3.1420 \(\int \frac{(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}-\frac{(318-1783 x) (2 x+3)}{54 \sqrt{3 x^2+2}}-\frac{2027}{81} \sqrt{3 x^2+2}-\frac{16 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^3)/(18*(2 + 3*x^2)^(3/2)) - ((318 - 1783*x)*(3 + 2*x))/(
54*Sqrt[2 + 3*x^2]) - (2027*Sqrt[2 + 3*x^2])/81 - (16*ArcSinh[Sqrt[3/2]*x])/(9*S
qrt[3])

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Rubi [A]  time = 0.140814, antiderivative size = 87, 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 \[ -\frac{7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}-\frac{(318-1783 x) (2 x+3)}{54 \sqrt{3 x^2+2}}-\frac{2027}{81} \sqrt{3 x^2+2}-\frac{16 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{9 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^3)/(18*(2 + 3*x^2)^(3/2)) - ((318 - 1783*x)*(3 + 2*x))/(
54*Sqrt[2 + 3*x^2]) - (2027*Sqrt[2 + 3*x^2])/81 - (16*ArcSinh[Sqrt[3/2]*x])/(9*S
qrt[3])

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Rubi in Sympy [A]  time = 13.703, size = 78, normalized size = 0.9 \[ - \frac{\left (- 14264 x + 2544\right ) \left (2 x + 3\right )}{432 \sqrt{3 x^{2} + 2}} - \frac{\left (- 98 x + 28\right ) \left (2 x + 3\right )^{3}}{36 \left (3 x^{2} + 2\right )^{\frac{3}{2}}} - \frac{2027 \sqrt{3 x^{2} + 2}}{81} - \frac{16 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-14264*x + 2544)*(2*x + 3)/(432*sqrt(3*x**2 + 2)) - (-98*x + 28)*(2*x + 3)**3/
(36*(3*x**2 + 2)**(3/2)) - 2027*sqrt(3*x**2 + 2)/81 - 16*sqrt(3)*asinh(sqrt(6)*x
/2)/27

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Mathematica [A]  time = 0.108576, size = 55, normalized size = 0.63 \[ \frac{1}{162} \left (-\frac{864 x^4-57285 x^3+16560 x^2-33381 x+25342}{\left (3 x^2+2\right )^{3/2}}-96 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((25342 - 33381*x + 16560*x^2 - 57285*x^3 + 864*x^4)/(2 + 3*x^2)^(3/2)) - 96*S
qrt[3]*ArcSinh[Sqrt[3/2]*x])/162

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Maple [A]  time = 0.01, size = 91, normalized size = 1.1 \[ -{\frac{57\,x}{2} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{2111\,x}{18}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{12671}{81} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{920\,{x}^{2}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,{x}^{3}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{16\,\sqrt{3}}{27}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{16\,{x}^{4}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-57/2*x/(3*x^2+2)^(3/2)+2111/18*x/(3*x^2+2)^(1/2)-12671/81/(3*x^2+2)^(3/2)-920/9
*x^2/(3*x^2+2)^(3/2)+16/9*x^3/(3*x^2+2)^(3/2)-16/27*arcsinh(1/2*x*6^(1/2))*3^(1/
2)-16/3*x^4/(3*x^2+2)^(3/2)

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Maxima [A]  time = 0.762621, size = 142, normalized size = 1.63 \[ -\frac{16 \, x^{4}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{16}{27} \, x{\left (\frac{9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{4}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\right )} - \frac{16}{27} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{6269 \, x}{54 \, \sqrt{3 \, x^{2} + 2}} - \frac{920 \, x^{2}}{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{57 \, x}{2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{12671}{81 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/3*x^4/(3*x^2 + 2)^(3/2) + 16/27*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(
3/2)) - 16/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 6269/54*x/sqrt(3*x^2 + 2) - 920/9
*x^2/(3*x^2 + 2)^(3/2) - 57/2*x/(3*x^2 + 2)^(3/2) - 12671/81/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 0.275268, size = 126, normalized size = 1.45 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (864 \, x^{4} - 57285 \, x^{3} + 16560 \, x^{2} - 33381 \, x + 25342\right )} \sqrt{3 \, x^{2} + 2} - 144 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} + 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )}}{486 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/486*sqrt(3)*(sqrt(3)*(864*x^4 - 57285*x^3 + 16560*x^2 - 33381*x + 25342)*sqrt
(3*x^2 + 2) - 144*(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)**4/(3*x**2+2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.319896, size = 70, normalized size = 0.8 \[ \frac{16}{27} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{9 \,{\left ({\left ({\left (96 \, x - 6365\right )} x + 1840\right )} x - 3709\right )} x + 25342}{162 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16/27*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(9*(((96*x - 6365)*x + 18
40)*x - 3709)*x + 25342)/(3*x^2 + 2)^(3/2)