Optimal. Leaf size=180 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}+\frac{1948963 \sqrt{1-2 x} \sqrt{5 x+3}}{8297856 (3 x+2)}-\frac{12371 \sqrt{1-2 x} \sqrt{5 x+3}}{592704 (3 x+2)^2}-\frac{14831 \sqrt{1-2 x} \sqrt{5 x+3}}{105840 (3 x+2)^3}+\frac{437 \sqrt{1-2 x} \sqrt{5 x+3}}{17640 (3 x+2)^4}-\frac{933031 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{307328 \sqrt{7}} \]
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Rubi [A] time = 0.0637625, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {98, 149, 151, 12, 93, 204} \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}+\frac{1948963 \sqrt{1-2 x} \sqrt{5 x+3}}{8297856 (3 x+2)}-\frac{12371 \sqrt{1-2 x} \sqrt{5 x+3}}{592704 (3 x+2)^2}-\frac{14831 \sqrt{1-2 x} \sqrt{5 x+3}}{105840 (3 x+2)^3}+\frac{437 \sqrt{1-2 x} \sqrt{5 x+3}}{17640 (3 x+2)^4}-\frac{933031 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{307328 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 149
Rule 151
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^6} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac{1}{105} \int \frac{\left (-\frac{981}{2}-845 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^5} \, dx\\ &=\frac{437 \sqrt{1-2 x} \sqrt{3+5 x}}{17640 (2+3 x)^4}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac{\int \frac{-\frac{263381}{4}-111745 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{8820}\\ &=\frac{437 \sqrt{1-2 x} \sqrt{3+5 x}}{17640 (2+3 x)^4}-\frac{14831 \sqrt{1-2 x} \sqrt{3+5 x}}{105840 (2+3 x)^3}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac{\int \frac{-\frac{2624125}{8}-519085 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{185220}\\ &=\frac{437 \sqrt{1-2 x} \sqrt{3+5 x}}{17640 (2+3 x)^4}-\frac{14831 \sqrt{1-2 x} \sqrt{3+5 x}}{105840 (2+3 x)^3}-\frac{12371 \sqrt{1-2 x} \sqrt{3+5 x}}{592704 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac{\int \frac{-\frac{28511035}{16}-\frac{2164925 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{2593080}\\ &=\frac{437 \sqrt{1-2 x} \sqrt{3+5 x}}{17640 (2+3 x)^4}-\frac{14831 \sqrt{1-2 x} \sqrt{3+5 x}}{105840 (2+3 x)^3}-\frac{12371 \sqrt{1-2 x} \sqrt{3+5 x}}{592704 (2+3 x)^2}+\frac{1948963 \sqrt{1-2 x} \sqrt{3+5 x}}{8297856 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac{\int -\frac{881714295}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{18151560}\\ &=\frac{437 \sqrt{1-2 x} \sqrt{3+5 x}}{17640 (2+3 x)^4}-\frac{14831 \sqrt{1-2 x} \sqrt{3+5 x}}{105840 (2+3 x)^3}-\frac{12371 \sqrt{1-2 x} \sqrt{3+5 x}}{592704 (2+3 x)^2}+\frac{1948963 \sqrt{1-2 x} \sqrt{3+5 x}}{8297856 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}+\frac{933031 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{614656}\\ &=\frac{437 \sqrt{1-2 x} \sqrt{3+5 x}}{17640 (2+3 x)^4}-\frac{14831 \sqrt{1-2 x} \sqrt{3+5 x}}{105840 (2+3 x)^3}-\frac{12371 \sqrt{1-2 x} \sqrt{3+5 x}}{592704 (2+3 x)^2}+\frac{1948963 \sqrt{1-2 x} \sqrt{3+5 x}}{8297856 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}+\frac{933031 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{307328}\\ &=\frac{437 \sqrt{1-2 x} \sqrt{3+5 x}}{17640 (2+3 x)^4}-\frac{14831 \sqrt{1-2 x} \sqrt{3+5 x}}{105840 (2+3 x)^3}-\frac{12371 \sqrt{1-2 x} \sqrt{3+5 x}}{592704 (2+3 x)^2}+\frac{1948963 \sqrt{1-2 x} \sqrt{3+5 x}}{8297856 (2+3 x)}+\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac{933031 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{307328 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.062893, size = 84, normalized size = 0.47 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (87703335 x^4+231277650 x^3+222865988 x^2+93291272 x+14330592\right )}{(3 x+2)^5}-13995465 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{32269440} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{64538880\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3400897995\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+11336326650\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+15115102200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+1227846690\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+10076734800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+3237887100\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+3358911600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3120123832\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+447854880\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1306077808\,x\sqrt{-10\,{x}^{2}-x+3}+200628288\,\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 = 3.22107, size = 248, normalized size = 1.38 \begin{align*} \frac{933031}{4302592} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{315 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{239 \, \sqrt{-10 \, x^{2} - x + 3}}{5880 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{14831 \, \sqrt{-10 \, x^{2} - x + 3}}{105840 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac{12371 \, \sqrt{-10 \, x^{2} - x + 3}}{592704 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{1948963 \, \sqrt{-10 \, x^{2} - x + 3}}{8297856 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88782, size = 437, normalized size = 2.43 \begin{align*} -\frac{13995465 \, \sqrt{7}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 (87703335 \, x^{4} + 231277650 \, x^{3} + 222865988 \, x^{2} + 93291272 \, x + 14330592\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{64538880 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 = 4.25396, size = 594, normalized size = 3.3 \begin{align*} \frac{933031}{43025920} \, \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{1331 \,{\left (2103 \, \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 )}^{9} + 2747920 \, \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} + 1406935040 \, \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} - 74141312000 \, \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} - 10228753920000 \, \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 )}}{460992 \,{\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 )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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