3.2447 \(\int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx\)

Optimal. Leaf size=204 \[ -\frac{1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+\frac{34}{99} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}}{320760}+\frac{91087 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{311040}-\frac{637609 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{22394880}+\frac{637609 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{214990848}-\frac{637609 (6 x+5) \sqrt{3 x^2+5 x+2}}{1719926784}+\frac{637609 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3439853568 \sqrt{3}} \]

[Out]

(-637609*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/1719926784 + (637609*(5 + 6*x)*(2 + 5*
x + 3*x^2)^(3/2))/214990848 - (637609*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/2239488
0 + (91087*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/311040 + (34*(3 + 2*x)^2*(2 + 5*x
+ 3*x^2)^(9/2))/99 - ((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2))/36 + ((863825 + 39079
8*x)*(2 + 5*x + 3*x^2)^(9/2))/320760 + (637609*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt
[2 + 5*x + 3*x^2])])/(3439853568*Sqrt[3])

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Rubi [A]  time = 0.301058, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+\frac{34}{99} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}}{320760}+\frac{91087 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{311040}-\frac{637609 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{22394880}+\frac{637609 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{214990848}-\frac{637609 (6 x+5) \sqrt{3 x^2+5 x+2}}{1719926784}+\frac{637609 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{3439853568 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-637609*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/1719926784 + (637609*(5 + 6*x)*(2 + 5*
x + 3*x^2)^(3/2))/214990848 - (637609*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/2239488
0 + (91087*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/311040 + (34*(3 + 2*x)^2*(2 + 5*x
+ 3*x^2)^(9/2))/99 - ((3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2))/36 + ((863825 + 39079
8*x)*(2 + 5*x + 3*x^2)^(9/2))/320760 + (637609*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt
[2 + 5*x + 3*x^2])])/(3439853568*Sqrt[3])

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Rubi in Sympy [A]  time = 30.2354, size = 189, normalized size = 0.93 \[ - \frac{\left (2 x + 3\right )^{3} \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{36} + \frac{34 \left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{99} + \frac{91087 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{311040} - \frac{637609 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{22394880} + \frac{637609 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{214990848} - \frac{637609 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{1719926784} + \frac{\left (1172394 x + 2591475\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{962280} + \frac{637609 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{10319560704} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**3*(3*x**2 + 5*x + 2)**(9/2)/36 + 34*(2*x + 3)**2*(3*x**2 + 5*x + 2)*
*(9/2)/99 + 91087*(6*x + 5)*(3*x**2 + 5*x + 2)**(7/2)/311040 - 637609*(6*x + 5)*
(3*x**2 + 5*x + 2)**(5/2)/22394880 + 637609*(6*x + 5)*(3*x**2 + 5*x + 2)**(3/2)/
214990848 - 637609*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)/1719926784 + (1172394*x + 25
91475)*(3*x**2 + 5*x + 2)**(9/2)/962280 + 637609*sqrt(3)*atanh(sqrt(3)*(6*x + 5)
/(6*sqrt(3*x**2 + 5*x + 2)))/10319560704

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Mathematica [A]  time = 0.137757, size = 105, normalized size = 0.51 \[ \frac{35068495 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (1702727516160 x^{11}[phone] x^{10}-15591566278656 x^9-235832896880640 x^8-866110416795648 x^7-1766184385305600 x^6-2298912734198016 x^5-1992318117275520 x^4-1149328734822000 x^3-425035984788120 x^2-91318722047870 x-8675936123685\right )}{567575838720} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-8675936123685 - 91318722047870*x - 425035984788120*x
^2 - 1149328734822000*x^3 - 1992318117275520*x^4 - 2298912734198016*x^5 - 176618
4385305600*x^6 - 866110416795648*x^7 - 235832896880640*x^8 - 15591566278656*x^9
+ 8487838679040*x^10 + 1702727516160*x^11) + 35068495*Sqrt[3]*Log[-5 - 6*x - 2*S
qrt[6 + 15*x + 9*x^2]])/567575838720

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Maple [A]  time = 0.011, size = 170, normalized size = 0.8 \[{\frac{455435+546522\,x}{311040} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{3188045+3825654\,x}{22394880} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{3188045+3825654\,x}{214990848} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{3188045+3825654\,x}{1719926784}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{637609\,\sqrt{3}}{10319560704}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{322939}{64152} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{22807\,x}{5940} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{37\,{x}^{2}}{99} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}-{\frac{2\,{x}^{3}}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

91087/311040*(5+6*x)*(3*x^2+5*x+2)^(7/2)-637609/22394880*(5+6*x)*(3*x^2+5*x+2)^(
5/2)+637609/214990848*(5+6*x)*(3*x^2+5*x+2)^(3/2)-637609/1719926784*(5+6*x)*(3*x
^2+5*x+2)^(1/2)+637609/10319560704*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))
*3^(1/2)+322939/64152*(3*x^2+5*x+2)^(9/2)+22807/5940*x*(3*x^2+5*x+2)^(9/2)+37/99
*x^2*(3*x^2+5*x+2)^(9/2)-2/9*x^3*(3*x^2+5*x+2)^(9/2)

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Maxima [A]  time = 0.803025, size = 281, normalized size = 1.38 \[ -\frac{2}{9} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{3} + \frac{37}{99} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{2} + \frac{22807}{5940} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x + \frac{322939}{64152} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{91087}{51840} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{91087}{62208} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{637609}{3732480} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{637609}{4478976} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{637609}{35831808} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{3188045}{214990848} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{637609}{286654464} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{637609}{10319560704} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{3188045}{1719926784} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(3*x^2 + 5*x + 2)^(9/2)*x^3 + 37/99*(3*x^2 + 5*x + 2)^(9/2)*x^2 + 22807/594
0*(3*x^2 + 5*x + 2)^(9/2)*x + 322939/64152*(3*x^2 + 5*x + 2)^(9/2) + 91087/51840
*(3*x^2 + 5*x + 2)^(7/2)*x + 91087/62208*(3*x^2 + 5*x + 2)^(7/2) - 637609/373248
0*(3*x^2 + 5*x + 2)^(5/2)*x - 637609/4478976*(3*x^2 + 5*x + 2)^(5/2) + 637609/35
831808*(3*x^2 + 5*x + 2)^(3/2)*x + 3188045/214990848*(3*x^2 + 5*x + 2)^(3/2) - 6
37609/286654464*sqrt(3*x^2 + 5*x + 2)*x + 637609/10319560704*sqrt(3)*log(2*sqrt(
3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 3188045/1719926784*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.284343, size = 155, normalized size = 0.76 \[ -\frac{1}{1135151677440} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (1702727516160 \, x^{11} + 8487838679040 \, x^{10} - 15591566278656 \, x^{9} - 235832896880640 \, x^{8} - 866110416795648 \, x^{7} - 1766184385305600 \, x^{6} - 2298912734198016 \, x^{5} - 1992318117275520 \, x^{4} - 1149328734822000 \, x^{3} - 425035984788120 \, x^{2} - 91318722047870 \, x - 8675936123685\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 35068495 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1135151677440*sqrt(3)*(4*sqrt(3)*(1702727516160*x^11 + 8487838679040*x^10 - 1
5591566278656*x^9 - 235832896880640*x^8 - 866110416795648*x^7 - 1766184385305600
*x^6 - 2298912734198016*x^5 - 1992318117275520*x^4 - 1149328734822000*x^3 - 4250
35984788120*x^2 - 91318722047870*x - 8675936123685)*sqrt(3*x^2 + 5*x + 2) - 3506
8495*log(sqrt(3)*(72*x^2 + 120*x + 49) + 12*sqrt(3*x^2 + 5*x + 2)*(6*x + 5)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 10044 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 40698 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 93965 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 135392 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 124716 x^{5} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 71336 x^{6} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 22247 x^{7} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1710 x^{8} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 972 x^{9} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 216 x^{10} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 1080 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-10044*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-40698*x**2*sqrt(3*x**2
+ 5*x + 2), x) - Integral(-93965*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-13
5392*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-124716*x**5*sqrt(3*x**2 + 5*x +
2), x) - Integral(-71336*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-22247*x**7
*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1710*x**8*sqrt(3*x**2 + 5*x + 2), x) - I
ntegral(972*x**9*sqrt(3*x**2 + 5*x + 2), x) - Integral(216*x**10*sqrt(3*x**2 + 5
*x + 2), x) - Integral(-1080*sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.285338, size = 140, normalized size = 0.69 \[ -\frac{1}{94595973120} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \,{\left (54 \,{\left (20 \,{\left (66 \, x + 329\right )} x - 12087\right )} x - 9872495\right )} x - 1740351757\right )} x - 7097898925\right )} x - 332597328443\right )} x - 1729442810135\right )} x - 7981449547375\right )} x - 17709832699505\right )} x - 45659361023935\right )} x - 8675936123685\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{637609}{10319560704} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/94595973120*(2*(12*(6*(8*(6*(36*(2*(48*(54*(20*(66*x + 329)*x - 12087)*x - 98
72495)*x - 1740351757)*x - 7097898925)*x - 332597328443)*x - 1729442810135)*x -
7981449547375)*x - 17709832699505)*x - 45659361023935)*x - 8675936123685)*sqrt(3
*x^2 + 5*x + 2) - 637609/10319560704*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2)) - 5))