Optimal. Leaf size=173 \[ -\frac{4427 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{84 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt{2 x+3}}-\frac{1}{210} (136-2493 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}+\frac{2411 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{60 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.102798, 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 = {812, 814, 843, 718, 424, 419} \[ -\frac{(x+47) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt{2 x+3}}-\frac{1}{210} (136-2493 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}-\frac{4427 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{84 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{2411 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{60 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 812
Rule 814
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)^{3/2}} \, dx &=-\frac{(47+x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{3+2 x}}-\frac{3}{14} \int \frac{(-231-277 x) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx\\ &=-\frac{1}{210} (136-2493 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{3+2 x}}+\frac{1}{420} \int \frac{14248+16877 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{1}{210} (136-2493 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{3+2 x}}+\frac{2411}{120} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx-\frac{4427}{168} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{1}{210} (136-2493 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{3+2 x}}+\frac{\left (2411 \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 )}{60 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{\left (4427 \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 )}{84 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{1}{210} (136-2493 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}-\frac{(47+x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt{3+2 x}}+\frac{2411 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{60 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{4427 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{84 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.346417, size = 192, normalized size = 1.11 \[ \frac{-3596 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{3/2} \sqrt{\frac{3 x+2}{2 x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )-1620 x^5+8208 x^4+18846 x^3+53340 x^2+73094 x+16877 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{3/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+28772}{1260 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 146, normalized size = 0.8 \begin{align*} -{\frac{1}{75600\,{x}^{3}+239400\,{x}^{2}+239400\,x+75600}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 5258\,\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 ) +16877\,\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 ) +16200\,{x}^{5}-82080\,{x}^{4}-188460\,{x}^{3}+479220\,{x}^{2}+956760\,x+387360 \right ) } \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{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{{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}{4 \, x^{2} + 12 \, x + 9}, 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{10 \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{23 x \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{10 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int \frac{3 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, 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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (x - 5\right )}}{{\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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