Optimal. Leaf size=191 \[ -\frac{6584 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{36015}+\frac{220076 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 \sqrt{3 x+2}}+\frac{3184 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{3/2}}+\frac{74 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{5/2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^{7/2}}-\frac{220076 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36015} \]
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Rubi [A] time = 0.0684654, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 152, 158, 113, 119} \[ \frac{220076 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 \sqrt{3 x+2}}+\frac{3184 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{3/2}}+\frac{74 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{5/2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^{7/2}}-\frac{6584 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36015}-\frac{220076 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36015} \]
Antiderivative was successfully verified.
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Rule 97
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^{9/2}} \, dx &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{-\frac{1}{2}-10 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}+\frac{4}{735} \int \frac{213-\frac{555 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}+\frac{3184 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 (2+3 x)^{3/2}}+\frac{8 \int \frac{\frac{39099}{4}-5970 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{15435}\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}+\frac{3184 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 (2+3 x)^{3/2}}+\frac{220076 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 \sqrt{2+3 x}}+\frac{16 \int \frac{\frac{261165}{2}+\frac{825285 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{108045}\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}+\frac{3184 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 (2+3 x)^{3/2}}+\frac{220076 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 \sqrt{2+3 x}}+\frac{36212 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{36015}+\frac{220076 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{36015}\\ &=-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^{7/2}}+\frac{74 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{5/2}}+\frac{3184 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 (2+3 x)^{3/2}}+\frac{220076 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 \sqrt{2+3 x}}-\frac{220076 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36015}-\frac{6584 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36015}\\ \end{align*}
Mathematica [A] time = 0.225068, size = 106, normalized size = 0.55 \[ \frac{4 \left (\sqrt{2} \left (55019 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-27860 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2971026 x^3+6042348 x^2+4100535 x+926791\right )}{2 (3 x+2)^{7/2}}\right )}{108045} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.032, size = 409, normalized size = 2.1 \begin{align*}{\frac{2}{1080450\,{x}^{2}+108045\,x-324135} \left ( 1504440\,\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}-2971026\,\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}+3008880\,\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}-5942052\,\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}+2005920\,\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}-3961368\,\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}+445760\,\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 ) -880304\,\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 ) +89130780\,{x}^{5}+190183518\,{x}^{4}+114403860\,{x}^{3}-14275797\,{x}^{2}-34124442\,x-8341119 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{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{\sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{9}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, 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{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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