Optimal. Leaf size=229 \[ \frac{30836 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2 (47 x+37)}{5 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{190792 \sqrt{3 x^2+5 x+2}}{1875 \sqrt{2 x+3}}+\frac{61672 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^{3/2}}+\frac{12 (737 x+652)}{25 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}}-\frac{95396 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.157071, 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 = {822, 834, 843, 718, 424, 419} \[ -\frac{2 (47 x+37)}{5 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{190792 \sqrt{3 x^2+5 x+2}}{1875 \sqrt{2 x+3}}+\frac{61672 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^{3/2}}+\frac{12 (737 x+652)}{25 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}}+\frac{30836 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{95396 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{15} \int \frac{1140+987 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (652+737 x)}{25 (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}}+\frac{4}{75} \int \frac{18285+19899 x}{(3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (652+737 x)}{25 (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}}+\frac{61672 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}-\frac{8 \int \frac{-34149-\frac{69381 x}{2}}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx}{1125}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (652+737 x)}{25 (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}}+\frac{61672 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}+\frac{190792 \sqrt{2+5 x+3 x^2}}{1875 \sqrt{3+2 x}}+\frac{16 \int \frac{-\frac{148509}{4}-\frac{214641 x}{4}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{5625}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (652+737 x)}{25 (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}}+\frac{61672 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}+\frac{190792 \sqrt{2+5 x+3 x^2}}{1875 \sqrt{3+2 x}}-\frac{47698}{625} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{15418}{125} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (652+737 x)}{25 (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}}+\frac{61672 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}+\frac{190792 \sqrt{2+5 x+3 x^2}}{1875 \sqrt{3+2 x}}-\frac{\left (95396 \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 )}{625 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (30836 \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 )}{125 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (652+737 x)}{25 (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}}+\frac{61672 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}+\frac{190792 \sqrt{2+5 x+3 x^2}}{1875 \sqrt{3+2 x}}-\frac{95396 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{30836 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.562006, size = 222, normalized size = 0.97 \[ \frac{2 \left (-2 (2 x+3) \left (3 x^2+5 x+2\right ) \left (-722 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+47698 \left (3 x^2+5 x+2\right )+23849 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )+1717128 x^5+9687072 x^4+21265294 x^3+22647906 x^2+11683203 x+2334397\right )}{1875 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 401, normalized size = 1.8 \begin{align*}{\frac{2}{9375\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 143094\,\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}+88176\,\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}+453131\,\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}+279224\,\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}+453131\,\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}+279224\,\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}+143094\,\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 ) +88176\,\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 ) +8585640\,{x}^{5}+48435360\,{x}^{4}+106326470\,{x}^{3}+113239530\,{x}^{2}+58416015\,x+11671985 \right ) \left ( 3+2\,x \right ) ^{-{\frac{3}{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{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (2 \, x + 3\right )}^{\frac{5}{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 )}}{216 \, x^{9} + 2052 \, x^{8} + 8550 \, x^{7} + 20503 \, x^{6} + 31179 \, x^{5} + 31179 \, x^{4} + 20503 \, x^{3} + 8550 \, x^{2} + 2052 \, x + 216}, 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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