3.1419 \(\int \frac{(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}-\frac{5 (16-421 x) (2 x+3)^2}{54 \sqrt{3 x^2+2}}-\frac{50}{81} (93 x+299) \sqrt{3 x^2+2}+\frac{1600 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^4)/(18*(2 + 3*x^2)^(3/2)) - (5*(16 - 421*x)*(3 + 2*x)^2)
/(54*Sqrt[2 + 3*x^2]) - (50*(299 + 93*x)*Sqrt[2 + 3*x^2])/81 + (1600*ArcSinh[Sqr
t[3/2]*x])/(27*Sqrt[3])

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Rubi [A]  time = 0.165924, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}-\frac{5 (16-421 x) (2 x+3)^2}{54 \sqrt{3 x^2+2}}-\frac{50}{81} (93 x+299) \sqrt{3 x^2+2}+\frac{1600 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^4)/(18*(2 + 3*x^2)^(3/2)) - (5*(16 - 421*x)*(3 + 2*x)^2)
/(54*Sqrt[2 + 3*x^2]) - (50*(299 + 93*x)*Sqrt[2 + 3*x^2])/81 + (1600*ArcSinh[Sqr
t[3/2]*x])/(27*Sqrt[3])

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Rubi in Sympy [A]  time = 15.3712, size = 82, normalized size = 0.87 \[ - \frac{\left (- 16840 x + 640\right ) \left (2 x + 3\right )^{2}}{432 \sqrt{3 x^{2} + 2}} - \frac{\left (- 98 x + 28\right ) \left (2 x + 3\right )^{4}}{36 \left (3 x^{2} + 2\right )^{\frac{3}{2}}} - \frac{\left (148800 x + 478400\right ) \sqrt{3 x^{2} + 2}}{2592} + \frac{1600 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-16840*x + 640)*(2*x + 3)**2/(432*sqrt(3*x**2 + 2)) - (-98*x + 28)*(2*x + 3)**
4/(36*(3*x**2 + 2)**(3/2)) - (148800*x + 478400)*sqrt(3*x**2 + 2)/2592 + 1600*sq
rt(3)*asinh(sqrt(6)*x/2)/81

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Mathematica [A]  time = 0.124934, size = 60, normalized size = 0.64 \[ \frac{1}{162} \left (3200 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{864 x^5+4320 x^4-183945 x^3+147600 x^2-79215 x+134126}{\left (3 x^2+2\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((134126 - 79215*x + 147600*x^2 - 183945*x^3 + 4320*x^4 + 864*x^5)/(2 + 3*x^2)
^(3/2)) + 3200*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/162

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Maple [A]  time = 0.01, size = 105, normalized size = 1.1 \[ -{\frac{615\,x}{2} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{21505\,x}{54}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{67063}{81} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{8200\,{x}^{2}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{1600\,{x}^{3}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{1600\,\sqrt{3}}{81}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{80\,{x}^{4}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{16\,{x}^{5}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-615/2*x/(3*x^2+2)^(3/2)+21505/54*x/(3*x^2+2)^(1/2)-67063/81/(3*x^2+2)^(3/2)-820
0/9*x^2/(3*x^2+2)^(3/2)-1600/27*x^3/(3*x^2+2)^(3/2)+1600/81*arcsinh(1/2*x*6^(1/2
))*3^(1/2)-80/3*x^4/(3*x^2+2)^(3/2)-16/3*x^5/(3*x^2+2)^(3/2)

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Maxima [A]  time = 0.751203, size = 161, normalized size = 1.71 \[ -\frac{16 \, x^{5}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{80 \, x^{4}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{1600}{81} \, x{\left (\frac{9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{4}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\right )} + \frac{1600}{81} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{70915 \, x}{162 \, \sqrt{3 \, x^{2} + 2}} - \frac{8200 \, x^{2}}{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{615 \, x}{2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{67063}{81 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/3*x^5/(3*x^2 + 2)^(3/2) - 80/3*x^4/(3*x^2 + 2)^(3/2) - 1600/81*x*(9*x^2/(3*x
^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) + 1600/81*sqrt(3)*arcsinh(1/2*sqrt(6)*x) +
70915/162*x/sqrt(3*x^2 + 2) - 8200/9*x^2/(3*x^2 + 2)^(3/2) - 615/2*x/(3*x^2 + 2)
^(3/2) - 67063/81/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 0.276012, size = 132, normalized size = 1.4 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (864 \, x^{5} + 4320 \, x^{4} - 183945 \, x^{3} + 147600 \, x^{2} - 79215 \, x + 134126\right )} \sqrt{3 \, x^{2} + 2} - 4800 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )}}{486 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/486*sqrt(3)*(sqrt(3)*(864*x^5 + 4320*x^4 - 183945*x^3 + 147600*x^2 - 79215*x
+ 134126)*sqrt(3*x^2 + 2) - 4800*(9*x^4 + 12*x^2 + 4)*log(-sqrt(3)*(3*x^2 + 1) -
3*sqrt(3*x^2 + 2)*x))/(9*x^4 + 12*x^2 + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.323747, size = 74, normalized size = 0.79 \[ -\frac{1600}{81} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{3 \,{\left ({\left ({\left (288 \,{\left (x + 5\right )} x - 61315\right )} x + 49200\right )} x - 26405\right )} x + 134126}{162 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1600/81*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(3*(((288*(x + 5)*x -
61315)*x + 49200)*x - 26405)*x + 134126)/(3*x^2 + 2)^(3/2)