Optimal. Leaf size=229 \[ \frac{332459 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{972972 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{886 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{1287}+\frac{\sqrt{2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}}{27027}-\frac{\sqrt{2 x+3} (783711 x+486863) \sqrt{3 x^2+5 x+2}}{2432430}-\frac{152657 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{694980 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.162777, 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 = {832, 814, 843, 718, 424, 419} \[ -\frac{2}{39} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{886 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{1287}+\frac{\sqrt{2 x+3} (50771 x+43822) \left (3 x^2+5 x+2\right )^{3/2}}{27027}-\frac{\sqrt{2 x+3} (783711 x+486863) \sqrt{3 x^2+5 x+2}}{2432430}+\frac{332459 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{972972 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{152657 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{694980 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac{2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac{2}{39} \int \sqrt{3+2 x} \left (336+\frac{443 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{886 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac{2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac{4 \int \frac{\left (\frac{31531}{4}+\frac{21759 x}{4}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx}{1287}\\ &=\frac{\sqrt{3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac{886 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac{2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{2 \int \frac{\left (\frac{161889}{2}+\frac{261237 x}{4}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{81081}\\ &=-\frac{\sqrt{3+2 x} (486863+783711 x) \sqrt{2+5 x+3 x^2}}{2432430}+\frac{\sqrt{3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac{886 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac{2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac{\int \frac{-\frac{2315253}{4}-\frac{3205797 x}{4}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{3648645}\\ &=-\frac{\sqrt{3+2 x} (486863+783711 x) \sqrt{2+5 x+3 x^2}}{2432430}+\frac{\sqrt{3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac{886 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac{2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{152657 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{1389960}+\frac{332459 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{1945944}\\ &=-\frac{\sqrt{3+2 x} (486863+783711 x) \sqrt{2+5 x+3 x^2}}{2432430}+\frac{\sqrt{3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac{886 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac{2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{\left (152657 \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 )}{694980 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (332459 \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 )}{972972 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{\sqrt{3+2 x} (486863+783711 x) \sqrt{2+5 x+3 x^2}}{2432430}+\frac{\sqrt{3+2 x} (43822+50771 x) \left (2+5 x+3 x^2\right )^{3/2}}{27027}+\frac{886 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{1287}-\frac{2}{39} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{152657 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{694980 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{332459 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{972972 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.35854, size = 213, normalized size = 0.93 \[ -\frac{-71222 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (40415760 x^8+52050600 x^7-895236300 x^6-4079217510 x^5-7944858702 x^4-8470029969 x^3-5141306625 x^2-1668494576 x-224705588\right ) \sqrt{2 x+3}+1068599 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{14594580 (2 x+3) \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 161, normalized size = 0.7 \begin{align*}{\frac{1}{875674800\,{x}^{3}+2772970200\,{x}^{2}+2772970200\,x+875674800}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -808315200\,{x}^{8}-1041012000\,{x}^{7}+17904726000\,{x}^{6}+81584350200\,{x}^{5}+593696\,\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 ) +1068599\,\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 ) +158897174040\,{x}^{4}+169400599380\,{x}^{3}+102890248440\,{x}^{2}+33476751420\,x+4536855720 \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{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (2 \, x + 3\right )}^{\frac{3}{2}}{\left (x - 5\right )}\,{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 (-{\left (6 \, x^{4} - 11 \, x^{3} - 76 \, x^{2} - 89 \, x - 30\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, 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 - 30 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 89 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 76 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 11 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 6 x^{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 -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (2 \, x + 3\right )}^{\frac{3}{2}}{\left (x - 5\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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