3.1384 \(\int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=88 \[ \frac{1}{12} (4-x) \left (3 x^2+2\right )^{7/2}+\frac{91}{36} x \left (3 x^2+2\right )^{5/2}+\frac{455}{72} x \left (3 x^2+2\right )^{3/2}+\frac{455}{24} x \sqrt{3 x^2+2}+\frac{455 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{12 \sqrt{3}} \]

[Out]

(455*x*Sqrt[2 + 3*x^2])/24 + (455*x*(2 + 3*x^2)^(3/2))/72 + (91*x*(2 + 3*x^2)^(5
/2))/36 + ((4 - x)*(2 + 3*x^2)^(7/2))/12 + (455*ArcSinh[Sqrt[3/2]*x])/(12*Sqrt[3
])

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Rubi [A]  time = 0.0709498, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{1}{12} (4-x) \left (3 x^2+2\right )^{7/2}+\frac{91}{36} x \left (3 x^2+2\right )^{5/2}+\frac{455}{72} x \left (3 x^2+2\right )^{3/2}+\frac{455}{24} x \sqrt{3 x^2+2}+\frac{455 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{12 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(455*x*Sqrt[2 + 3*x^2])/24 + (455*x*(2 + 3*x^2)^(3/2))/72 + (91*x*(2 + 3*x^2)^(5
/2))/36 + ((4 - x)*(2 + 3*x^2)^(7/2))/12 + (455*ArcSinh[Sqrt[3/2]*x])/(12*Sqrt[3
])

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Rubi in Sympy [A]  time = 6.19321, size = 80, normalized size = 0.91 \[ \frac{91 x \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{36} + \frac{455 x \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{72} + \frac{455 x \sqrt{3 x^{2} + 2}}{24} + \frac{\left (- 14 x + 56\right ) \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{168} + \frac{455 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

91*x*(3*x**2 + 2)**(5/2)/36 + 455*x*(3*x**2 + 2)**(3/2)/72 + 455*x*sqrt(3*x**2 +
2)/24 + (-14*x + 56)*(3*x**2 + 2)**(7/2)/168 + 455*sqrt(3)*asinh(sqrt(6)*x/2)/3
6

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Mathematica [A]  time = 0.0625426, size = 70, normalized size = 0.8 \[ \frac{1}{72} \left (910 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-3 \sqrt{3 x^2+2} \left (54 x^7-216 x^6-438 x^5-432 x^4-1111 x^3-288 x^2-985 x-64\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[2 + 3*x^2]*(-64 - 985*x - 288*x^2 - 1111*x^3 - 432*x^4 - 438*x^5 - 216*
x^6 + 54*x^7) + 910*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/72

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Maple [A]  time = 0.007, size = 73, normalized size = 0.8 \[{\frac{91\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{455\,x}{72} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{455\,x}{24}\sqrt{3\,{x}^{2}+2}}+{\frac{455\,\sqrt{3}}{36}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{1}{3} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}-{\frac{x}{12} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

91/36*x*(3*x^2+2)^(5/2)+455/72*x*(3*x^2+2)^(3/2)+455/24*x*(3*x^2+2)^(1/2)+455/36
*arcsinh(1/2*x*6^(1/2))*3^(1/2)+1/3*(3*x^2+2)^(7/2)-1/12*x*(3*x^2+2)^(7/2)

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Maxima [A]  time = 0.772212, size = 97, normalized size = 1.1 \[ -\frac{1}{12} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x + \frac{1}{3} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} + \frac{91}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{455}{72} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{455}{24} \, \sqrt{3 \, x^{2} + 2} x + \frac{455}{36} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(3*x^2 + 2)^(7/2)*x + 1/3*(3*x^2 + 2)^(7/2) + 91/36*(3*x^2 + 2)^(5/2)*x +
455/72*(3*x^2 + 2)^(3/2)*x + 455/24*sqrt(3*x^2 + 2)*x + 455/36*sqrt(3)*arcsinh(1
/2*sqrt(6)*x)

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Fricas [A]  time = 0.277657, size = 111, normalized size = 1.26 \[ -\frac{1}{72} \, \sqrt{3}{\left (\sqrt{3}{\left (54 \, x^{7} - 216 \, x^{6} - 438 \, x^{5} - 432 \, x^{4} - 1111 \, x^{3} - 288 \, x^{2} - 985 \, x - 64\right )} \sqrt{3 \, x^{2} + 2} - 455 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*sqrt(3)*(sqrt(3)*(54*x^7 - 216*x^6 - 438*x^5 - 432*x^4 - 1111*x^3 - 288*x^
2 - 985*x - 64)*sqrt(3*x^2 + 2) - 455*log(-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 +
2)*x))

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Sympy [A]  time = 171.102, size = 143, normalized size = 1.62 \[ - \frac{9 x^{7} \sqrt{3 x^{2} + 2}}{4} + 9 x^{6} \sqrt{3 x^{2} + 2} + \frac{73 x^{5} \sqrt{3 x^{2} + 2}}{4} + 18 x^{4} \sqrt{3 x^{2} + 2} + \frac{1111 x^{3} \sqrt{3 x^{2} + 2}}{24} + 12 x^{2} \sqrt{3 x^{2} + 2} + \frac{985 x \sqrt{3 x^{2} + 2}}{24} + \frac{8 \sqrt{3 x^{2} + 2}}{3} + \frac{455 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*x**7*sqrt(3*x**2 + 2)/4 + 9*x**6*sqrt(3*x**2 + 2) + 73*x**5*sqrt(3*x**2 + 2)/
4 + 18*x**4*sqrt(3*x**2 + 2) + 1111*x**3*sqrt(3*x**2 + 2)/24 + 12*x**2*sqrt(3*x*
*2 + 2) + 985*x*sqrt(3*x**2 + 2)/24 + 8*sqrt(3*x**2 + 2)/3 + 455*sqrt(3)*asinh(s
qrt(6)*x/2)/36

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GIAC/XCAS [A]  time = 0.299493, size = 85, normalized size = 0.97 \[ -\frac{1}{24} \,{\left ({\left ({\left ({\left (6 \,{\left ({\left (9 \,{\left (x - 4\right )} x - 73\right )} x - 72\right )} x - 1111\right )} x - 288\right )} x - 985\right )} x - 64\right )} \sqrt{3 \, x^{2} + 2} - \frac{455}{36} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*((((6*((9*(x - 4)*x - 73)*x - 72)*x - 1111)*x - 288)*x - 985)*x - 64)*sqrt
(3*x^2 + 2) - 455/36*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2))