3.1368 \(\int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=138 \[ -\frac{1}{27} \left (3 x^2+2\right )^{5/2} (2 x+3)^4+\frac{13}{36} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac{4421 \left (3 x^2+2\right )^{5/2} (2 x+3)^2}{2268}+\frac{(226755 x+661583) \left (3 x^2+2\right )^{5/2}}{17010}+\frac{2777}{36} x \left (3 x^2+2\right )^{3/2}+\frac{2777}{12} x \sqrt{3 x^2+2}+\frac{2777 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(2777*x*Sqrt[2 + 3*x^2])/12 + (2777*x*(2 + 3*x^2)^(3/2))/36 + (4421*(3 + 2*x)^2*
(2 + 3*x^2)^(5/2))/2268 + (13*(3 + 2*x)^3*(2 + 3*x^2)^(5/2))/36 - ((3 + 2*x)^4*(
2 + 3*x^2)^(5/2))/27 + ((661583 + 226755*x)*(2 + 3*x^2)^(5/2))/17010 + (2777*Arc
Sinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi [A]  time = 0.228694, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{27} \left (3 x^2+2\right )^{5/2} (2 x+3)^4+\frac{13}{36} \left (3 x^2+2\right )^{5/2} (2 x+3)^3+\frac{4421 \left (3 x^2+2\right )^{5/2} (2 x+3)^2}{2268}+\frac{(226755 x+661583) \left (3 x^2+2\right )^{5/2}}{17010}+\frac{2777}{36} x \left (3 x^2+2\right )^{3/2}+\frac{2777}{12} x \sqrt{3 x^2+2}+\frac{2777 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(2777*x*Sqrt[2 + 3*x^2])/12 + (2777*x*(2 + 3*x^2)^(3/2))/36 + (4421*(3 + 2*x)^2*
(2 + 3*x^2)^(5/2))/2268 + (13*(3 + 2*x)^3*(2 + 3*x^2)^(5/2))/36 - ((3 + 2*x)^4*(
2 + 3*x^2)^(5/2))/27 + ((661583 + 226755*x)*(2 + 3*x^2)^(5/2))/17010 + (2777*Arc
Sinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi in Sympy [A]  time = 19.5155, size = 124, normalized size = 0.9 \[ \frac{2777 x \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{36} + \frac{2777 x \sqrt{3 x^{2} + 2}}{12} - \frac{\left (2 x + 3\right )^{4} \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{27} + \frac{13 \left (2 x + 3\right )^{3} \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{36} + \frac{4421 \left (2 x + 3\right )^{2} \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{2268} + \frac{\left (16326360 x + 47633976\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{1224720} + \frac{2777 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2777*x*(3*x**2 + 2)**(3/2)/36 + 2777*x*sqrt(3*x**2 + 2)/12 - (2*x + 3)**4*(3*x**
2 + 2)**(5/2)/27 + 13*(2*x + 3)**3*(3*x**2 + 2)**(5/2)/36 + 4421*(2*x + 3)**2*(3
*x**2 + 2)**(5/2)/2268 + (16326360*x + 47633976)*(3*x**2 + 2)**(5/2)/1224720 + 2
777*sqrt(3)*asinh(sqrt(6)*x/2)/18

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Mathematica [A]  time = 0.0939425, size = 75, normalized size = 0.54 \[ \frac{\sqrt{3 x^2+2} \left (-181440 x^8-204120 x^7+3676320 x^6+14492520 x^5+24490404 x^4+27468315 x^3+27537072 x^2+19683405 x+8598544\right )}{34020}+\frac{2777 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + 3*x^2]*(8598544 + 19683405*x + 27537072*x^2 + 27468315*x^3 + 24490404*
x^4 + 14492520*x^5 + 3676320*x^6 - 204120*x^7 - 181440*x^8))/34020 + (2777*ArcSi
nh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Maple [A]  time = 0.018, size = 103, normalized size = 0.8 \[{\frac{2777\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{2777\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{2777\,\sqrt{3}}{18}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{537409}{8505} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{434\,x}{9} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{7256\,{x}^{2}}{567} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{x}^{3}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}-{\frac{16\,{x}^{4}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2777/36*x*(3*x^2+2)^(3/2)+2777/12*x*(3*x^2+2)^(1/2)+2777/18*arcsinh(1/2*x*6^(1/2
))*3^(1/2)+537409/8505*(3*x^2+2)^(5/2)+434/9*x*(3*x^2+2)^(5/2)+7256/567*x^2*(3*x
^2+2)^(5/2)-2/3*x^3*(3*x^2+2)^(5/2)-16/27*x^4*(3*x^2+2)^(5/2)

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Maxima [A]  time = 0.768092, size = 138, normalized size = 1. \[ -\frac{16}{27} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{4} - \frac{2}{3} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{3} + \frac{7256}{567} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x^{2} + \frac{434}{9} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{537409}{8505} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} + \frac{2777}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{2777}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{2777}{18} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/27*(3*x^2 + 2)^(5/2)*x^4 - 2/3*(3*x^2 + 2)^(5/2)*x^3 + 7256/567*(3*x^2 + 2)^
(5/2)*x^2 + 434/9*(3*x^2 + 2)^(5/2)*x + 537409/8505*(3*x^2 + 2)^(5/2) + 2777/36*
(3*x^2 + 2)^(3/2)*x + 2777/12*sqrt(3*x^2 + 2)*x + 2777/18*sqrt(3)*arcsinh(1/2*sq
rt(6)*x)

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Fricas [A]  time = 0.288293, size = 117, normalized size = 0.85 \[ -\frac{1}{102060} \, \sqrt{3}{\left (\sqrt{3}{\left (181440 \, x^{8} + 204120 \, x^{7} - 3676320 \, x^{6} - 14492520 \, x^{5} - 24490404 \, x^{4} - 27468315 \, x^{3} - 27537072 \, x^{2} - 19683405 \, x - 8598544\right )} \sqrt{3 \, x^{2} + 2} - 7872795 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/102060*sqrt(3)*(sqrt(3)*(181440*x^8 + 204120*x^7 - 3676320*x^6 - 14492520*x^5
- 24490404*x^4 - 27468315*x^3 - 27537072*x^2 - 19683405*x - 8598544)*sqrt(3*x^2
+ 2) - 7872795*log(-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))

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Sympy [A]  time = 107.144, size = 162, normalized size = 1.17 \[ - \frac{16 x^{8} \sqrt{3 x^{2} + 2}}{3} - 6 x^{7} \sqrt{3 x^{2} + 2} + \frac{6808 x^{6} \sqrt{3 x^{2} + 2}}{63} + 426 x^{5} \sqrt{3 x^{2} + 2} + \frac{226763 x^{4} \sqrt{3 x^{2} + 2}}{315} + \frac{9689 x^{3} \sqrt{3 x^{2} + 2}}{12} + \frac{2294756 x^{2} \sqrt{3 x^{2} + 2}}{2835} + \frac{6943 x \sqrt{3 x^{2} + 2}}{12} + \frac{2149636 \sqrt{3 x^{2} + 2}}{8505} + \frac{2777 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*x**8*sqrt(3*x**2 + 2)/3 - 6*x**7*sqrt(3*x**2 + 2) + 6808*x**6*sqrt(3*x**2 +
2)/63 + 426*x**5*sqrt(3*x**2 + 2) + 226763*x**4*sqrt(3*x**2 + 2)/315 + 9689*x**3
*sqrt(3*x**2 + 2)/12 + 2294756*x**2*sqrt(3*x**2 + 2)/2835 + 6943*x*sqrt(3*x**2 +
2)/12 + 2149636*sqrt(3*x**2 + 2)/8505 + 2777*sqrt(3)*asinh(sqrt(6)*x/2)/18

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GIAC/XCAS [A]  time = 0.309926, size = 97, normalized size = 0.7 \[ -\frac{1}{34020} \,{\left (3 \,{\left ({\left (9 \,{\left (4 \,{\left (10 \,{\left ({\left (21 \,{\left (8 \, x + 9\right )} x - 3404\right )} x - 13419\right )} x - 226763\right )} x - 1017345\right )} x - 9179024\right )} x - 6561135\right )} x - 8598544\right )} \sqrt{3 \, x^{2} + 2} - \frac{2777}{18} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/34020*(3*((9*(4*(10*((21*(8*x + 9)*x - 3404)*x - 13419)*x - 226763)*x - 10173
45)*x - 9179024)*x - 6561135)*x - 8598544)*sqrt(3*x^2 + 2) - 2777/18*sqrt(3)*ln(
-sqrt(3)*x + sqrt(3*x^2 + 2))