Optimal. Leaf size=125 \[ -\frac{178 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{125 \sqrt{33}}-\frac{2 \sqrt{1-2 x} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}-\frac{194 \sqrt{1-2 x} \sqrt{3 x+2}}{825 \sqrt{5 x+3}}+\frac{458 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{125 \sqrt{33}} \]
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Rubi [A] time = 0.0404359, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 150, 158, 113, 119} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{3/2}}{15 (5 x+3)^{3/2}}-\frac{194 \sqrt{1-2 x} \sqrt{3 x+2}}{825 \sqrt{5 x+3}}-\frac{178 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{125 \sqrt{33}}+\frac{458 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{125 \sqrt{33}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 150
Rule 158
Rule 113
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x} (2+3 x)^{3/2}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{\left (\frac{5}{2}-12 x\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{194 \sqrt{1-2 x} \sqrt{2+3 x}}{825 \sqrt{3+5 x}}+\frac{4}{825} \int \frac{-\frac{237}{4}-\frac{687 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{194 \sqrt{1-2 x} \sqrt{2+3 x}}{825 \sqrt{3+5 x}}-\frac{458 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1375}+\frac{89}{125} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{194 \sqrt{1-2 x} \sqrt{2+3 x}}{825 \sqrt{3+5 x}}+\frac{458 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{125 \sqrt{33}}-\frac{178 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{125 \sqrt{33}}\\ \end{align*}
Mathematica [A] time = 0.271123, size = 97, normalized size = 0.78 \[ \frac{3395 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} (650 x+401)}{(5 x+3)^{3/2}}-458 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{4125} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 219, normalized size = 1.8 \begin{align*} -{\frac{1}{24750\,{x}^{2}+4125\,x-8250} \left ( 16975\,\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}-2290\,\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}+10185\,\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 ) -1374\,\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 ) +39000\,{x}^{3}+30560\,{x}^{2}-8990\,x-8020 \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{3}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\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{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, 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{3}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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