Optimal. Leaf size=156 \[ -\frac{2281 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{6875 \sqrt{33}}-\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}-\frac{536 \sqrt{1-2 x} (3 x+2)^{3/2}}{9075 \sqrt{5 x+3}}-\frac{487 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{46159 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6875 \sqrt{33}} \]
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Rubi [A] time = 0.0506745, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}-\frac{536 \sqrt{1-2 x} (3 x+2)^{3/2}}{9075 \sqrt{5 x+3}}-\frac{487 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{15125}-\frac{2281 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6875 \sqrt{33}}-\frac{46159 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6875 \sqrt{33}} \]
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)^{7/2}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac{2}{165} \int \frac{\left (-\frac{221}{2}-\frac{279 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac{536 \sqrt{1-2 x} (2+3 x)^{3/2}}{9075 \sqrt{3+5 x}}-\frac{4 \int \frac{\left (-\frac{4275}{2}-\frac{4383 x}{4}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{9075}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac{536 \sqrt{1-2 x} (2+3 x)^{3/2}}{9075 \sqrt{3+5 x}}-\frac{487 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{15125}+\frac{4 \int \frac{\frac{543681}{8}+\frac{415431 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{136125}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac{536 \sqrt{1-2 x} (2+3 x)^{3/2}}{9075 \sqrt{3+5 x}}-\frac{487 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{15125}+\frac{2281 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{13750}+\frac{46159 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{75625}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac{536 \sqrt{1-2 x} (2+3 x)^{3/2}}{9075 \sqrt{3+5 x}}-\frac{487 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{15125}-\frac{46159 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6875 \sqrt{33}}-\frac{2281 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6875 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.19047, size = 102, normalized size = 0.65 \[ \frac{-17045 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (81675 x^2+101350 x+31429\right )}{(5 x+3)^{3/2}}+92318 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{453750} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 224, normalized size = 1.4 \begin{align*}{\frac{1}{2722500\,{x}^{2}+453750\,x-907500} \left ( 85225\,\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}-461590\,\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}+51135\,\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 ) -276954\,\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 ) -4900500\,{x}^{4}-6897750\,{x}^{3}-1265740\,{x}^{2}+1712710\,x+628580 \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{7}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{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 (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{250 \, x^{4} + 325 \, x^{3} + 45 \, x^{2} - 81 \, 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{7}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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