3.2454 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=190 \[ -\frac{(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}+\frac{7 (414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^2}-\frac{7 (1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)}-\frac{7 (37375-78054 x) \sqrt{3 x^2+5 x+2}}{6144}+\frac{2776697 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{12288 \sqrt{3}}-\frac{59745 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024} \]

[Out]

(-7*(37375 - 78054*x)*Sqrt[2 + 5*x + 3*x^2])/6144 - (7*(5713 + 1652*x)*(2 + 5*x
+ 3*x^2)^(3/2))/(768*(3 + 2*x)) + (7*(1171 + 414*x)*(2 + 5*x + 3*x^2)^(5/2))/(96
0*(3 + 2*x)^2) - ((37 + 3*x)*(2 + 5*x + 3*x^2)^(7/2))/(30*(3 + 2*x)^3) + (277669
7*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(12288*Sqrt[3]) - (59745
*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

_______________________________________________________________________________________

Rubi [A]  time = 0.393659, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}+\frac{7 (414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^2}-\frac{7 (1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)}-\frac{7 (37375-78054 x) \sqrt{3 x^2+5 x+2}}{6144}+\frac{2776697 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{12288 \sqrt{3}}-\frac{59745 \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{1024} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^4,x]

[Out]

(-7*(37375 - 78054*x)*Sqrt[2 + 5*x + 3*x^2])/6144 - (7*(5713 + 1652*x)*(2 + 5*x
+ 3*x^2)^(3/2))/(768*(3 + 2*x)) + (7*(1171 + 414*x)*(2 + 5*x + 3*x^2)^(5/2))/(96
0*(3 + 2*x)^2) - ((37 + 3*x)*(2 + 5*x + 3*x^2)^(7/2))/(30*(3 + 2*x)^3) + (277669
7*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(12288*Sqrt[3]) - (59745
*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 49.292, size = 175, normalized size = 0.92 \[ - \frac{7 \left (- 3746592 x + 1794000\right ) \sqrt{3 x^{2} + 5 x + 2}}{294912} + \frac{2776697 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{36864} + \frac{59745 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{1024} - \frac{7 \left (39648 x + 137112\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{18432 \left (2 x + 3\right )} + \frac{7 \left (1656 x + 4684\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{3840 \left (2 x + 3\right )^{2}} - \frac{\left (6 x + 74\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{60 \left (2 x + 3\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**4,x)

[Out]

-7*(-3746592*x + 1794000)*sqrt(3*x**2 + 5*x + 2)/294912 + 2776697*sqrt(3)*atanh(
sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/36864 + 59745*sqrt(5)*atanh(sqrt(5
)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/1024 - 7*(39648*x + 137112)*(3*x**2 +
5*x + 2)**(3/2)/(18432*(2*x + 3)) + 7*(1656*x + 4684)*(3*x**2 + 5*x + 2)**(5/2)/
(3840*(2*x + 3)**2) - (6*x + 74)*(3*x**2 + 5*x + 2)**(7/2)/(60*(2*x + 3)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.241901, size = 141, normalized size = 0.74 \[ \frac{2150820 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+2776697 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{6 \sqrt{3 x^2+5 x+2} \left (82944 x^7-231552 x^6-1266816 x^5-3277520 x^4+746240 x^3+44770416 x^2+98927312 x+61268351\right )}{5 (2 x+3)^3}-2150820 \sqrt{5} \log (2 x+3)}{36864} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^4,x]

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(61268351 + 98927312*x + 44770416*x^2 + 746240*x^3 -
3277520*x^4 - 1266816*x^5 - 231552*x^6 + 82944*x^7))/(5*(3 + 2*x)^3) - 2150820*S
qrt[5]*Log[3 + 2*x] + 2150820*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x + 3*
x^2]] + 2776697*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/36864

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 274, normalized size = 1.4 \[ -{\frac{1707}{200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{11949}{800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{3983}{128} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{59745}{1024}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13}{120} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}+{\frac{57}{200} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{96}{25} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{240+288\,x}{25} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}+{\frac{6265+7518\,x}{400} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{22645+27174\,x}{768} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{455315+546378\,x}{6144}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{2776697\,\sqrt{3}}{36864}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{59745\,\sqrt{5}}{1024}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^4,x)

[Out]

-1707/200*(3*(x+3/2)^2-4*x-19/4)^(7/2)-11949/800*(3*(x+3/2)^2-4*x-19/4)^(5/2)-39
83/128*(3*(x+3/2)^2-4*x-19/4)^(3/2)-59745/1024*(12*(x+3/2)^2-16*x-19)^(1/2)-13/1
20/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)+57/200/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4
)^(9/2)-96/25/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)+48/25*(5+6*x)*(3*(x+3/2)^2-4*
x-19/4)^(7/2)+1253/400*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+4529/768*(5+6*x)*(3*
(x+3/2)^2-4*x-19/4)^(3/2)+91063/6144*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+277669
7/36864*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+59745/102
4*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.812376, size = 336, normalized size = 1.77 \[ -\frac{171}{200} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{15 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac{57 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{50 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{3759}{200} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{581}{800} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{48 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{5 \,{\left (2 \, x + 3\right )}} + \frac{4529}{128} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{1253}{768} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{91063}{1024} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{2776697}{36864} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{59745}{1024} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{261625}{6144} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^4,x, algorithm="maxima")

[Out]

-171/200*(3*x^2 + 5*x + 2)^(7/2) - 13/15*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2
+ 54*x + 27) + 57/50*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) + 3759/200*(3*x
^2 + 5*x + 2)^(5/2)*x + 581/800*(3*x^2 + 5*x + 2)^(5/2) - 48/5*(3*x^2 + 5*x + 2)
^(7/2)/(2*x + 3) + 4529/128*(3*x^2 + 5*x + 2)^(3/2)*x - 1253/768*(3*x^2 + 5*x +
2)^(3/2) + 91063/1024*sqrt(3*x^2 + 5*x + 2)*x + 2776697/36864*sqrt(3)*log(sqrt(3
)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 59745/1024*sqrt(5)*log(sqrt(5)*sqrt(3*x^2
+ 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 261625/6144*sqrt(3*x^2 + 5*x
+ 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.295725, size = 266, normalized size = 1.4 \[ \frac{\sqrt{3}{\left (3584700 \, \sqrt{5} \sqrt{3}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, \sqrt{3}{\left (82944 \, x^{7} - 231552 \, x^{6} - 1266816 \, x^{5} - 3277520 \, x^{4} + 746240 \, x^{3} + 44770416 \, x^{2} + 98927312 \, x + 61268351\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 13883485 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{368640 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^4,x, algorithm="fricas")

[Out]

1/368640*sqrt(3)*(3584700*sqrt(5)*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-(4*s
qrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) - 124*x^2 - 212*x - 89)/(4*x^2 + 12*x + 9
)) - 4*sqrt(3)*(82944*x^7 - 231552*x^6 - 1266816*x^5 - 3277520*x^4 + 746240*x^3
+ 44770416*x^2 + 98927312*x + 61268351)*sqrt(3*x^2 + 5*x + 2) + 13883485*(8*x^3
+ 36*x^2 + 54*x + 27)*log(sqrt(3)*(72*x^2 + 120*x + 49) + 12*sqrt(3*x^2 + 5*x +
2)*(6*x + 5)))/(8*x^3 + 36*x^2 + 54*x + 27)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{40 \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{292 x \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{870 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{1339 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{1090 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{396 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac{27 x^{7} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**4,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81)
, x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 21
6*x + 81), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 2
16*x**2 + 216*x + 81), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(16*x**4
+ 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x +
2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-396*x**5*sqrt(3*
x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(27*x*
*7*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^4,x, algorithm="giac")

[Out]

undef