Optimal. Leaf size=175 \[ \frac{1327 \sqrt{3} \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{700 \sqrt{3 x^2+5 x+2}}+\frac{(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}+\frac{(6179 x+8561) \sqrt{3 x^2+5 x+2}}{1750 (2 x+3)^{3/2}}-\frac{721 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{500 \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.099538, 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 = {810, 843, 718, 424, 419} \[ \frac{(347 x+358) \left (3 x^2+5 x+2\right )^{3/2}}{175 (2 x+3)^{7/2}}+\frac{(6179 x+8561) \sqrt{3 x^2+5 x+2}}{1750 (2 x+3)^{3/2}}+\frac{1327 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{700 \sqrt{3 x^2+5 x+2}}-\frac{721 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{500 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx &=\frac{(358+347 x) \left (2+5 x+3 x^2\right )^{3/2}}{175 (3+2 x)^{7/2}}-\frac{3}{350} \int \frac{(250+261 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{5/2}} \, dx\\ &=\frac{(8561+6179 x) \sqrt{2+5 x+3 x^2}}{1750 (3+2 x)^{3/2}}+\frac{(358+347 x) \left (2+5 x+3 x^2\right )^{3/2}}{175 (3+2 x)^{7/2}}+\frac{\int \frac{-12759-15141 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{3500}\\ &=\frac{(8561+6179 x) \sqrt{2+5 x+3 x^2}}{1750 (3+2 x)^{3/2}}+\frac{(358+347 x) \left (2+5 x+3 x^2\right )^{3/2}}{175 (3+2 x)^{7/2}}-\frac{2163 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{1000}+\frac{3981 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{1400}\\ &=\frac{(8561+6179 x) \sqrt{2+5 x+3 x^2}}{1750 (3+2 x)^{3/2}}+\frac{(358+347 x) \left (2+5 x+3 x^2\right )^{3/2}}{175 (3+2 x)^{7/2}}-\frac{\left (721 \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 )}{500 \sqrt{2+5 x+3 x^2}}+\frac{\left (1327 \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 )}{700 \sqrt{2+5 x+3 x^2}}\\ &=\frac{(8561+6179 x) \sqrt{2+5 x+3 x^2}}{1750 (3+2 x)^{3/2}}+\frac{(358+347 x) \left (2+5 x+3 x^2\right )^{3/2}}{175 (3+2 x)^{7/2}}-\frac{721 \sqrt{3} \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{500 \sqrt{2+5 x+3 x^2}}+\frac{1327 \sqrt{3} \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{700 \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.358465, size = 192, normalized size = 1.1 \[ -\frac{-1066 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{9/2} \sqrt{\frac{3 x+2}{2 x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+31500 x^5+323760 x^4+1009230 x^3+1386750 x^2+878020 x+5047 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{9/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+208240}{3500 (2 x+3)^{7/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 389, normalized size = 2.2 \begin{align*}{\frac{1}{35000} \left ( 12704\,\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}+40376\,\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}+57168\,\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}+181692\,\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}+85752\,\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}+272538\,\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}+42876\,\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 ) +136269\,\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 ) +2107560\,{x}^{5}+11701520\,{x}^{4}+26044220\,{x}^{3}+28830120\,{x}^{2}+15748220\,x+3368360 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}} \left ( 3+2\,x \right ) ^{-{\frac{7}{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{3}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{9}{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 (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}{32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243}, 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{3}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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