Optimal. Leaf size=135 \[ \frac{(5 x+3)^{3/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{101 (5 x+3)^{3/2} (3 x+2)^2}{22 \sqrt{1-2 x}}-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (28200 x+59719)}{3520}-\frac{4246733 \sqrt{1-2 x} \sqrt{5 x+3}}{14080}+\frac{4246733 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280 \sqrt{10}} \]
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Rubi [A] time = 0.0372092, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 150, 147, 50, 54, 216} \[ \frac{(5 x+3)^{3/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{101 (5 x+3)^{3/2} (3 x+2)^2}{22 \sqrt{1-2 x}}-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (28200 x+59719)}{3520}-\frac{4246733 \sqrt{1-2 x} \sqrt{5 x+3}}{14080}+\frac{4246733 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 147
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^2 \sqrt{3+5 x} \left (42+\frac{135 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-\frac{9969}{2}-\frac{31725 x}{4}\right ) (2+3 x) \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac{4246733 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{7040}\\ &=-\frac{4246733 \sqrt{1-2 x} \sqrt{3+5 x}}{14080}-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac{4246733 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2560}\\ &=-\frac{4246733 \sqrt{1-2 x} \sqrt{3+5 x}}{14080}-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac{4246733 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1280 \sqrt{5}}\\ &=-\frac{4246733 \sqrt{1-2 x} \sqrt{3+5 x}}{14080}-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac{4246733 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1280 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.0697551, size = 79, normalized size = 0.59 \[ \frac{12740199 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (86400 x^4+447120 x^3+1544724 x^2-5349344 x+1925361\right )}{38400 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 154, normalized size = 1.1 \begin{align*}{\frac{1}{76800\, \left ( 2\,x-1 \right ) ^{2}} \left ( -1728000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+50960796\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-8942400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-50960796\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-30894480\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+12740199\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +106986880\,x\sqrt{-10\,{x}^{2}-x+3}-38507220\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 3.65015, size = 285, normalized size = 2.11 \begin{align*} \frac{428267}{2560} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{35937}{25600} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) + \frac{9}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{297}{64} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x + \frac{6237}{1280} \, \sqrt{10 \, x^{2} - 21 \, x + 8} - \frac{6237}{128} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{343 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{48 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{189 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (2 \, x - 1\right )}} + \frac{3773 \, \sqrt{-10 \, x^{2} - x + 3}}{96 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{3479 \, \sqrt{-10 \, x^{2} - x + 3}}{6 \,{\left (2 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51703, size = 324, normalized size = 2.4 \begin{align*} -\frac{12740199 \, \sqrt{10}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (86400 \, x^{4} + 447120 \, x^{3} + 1544724 \, x^{2} - 5349344 \, x + 1925361\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{76800 \,{\left (4 \, x^{2} - 4 \, x + 1\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 [A] time = 1.48061, size = 131, normalized size = 0.97 \begin{align*} \frac{4246733}{12800} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (27 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 111 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 8579 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 8493466 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 140142189 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{480000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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