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 = 0.445394, 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 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{180 (3 x+2)^5}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac{137752591 \sqrt{1-2 x} \sqrt{5 x+3}}{14224896 (3 x+2)}+\frac{1316353 \sqrt{1-2 x} \sqrt{5 x+3}}{1016064 (3 x+2)^2}+\frac{37333 \sqrt{1-2 x} \sqrt{5 x+3}}{181440 (3 x+2)^3}-\frac{7591 \sqrt{1-2 x} \sqrt{5 x+3}}{30240 (3 x+2)^4}-\frac{19457889 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^7,x]
[Out]
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Rubi in Sympy [A] time = 43.9276, size = 190, normalized size = 0.91 \[ - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1260 \left (3 x + 2\right )^{5}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{18 \left (3 x + 2\right )^{6}} + \frac{137752591 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{14224896 \left (3 x + 2\right )} + \frac{1316353 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1016064 \left (3 x + 2\right )^{2}} + \frac{37333 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{181440 \left (3 x + 2\right )^{3}} + \frac{311 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4320 \left (3 x + 2\right )^{4}} - \frac{19457889 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1229312} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**7,x)
[Out]
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Mathematica [A] time = 0.137596, size = 92, normalized size = 0.44 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (2066288865 x^5+6979774260 x^4+9434103472 x^3+6379024416 x^2+2157325040 x+291805632\right )}{(3 x+2)^6}-97289445 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{12293120} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^7,x]
[Out]
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Maple [B] time = 0.02, size = 346, normalized size = 1.7 \[{\frac{1}{12293120\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 70924005405\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}+283696021620\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}+472826702700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+28928044110\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+420290402400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+97716839640\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+210145201200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+132077448608\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+56038720320\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+89306341824\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+6226524480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +30202550560\,x\sqrt{-10\,{x}^{2}-x+3}+4085278848\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^7,x)
[Out]
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Maxima [A] time = 1.51457, size = 369, normalized size = 1.77 \[ \frac{3652535}{921984} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{14 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{140 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{1329 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1568 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{49173 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21952 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{2191521 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{29749665}{614656} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{19457889}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{26211867}{1229312} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{8670839 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3687936 \,{\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228916, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (2066288865 \, x^{5} + 6979774260 \, x^{4} + 9434103472 \, x^{3} + 6379024416 \, x^{2} + 2157325040 \, x + 291805632\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 97289445 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{12293120 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^7,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**7,x)
[Out]
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GIAC/XCAS [A] time = 0.598949, size = 676, normalized size = 3.23 \[ \frac{19457889}{24586240} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{14641 \,{\left (1329 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 2108680 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} - 1434500480 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 382530534400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 46289743360000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 2287257907200000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{87808 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^7,x, algorithm="giac")
[Out]