3.2448 \(\int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2} \, dx\)

Optimal. Leaf size=179 \[ -\frac{1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}}{26730}+\frac{5627 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{25920}-\frac{39389 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1866240}+\frac{39389 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{17915904}-\frac{39389 (6 x+5) \sqrt{3 x^2+5 x+2}}{143327232}+\frac{39389 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{286654464 \sqrt{3}} \]

[Out]

(-39389*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/143327232 + (39389*(5 + 6*x)*(2 + 5*x +
3*x^2)^(3/2))/17915904 - (39389*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/1866240 + (5
627*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/25920 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9
/2))/33 + ((47425 + 20358*x)*(2 + 5*x + 3*x^2)^(9/2))/26730 + (39389*ArcTanh[(5
+ 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(286654464*Sqrt[3])

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Rubi [A]  time = 0.215449, antiderivative size = 179, 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 \[ -\frac{1}{33} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(20358 x+47425) \left (3 x^2+5 x+2\right )^{9/2}}{26730}+\frac{5627 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{25920}-\frac{39389 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{1866240}+\frac{39389 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{17915904}-\frac{39389 (6 x+5) \sqrt{3 x^2+5 x+2}}{143327232}+\frac{39389 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{286654464 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-39389*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/143327232 + (39389*(5 + 6*x)*(2 + 5*x +
3*x^2)^(3/2))/17915904 - (39389*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/1866240 + (5
627*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/25920 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(9
/2))/33 + ((47425 + 20358*x)*(2 + 5*x + 3*x^2)^(9/2))/26730 + (39389*ArcTanh[(5
+ 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(286654464*Sqrt[3])

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Rubi in Sympy [A]  time = 20.9192, size = 165, normalized size = 0.92 \[ - \frac{\left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{33} + \frac{5627 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{25920} - \frac{39389 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{1866240} + \frac{39389 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{17915904} - \frac{39389 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{143327232} + \frac{\left (20358 x + 47425\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{26730} + \frac{39389 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{859963392} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**2*(3*x**2 + 5*x + 2)**(9/2)/33 + 5627*(6*x + 5)*(3*x**2 + 5*x + 2)**
(7/2)/25920 - 39389*(6*x + 5)*(3*x**2 + 5*x + 2)**(5/2)/1866240 + 39389*(6*x + 5
)*(3*x**2 + 5*x + 2)**(3/2)/17915904 - 39389*(6*x + 5)*sqrt(3*x**2 + 5*x + 2)/14
3327232 + (20358*x + 47425)*(3*x**2 + 5*x + 2)**(9/2)/26730 + 39389*sqrt(3)*atan
h(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/859963392

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Mathematica [A]  time = 0.145301, size = 100, normalized size = 0.56 \[ \frac{2166395 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (77396705280 x^{10}[phone] x^9-1156531322880 x^8-9116575930368 x^7-25723491978240 x^6-41190616509696 x^5-41472321125760 x^4-26847121235760 x^3-10882383306360 x^2-2519542755670 x-254668717065\right )}{47297986560} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-254668717065 - 2519542755670*x - 10882383306360*x^2
- 26847121235760*x^3 - 41472321125760*x^4 - 41190616509696*x^5 - 25723491978240*
x^6 - 9116575930368*x^7 - 1156531322880*x^8 + 261858852864*x^9 + 77396705280*x^1
0) + 2166395*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/47297986560

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Maple [A]  time = 0.01, size = 153, normalized size = 0.9 \[{\frac{28135+33762\,x}{25920} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{196945+236334\,x}{1866240} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{196945+236334\,x}{17915904} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{196945+236334\,x}{143327232}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{39389\,\sqrt{3}}{859963392}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{8027}{5346} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{197\,x}{495} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}-{\frac{4\,{x}^{2}}{33} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5627/25920*(5+6*x)*(3*x^2+5*x+2)^(7/2)-39389/1866240*(5+6*x)*(3*x^2+5*x+2)^(5/2)
+39389/17915904*(5+6*x)*(3*x^2+5*x+2)^(3/2)-39389/143327232*(5+6*x)*(3*x^2+5*x+2
)^(1/2)+39389/859963392*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+80
27/5346*(3*x^2+5*x+2)^(9/2)+197/495*x*(3*x^2+5*x+2)^(9/2)-4/33*x^2*(3*x^2+5*x+2)
^(9/2)

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Maxima [A]  time = 0.7951, size = 258, normalized size = 1.44 \[ -\frac{4}{33} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{2} + \frac{197}{495} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x + \frac{8027}{5346} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{5627}{4320} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{5627}{5184} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{39389}{311040} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{39389}{373248} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{39389}{2985984} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{196945}{17915904} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{39389}{23887872} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{39389}{859963392} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{196945}{143327232} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/33*(3*x^2 + 5*x + 2)^(9/2)*x^2 + 197/495*(3*x^2 + 5*x + 2)^(9/2)*x + 8027/534
6*(3*x^2 + 5*x + 2)^(9/2) + 5627/4320*(3*x^2 + 5*x + 2)^(7/2)*x + 5627/5184*(3*x
^2 + 5*x + 2)^(7/2) - 39389/311040*(3*x^2 + 5*x + 2)^(5/2)*x - 39389/373248*(3*x
^2 + 5*x + 2)^(5/2) + 39389/2985984*(3*x^2 + 5*x + 2)^(3/2)*x + 196945/17915904*
(3*x^2 + 5*x + 2)^(3/2) - 39389/23887872*sqrt(3*x^2 + 5*x + 2)*x + 39389/8599633
92*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 196945/143327232*sqr
t(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.277968, size = 149, normalized size = 0.83 \[ -\frac{1}{94595973120} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (77396705280 \, x^{10} + 261858852864 \, x^{9} - 1156531322880 \, x^{8} - 9116575930368 \, x^{7} - 25723491978240 \, x^{6} - 41190616509696 \, x^{5} - 41472321125760 \, x^{4} - 26847121235760 \, x^{3} - 10882383306360 \, x^{2} - 2519542755670 \, x - 254668717065\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 2166395 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/94595973120*sqrt(3)*(4*sqrt(3)*(77396705280*x^10 + 261858852864*x^9 - 1156531
322880*x^8 - 9116575930368*x^7 - 25723491978240*x^6 - 41190616509696*x^5 - 41472
321125760*x^4 - 26847121235760*x^3 - 10882383306360*x^2 - 2519542755670*x - 2546
68717065)*sqrt(3*x^2 + 5*x + 2) - 2166395*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. \[ - \int \left (- 3108 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 11494 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 23659 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 29358 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 22000 x^{5} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 9112 x^{6} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1341 x^{7} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 324 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 108 x^{9} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 360 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-3108*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-11494*x**2*sqrt(3*x**2
+ 5*x + 2), x) - Integral(-23659*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-293
58*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-22000*x**5*sqrt(3*x**2 + 5*x + 2)
, x) - Integral(-9112*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1341*x**7*sqrt
(3*x**2 + 5*x + 2), x) - Integral(324*x**8*sqrt(3*x**2 + 5*x + 2), x) - Integral
(108*x**9*sqrt(3*x**2 + 5*x + 2), x) - Integral(-360*sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.268219, size = 134, normalized size = 0.75 \[ -\frac{1}{7882997760} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \,{\left (54 \,{\left (60 \, x + 203\right )} x - 48415\right )} x - 18318737\right )} x - 103376945\right )} x - 5959290583\right )} x - 36000278755\right )} x - 186438341915\right )} x - 453432637765\right )} x - 1259771377835\right )} x - 254668717065\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{39389}{859963392} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7882997760*(2*(12*(6*(8*(6*(36*(2*(48*(54*(60*x + 203)*x - 48415)*x - 1831873
7)*x - 103376945)*x - 5959290583)*x - 36000278755)*x - 186438341915)*x - 4534326
37765)*x - 1259771377835)*x - 254668717065)*sqrt(3*x^2 + 5*x + 2) - 39389/859963
392*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))