Optimal. Leaf size=209 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{180 (3 x+2)^5}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac{137752591 \sqrt{1-2 x} \sqrt{5 x+3}}{14224896 (3 x+2)}+\frac{1316353 \sqrt{1-2 x} \sqrt{5 x+3}}{1016064 (3 x+2)^2}+\frac{37333 \sqrt{1-2 x} \sqrt{5 x+3}}{181440 (3 x+2)^3}-\frac{7591 \sqrt{1-2 x} \sqrt{5 x+3}}{30240 (3 x+2)^4}-\frac{19457889 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]
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Rubi [A] time = 0.0799882, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{180 (3 x+2)^5}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac{137752591 \sqrt{1-2 x} \sqrt{5 x+3}}{14224896 (3 x+2)}+\frac{1316353 \sqrt{1-2 x} \sqrt{5 x+3}}{1016064 (3 x+2)^2}+\frac{37333 \sqrt{1-2 x} \sqrt{5 x+3}}{181440 (3 x+2)^3}-\frac{7591 \sqrt{1-2 x} \sqrt{5 x+3}}{30240 (3 x+2)^4}-\frac{19457889 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 151
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{1}{18} \int \frac{\left (-\frac{3}{2}-30 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^6} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac{1}{270} \int \frac{\sqrt{3+5 x} \left (-\frac{3951}{4}+1365 x\right )}{\sqrt{1-2 x} (2+3 x)^5} \, dx\\ &=-\frac{7591 \sqrt{1-2 x} \sqrt{3+5 x}}{30240 (2+3 x)^4}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac{\int \frac{-\frac{153051}{8}+\frac{40605 x}{2}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{22680}\\ &=-\frac{7591 \sqrt{1-2 x} \sqrt{3+5 x}}{30240 (2+3 x)^4}+\frac{37333 \sqrt{1-2 x} \sqrt{3+5 x}}{181440 (2+3 x)^3}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac{\int \frac{-\frac{25165875}{16}+\frac{3919965 x}{2}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{476280}\\ &=-\frac{7591 \sqrt{1-2 x} \sqrt{3+5 x}}{30240 (2+3 x)^4}+\frac{37333 \sqrt{1-2 x} \sqrt{3+5 x}}{181440 (2+3 x)^3}+\frac{1316353 \sqrt{1-2 x} \sqrt{3+5 x}}{1016064 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac{\int \frac{-\frac{2978446485}{32}+\frac{691085325 x}{8}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{6667920}\\ &=-\frac{7591 \sqrt{1-2 x} \sqrt{3+5 x}}{30240 (2+3 x)^4}+\frac{37333 \sqrt{1-2 x} \sqrt{3+5 x}}{181440 (2+3 x)^3}+\frac{1316353 \sqrt{1-2 x} \sqrt{3+5 x}}{1016064 (2+3 x)^2}+\frac{137752591 \sqrt{1-2 x} \sqrt{3+5 x}}{14224896 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac{\int -\frac{165489345945}{64 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{46675440}\\ &=-\frac{7591 \sqrt{1-2 x} \sqrt{3+5 x}}{30240 (2+3 x)^4}+\frac{37333 \sqrt{1-2 x} \sqrt{3+5 x}}{181440 (2+3 x)^3}+\frac{1316353 \sqrt{1-2 x} \sqrt{3+5 x}}{1016064 (2+3 x)^2}+\frac{137752591 \sqrt{1-2 x} \sqrt{3+5 x}}{14224896 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}+\frac{19457889 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{351232}\\ &=-\frac{7591 \sqrt{1-2 x} \sqrt{3+5 x}}{30240 (2+3 x)^4}+\frac{37333 \sqrt{1-2 x} \sqrt{3+5 x}}{181440 (2+3 x)^3}+\frac{1316353 \sqrt{1-2 x} \sqrt{3+5 x}}{1016064 (2+3 x)^2}+\frac{137752591 \sqrt{1-2 x} \sqrt{3+5 x}}{14224896 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}+\frac{19457889 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{175616}\\ &=-\frac{7591 \sqrt{1-2 x} \sqrt{3+5 x}}{30240 (2+3 x)^4}+\frac{37333 \sqrt{1-2 x} \sqrt{3+5 x}}{181440 (2+3 x)^3}+\frac{1316353 \sqrt{1-2 x} \sqrt{3+5 x}}{1016064 (2+3 x)^2}+\frac{137752591 \sqrt{1-2 x} \sqrt{3+5 x}}{14224896 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{180 (2+3 x)^5}-\frac{19457889 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{175616 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.142647, size = 138, normalized size = 0.66 \[ \frac{1}{280} \left (\frac{2215 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )}{(3 x+2)^4}-43923 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{21952}+\frac{74 (1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^5}+\frac{20 (1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^6}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{12293120\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 70924005405\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+283696021620\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+472826702700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+28928044110\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+420290402400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+97716839640\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+210145201200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+132077448608\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+56038720320\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+89306341824\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+6226524480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +30202550560\,x\sqrt{-10\,{x}^{2}-x+3}+4085278848\,\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 = 2.19768, size = 369, normalized size = 1.77 \begin{align*} \frac{3652535}{921984} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{14 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{140 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{1329 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1568 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{49173 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21952 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{2191521 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{29749665}{614656} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{19457889}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{26211867}{1229312} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{8670839 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3687936 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57887, size = 505, normalized size = 2.42 \begin{align*} -\frac{97289445 \, \sqrt{7}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 (2066288865 \, x^{5} + 6979774260 \, x^{4} + 9434103472 \, x^{3} + 6379024416 \, x^{2} + 2157325040 \, x + 291805632\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{12293120 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 = 4.99862, size = 676, normalized size = 3.23 \begin{align*} \frac{19457889}{24586240} \, \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{14641 \,{\left (1329 \, \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 )}^{11} + 2108680 \, \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} - 1434500480 \, \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} - 382530534400 \, \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} - 46289743360000 \, \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} - 2287257907200000 \, \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 )}}{87808 \,{\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 )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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