3.1386 \(\int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx\)

Optimal. Leaf size=112 \[ \frac{1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}+\frac{7}{96} (130-53 x) \left (3 x^2+2\right )^{3/2}+\frac{7}{64} (2275-691 x) \sqrt{3 x^2+2}-\frac{15925}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{162673 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{128 \sqrt{3}} \]

[Out]

(7*(2275 - 691*x)*Sqrt[2 + 3*x^2])/64 + (7*(130 - 53*x)*(2 + 3*x^2)^(3/2))/96 +
((39 - 5*x)*(2 + 3*x^2)^(5/2))/60 - (162673*ArcSinh[Sqrt[3/2]*x])/(128*Sqrt[3])
- (15925*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/128

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Rubi [A]  time = 0.230658, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}+\frac{7}{96} (130-53 x) \left (3 x^2+2\right )^{3/2}+\frac{7}{64} (2275-691 x) \sqrt{3 x^2+2}-\frac{15925}{128} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{162673 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{128 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(7*(2275 - 691*x)*Sqrt[2 + 3*x^2])/64 + (7*(130 - 53*x)*(2 + 3*x^2)^(3/2))/96 +
((39 - 5*x)*(2 + 3*x^2)^(5/2))/60 - (162673*ArcSinh[Sqrt[3/2]*x])/(128*Sqrt[3])
- (15925*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/128

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Rubi in Sympy [A]  time = 22.683, size = 100, normalized size = 0.89 \[ \frac{\left (- 6268752 x + 20638800\right ) \sqrt{3 x^{2} + 2}}{82944} + \frac{\left (- 40068 x + 98280\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{10368} + \frac{\left (- 30 x + 234\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{360} - \frac{162673 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{384} - \frac{15925 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-6268752*x + 20638800)*sqrt(3*x**2 + 2)/82944 + (-40068*x + 98280)*(3*x**2 + 2)
**(3/2)/10368 + (-30*x + 234)*(3*x**2 + 2)**(5/2)/360 - 162673*sqrt(3)*asinh(sqr
t(6)*x/2)/384 - 15925*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/
128

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Mathematica [A]  time = 0.10552, size = 158, normalized size = 1.41 \[ \frac{69576 \sqrt{3 x^2+2} x^2-160590 \sqrt{3 x^2+2} x+519142 \sqrt{3 x^2+2}-238875 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-1440 \sqrt{3 x^2+2} x^5+11232 \sqrt{3 x^2+2} x^4-24180 \sqrt{3 x^2+2} x^3+238875 \sqrt{35} \log (2 x+3)-813365 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{1920} \]

Antiderivative was successfully verified.

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

[Out]

(519142*Sqrt[2 + 3*x^2] - 160590*x*Sqrt[2 + 3*x^2] + 69576*x^2*Sqrt[2 + 3*x^2] -
24180*x^3*Sqrt[2 + 3*x^2] + 11232*x^4*Sqrt[2 + 3*x^2] - 1440*x^5*Sqrt[2 + 3*x^2
] - 813365*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + 238875*Sqrt[35]*Log[3 + 2*x] - 238875*
Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/1920

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Maple [A]  time = 0.011, size = 162, normalized size = 1.5 \[ -{\frac{x}{12} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}-{\frac{5\,x}{24} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{5\,x}{8}\sqrt{3\,{x}^{2}+2}}-{\frac{162673\,\sqrt{3}}{384}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{13}{20} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{117\,x}{32} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{4797\,x}{64}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{455}{48} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{15925}{128}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{15925\,\sqrt{35}}{128}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*x*(3*x^2+2)^(5/2)-5/24*x*(3*x^2+2)^(3/2)-5/8*x*(3*x^2+2)^(1/2)-162673/384*
arcsinh(1/2*x*6^(1/2))*3^(1/2)+13/20*(3*(x+3/2)^2-9*x-19/4)^(5/2)-117/32*x*(3*(x
+3/2)^2-9*x-19/4)^(3/2)-4797/64*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+455/48*(3*(x+3/2)
^2-9*x-19/4)^(3/2)+15925/128*(12*(x+3/2)^2-36*x-19)^(1/2)-15925/128*35^(1/2)*arc
tanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]  time = 0.777375, size = 157, normalized size = 1.4 \[ -\frac{1}{12} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{13}{20} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{371}{96} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{455}{48} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{4837}{64} \, \sqrt{3 \, x^{2} + 2} x - \frac{162673}{384} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{15925}{128} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{15925}{64} \, \sqrt{3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(3*x^2 + 2)^(5/2)*x + 13/20*(3*x^2 + 2)^(5/2) - 371/96*(3*x^2 + 2)^(3/2)*x
+ 455/48*(3*x^2 + 2)^(3/2) - 4837/64*sqrt(3*x^2 + 2)*x - 162673/384*sqrt(3)*arc
sinh(1/2*sqrt(6)*x) + 15925/128*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/
3*sqrt(6)/abs(2*x + 3)) + 15925/64*sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.288632, size = 167, normalized size = 1.49 \[ -\frac{1}{11520} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (720 \, x^{5} - 5616 \, x^{4} + 12090 \, x^{3} - 34788 \, x^{2} + 80295 \, x - 259571\right )} \sqrt{3 \, x^{2} + 2} - 238875 \, \sqrt{35} \sqrt{3} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 2440095 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} + 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/11520*sqrt(3)*(4*sqrt(3)*(720*x^5 - 5616*x^4 + 12090*x^3 - 34788*x^2 + 80295*
x - 259571)*sqrt(3*x^2 + 2) - 238875*sqrt(35)*sqrt(3)*log(-(sqrt(35)*sqrt(3*x^2
+ 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 2440095*log(-sqrt(3)*
(3*x^2 + 1) + 3*sqrt(3*x^2 + 2)*x))

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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*x**2+2)**(5/2)/(3+2*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.324534, size = 169, normalized size = 1.51 \[ -\frac{1}{960} \,{\left (3 \,{\left (2 \,{\left ({\left (24 \,{\left (5 \, x - 39\right )} x + 2015\right )} x - 5798\right )} x + 26765\right )} x - 259571\right )} \sqrt{3 \, x^{2} + 2} + \frac{162673}{384} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{15925}{128} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/960*(3*(2*((24*(5*x - 39)*x + 2015)*x - 5798)*x + 26765)*x - 259571)*sqrt(3*x
^2 + 2) + 162673/384*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 15925/128*sqrt(3
5)*ln(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x
- sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2)))