Optimal. Leaf size=249 \[ -\frac{37904696 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{47647845 \sqrt{33}}-\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{5 x+3}}{524126295 \sqrt{3 x+2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac{13022 \sqrt{1-2 x} \sqrt{5 x+3}}{305613 (3 x+2)^{7/2}}-\frac{1305025844 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}} \]
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Rubi [A] time = 0.0971437, 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} \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{5 x+3}}{524126295 \sqrt{3 x+2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac{13022 \sqrt{1-2 x} \sqrt{5 x+3}}{305613 (3 x+2)^{7/2}}-\frac{37904696 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}}-\frac{1305025844 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}} \]
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{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{2}{33} \int \frac{\left (\frac{19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{11/2}} \, dx\\ &=-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{4 \int \frac{\left (-\frac{189}{4}-\frac{5025 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx}{6237}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{8 \int \frac{-\frac{676497}{8}-185700 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{916839}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{16 \int \frac{\frac{1286433}{2}-\frac{14125635 x}{8}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{32089365}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 (2+3 x)^{3/2}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{32 \int \frac{\frac{687512943}{16}-\frac{109221165 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{673876665}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{3+5 x}}{524126295 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{64 \int \frac{\frac{2319498765}{4}+\frac{14681540745 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4717136655}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{3+5 x}}{524126295 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{18952348 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{47647845}+\frac{1305025844 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{524126295}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{3+5 x}}{524126295 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac{1305025844 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}}-\frac{37904696 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.300951, size = 112, normalized size = 0.45 \[ \frac{-10873573760 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (158560640046 x^5+534040213536 x^4+719808574005 x^3+484598540169 x^2+162787885893 x+21813966691\right )}{(3 x+2)^{11/2}}+20880413504 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{12579031080 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.036, size = 599, normalized size = 2.4 \begin{align*}{\frac{2}{15723788850\,{x}^{2}+1572378885\,x-4717136655} \left ( 82571200740\,\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}-158560640046\,\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}+275237335800\,\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}-528535466820\,\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}+366983114400\,\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}-704713955760\,\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}+244655409600\,\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}-469809303840\,\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}+4756819201380\,{x}^{7}+81551803200\,\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}-156603101280\,\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}+16496888326218\,{x}^{6}+10873573760\,\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 ) -20880413504\,\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 ) +21769332100344\,{x}^{5}+11891020005261\,{x}^{4}-140844968748\,{x}^{3}-3218604203112\,{x}^{2}-1399649072964\,x-196325700219 \right ) \sqrt{1-2\,x}\sqrt{3+5\,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{5}{2}} \sqrt{-2 \, x + 1}}{{\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 (25 \, x^{2} + 30 \, x + 9\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{5}{2}} \sqrt{-2 \, x + 1}}{{\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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