Optimal. Leaf size=249 \[ -\frac{43537016 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{6806835 \sqrt{33}}+\frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{297 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}+\frac{1446357824 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 \sqrt{3 x+2}}+\frac{20799916 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{3/2}}+\frac{442076 \sqrt{1-2 x} \sqrt{5 x+3}}{1528065 (3 x+2)^{5/2}}-\frac{12872 \sqrt{1-2 x} \sqrt{5 x+3}}{43659 (3 x+2)^{7/2}}-\frac{1446357824 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835 \sqrt{33}} \]
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Rubi [A] time = 0.0984128, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 152, 158, 113, 119} \[ \frac{74 \sqrt{1-2 x} (5 x+3)^{3/2}}{297 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}+\frac{1446357824 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 \sqrt{3 x+2}}+\frac{20799916 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{3/2}}+\frac{442076 \sqrt{1-2 x} \sqrt{5 x+3}}{1528065 (3 x+2)^{5/2}}-\frac{12872 \sqrt{1-2 x} \sqrt{5 x+3}}{43659 (3 x+2)^{7/2}}-\frac{43537016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835 \sqrt{33}}-\frac{1446357824 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{2}{33} \int \frac{\left (-\frac{3}{2}-30 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac{4}{891} \int \frac{\sqrt{3+5 x} \left (-864+\frac{2235 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx\\ &=-\frac{12872 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac{8 \int \frac{-\frac{67269}{4}+\frac{64875 x}{4}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{130977}\\ &=-\frac{12872 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{7/2}}+\frac{442076 \sqrt{1-2 x} \sqrt{3+5 x}}{1528065 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac{16 \int \frac{-\frac{8968797}{8}+\frac{4973355 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{4584195}\\ &=-\frac{12872 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{7/2}}+\frac{442076 \sqrt{1-2 x} \sqrt{3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac{20799916 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac{32 \int \frac{-\frac{193192407}{4}+\frac{233999055 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{96268095}\\ &=-\frac{12872 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{7/2}}+\frac{442076 \sqrt{1-2 x} \sqrt{3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac{20799916 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac{1446357824 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac{64 \int \frac{-\frac{10301685885}{16}-1016970345 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{673876665}\\ &=-\frac{12872 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{7/2}}+\frac{442076 \sqrt{1-2 x} \sqrt{3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac{20799916 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac{1446357824 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}+\frac{21768508 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{6806835}+\frac{1446357824 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{74875185}\\ &=-\frac{12872 \sqrt{1-2 x} \sqrt{3+5 x}}{43659 (2+3 x)^{7/2}}+\frac{442076 \sqrt{1-2 x} \sqrt{3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac{20799916 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac{1446357824 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{74 \sqrt{1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac{1446357824 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835 \sqrt{33}}-\frac{43537016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6806835 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.293594, size = 112, normalized size = 0.45 \[ \frac{-5823976480 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{24 \sqrt{2-4 x} \sqrt{5 x+3} \left (175732475616 x^5+591671694906 x^4+797050394730 x^3+537061687749 x^2+180988667568 x+24398176891\right )}{(3 x+2)^{11/2}}+11570862592 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{898502220 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 599, normalized size = 2.4 \begin{align*}{\frac{2}{2246255550\,{x}^{2}+224625555\,x-673876665} \left ( 88451642790\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{5}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-175732475616\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{5}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+294838809300\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}-585774918720\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}+393118412400\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-781033224960\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+262078941600\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-520688816640\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+5271974268480\,{x}^{7}+87359647200\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-173562938880\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+18277348274028\,{x}^{6}+11647952960\,\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 ) -23141725184\,\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 ) +24104934646074\,{x}^{5}+13177956562506\,{x}^{4}-132608462283\,{x}^{3}-3558643880307\,{x}^{2}-1555703477439\,x-219583592019 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{11}{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 (5 \, x + 3\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{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 (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128}, 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{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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