Optimal. Leaf size=115 \[ -\frac{2615 \sqrt{1-2 x}}{28 \sqrt{5 x+3}}+\frac{173 \sqrt{1-2 x}}{28 (3 x+2) \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 \sqrt{5 x+3}}+\frac{17951 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]
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Rubi [A] time = 0.034644, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {99, 151, 152, 12, 93, 204} \[ -\frac{2615 \sqrt{1-2 x}}{28 \sqrt{5 x+3}}+\frac{173 \sqrt{1-2 x}}{28 (3 x+2) \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 \sqrt{5 x+3}}+\frac{17951 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 151
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 \sqrt{3+5 x}}-\frac{1}{2} \int \frac{-\frac{31}{2}+20 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 \sqrt{3+5 x}}+\frac{173 \sqrt{1-2 x}}{28 (2+3 x) \sqrt{3+5 x}}-\frac{1}{14} \int \frac{-\frac{3677}{4}+865 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{2615 \sqrt{1-2 x}}{28 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 \sqrt{3+5 x}}+\frac{173 \sqrt{1-2 x}}{28 (2+3 x) \sqrt{3+5 x}}+\frac{1}{77} \int -\frac{197461}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{2615 \sqrt{1-2 x}}{28 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 \sqrt{3+5 x}}+\frac{173 \sqrt{1-2 x}}{28 (2+3 x) \sqrt{3+5 x}}-\frac{17951}{56} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{2615 \sqrt{1-2 x}}{28 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 \sqrt{3+5 x}}+\frac{173 \sqrt{1-2 x}}{28 (2+3 x) \sqrt{3+5 x}}-\frac{17951}{28} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{2615 \sqrt{1-2 x}}{28 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 \sqrt{3+5 x}}+\frac{173 \sqrt{1-2 x}}{28 (2+3 x) \sqrt{3+5 x}}+\frac{17951 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{28 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.049807, size = 74, normalized size = 0.64 \[ \frac{1}{196} \left (17951 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\frac{7 \sqrt{1-2 x} \left (23535 x^2+30861 x+10100\right )}{(3 x+2)^2 \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 202, normalized size = 1.8 \begin{align*} -{\frac{1}{392\, \left ( 2+3\,x \right ) ^{2}} \left ( 807795\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+1561737\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1005256\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+329490\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+215412\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +432054\,x\sqrt{-10\,{x}^{2}-x+3}+141400\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.11925, size = 193, normalized size = 1.68 \begin{align*} -\frac{17951}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2615 \, x}{14 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{8191}{84 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{7}{6 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{169}{12 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31938, size = 305, normalized size = 2.65 \begin{align*} \frac{17951 \, \sqrt{7}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\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 (23535 \, x^{2} + 30861 \, x + 10100\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{392 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\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 = 2.51949, size = 427, normalized size = 3.71 \begin{align*} -\frac{1}{3920} \, \sqrt{5}{\left (17951 \, \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 )} + 9800 \, \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{9240 \, \sqrt{2}{\left (313 \,{\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} + \frac{69160 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{276640 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\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 )}^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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