Optimal. Leaf size=173 \[ \frac{748 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{2 x+3} (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 \sqrt{2 x+3} (2139 x+1759)}{15 \sqrt{3 x^2+5 x+2}}-\frac{2852 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.102819, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {820, 822, 843, 718, 424, 419} \[ -\frac{2 \sqrt{2 x+3} (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 \sqrt{2 x+3} (2139 x+1759)}{15 \sqrt{3 x^2+5 x+2}}+\frac{748 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2852 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 820
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} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{3} \int \frac{181+105 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 \sqrt{3+2 x} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 \sqrt{3+2 x} (1759+2139 x)}{15 \sqrt{2+5 x+3 x^2}}+\frac{4}{15} \int \frac{-1806-2139 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 \sqrt{3+2 x} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 \sqrt{3+2 x} (1759+2139 x)}{15 \sqrt{2+5 x+3 x^2}}-\frac{1426}{5} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx+374 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 \sqrt{3+2 x} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 \sqrt{3+2 x} (1759+2139 x)}{15 \sqrt{2+5 x+3 x^2}}-\frac{\left (2852 \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 )}{5 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (748 \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 )}{\sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{2 \sqrt{3+2 x} (29+35 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 \sqrt{3+2 x} (1759+2139 x)}{15 \sqrt{2+5 x+3 x^2}}-\frac{2852 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{5 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{748 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.360602, size = 196, normalized size = 1.13 \[ -\frac{-\frac{608 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}+\frac{5704 \left (3 x^2+5 x+2\right )}{\sqrt{2 x+3}}-\frac{6 \sqrt{2 x+3} \left (4278 x^3+10648 x^2+8657 x+2297\right )}{3 x^2+5 x+2}+\frac{2852 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}}{15 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 308, normalized size = 1.8 \begin{align*}{\frac{2}{75\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}} \left ( 666\,\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}+2139\,\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}+1110\,\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}+3565\,\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}+444\,\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 ) +1426\,\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 ) +128340\,{x}^{4}+511950\,{x}^{3}+738870\,{x}^{2}+458475\,x+103365 \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{\sqrt{2 \, x + 3}{\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\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 )}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, 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{5 \sqrt{2 x + 3}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \frac{x \sqrt{2 x + 3}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \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{\sqrt{2 \, x + 3}{\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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