Optimal. Leaf size=151 \[ \frac{3 \sqrt{5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}+\frac{247 \sqrt{5 x+3} (1-2 x)^{5/2}}{168 (3 x+2)^3}+\frac{13585 \sqrt{5 x+3} (1-2 x)^{3/2}}{672 (3 x+2)^2}+\frac{149435 \sqrt{5 x+3} \sqrt{1-2 x}}{448 (3 x+2)}-\frac{1643785 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]
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Rubi [A] time = 0.0393694, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{3 \sqrt{5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4}+\frac{247 \sqrt{5 x+3} (1-2 x)^{5/2}}{168 (3 x+2)^3}+\frac{13585 \sqrt{5 x+3} (1-2 x)^{3/2}}{672 (3 x+2)^2}+\frac{149435 \sqrt{5 x+3} \sqrt{1-2 x}}{448 (3 x+2)}-\frac{1643785 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 \sqrt{3+5 x}} \, dx &=\frac{3 (1-2 x)^{7/2} \sqrt{3+5 x}}{28 (2+3 x)^4}+\frac{247}{56} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{3 (1-2 x)^{7/2} \sqrt{3+5 x}}{28 (2+3 x)^4}+\frac{247 (1-2 x)^{5/2} \sqrt{3+5 x}}{168 (2+3 x)^3}+\frac{13585}{336} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{3 (1-2 x)^{7/2} \sqrt{3+5 x}}{28 (2+3 x)^4}+\frac{247 (1-2 x)^{5/2} \sqrt{3+5 x}}{168 (2+3 x)^3}+\frac{13585 (1-2 x)^{3/2} \sqrt{3+5 x}}{672 (2+3 x)^2}+\frac{149435}{448} \int \frac{\sqrt{1-2 x}}{(2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{3 (1-2 x)^{7/2} \sqrt{3+5 x}}{28 (2+3 x)^4}+\frac{247 (1-2 x)^{5/2} \sqrt{3+5 x}}{168 (2+3 x)^3}+\frac{13585 (1-2 x)^{3/2} \sqrt{3+5 x}}{672 (2+3 x)^2}+\frac{149435 \sqrt{1-2 x} \sqrt{3+5 x}}{448 (2+3 x)}+\frac{1643785}{896} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{3 (1-2 x)^{7/2} \sqrt{3+5 x}}{28 (2+3 x)^4}+\frac{247 (1-2 x)^{5/2} \sqrt{3+5 x}}{168 (2+3 x)^3}+\frac{13585 (1-2 x)^{3/2} \sqrt{3+5 x}}{672 (2+3 x)^2}+\frac{149435 \sqrt{1-2 x} \sqrt{3+5 x}}{448 (2+3 x)}+\frac{1643785}{448} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{3 (1-2 x)^{7/2} \sqrt{3+5 x}}{28 (2+3 x)^4}+\frac{247 (1-2 x)^{5/2} \sqrt{3+5 x}}{168 (2+3 x)^3}+\frac{13585 (1-2 x)^{3/2} \sqrt{3+5 x}}{672 (2+3 x)^2}+\frac{149435 \sqrt{1-2 x} \sqrt{3+5 x}}{448 (2+3 x)}-\frac{1643785 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{448 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0947234, size = 104, normalized size = 0.69 \[ \frac{247 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (15707 x^2+21638 x+7488\right )}{(3 x+2)^3}-19965 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{9408}+\frac{3 \sqrt{5 x+3} (1-2 x)^{7/2}}{28 (3 x+2)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 250, normalized size = 1.7 \begin{align*}{\frac{1}{18816\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 399439755\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+1065172680\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+1065172680\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+162928290\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+473410080\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+333126416\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+78901680\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +227316824\,x\sqrt{-10\,{x}^{2}-x+3}+51789024\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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 [A] time = 2.91958, size = 193, normalized size = 1.28 \begin{align*} \frac{1643785}{6272} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{36 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1477 \, \sqrt{-10 \, x^{2} - x + 3}}{216 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{37091 \, \sqrt{-10 \, x^{2} - x + 3}}{864 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{3879245 \, \sqrt{-10 \, x^{2} - x + 3}}{12096 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91738, size = 373, normalized size = 2.47 \begin{align*} -\frac{4931355 \, \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 (11637735 \, x^{3} + 23794744 \, x^{2} + 16236916 \, x + 3699216\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{18816 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 = 3.65008, size = 512, normalized size = 3.39 \begin{align*} \frac{328757}{12544} \, \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{6655 \,{\left (1947 \, \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 )}^{7} + 1009736 \, \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} + 213012800 \, \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} + 16266432000 \, \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 )}}{672 \,{\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 )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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