Optimal. Leaf size=103 \[ \frac{2495 \sqrt{1-2 x}}{33 \sqrt{5 x+3}}-\frac{25 \sqrt{1-2 x}}{3 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^{3/2}}-\frac{519 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]
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Rubi [A] time = 0.0296518, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {99, 152, 12, 93, 204} \[ \frac{2495 \sqrt{1-2 x}}{33 \sqrt{5 x+3}}-\frac{25 \sqrt{1-2 x}}{3 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^{3/2}}-\frac{519 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx &=\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}}-\int \frac{-\frac{31}{2}+20 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{25 \sqrt{1-2 x}}{3 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}}+\frac{2}{33} \int \frac{-\frac{3509}{4}+825 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{25 \sqrt{1-2 x}}{3 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}}+\frac{2495 \sqrt{1-2 x}}{33 \sqrt{3+5 x}}-\frac{4}{363} \int -\frac{188397}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{25 \sqrt{1-2 x}}{3 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}}+\frac{2495 \sqrt{1-2 x}}{33 \sqrt{3+5 x}}+\frac{519}{2} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{25 \sqrt{1-2 x}}{3 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}}+\frac{2495 \sqrt{1-2 x}}{33 \sqrt{3+5 x}}+519 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{25 \sqrt{1-2 x}}{3 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}}+\frac{2495 \sqrt{1-2 x}}{33 \sqrt{3+5 x}}-\frac{519 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{\sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0409544, size = 93, normalized size = 0.9 \[ \frac{7 \sqrt{1-2 x} \left (37425 x^2+46580 x+14453\right )-17127 \sqrt{7} \sqrt{5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{231 (3 x+2) (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 202, normalized size = 2. \begin{align*}{\frac{1}{924+1386\,x} \left ( 1284525\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+2397780\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1490049\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+523950\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+308286\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +652120\,x\sqrt{-10\,{x}^{2}-x+3}+202342\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \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 [A] time = 1.59784, size = 163, normalized size = 1.58 \begin{align*} \frac{519}{14} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{4990 \, x}{33 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2605}{33 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{38 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{49}{9 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{185}{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51992, size = 309, normalized size = 3. \begin{align*} -\frac{17127 \, \sqrt{7}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \,{\left (37425 \, x^{2} + 46580 \, x + 14453\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{462 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\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 [B] time = 3.0895, size = 429, normalized size = 4.17 \begin{align*} -\frac{1}{18480} \, \sqrt{5}{\left (35 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 68508 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 55440 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{3659040 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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