Optimal. Leaf size=175 \[ \frac{916 \sqrt{3} \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 \sqrt{2 x+3} (2607 x+2152)}{25 \sqrt{3 x^2+5 x+2}}-\frac{3476 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.108425, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {822, 843, 718, 424, 419} \[ -\frac{2 \sqrt{2 x+3} (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 \sqrt{2 x+3} (2607 x+2152)}{25 \sqrt{3 x^2+5 x+2}}+\frac{916 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}}-\frac{3476 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{5-x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 \sqrt{3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{15} \int \frac{696+423 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 \sqrt{3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 \sqrt{3+2 x} (2152+2607 x)}{25 \sqrt{2+5 x+3 x^2}}+\frac{4}{75} \int \frac{-6579-7821 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 \sqrt{3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 \sqrt{3+2 x} (2152+2607 x)}{25 \sqrt{2+5 x+3 x^2}}-\frac{5214}{25} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{1374}{5} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 \sqrt{3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 \sqrt{3+2 x} (2152+2607 x)}{25 \sqrt{2+5 x+3 x^2}}-\frac{\left (3476 \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 )}{25 \sqrt{2+5 x+3 x^2}}+\frac{\left (916 \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 )}{5 \sqrt{2+5 x+3 x^2}}\\ &=-\frac{2 \sqrt{3+2 x} (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 \sqrt{3+2 x} (2152+2607 x)}{25 \sqrt{2+5 x+3 x^2}}-\frac{3476 \sqrt{3} \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{25 \sqrt{2+5 x+3 x^2}}+\frac{916 \sqrt{3} \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{5 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.341815, size = 196, normalized size = 1.12 \[ \frac{\frac{728 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}-\frac{6952 \left (3 x^2+5 x+2\right )}{\sqrt{2 x+3}}+\frac{2 \sqrt{2 x+3} \left (15642 x^3+38982 x^2+31713 x+8423\right )}{3 x^2+5 x+2}-\frac{3476 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}}{25 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 308, normalized size = 1.8 \begin{align*}{\frac{2}{125\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}} \left ( 828\,\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}+2607\,\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}+1380\,\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}+4345\,\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}+552\,\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 ) +1738\,\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 ) +156420\,{x}^{4}+624450\,{x}^{3}+901860\,{x}^{2}+559925\,x+126345 \right ) \sqrt{3\,{x}^{2}+5\,x+2}{\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}} \sqrt{2 \, x + 3}}\,{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 )}}{54 \, x^{7} + 351 \, x^{6} + 963 \, x^{5} + 1447 \, x^{4} + 1287 \, x^{3} + 678 \, x^{2} + 196 \, x + 24}, 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{x}{9 x^{4} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{9 x^{4} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, 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{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} \sqrt{2 \, x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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