3.1391 \(\int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx\)

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]

(-9*(8575 + 2643*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)) + ((6637 + 8193*x)*(2 + 3
*x^2)^(3/2))/(9800*(3 + 2*x)^3) + ((23 + 76*x)*(2 + 3*x^2)^(5/2))/(140*(3 + 2*x)
^5) + (63*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + (789723*ArcTanh[(4 - 9*x)/(Sqrt[35]
*Sqrt[2 + 3*x^2])])/(39200*Sqrt[35])

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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]

(-9*(8575 + 2643*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)) + ((6637 + 8193*x)*(2 + 3
*x^2)^(3/2))/(9800*(3 + 2*x)^3) + ((23 + 76*x)*(2 + 3*x^2)^(5/2))/(140*(3 + 2*x)
^5) + (63*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + (789723*ArcTanh[(4 - 9*x)/(Sqrt[35]
*Sqrt[2 + 3*x^2])])/(39200*Sqrt[35])

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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]

63*sqrt(3)*asinh(sqrt(6)*x/2)/32 + 789723*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35
*sqrt(3*x**2 + 2)))/1372000 - (3044736*x + 9878400)*sqrt(3*x**2 + 2)/(2508800*(2
*x + 3)) + (786528*x + 637152)*(3*x**2 + 2)**(3/2)/(940800*(2*x + 3)**3) + (1520
*x + 460)*(3*x**2 + 2)**(5/2)/(2800*(2*x + 3)**5)

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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]

((-70*Sqrt[2 + 3*x^2]*(5999363 + 17940463*x + 20911298*x^2 + 11367738*x^3 + 2740
188*x^4 + 88200*x^5))/(3 + 2*x)^5 + 2701125*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] - 78972
3*Sqrt[35]*Log[3 + 2*x] + 789723*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x
^2])])/1372000

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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]

-377133/525218750/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)+248967/686000*x*(3*(x+3/2
)^2-9*x-19/4)^(1/2)+789723/1372000*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x
+3/2)^2-36*x-19)^(1/2))+1131399/525218750*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)-13/5600
/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)+63/32*arcsinh(1/2*x*6^(1/2))*3^(1/2)-11/
24500/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(7/2)-521/857500/(x+3/2)^3*(3*(x+3/2)^2-9
*x-19/4)^(7/2)+267723/12005000*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-2241/30012500/(x+3
/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)-263241/6002500*(3*(x+3/2)^2-9*x-19/4)^(3/2)-7
89723/1372000*(12*(x+3/2)^2-36*x-19)^(1/2)-789723/262609375*(3*(x+3/2)^2-9*x-19/
4)^(5/2)

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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]

6723/30012500*(3*x^2 + 2)^(5/2) - 13/175*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 7
20*x^3 + 1080*x^2 + 810*x + 243) - 44/6125*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 +
216*x^2 + 216*x + 81) - 1042/214375*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 2
7) - 2241/7503125*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 267723/12005000*(3*x^2
+ 2)^(3/2)*x - 263241/6002500*(3*x^2 + 2)^(3/2) - 377133/30012500*(3*x^2 + 2)^(5
/2)/(2*x + 3) + 248967/686000*sqrt(3*x^2 + 2)*x + 63/32*sqrt(3)*arcsinh(1/2*sqrt
(6)*x) - 789723/1372000*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6
)/abs(2*x + 3)) - 789723/686000*sqrt(3*x^2 + 2)

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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]

1/2744000*sqrt(35)*(77175*sqrt(35)*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^
2 + 810*x + 243)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - 4*sqrt(35)*(88200
*x^5 + 2740188*x^4 + 11367738*x^3 + 20911298*x^2 + 17940463*x + 5999363)*sqrt(3*
x^2 + 2) + 789723*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log(-(sq
rt(35)*(93*x^2 - 36*x + 43) - 35*sqrt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))
/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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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((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**6,x)

[Out]

Timed out

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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]

-63/32*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 789723/1372000*sqrt(35)*ln(-ab
s(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(3
5) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/64*sqrt(3*x^2 + 2) - 3/156800*(3103461*
(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 28143036*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))
^8 + 283092753*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 328235733*sqrt(3)*(sqrt(3)*x -
sqrt(3*x^2 + 2))^6 - 360132696*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 774358774*sqrt(
3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 1736218428*(sqrt(3)*x - sqrt(3*x^2 + 2))^3
- 495467552*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 199787184*sqrt(3)*x - 1108
6336*sqrt(3) - 199787184*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^5