Optimal. Leaf size=191 \[ -\frac{265648 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{2268945}+\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{147 (3 x+2)^{7/2}}+\frac{816622 \sqrt{1-2 x} \sqrt{5 x+3}}{2268945 \sqrt{3 x+2}}-\frac{101902 \sqrt{1-2 x} \sqrt{5 x+3}}{324135 (3 x+2)^{3/2}}+\frac{676 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{5/2}}-\frac{816622 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2268945} \]
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Rubi [A] time = 0.0672227, 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 = {98, 150, 152, 158, 113, 119} \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{147 (3 x+2)^{7/2}}+\frac{816622 \sqrt{1-2 x} \sqrt{5 x+3}}{2268945 \sqrt{3 x+2}}-\frac{101902 \sqrt{1-2 x} \sqrt{5 x+3}}{324135 (3 x+2)^{3/2}}+\frac{676 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{5/2}}-\frac{265648 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2268945}-\frac{816622 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2268945} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx &=\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac{2}{147} \int \frac{\left (-342-\frac{1195 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx\\ &=\frac{676 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{5/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac{4 \int \frac{-\frac{115673}{4}-\frac{198985 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{15435}\\ &=\frac{676 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{5/2}}-\frac{101902 \sqrt{1-2 x} \sqrt{3+5 x}}{324135 (2+3 x)^{3/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac{8 \int \frac{-\frac{475777}{8}-\frac{254755 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{324135}\\ &=\frac{676 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{5/2}}-\frac{101902 \sqrt{1-2 x} \sqrt{3+5 x}}{324135 (2+3 x)^{3/2}}+\frac{816622 \sqrt{1-2 x} \sqrt{3+5 x}}{2268945 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac{16 \int \frac{-\frac{1955465}{8}-\frac{2041555 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2268945}\\ &=\frac{676 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{5/2}}-\frac{101902 \sqrt{1-2 x} \sqrt{3+5 x}}{324135 (2+3 x)^{3/2}}+\frac{816622 \sqrt{1-2 x} \sqrt{3+5 x}}{2268945 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}+\frac{816622 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2268945}+\frac{1461064 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2268945}\\ &=\frac{676 \sqrt{1-2 x} \sqrt{3+5 x}}{15435 (2+3 x)^{5/2}}-\frac{101902 \sqrt{1-2 x} \sqrt{3+5 x}}{324135 (2+3 x)^{3/2}}+\frac{816622 \sqrt{1-2 x} \sqrt{3+5 x}}{2268945 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{147 (2+3 x)^{7/2}}-\frac{816622 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2268945}-\frac{265648 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2268945}\\ \end{align*}
Mathematica [A] time = 0.158384, size = 104, normalized size = 0.54 \[ \frac{2 \left (\sqrt{2} \left (1783285 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+408311 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (11024397 x^3+18838881 x^2+10645545 x+1985537\right )}{(3 x+2)^{7/2}}\right )}{6806835} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 409, normalized size = 2.1 \begin{align*} -{\frac{2}{68068350\,{x}^{2}+6806835\,x-20420505} \left ( 48148695\,\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}+11024397\,\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}+96297390\,\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}+22048794\,\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}+64198260\,\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}+14699196\,\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}+14266280\,\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 ) +3266488\,\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 ) -330731910\,{x}^{5}-598239621\,{x}^{4}-276663420\,{x}^{3}+78047184\,{x}^{2}+89853294\,x+17869833 \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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}\,{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}}{486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, 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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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