Optimal. Leaf size=160 \[ -\frac{28174 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{28125}-\frac{2 \sqrt{3 x+2} (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{24}{125} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{3028 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{5625}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{28125} \]
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Rubi [A] time = 0.0508289, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{2 \sqrt{3 x+2} (1-2 x)^{5/2}}{5 \sqrt{5 x+3}}-\frac{24}{125} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}-\frac{3028 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{5625}-\frac{28174 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{28125}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{28125} \]
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} \sqrt{2+3 x}}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{\left (-\frac{17}{2}-18 x\right ) (1-2 x)^{3/2}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{5 \sqrt{3+5 x}}-\frac{24}{125} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{4}{375} \int \frac{\left (-\frac{1887}{4}-\frac{2271 x}{2}\right ) \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{5 \sqrt{3+5 x}}-\frac{3028 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{5625}-\frac{24}{125} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{8 \int \frac{-\frac{53121}{8}-\frac{60873 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{16875}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{5 \sqrt{3+5 x}}-\frac{3028 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{5625}-\frac{24}{125} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{81164 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{28125}+\frac{154957 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{28125}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{2+3 x}}{5 \sqrt{3+5 x}}-\frac{3028 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{5625}-\frac{24}{125} (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{81164 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{28125}-\frac{28174 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{28125}\\ \end{align*}
Mathematica [A] time = 0.279516, size = 102, normalized size = 0.64 \[ \frac{546035 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (900 x^2-2530 x-7287\right )}{\sqrt{5 x+3}}-81164 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{84375} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 145, normalized size = 0.9 \begin{align*} -{\frac{1}{2531250\,{x}^{3}+1940625\,{x}^{2}-590625\,x-506250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 546035\,\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 ) -81164\,\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 ) -162000\,{x}^{4}+428400\,{x}^{3}+1441560\,{x}^{2}+66810\,x-437220 \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{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, 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 (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, 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{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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