Optimal. Leaf size=205 \[ \frac{306175 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{1512 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{(x+21) \left (3 x^2+5 x+2\right )^{5/2}}{9 (2 x+3)^{3/2}}+\frac{5 (121 x+745) \left (3 x^2+5 x+2\right )^{3/2}}{126 \sqrt{2 x+3}}+\frac{5}{756} (326-6957 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}-\frac{33335 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{216 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.130216, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{(x+21) \left (3 x^2+5 x+2\right )^{5/2}}{9 (2 x+3)^{3/2}}+\frac{5 (121 x+745) \left (3 x^2+5 x+2\right )^{3/2}}{126 \sqrt{2 x+3}}+\frac{5}{756} (326-6957 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}+\frac{306175 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1512 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{33335 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{216 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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 )^{5/2}}{(3+2 x)^{5/2}} \, dx &=-\frac{(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac{5}{54} \int \frac{(-303-363 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx\\ &=\frac{5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt{3+2 x}}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}+\frac{5}{252} \int \frac{(-9723-11595 x) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx\\ &=\frac{5}{756} (326-6957 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt{3+2 x}}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac{\int \frac{590790+700035 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{4536}\\ &=\frac{5}{756} (326-6957 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt{3+2 x}}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac{33335}{432} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{306175 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{3024}\\ &=\frac{5}{756} (326-6957 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt{3+2 x}}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac{\left (33335 \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 )}{216 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (306175 \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 )}{1512 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{5}{756} (326-6957 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt{3+2 x}}-\frac{(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac{33335 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{216 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{306175 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{1512 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.347595, size = 202, normalized size = 0.99 \[ -\frac{-49640 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{5/2} \sqrt{\frac{3 x+2}{2 x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+13608 x^7-38232 x^6-234684 x^5-561564 x^4+120594 x^3+2607724 x^2+3207982 x+233345 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{5/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+1099572}{4536 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 223, normalized size = 1.1 \begin{align*}{\frac{1}{9072} \left ( -27216\,{x}^{7}+29132\,\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}+93338\,\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}+76464\,{x}^{6}+43698\,\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 ) +140007\,\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 ) +469368\,{x}^{5}+1123128\,{x}^{4}+5359092\,{x}^{3}+12518772\,{x}^{2}+11318256\,x+3401136 \right ) \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+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{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\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{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}{8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27}, 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{20 \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\, dx - \int - \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\, dx - \int - \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\, dx - \int - \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\, dx - \int - \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \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{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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