3.1394 \(\int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^9} \, dx\)

Optimal. Leaf size=158 \[ -\frac{773 \left (3 x^2+2\right )^{7/2}}{68600 (2 x+3)^7}-\frac{13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}-\frac{233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{171500 (2 x+3)^6}-\frac{699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{2401000 (2 x+3)^4}-\frac{6291 (4-9 x) \sqrt{3 x^2+2}}{84035000 (2 x+3)^2}-\frac{18873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42017500 \sqrt{35}} \]

[Out]

(-6291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(84035000*(3 + 2*x)^2) - (699*(4 - 9*x)*(2 + 3
*x^2)^(3/2))/(2401000*(3 + 2*x)^4) - (233*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(171500*(
3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(280*(3 + 2*x)^8) - (773*(2 + 3*x^2)^(7/2))
/(68600*(3 + 2*x)^7) - (18873*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(42
017500*Sqrt[35])

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Rubi [A]  time = 0.226547, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{773 \left (3 x^2+2\right )^{7/2}}{68600 (2 x+3)^7}-\frac{13 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}-\frac{233 (4-9 x) \left (3 x^2+2\right )^{5/2}}{171500 (2 x+3)^6}-\frac{699 (4-9 x) \left (3 x^2+2\right )^{3/2}}{2401000 (2 x+3)^4}-\frac{6291 (4-9 x) \sqrt{3 x^2+2}}{84035000 (2 x+3)^2}-\frac{18873 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42017500 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^9,x]

[Out]

(-6291*(4 - 9*x)*Sqrt[2 + 3*x^2])/(84035000*(3 + 2*x)^2) - (699*(4 - 9*x)*(2 + 3
*x^2)^(3/2))/(2401000*(3 + 2*x)^4) - (233*(4 - 9*x)*(2 + 3*x^2)^(5/2))/(171500*(
3 + 2*x)^6) - (13*(2 + 3*x^2)^(7/2))/(280*(3 + 2*x)^8) - (773*(2 + 3*x^2)^(7/2))
/(68600*(3 + 2*x)^7) - (18873*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(42
017500*Sqrt[35])

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Rubi in Sympy [A]  time = 28.5442, size = 150, normalized size = 0.95 \[ - \frac{6291 \left (- 18 x + 8\right ) \sqrt{3 x^{2} + 2}}{168070000 \left (2 x + 3\right )^{2}} - \frac{699 \left (- 18 x + 8\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{4802000 \left (2 x + 3\right )^{4}} - \frac{233 \left (- 18 x + 8\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{343000 \left (2 x + 3\right )^{6}} - \frac{18873 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{1470612500} - \frac{773 \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{68600 \left (2 x + 3\right )^{7}} - \frac{13 \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{280 \left (2 x + 3\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**9,x)

[Out]

-6291*(-18*x + 8)*sqrt(3*x**2 + 2)/(168070000*(2*x + 3)**2) - 699*(-18*x + 8)*(3
*x**2 + 2)**(3/2)/(4802000*(2*x + 3)**4) - 233*(-18*x + 8)*(3*x**2 + 2)**(5/2)/(
343000*(2*x + 3)**6) - 18873*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2
+ 2)))/1470612500 - 773*(3*x**2 + 2)**(7/2)/(68600*(2*x + 3)**7) - 13*(3*x**2 +
2)**(7/2)/(280*(2*x + 3)**8)

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Mathematica [A]  time = 0.190563, size = 105, normalized size = 0.66 \[ \frac{-37746 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+\frac{35 \sqrt{3 x^2+2} \left (49626 x^7+2206008 x^6+210306726 x^5+33613440 x^4+226355535 x^3-178164896 x^2-38788883 x-104577556\right )}{(2 x+3)^8}+37746 \sqrt{35} \log (2 x+3)}{2941225000} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^9,x]

[Out]

((35*Sqrt[2 + 3*x^2]*(-104577556 - 38788883*x - 178164896*x^2 + 226355535*x^3 +
33613440*x^4 + 210306726*x^5 + 2206008*x^6 + 49626*x^7))/(3 + 2*x)^8 + 37746*Sqr
t[35]*Log[3 + 2*x] - 37746*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])
/2941225000

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Maple [B]  time = 0.039, size = 299, normalized size = 1.9 \[ -{\frac{26214597}{9007501562500} \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{169857\,x}{2941225000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{18873\,\sqrt{35}}{1470612500}{\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{78643791\,x}{9007501562500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{2097}{96040000} \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{13}{71680} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-8}}-{\frac{233}{5488000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{20271}{1680700000} \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{207603}{29412250000} \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{773}{8780800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-7}}+{\frac{2208141\,x}{102942875000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2258469}{514714375000} \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{12582}{12867859375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{18873}{1470612500}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{150984}{2251875390625} \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)^9,x)

[Out]

-26214597/9007501562500/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)+169857/2941225000*x
*(3*(x+3/2)^2-9*x-19/4)^(1/2)-18873/1470612500*35^(1/2)*arctanh(2/35*(4-9*x)*35^
(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))[phone]/9007501562500*x*(3*(x+3/2)^2-9*x-19
/4)^(5/2)-2097/96040000/(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)-13/71680/(x+3/2)^
8*(3*(x+3/2)^2-9*x-19/4)^(7/2)-233/5488000/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(7/2
)-20271/1680700000/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(7/2)-207603/29412250000/(x+
3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)-773/8780800/(x+3/2)^7*(3*(x+3/2)^2-9*x-19/4)
^(7/2)+2208141/102942875000*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-2258469/514714375000/
(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(7/2)+12582/12867859375*(3*(x+3/2)^2-9*x-19/4)^
(3/2)+18873/1470612500*(12*(x+3/2)^2-36*x-19)^(1/2)+150984/2251875390625*(3*(x+3
/2)^2-9*x-19/4)^(5/2)

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Maxima [A]  time = 0.788931, size = 508, normalized size = 3.22 \[ \frac{6775407}{514714375000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{280 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac{773 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{68600 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{233 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{85750 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{2097 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{3001250 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{20271 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{105043750 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{207603 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{3676531250 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{2258469 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{128678593750 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{2208141}{102942875000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{12582}{12867859375} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{26214597 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{514714375000 \,{\left (2 \, x + 3\right )}} + \frac{169857}{2941225000} \, \sqrt{3 \, x^{2} + 2} x + \frac{18873}{1470612500} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{18873}{735306250} \, \sqrt{3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(5/2)*(x - 5)/(2*x + 3)^9,x, algorithm="maxima")

[Out]

6775407/514714375000*(3*x^2 + 2)^(5/2) - 13/280*(3*x^2 + 2)^(7/2)/(256*x^8 + 307
2*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6
561) - 773/68600*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 +
22680*x^3 + 20412*x^2 + 10206*x + 2187) - 233/85750*(3*x^2 + 2)^(7/2)/(64*x^6 +
576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 2097/3001250*(3*x^2 +
2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 20271/10504375
0*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 207603/3676531250
*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2258469/128678593750*(3*x^2 +
2)^(7/2)/(4*x^2 + 12*x + 9) + 2208141/102942875000*(3*x^2 + 2)^(3/2)*x + 12582/1
2867859375*(3*x^2 + 2)^(3/2) - 26214597/514714375000*(3*x^2 + 2)^(5/2)/(2*x + 3)
+ 169857/2941225000*sqrt(3*x^2 + 2)*x + 18873/1470612500*sqrt(35)*arcsinh(3/2*s
qrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 18873/735306250*sqrt(3*x^2 +
2)

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Fricas [A]  time = 0.290818, size = 248, normalized size = 1.57 \[ \frac{\sqrt{35}{\left (\sqrt{35}{\left (49626 \, x^{7} + 2206008 \, x^{6} + 210306726 \, x^{5} + 33613440 \, x^{4} + 226355535 \, x^{3} - 178164896 \, x^{2} - 38788883 \, x - 104577556\right )} \sqrt{3 \, x^{2} + 2} + 18873 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{2941225000 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(5/2)*(x - 5)/(2*x + 3)^9,x, algorithm="fricas")

[Out]

1/2941225000*sqrt(35)*(sqrt(35)*(49626*x^7 + 2206008*x^6 + 210306726*x^5 + 33613
440*x^4 + 226355535*x^3 - 178164896*x^2 - 38788883*x - 104577556)*sqrt(3*x^2 + 2
) + 18873*(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 +
81648*x^2 + 34992*x + 6561)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2
+ 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^
5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)

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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)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.323414, size = 612, normalized size = 3.87 \[ \frac{18873}{1470612500} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) - \frac{27 \,{\left (178944 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{15} + 46043740 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{14} + 30787400 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{13} + 191125270 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{12} - 3328877720 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} - 2893694188 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} - 13787031160 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 522152825 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} - 28541438480 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} + 10194100560 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 23140527424 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} - 4295198880 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} + 1726278400 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} - 3033847040 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 39843840 \, \sqrt{3} x - 470528 \, \sqrt{3} - 39843840 \, \sqrt{3 \, x^{2} + 2}\right )}}{10756480000 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(5/2)*(x - 5)/(2*x + 3)^9,x, algorithm="giac")

[Out]

18873/1470612500*sqrt(35)*ln(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3
*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 27/107564
80000*(178944*(sqrt(3)*x - sqrt(3*x^2 + 2))^15 + 46043740*sqrt(3)*(sqrt(3)*x - s
qrt(3*x^2 + 2))^14 + 30787400*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 + 191125270*sqrt(
3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^12 - 3328877720*(sqrt(3)*x - sqrt(3*x^2 + 2))^1
1 - 2893694188*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 - 13787031160*(sqrt(3)*x
- sqrt(3*x^2 + 2))^9 + 522152825*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 - 2854
1438480*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 10194100560*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 2))^6 - 23140527424*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 4295198880*sqrt(3)
*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 1726278400*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 -
3033847040*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 39843840*sqrt(3)*x - 470528
*sqrt(3) - 39843840*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3
)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^8