Optimal. Leaf size=137 \[ \frac{608185 \sqrt{1-2 x}}{924 \sqrt{5 x+3}}-\frac{6095 \sqrt{1-2 x}}{84 (5 x+3)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (3 x+2) (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]
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Rubi [A] time = 0.0469776, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{608185 \sqrt{1-2 x}}{924 \sqrt{5 x+3}}-\frac{6095 \sqrt{1-2 x}}{84 (5 x+3)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (3 x+2) (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]
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)^{5/2}} \, dx &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}-\frac{1}{2} \int \frac{-\frac{41}{2}+30 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}-\frac{1}{14} \int \frac{-\frac{7577}{4}+2430 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{1}{231} \int \frac{-\frac{855283}{8}+\frac{201135 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{608185 \sqrt{1-2 x}}{924 \sqrt{3+5 x}}-\frac{2 \int -\frac{45924219}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2541}\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{608185 \sqrt{1-2 x}}{924 \sqrt{3+5 x}}+\frac{126513}{56} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{608185 \sqrt{1-2 x}}{924 \sqrt{3+5 x}}+\frac{126513}{28} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{608185 \sqrt{1-2 x}}{924 \sqrt{3+5 x}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{28 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0643506, size = 79, normalized size = 0.58 \[ \frac{\sqrt{1-2 x} \left (27368325 x^3+52308690 x^2+33277877 x+7046540\right )}{924 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 250, normalized size = 1.8 \begin{align*}{\frac{1}{12936\, \left ( 2+3\,x \right ) ^{2}} \left ( 939359025\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+2379709530\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+2258636589\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+383156550\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+951883812\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+732321660\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+150297444\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +465890278\,x\sqrt{-10\,{x}^{2}-x+3}+98651560\,\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 = 1.95239, size = 232, normalized size = 1.69 \begin{align*} \frac{126513}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{608185 \, x}{462 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{635003}{924 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1985 \, x}{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{49}{18 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{1645}{36 \,{\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{6433}{36 \,{\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 = 1.59723, size = 378, normalized size = 2.76 \begin{align*} -\frac{4174929 \, \sqrt{7}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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 (27368325 \, x^{3} + 52308690 \, x^{2} + 33277877 \, x + 7046540\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{12936 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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 = 2.64839, size = 509, normalized size = 3.72 \begin{align*} -\frac{1}{129360} \, \sqrt{5}{\left (1225 \, \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} - 4174929 \, \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 )} - 2910600 \, \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{2744280 \, \sqrt{2}{\left (151 \,{\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{36120 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{144480 \, \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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