Optimal. Leaf size=172 \[ \frac{1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}+\frac{37}{360} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac{4783 \sqrt{1-2 x} (5 x+3)^{5/2}}{32400}-\frac{14557 \sqrt{1-2 x} (5 x+3)^{3/2}}{28800}-\frac{1994287 \sqrt{1-2 x} \sqrt{5 x+3}}{3110400}+\frac{109715471 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9331200 \sqrt{10}}+\frac{98}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
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Rubi [A] time = 0.0802862, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {101, 154, 157, 54, 216, 93, 204} \[ \frac{1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}+\frac{37}{360} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac{4783 \sqrt{1-2 x} (5 x+3)^{5/2}}{32400}-\frac{14557 \sqrt{1-2 x} (5 x+3)^{3/2}}{28800}-\frac{1994287 \sqrt{1-2 x} \sqrt{5 x+3}}{3110400}+\frac{109715471 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9331200 \sqrt{10}}+\frac{98}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx &=\frac{1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{15} \int \frac{\left (-50-\frac{185 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=\frac{37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{900} \int \frac{\left (-\frac{4705}{2}-\frac{23915 x}{4}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=\frac{4783 \sqrt{1-2 x} (3+5 x)^{5/2}}{32400}+\frac{37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{\int \frac{\left (\frac{30935}{4}-\frac{1965195 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)} \, dx}{40500}\\ &=-\frac{14557 \sqrt{1-2 x} (3+5 x)^{3/2}}{28800}+\frac{4783 \sqrt{1-2 x} (3+5 x)^{5/2}}{32400}+\frac{37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{\int \frac{\sqrt{3+5 x} \left (\frac{15459435}{8}+\frac{29914305 x}{16}\right )}{\sqrt{1-2 x} (2+3 x)} \, dx}{486000}\\ &=-\frac{1994287 \sqrt{1-2 x} \sqrt{3+5 x}}{3110400}-\frac{14557 \sqrt{1-2 x} (3+5 x)^{3/2}}{28800}+\frac{4783 \sqrt{1-2 x} (3+5 x)^{5/2}}{32400}+\frac{37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{\int \frac{-\frac{526625355}{16}-\frac{1645732065 x}{32}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2916000}\\ &=-\frac{1994287 \sqrt{1-2 x} \sqrt{3+5 x}}{3110400}-\frac{14557 \sqrt{1-2 x} (3+5 x)^{3/2}}{28800}+\frac{4783 \sqrt{1-2 x} (3+5 x)^{5/2}}{32400}+\frac{37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{343}{729} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx+\frac{109715471 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{18662400}\\ &=-\frac{1994287 \sqrt{1-2 x} \sqrt{3+5 x}}{3110400}-\frac{14557 \sqrt{1-2 x} (3+5 x)^{3/2}}{28800}+\frac{4783 \sqrt{1-2 x} (3+5 x)^{5/2}}{32400}+\frac{37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac{686}{729} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{109715471 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{9331200 \sqrt{5}}\\ &=-\frac{1994287 \sqrt{1-2 x} \sqrt{3+5 x}}{3110400}-\frac{14557 \sqrt{1-2 x} (3+5 x)^{3/2}}{28800}+\frac{4783 \sqrt{1-2 x} (3+5 x)^{5/2}}{32400}+\frac{37}{360} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{1}{15} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{109715471 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{9331200 \sqrt{10}}+\frac{98}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0940488, size = 115, normalized size = 0.67 \[ \frac{30 \sqrt{5 x+3} \left (-41472000 x^5+44409600 x^4+12050880 x^3-28956360 x^2+4176026 x+2165117\right )-109715471 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+12544000 \sqrt{7-14 x} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{93312000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 149, normalized size = 0.9 \begin{align*}{\frac{1}{186624000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1244160000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-710208000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-716630400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+109715471\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -12544000\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +510375600\,x\sqrt{-10\,{x}^{2}-x+3}+129907020\,\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 = 2.52454, size = 151, normalized size = 0.88 \begin{align*} \frac{1}{15} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{37}{72} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{787}{12960} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{79439}{51840} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{109715471}{186624000} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{49}{729} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{865517}{3110400} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81688, size = 423, normalized size = 2.46 \begin{align*} \frac{1}{3110400} \,{\left (20736000 \, x^{4} - 11836800 \, x^{3} - 11943840 \, x^{2} + 8506260 \, x + 2165117\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + \frac{49}{729} \, \sqrt{7} \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 ) - \frac{109715471}{186624000} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\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 [A] time = 2.19966, size = 286, normalized size = 1.66 \begin{align*} -\frac{49}{7290} \, \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{1}{77760000} \,{\left (12 \,{\left (8 \,{\left (36 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} - 713 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 112817 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 655065 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 9971435 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{109715471}{186624000} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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