Optimal. Leaf size=222 \[ -\frac{310399 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{382725}-\frac{40}{81} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{2108 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1701}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{64628 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{8505}-\frac{310399 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{76545}-\frac{25111 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{382725} \]
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Rubi [A] time = 0.0809355, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ -\frac{40}{81} (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{2108 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{1701}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{3 \sqrt{3 x+2}}+\frac{64628 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{8505}-\frac{310399 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{76545}-\frac{310399 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{382725}-\frac{25111 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{382725} \]
Antiderivative was successfully verified.
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Rule 97
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}+\frac{2}{3} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{40}{81} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{4}{405} \int \frac{\left (\frac{1625}{4}-\frac{13175 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{2108 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1701}-\frac{40}{81} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{8 \int \frac{\left (\frac{1409075}{8}-\frac{1211775 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{42525}\\ &=\frac{64628 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{8505}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{2108 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1701}-\frac{40}{81} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{8 \int \frac{\left (\frac{2027475}{8}-\frac{23279925 x}{8}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{637875}\\ &=-\frac{310399 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{76545}+\frac{64628 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{8505}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{2108 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1701}-\frac{40}{81} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{8 \int \frac{\frac{53475825}{16}+\frac{1883325 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5740875}\\ &=-\frac{310399 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{76545}+\frac{64628 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{8505}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{2108 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1701}-\frac{40}{81} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{25111 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{382725}+\frac{3414389 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{765450}\\ &=-\frac{310399 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{76545}+\frac{64628 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{8505}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{3 \sqrt{2+3 x}}-\frac{2108 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{1701}-\frac{40}{81} (1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{25111 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{382725}-\frac{310399 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{382725}\\ \end{align*}
Mathematica [A] time = 0.301356, size = 112, normalized size = 0.5 \[ \frac{10192945 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (567000 x^4-386100 x^3-259650 x^2+245751 x+21964\right )}{\sqrt{3 x+2}}+50222 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2296350} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 155, normalized size = 0.7 \begin{align*} -{\frac{1}{68890500\,{x}^{3}+52816050\,{x}^{2}-16074450\,x-13778100}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -170100000\,{x}^{6}+10192945\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +50222\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +98820000\,{x}^{5}+140508000\,{x}^{4}-100684800\,{x}^{3}-37330230\,{x}^{2}+21458670\,x+1976760 \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 (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\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 (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, 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 (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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