3.2433 \(\int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=156 \[ -\frac{1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac{(4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}}{4536}+\frac{4507 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{15552}-\frac{22535 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{746496}+\frac{22535 (6 x+5) \sqrt{3 x^2+5 x+2}}{5971968}-\frac{22535 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{11943936 \sqrt{3}} \]

[Out]

(22535*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5971968 - (22535*(5 + 6*x)*(2 + 5*x + 3*
x^2)^(3/2))/746496 + (4507*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/15552 - ((3 + 2*x)
^2*(2 + 5*x + 3*x^2)^(7/2))/27 + ((10211 + 4298*x)*(2 + 5*x + 3*x^2)^(7/2))/4536
- (22535*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(11943936*Sqrt[3
])

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Rubi [A]  time = 0.186419, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac{(4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}}{4536}+\frac{4507 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{15552}-\frac{22535 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{746496}+\frac{22535 (6 x+5) \sqrt{3 x^2+5 x+2}}{5971968}-\frac{22535 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{11943936 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(22535*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5971968 - (22535*(5 + 6*x)*(2 + 5*x + 3*
x^2)^(3/2))/746496 + (4507*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/15552 - ((3 + 2*x)
^2*(2 + 5*x + 3*x^2)^(7/2))/27 + ((10211 + 4298*x)*(2 + 5*x + 3*x^2)^(7/2))/4536
- (22535*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(11943936*Sqrt[3
])

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Rubi in Sympy [A]  time = 19.408, size = 143, normalized size = 0.92 \[ - \frac{\left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{27} + \frac{4507 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{15552} - \frac{22535 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{746496} + \frac{22535 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{5971968} + \frac{\left (12894 x + 30633\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{13608} - \frac{22535 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{35831808} \]

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)**(5/2),x)

[Out]

-(2*x + 3)**2*(3*x**2 + 5*x + 2)**(7/2)/27 + 4507*(6*x + 5)*(3*x**2 + 5*x + 2)**
(5/2)/15552 - 22535*(6*x + 5)*(3*x**2 + 5*x + 2)**(3/2)/746496 + 22535*(6*x + 5)
*sqrt(3*x**2 + 5*x + 2)/5971968 + (12894*x + 30633)*(3*x**2 + 5*x + 2)**(7/2)/13
608 - 22535*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/35831808

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Mathematica [A]  time = 0.113095, size = 90, normalized size = 0.58 \[ \frac{-157745 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (167215104 x^8+268240896 x^7-3275873280 x^6-15455860992 x^5-30355761024 x^4-32476001904 x^3-19762157208 x^2-6434937470 x-871825317\right )}{250822656} \]

Antiderivative was successfully verified.

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-871825317 - 6434937470*x - 19762157208*x^2 - 3247600
1904*x^3 - 30355761024*x^4 - 15455860992*x^5 - 3275873280*x^6 + 268240896*x^7 +
167215104*x^8) - 157745*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/250822
656

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Maple [A]  time = 0.009, size = 134, normalized size = 0.9 \[{\frac{22535+27042\,x}{15552} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}-{\frac{112675+135210\,x}{746496} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{112675+135210\,x}{5971968}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{22535\,\sqrt{3}}{35831808}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{8699}{4536} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}+{\frac{163\,x}{324} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{4\,{x}^{2}}{27} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4507/15552*(5+6*x)*(3*x^2+5*x+2)^(5/2)-22535/746496*(5+6*x)*(3*x^2+5*x+2)^(3/2)+
22535/5971968*(5+6*x)*(3*x^2+5*x+2)^(1/2)-22535/35831808*ln(1/3*(5/2+3*x)*3^(1/2
)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+8699/4536*(3*x^2+5*x+2)^(7/2)+163/324*x*(3*x^2+5*
x+2)^(7/2)-4/27*x^2*(3*x^2+5*x+2)^(7/2)

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Maxima [A]  time = 0.767859, size = 219, normalized size = 1.4 \[ -\frac{4}{27} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x^{2} + \frac{163}{324} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{8699}{4536} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} + \frac{4507}{2592} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{22535}{15552} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{22535}{124416} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{112675}{746496} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{22535}{995328} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{22535}{35831808} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{112675}{5971968} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/27*(3*x^2 + 5*x + 2)^(7/2)*x^2 + 163/324*(3*x^2 + 5*x + 2)^(7/2)*x + 8699/453
6*(3*x^2 + 5*x + 2)^(7/2) + 4507/2592*(3*x^2 + 5*x + 2)^(5/2)*x + 22535/15552*(3
*x^2 + 5*x + 2)^(5/2) - 22535/124416*(3*x^2 + 5*x + 2)^(3/2)*x - 112675/746496*(
3*x^2 + 5*x + 2)^(3/2) + 22535/995328*sqrt(3*x^2 + 5*x + 2)*x - 22535/35831808*s
qrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 112675/5971968*sqrt(3*x^
2 + 5*x + 2)

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Fricas [A]  time = 0.275611, size = 135, normalized size = 0.87 \[ -\frac{1}{501645312} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (167215104 \, x^{8} + 268240896 \, x^{7} - 3275873280 \, x^{6} - 15455860992 \, x^{5} - 30355761024 \, x^{4} - 32476001904 \, x^{3} - 19762157208 \, x^{2} - 6434937470 \, x - 871825317\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 157745 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/501645312*sqrt(3)*(4*sqrt(3)*(167215104*x^8 + 268240896*x^7 - 3275873280*x^6
- 15455860992*x^5 - 30355761024*x^4 - 32476001904*x^3 - 19762157208*x^2 - 643493
7470*x - 871825317)*sqrt(3*x^2 + 5*x + 2) - 157745*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. \[ - \int \left (- 1104 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2717 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3381 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2151 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 551 x^{5} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 48 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 36 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 180 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-1104*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-2717*x**2*sqrt(3*x**2 +
5*x + 2), x) - Integral(-3381*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-2151*
x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-551*x**5*sqrt(3*x**2 + 5*x + 2), x)
- Integral(48*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(36*x**7*sqrt(3*x**2 + 5
*x + 2), x) - Integral(-180*sqrt(3*x**2 + 5*x + 2), x)

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GIAC/XCAS [A]  time = 0.271095, size = 120, normalized size = 0.77 \[ -\frac{1}{41803776} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (14 \,{\left (48 \, x + 77\right )} x - 13165\right )} x - 2236091\right )} x - 26350487\right )} x - 225527791\right )} x - 823423217\right )} x - 3217468735\right )} x - 871825317\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{22535}{35831808} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/41803776*(2*(12*(6*(8*(6*(36*(14*(48*x + 77)*x - 13165)*x - 2236091)*x - 2635
0487)*x - 225527791)*x - 823423217)*x - 3217468735)*x - 871825317)*sqrt(3*x^2 +
5*x + 2) + 22535/35831808*sqrt(3)*ln(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*
x + 2)) - 5))