Optimal. Leaf size=144 \[ -\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{5 x+3}}+\frac{44475 \sqrt{1-2 x}}{2744 (3 x+2) \sqrt{5 x+3}}+\frac{255 \sqrt{1-2 x}}{196 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{4616025 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
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Rubi [A] time = 0.0480451, antiderivative size = 144, 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 = {103, 151, 152, 12, 93, 204} \[ -\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{5 x+3}}+\frac{44475 \sqrt{1-2 x}}{2744 (3 x+2) \sqrt{5 x+3}}+\frac{255 \sqrt{1-2 x}}{196 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{4616025 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 103
Rule 151
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{1}{21} \int \frac{\frac{135}{2}-90 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{255 \sqrt{1-2 x}}{196 (2+3 x)^2 \sqrt{3+5 x}}+\frac{1}{294} \int \frac{\frac{24075}{4}-7650 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{255 \sqrt{1-2 x}}{196 (2+3 x)^2 \sqrt{3+5 x}}+\frac{44475 \sqrt{1-2 x}}{2744 (2+3 x) \sqrt{3+5 x}}+\frac{\int \frac{\frac{2837025}{8}-\frac{667125 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{2058}\\ &=-\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{255 \sqrt{1-2 x}}{196 (2+3 x)^2 \sqrt{3+5 x}}+\frac{44475 \sqrt{1-2 x}}{2744 (2+3 x) \sqrt{3+5 x}}-\frac{\int \frac{152328825}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{11319}\\ &=-\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{255 \sqrt{1-2 x}}{196 (2+3 x)^2 \sqrt{3+5 x}}+\frac{44475 \sqrt{1-2 x}}{2744 (2+3 x) \sqrt{3+5 x}}-\frac{4616025 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{5488}\\ &=-\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{255 \sqrt{1-2 x}}{196 (2+3 x)^2 \sqrt{3+5 x}}+\frac{44475 \sqrt{1-2 x}}{2744 (2+3 x) \sqrt{3+5 x}}-\frac{4616025 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2744}\\ &=-\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{255 \sqrt{1-2 x}}{196 (2+3 x)^2 \sqrt{3+5 x}}+\frac{44475 \sqrt{1-2 x}}{2744 (2+3 x) \sqrt{3+5 x}}+\frac{4616025 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2744 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0617622, size = 79, normalized size = 0.55 \[ \frac{50776275 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\frac{7 \sqrt{1-2 x} \left (199715625 x^3+395028225 x^2+260298990 x+57135248\right )}{(3 x+2)^3 \sqrt{5 x+3}}}{211288} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 250, normalized size = 1.7 \begin{align*} -{\frac{1}{422576\, \left ( 2+3\,x \right ) ^{3}} \left ( 6854797125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+17822472525\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+17365486050\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+2796018750\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7514888700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+5530395150\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1218630600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +3644185860\,x\sqrt{-10\,{x}^{2}-x+3}+799893472\,\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{4} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52712, size = 385, normalized size = 2.67 \begin{align*} \frac{50776275 \, \sqrt{7}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 (199715625 \, x^{3} + 395028225 \, x^{2} + 260298990 \, x + 57135248\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{422576 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.19329, size = 509, normalized size = 3.53 \begin{align*} -\frac{923205}{76832} \, \sqrt{70} \sqrt{10}{\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 )} - \frac{125}{22} \, \sqrt{10}{\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{7425 \,{\left (487 \, \sqrt{10}{\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 )}^{5} + 217280 \, \sqrt{10}{\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} + 25693248 \, \sqrt{10}{\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 )}\right )}}{1372 \,{\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 )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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