3.2439 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=165 \[ -\frac{(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}+\frac{5 (43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{48 (2 x+3)^2}-\frac{5 (343 x+736) \sqrt{3 x^2+5 x+2}}{64 (2 x+3)}+\frac{13505 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{256 \sqrt{3}}-\frac{3487}{256} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

[Out]

(-5*(736 + 343*x)*Sqrt[2 + 5*x + 3*x^2])/(64*(3 + 2*x)) + (5*(93 + 43*x)*(2 + 5*
x + 3*x^2)^(3/2))/(48*(3 + 2*x)^2) - ((8 + x)*(2 + 5*x + 3*x^2)^(5/2))/(6*(3 + 2
*x)^3) + (13505*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(256*Sqrt[
3]) - (3487*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/256

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Rubi [A]  time = 0.317012, antiderivative size = 165, 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 \[ -\frac{(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}+\frac{5 (43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{48 (2 x+3)^2}-\frac{5 (343 x+736) \sqrt{3 x^2+5 x+2}}{64 (2 x+3)}+\frac{13505 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{256 \sqrt{3}}-\frac{3487}{256} \sqrt{5} \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*(736 + 343*x)*Sqrt[2 + 5*x + 3*x^2])/(64*(3 + 2*x)) + (5*(93 + 43*x)*(2 + 5*
x + 3*x^2)^(3/2))/(48*(3 + 2*x)^2) - ((8 + x)*(2 + 5*x + 3*x^2)^(5/2))/(6*(3 + 2
*x)^3) + (13505*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(256*Sqrt[
3]) - (3487*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/256

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Rubi in Sympy [A]  time = 40.7274, size = 153, normalized size = 0.93 \[ \frac{13505 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{768} + \frac{3487 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{256} - \frac{5 \left (16464 x + 35328\right ) \sqrt{3 x^{2} + 5 x + 2}}{3072 \left (2 x + 3\right )} + \frac{5 \left (1032 x + 2232\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{1152 \left (2 x + 3\right )^{2}} - \frac{\left (6 x + 48\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{36 \left (2 x + 3\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13505*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/768 + 3487*sqr
t(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/256 - 5*(16464*x + 35
328)*sqrt(3*x**2 + 5*x + 2)/(3072*(2*x + 3)) + 5*(1032*x + 2232)*(3*x**2 + 5*x +
2)**(3/2)/(1152*(2*x + 3)**2) - (6*x + 48)*(3*x**2 + 5*x + 2)**(5/2)/(36*(2*x +
3)**3)

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Mathematica [A]  time = 0.183456, size = 129, normalized size = 0.78 \[ \frac{1}{768} \left (10461 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+13505 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{4 \sqrt{3 x^2+5 x+2} \left (288 x^5-1896 x^4+1944 x^3+64332 x^2+143533 x+89224\right )}{(2 x+3)^3}-10461 \sqrt{5} \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-4*Sqrt[2 + 5*x + 3*x^2]*(89224 + 143533*x + 64332*x^2 + 1944*x^3 - 1896*x^4 +
288*x^5))/(3 + 2*x)^3 - 10461*Sqrt[5]*Log[3 + 2*x] + 10461*Sqrt[5]*Log[-7 - 8*x
+ 2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]] + 13505*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15
*x + 9*x^2]])/768

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Maple [A]  time = 0.017, size = 237, normalized size = 1.4 \[ -{\frac{13}{120} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}+{\frac{67}{600} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{197}{125} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{3487}{1000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{1645+1974\,x}{240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{2215+2658\,x}{128}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{13505\,\sqrt{3}}{768}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }-{\frac{3487}{480} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{3487}{256}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}+{\frac{3487\,\sqrt{5}}{256}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }+{\frac{985+1182\,x}{250} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/120/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(7/2)+67/600/(x+3/2)^2*(3*(x+3/2)^2-4*x
-19/4)^(7/2)-197/125/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-3487/1000*(3*(x+3/2)^2
-4*x-19/4)^(5/2)+329/240*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)+443/128*(5+6*x)*(3
*(x+3/2)^2-4*x-19/4)^(1/2)+13505/768*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3/2)^2-4*x-1
9/4)^(1/2))*3^(1/2)-3487/480*(3*(x+3/2)^2-4*x-19/4)^(3/2)-3487/256*(12*(x+3/2)^2
-16*x-19)^(1/2)+3487/256*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16
*x-19)^(1/2))+197/250*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)

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Maxima [A]  time = 0.807796, size = 297, normalized size = 1.8 \[ -\frac{67}{200} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{15 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac{67 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{150 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{329}{40} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{197}{480} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{197 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}{50 \,{\left (2 \, x + 3\right )}} + \frac{1329}{64} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{13505}{768} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{3487}{256} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{159}{16} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

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)^4,x, algorithm="maxima")

[Out]

-67/200*(3*x^2 + 5*x + 2)^(5/2) - 13/15*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2
+ 54*x + 27) + 67/150*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) + 329/40*(3*x^2
+ 5*x + 2)^(3/2)*x - 197/480*(3*x^2 + 5*x + 2)^(3/2) - 197/50*(3*x^2 + 5*x + 2)
^(5/2)/(2*x + 3) + 1329/64*sqrt(3*x^2 + 5*x + 2)*x + 13505/768*sqrt(3)*log(sqrt(
3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 3487/256*sqrt(5)*log(sqrt(5)*sqrt(3*x^2
+ 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 159/16*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.289286, size = 252, normalized size = 1.53 \[ \frac{\sqrt{3}{\left (10461 \, \sqrt{5} \sqrt{3}{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 8 \, \sqrt{3}{\left (288 \, x^{5} - 1896 \, x^{4} + 1944 \, x^{3} + 64332 \, x^{2} + 143533 \, x + 89224\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 40515 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )}}{4608 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4608*sqrt(3)*(10461*sqrt(5)*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(-(4*sqrt(
5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) - 124*x^2 - 212*x - 89)/(4*x^2 + 12*x + 9)) -
8*sqrt(3)*(288*x^5 - 1896*x^4 + 1944*x^3 + 64332*x^2 + 143533*x + 89224)*sqrt(3
*x^2 + 5*x + 2) + 40515*(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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81)
, x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216
*x + 81), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 21
6*x**2 + 216*x + 81), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(16*x**4 +
96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/
(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(9*x**5*sqrt(3*x**2 +
5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x)

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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)^(5/2)*(x - 5)/(2*x + 3)^4,x, algorithm="giac")

[Out]

undef