-\frac{17598}{125} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{4818}{25} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (2054+2409 x)}{25 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}+\frac{23464 \sqrt{2+5 x+3 x^2}}{125 \sqrt{3+2 x}}-\frac{\left (11732 \sqrt{3} \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 )}{125 \sqrt{2+5 x+3 x^2}}+\frac{\left (3212 \sqrt{3} \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 )}{25 \sqrt{2+5 x+3 x^2}}\\ &=-\frac{2 (37+47 x)}{5 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (2054+2409 x)}{25 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}+\frac{23464 \sqrt{2+5 x+3 x^2}}{125 \sqrt{3+2 x}}-\frac{11732 \sqrt{3} \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{125 \sqrt{2+5 x+3 x^2}}+\frac{3212 \sqrt{3} \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{25 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.393207, size = 215, normalized size = 1.06 \[ \frac{2 \left (1048 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{2 x+3} \sqrt{\frac{3 x+2}{2 x+3}} \left (6 x^3+19 x^2+19 x+6\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+5 \left (14454 x^3+36414 x^2+29941 x+8031\right )-5866 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{2 x+3} \sqrt{\frac{3 x+2}{2 x+3}} \left (6 x^3+19 x^2+19 x+6\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )}{125 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 308, normalized size = 1.5 \begin{align*}{\frac{2}{625\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 3246\,\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}+8799\,\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}+5410\,\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}+14665\,\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}+2164\,\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 ) +5866\,\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 ) +527940\,{x}^{4}+2121150\,{x}^{3}+3080770\,{x}^{2}+1921725\,x+435415 \right ){\frac{1}{\sqrt{3+2\,x}}}} \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{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\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{\sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}{\left (x - 5\right )}}{108 \, x^{8} + 864 \, x^{7} + 2979 \, x^{6} + 5783 \, x^{5} + 6915 \, x^{4} + 5217 \, x^{3} + 2426 \, x^{2} + 636 \, x + 72}, 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{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\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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