Optimal. Leaf size=115 \[ -\frac{3895 \sqrt{5 x+3}}{7546 \sqrt{1-2 x}}+\frac{345 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)}+\frac{3 \sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}-\frac{12465 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
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Rubi [A] time = 0.0355607, 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 = {103, 151, 152, 12, 93, 204} \[ -\frac{3895 \sqrt{5 x+3}}{7546 \sqrt{1-2 x}}+\frac{345 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)}+\frac{3 \sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}-\frac{12465 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
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}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx &=\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{1}{14} \int \frac{\frac{35}{2}-60 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{345 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)}+\frac{1}{98} \int \frac{-\frac{445}{4}-1725 x}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{3895 \sqrt{3+5 x}}{7546 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{345 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)}-\frac{\int -\frac{137115}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{3773}\\ &=-\frac{3895 \sqrt{3+5 x}}{7546 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{345 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)}+\frac{12465 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2744}\\ &=-\frac{3895 \sqrt{3+5 x}}{7546 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{345 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)}+\frac{12465 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1372}\\ &=-\frac{3895 \sqrt{3+5 x}}{7546 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{345 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)}-\frac{12465 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1372 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0466568, size = 85, normalized size = 0.74 \[ \frac{-7 \sqrt{5 x+3} \left (70110 x^2+13785 x-25204\right )-137115 \sqrt{7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{105644 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 209, normalized size = 1.8 \begin{align*}{\frac{1}{211288\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) } \left ( 2468070\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+2056725\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-548460\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+981540\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-548460\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +192990\,x\sqrt{-10\,{x}^{2}-x+3}-352856\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \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}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7778, size = 306, normalized size = 2.66 \begin{align*} -\frac{137115 \, \sqrt{7}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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 (70110 \, x^{2} + 13785 \, x - 25204\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{211288 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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 = 3.33763, size = 382, normalized size = 3.32 \begin{align*} \frac{2493}{38416} \, \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{16 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{18865 \,{\left (2 \, x - 1\right )}} + \frac{297 \,{\left (9 \, \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} + 1640 \, \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 )}}{98 \,{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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