Optimal. Leaf size=218 \[ -\frac{5442127 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1663750 \sqrt{33}}+\frac{7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{217 (3 x+2)^{7/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3218 \sqrt{1-2 x} (3 x+2)^{5/2}}{19965 (5 x+3)^{3/2}}+\frac{110519 \sqrt{1-2 x} (3 x+2)^{3/2}}{1098075 \sqrt{5 x+3}}-\frac{5199979 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3660250}-\frac{90397364 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{831875 \sqrt{33}} \]
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Rubi [A] time = 0.0827446, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ \frac{7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{217 (3 x+2)^{7/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3218 \sqrt{1-2 x} (3 x+2)^{5/2}}{19965 (5 x+3)^{3/2}}+\frac{110519 \sqrt{1-2 x} (3 x+2)^{3/2}}{1098075 \sqrt{5 x+3}}-\frac{5199979 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3660250}-\frac{5442127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1663750 \sqrt{33}}-\frac{90397364 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{831875 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^{7/2} \left (\frac{345}{2}+306 x\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac{217 (2+3 x)^{7/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{363} \int \frac{\left (-\frac{21705}{2}-\frac{39393 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{3218 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac{217 (2+3 x)^{7/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{2 \int \frac{\left (-594474-\frac{4073679 x}{4}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac{3218 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac{217 (2+3 x)^{7/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{110519 \sqrt{1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt{3+5 x}}-\frac{4 \int \frac{\left (-\frac{86636925}{8}-\frac{140399433 x}{8}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3294225}\\ &=\frac{3218 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac{217 (2+3 x)^{7/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{110519 \sqrt{1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt{3+5 x}}-\frac{5199979 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3660250}+\frac{4 \int \frac{\frac{6181011531}{16}+610182207 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{49413375}\\ &=\frac{3218 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac{217 (2+3 x)^{7/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{110519 \sqrt{1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt{3+5 x}}-\frac{5199979 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3660250}+\frac{5442127 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3327500}+\frac{90397364 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{9150625}\\ &=\frac{3218 \sqrt{1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac{217 (2+3 x)^{7/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{110519 \sqrt{1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt{3+5 x}}-\frac{5199979 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3660250}-\frac{90397364 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{831875 \sqrt{33}}-\frac{5442127 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1663750 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.261288, size = 112, normalized size = 0.51 \[ \frac{-181999265 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{10 \sqrt{3 x+2} \left (177888150 x^4-1825153850 x^3-1696384053 x^2+89252928 x+246962693\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}+361589456 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{109807500} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 326, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( 219615000\,x-109807500 \right ) \left ( 6\,{x}^{2}+x-2 \right ) } \left ( 1819992650\,\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}-3615894560\,\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}+181999265\,\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}-361589456\,\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}-545997795\,\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 ) +1084768368\,\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 ) -5336644500\,{x}^{5}+51196852500\,{x}^{4}+87394598590\,{x}^{3}+31250093220\,{x}^{2}-9193939350\,x-4939253860 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,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 (3 \, x + 2\right )}^{\frac{11}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\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 (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1000 \, x^{6} + 300 \, x^{5} - 870 \, x^{4} - 179 \, x^{3} + 261 \, x^{2} + 27 \, x - 27}, 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 (3 \, x + 2\right )}^{\frac{11}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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