Optimal. Leaf size=133 \[ \frac{(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}+\frac{(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{9800 (2 x+3)^3}-\frac{9 (2643 x+8575) \sqrt{3 x^2+2}}{19600 (2 x+3)}+\frac{789723 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}+\frac{63}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.241289, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}+\frac{(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{9800 (2 x+3)^3}-\frac{9 (2643 x+8575) \sqrt{3 x^2+2}}{19600 (2 x+3)}+\frac{789723 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{39200 \sqrt{35}}+\frac{63}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 24.2521, size = 117, normalized size = 0.88 \[ \frac{63 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{32} + \frac{789723 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{1372000} - \frac{\left (3044736 x + 9878400\right ) \sqrt{3 x^{2} + 2}}{2508800 \left (2 x + 3\right )} + \frac{\left (786528 x + 637152\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{940800 \left (2 x + 3\right )^{3}} + \frac{\left (1520 x + 460\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{2800 \left (2 x + 3\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**6,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.176722, size = 112, normalized size = 0.84 \[ \frac{789723 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{70 \sqrt{3 x^2+2} \left (88200 x^5+2740188 x^4+11367738 x^3+20911298 x^2+17940463 x+5999363\right )}{(2 x+3)^5}-789723 \sqrt{35} \log (2 x+3)+2701125 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{1372000} \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.021, size = 248, normalized size = 1.9 \[ -{\frac{377133}{525218750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{248967\,x}{686000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{789723\,\sqrt{35}}{1372000}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{1131399\,x}{525218750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{13}{5600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}+{\frac{63\,\sqrt{3}}{32}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{11}{24500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{521}{857500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}+{\frac{267723\,x}{12005000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2241}{30012500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{263241}{6002500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{789723}{1372000}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{789723}{262609375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(3*x^2+2)^(5/2)/(2*x+3)^6,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.779823, size = 329, normalized size = 2.47 \[ \frac{6723}{30012500} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{175 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{44 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{6125 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{1042 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{214375 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{2241 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{7503125 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{267723}{12005000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x - \frac{263241}{6002500} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{377133 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{30012500 \,{\left (2 \, x + 3\right )}} + \frac{248967}{686000} \, \sqrt{3 \, x^{2} + 2} x + \frac{63}{32} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{789723}{1372000} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{789723}{686000} \, \sqrt{3 \, x^{2} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 2)^(5/2)*(x - 5)/(2*x + 3)^6,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.292701, size = 271, normalized size = 2.04 \[ \frac{\sqrt{35}{\left (77175 \, \sqrt{35} \sqrt{3}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - 4 \, \sqrt{35}{\left (88200 \, x^{5} + 2740188 \, x^{4} + 11367738 \, x^{3} + 20911298 \, x^{2} + 17940463 \, x + 5999363\right )} \sqrt{3 \, x^{2} + 2} + 789723 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} - 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{2744000 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 2)^(5/2)*(x - 5)/(2*x + 3)^6,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**6,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.318399, size = 474, normalized size = 3.56 \[ -\frac{63}{32} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{789723}{1372000} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{9}{64} \, \sqrt{3 \, x^{2} + 2} - \frac{3 \,{\left (3103461 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 28143036 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 283092753 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 328235733 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 360132696 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 774358774 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 1736218428 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 495467552 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 199787184 \, \sqrt{3} x - 11086336 \, \sqrt{3} - 199787184 \, \sqrt{3 \, x^{2} + 2}\right )}}{156800 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 2)^(5/2)*(x - 5)/(2*x + 3)^6,x, algorithm="giac")
[Out]