Optimal. Leaf size=158 \[ -\frac{184 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5 \sqrt{33}}+\frac{556 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{92 \sqrt{1-2 x} \sqrt{3 x+2}}{3 (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{3 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{556}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.052475, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ \frac{556 \sqrt{1-2 x} \sqrt{3 x+2}}{3 \sqrt{5 x+3}}-\frac{92 \sqrt{1-2 x} \sqrt{3 x+2}}{3 (5 x+3)^{3/2}}+\frac{14 \sqrt{1-2 x}}{3 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{184 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}-\frac{556}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 152
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{14 \sqrt{1-2 x}}{3 \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{2}{3} \int \frac{90-103 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{3 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{92 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}-\frac{4}{99} \int \frac{\frac{7359}{2}-2277 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{3 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{92 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{556 \sqrt{1-2 x} \sqrt{2+3 x}}{3 \sqrt{3+5 x}}+\frac{8 \int \frac{47916+\frac{151371 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1089}\\ &=\frac{14 \sqrt{1-2 x}}{3 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{92 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{556 \sqrt{1-2 x} \sqrt{2+3 x}}{3 \sqrt{3+5 x}}+\frac{92}{5} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{556}{5} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 \sqrt{1-2 x}}{3 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{92 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{556 \sqrt{1-2 x} \sqrt{2+3 x}}{3 \sqrt{3+5 x}}-\frac{556}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{184 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.127017, size = 100, normalized size = 0.63 \[ \frac{4}{15} \sqrt{2} \left (139 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-70 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{2 \sqrt{1-2 x} \left (4170 x^2+5144 x+1583\right )}{3 \sqrt{3 x+2} (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.023, size = 219, normalized size = 1.4 \begin{align*}{\frac{2}{90\,{x}^{2}+15\,x-30}\sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 700\,\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}-1390\,\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}+420\,\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 ) -834\,\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 ) +41700\,{x}^{3}+30590\,{x}^{2}-9890\,x-7915 \right ) \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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{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{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{1125 \, x^{5} + 3525 \, x^{4} + 4415 \, x^{3} + 2763 \, x^{2} + 864 \, x + 108}, 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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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