Optimal. Leaf size=146 \[ \frac{1}{360} (209-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{(25-5586 x) \left (3 x^2+5 x+2\right )^{3/2}}{3456}+\frac{(51455-106734 x) \sqrt{3 x^2+5 x+2}}{27648}-\frac{543811 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{55296 \sqrt{3}}+\frac{325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]
[Out]
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Rubi [A] time = 0.298556, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{1}{360} (209-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac{(25-5586 x) \left (3 x^2+5 x+2\right )^{3/2}}{3456}+\frac{(51455-106734 x) \sqrt{3 x^2+5 x+2}}{27648}-\frac{543811 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{55296 \sqrt{3}}+\frac{325}{128} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x),x]
[Out]
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Rubi in Sympy [A] time = 40.8677, size = 133, normalized size = 0.91 \[ \frac{\left (- 1280808 x + 617460\right ) \sqrt{3 x^{2} + 5 x + 2}}{331776} + \frac{\left (- 33516 x + 150\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{20736} + \frac{\left (- 30 x + 209\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{360} - \frac{543811 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{165888} - \frac{325 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x),x)
[Out]
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Mathematica [A] time = 0.134184, size = 122, normalized size = 0.84 \[ \frac{-2106000 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-2719055 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (103680 x^5-376704 x^4-1311120 x^3-1624872 x^2-583490 x-580299\right )+2106000 \sqrt{5} \log (2 x+3)}{829440} \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x),x]
[Out]
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Maple [B] time = 0.012, size = 239, normalized size = 1.6 \[ -{\frac{5+6\,x}{72} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{25+30\,x}{3456} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{25+30\,x}{27648}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{5\,\sqrt{3}}{165888}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{13}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{65+78\,x}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{1235+1482\,x}{384}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}-{\frac{7553\,\sqrt{3}}{2304}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{65}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{325}{128}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{325\,\sqrt{5}}{128}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x),x)
[Out]
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Maxima [A] time = 0.812973, size = 212, normalized size = 1.45 \[ -\frac{1}{12} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{209}{360} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{931}{576} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{25}{3456} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{17789}{4608} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{543811}{165888} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{325}{128} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{51455}{27648} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.288204, size = 188, normalized size = 1.29 \[ -\frac{1}{1658880} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (103680 \, x^{5} - 376704 \, x^{4} - 1311120 \, x^{3} - 1624872 \, x^{2} - 583490 \, x - 580299\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 702000 \, \sqrt{5} \sqrt{3} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 2719055 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x),x)
[Out]
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GIAC/XCAS [A] time = 0.309497, size = 197, normalized size = 1.35 \[ -\frac{1}{138240} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (8 \,{\left (30 \, x - 109\right )} x - 3035\right )} x - 67703\right )} x - 291745\right )} x - 580299\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{325}{128} \, \sqrt{5}{\rm ln}\left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac{543811}{165888} \, \sqrt{3}{\rm ln}\left ({\left | -6 \, \sqrt{3} x - 5 \, \sqrt{3} + 6 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3),x, algorithm="giac")
[Out]