Optimal. Leaf size=191 \[ -\frac{43214 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{6075}-\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 \sqrt{3 x+2}}+\frac{788}{135} \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}-\frac{43214 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1215}+\frac{116854 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6075} \]
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Rubi [A] time = 0.0673501, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 \sqrt{3 x+2}}+\frac{788}{135} \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}-\frac{43214 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{1215}-\frac{43214 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6075}+\frac{116854 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6075} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac{2}{9} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt{2+3 x}}-\frac{4}{27} \int \frac{\left (-210-\frac{2955 x}{2}\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{\sqrt{2+3 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt{2+3 x}}+\frac{788}{135} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{8 \int \frac{\left (\frac{48285}{4}-\frac{324105 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2025}\\ &=-\frac{43214 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1215}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt{2+3 x}}+\frac{788}{135} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{8 \int \frac{-\frac{338655}{8}-\frac{876405 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{18225}\\ &=-\frac{43214 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1215}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt{2+3 x}}+\frac{788}{135} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{116854 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{6075}+\frac{237677 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{6075}\\ &=-\frac{43214 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{1215}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt{2+3 x}}+\frac{788}{135} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}+\frac{116854 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6075}-\frac{43214 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6075}\\ \end{align*}
Mathematica [A] time = 0.18924, size = 107, normalized size = 0.56 \[ \frac{829885 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (1620 x^3-3906 x^2-23538 x-13231\right )}{(3 x+2)^{3/2}}-116854 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{18225} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 229, normalized size = 1.2 \begin{align*} -{\frac{1}{182250\,{x}^{2}+18225\,x-54675} \left ( 2489655\,\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}-350562\,\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}+1659770\,\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 ) -233708\,\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 ) -486000\,{x}^{5}+1123200\,{x}^{4}+7324380\,{x}^{3}+4323900\,{x}^{2}-1721490\,x-1190790 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{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{5}{2}}}{{\left (3 \, x + 2\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 (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}, 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{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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