Optimal. Leaf size=173 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt{5 x+3}}+\frac{3997345 \sqrt{1-2 x}}{4032 (3 x+2) \sqrt{5 x+3}}+\frac{22957 \sqrt{1-2 x}}{288 (3 x+2)^2 \sqrt{5 x+3}}+\frac{2051 \sqrt{1-2 x}}{216 (3 x+2)^3 \sqrt{5 x+3}}-\frac{181304825 \sqrt{1-2 x}}{12096 \sqrt{5 x+3}}+\frac{46095555 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]
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Rubi [A] time = 0.0609107, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {98, 149, 151, 152, 12, 93, 204} \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt{5 x+3}}+\frac{3997345 \sqrt{1-2 x}}{4032 (3 x+2) \sqrt{5 x+3}}+\frac{22957 \sqrt{1-2 x}}{288 (3 x+2)^2 \sqrt{5 x+3}}+\frac{2051 \sqrt{1-2 x}}{216 (3 x+2)^3 \sqrt{5 x+3}}-\frac{181304825 \sqrt{1-2 x}}{12096 \sqrt{5 x+3}}+\frac{46095555 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 151
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{1}{12} \int \frac{\left (\frac{425}{2}-194 x\right ) \sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{2051 \sqrt{1-2 x}}{216 (2+3 x)^3 \sqrt{3+5 x}}-\frac{1}{108} \int \frac{-\frac{81763}{4}+29601 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{2051 \sqrt{1-2 x}}{216 (2+3 x)^3 \sqrt{3+5 x}}+\frac{22957 \sqrt{1-2 x}}{288 (2+3 x)^2 \sqrt{3+5 x}}-\frac{\int \frac{-\frac{15125495}{8}+2410485 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx}{1512}\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{2051 \sqrt{1-2 x}}{216 (2+3 x)^3 \sqrt{3+5 x}}+\frac{22957 \sqrt{1-2 x}}{288 (2+3 x)^2 \sqrt{3+5 x}}+\frac{3997345 \sqrt{1-2 x}}{4032 (2+3 x) \sqrt{3+5 x}}-\frac{\int \frac{-\frac{1784763365}{16}+\frac{419721225 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{10584}\\ &=-\frac{181304825 \sqrt{1-2 x}}{12096 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{2051 \sqrt{1-2 x}}{216 (2+3 x)^3 \sqrt{3+5 x}}+\frac{22957 \sqrt{1-2 x}}{288 (2+3 x)^2 \sqrt{3+5 x}}+\frac{3997345 \sqrt{1-2 x}}{4032 (2+3 x) \sqrt{3+5 x}}+\frac{\int -\frac{95832658845}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{58212}\\ &=-\frac{181304825 \sqrt{1-2 x}}{12096 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{2051 \sqrt{1-2 x}}{216 (2+3 x)^3 \sqrt{3+5 x}}+\frac{22957 \sqrt{1-2 x}}{288 (2+3 x)^2 \sqrt{3+5 x}}+\frac{3997345 \sqrt{1-2 x}}{4032 (2+3 x) \sqrt{3+5 x}}-\frac{46095555}{896} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{181304825 \sqrt{1-2 x}}{12096 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{2051 \sqrt{1-2 x}}{216 (2+3 x)^3 \sqrt{3+5 x}}+\frac{22957 \sqrt{1-2 x}}{288 (2+3 x)^2 \sqrt{3+5 x}}+\frac{3997345 \sqrt{1-2 x}}{4032 (2+3 x) \sqrt{3+5 x}}-\frac{46095555}{448} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{181304825 \sqrt{1-2 x}}{12096 \sqrt{3+5 x}}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{2051 \sqrt{1-2 x}}{216 (2+3 x)^3 \sqrt{3+5 x}}+\frac{22957 \sqrt{1-2 x}}{288 (2+3 x)^2 \sqrt{3+5 x}}+\frac{3997345 \sqrt{1-2 x}}{4032 (2+3 x) \sqrt{3+5 x}}+\frac{46095555 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{448 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0696793, size = 84, normalized size = 0.49 \[ \frac{46095555 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\frac{7 \sqrt{1-2 x} \left (543914475 x^4+1438446565 x^3+1426133132 x^2+628209228 x+103735088\right )}{(3 x+2)^4 \sqrt{5 x+3}}}{3136} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 298, normalized size = 1.7 \begin{align*} -{\frac{1}{6272\, \left ( 2+3\,x \right ) ^{4}} \left ( 18668699775\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+60984419265\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+79653119040\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+7614802650\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+51995786040\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+20138251910\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+16963164240\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+19965863848\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+2212586640\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +8794929192\,x\sqrt{-10\,{x}^{2}-x+3}+1452291232\,\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 [B] time = 2.46005, size = 400, normalized size = 2.31 \begin{align*} -\frac{46095555}{6272} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{181304825 \, x}{6048 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{189299515}{12096 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343}{108 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{13181}{648 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{466361}{2592 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{1301839}{576 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83729, size = 443, normalized size = 2.56 \begin{align*} \frac{46095555 \, \sqrt{7}{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\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 (543914475 \, x^{4} + 1438446565 \, x^{3} + 1426133132 \, x^{2} + 628209228 \, x + 103735088\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{6272 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\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 = 3.76462, size = 591, normalized size = 3.42 \begin{align*} -\frac{9219111}{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{605}{2} \, \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{605 \,{\left (77025 \, \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} + 51138136 \, \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} + 12067876800 \, \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} + 984130112000 \, \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 )}}{224 \,{\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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