Optimal. Leaf size=158 \[ -\frac{15}{2} \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{(3 x+2)^{3/2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{34 \sqrt{3 x+2} (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}-\frac{225}{22} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-68 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.0531453, antiderivative size = 158, 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 = {97, 150, 154, 158, 113, 119} \[ \frac{(3 x+2)^{3/2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{34 \sqrt{3 x+2} (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}-\frac{225}{22} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{15}{2} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-68 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 154
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{\sqrt{2+3 x} \sqrt{3+5 x} \left (\frac{57}{2}+45 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{34 \sqrt{2+3 x} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-1974-\frac{6075 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{225}{22} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{34 \sqrt{2+3 x} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac{1}{297} \int \frac{\frac{255717}{4}+100980 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{225}{22} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{34 \sqrt{2+3 x} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac{45}{4} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+68 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{225}{22} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{34 \sqrt{2+3 x} (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{(2+3 x)^{3/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-68 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{15}{2} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}
Mathematica [A] time = 0.202569, size = 120, normalized size = 0.76 \[ -\frac{-137 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2 \sqrt{3 x+2} \sqrt{5 x+3} \left (30 x^2-302 x+105\right )+272 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{12 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 238, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( 360\,{x}^{3}+276\,{x}^{2}-84\,x-72 \right ) \left ( 2\,x-1 \right ) } \left ( 274\,\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}-544\,\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}-137\,\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 ) +272\,\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 ) -900\,{x}^{4}+7920\,{x}^{3}+7966\,{x}^{2}-366\,x-1260 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \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{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{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 (15 \, x^{2} + 19 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, 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{3}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{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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