3.1373 \(\int \frac{(5-x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx\)

Optimal. Leaf size=92 \[ \frac{1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}+\frac{1}{16} (455-123 x) \sqrt{3 x^2+2}-\frac{455}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{1529}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

((455 - 123*x)*Sqrt[2 + 3*x^2])/16 + ((26 - 3*x)*(2 + 3*x^2)^(3/2))/24 - (1529*S
qrt[3]*ArcSinh[Sqrt[3/2]*x])/32 - (455*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt
[2 + 3*x^2])])/32

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Rubi [A]  time = 0.178157, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{1}{24} (26-3 x) \left (3 x^2+2\right )^{3/2}+\frac{1}{16} (455-123 x) \sqrt{3 x^2+2}-\frac{455}{32} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )-\frac{1529}{32} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

((455 - 123*x)*Sqrt[2 + 3*x^2])/16 + ((26 - 3*x)*(2 + 3*x^2)^(3/2))/24 - (1529*S
qrt[3]*ArcSinh[Sqrt[3/2]*x])/32 - (455*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt
[2 + 3*x^2])])/32

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Rubi in Sympy [A]  time = 17.9975, size = 83, normalized size = 0.9 \[ \frac{\left (- 8856 x + 32760\right ) \sqrt{3 x^{2} + 2}}{1152} + \frac{\left (- 18 x + 156\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{144} - \frac{1529 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{32} - \frac{455 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-8856*x + 32760)*sqrt(3*x**2 + 2)/1152 + (-18*x + 156)*(3*x**2 + 2)**(3/2)/144
- 1529*sqrt(3)*asinh(sqrt(6)*x/2)/32 - 455*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(3
5*sqrt(3*x**2 + 2)))/32

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Mathematica [A]  time = 0.0923913, size = 126, normalized size = 1.37 \[ \frac{1}{96} \left (312 \sqrt{3 x^2+2} x^2-762 \sqrt{3 x^2+2} x+2938 \sqrt{3 x^2+2}-1365 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-36 \sqrt{3 x^2+2} x^3+1365 \sqrt{35} \log (2 x+3)-4587 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2938*Sqrt[2 + 3*x^2] - 762*x*Sqrt[2 + 3*x^2] + 312*x^2*Sqrt[2 + 3*x^2] - 36*x^3
*Sqrt[2 + 3*x^2] - 4587*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + 1365*Sqrt[35]*Log[3 + 2*x
] - 1365*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/96

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Maple [A]  time = 0.011, size = 117, normalized size = 1.3 \[ -{\frac{x}{8} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}-{\frac{3\,x}{8}\sqrt{3\,{x}^{2}+2}}-{\frac{1529\,\sqrt{3}}{32}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{13}{12} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{117\,x}{16}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{455}{32}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{455\,\sqrt{35}}{32}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*x*(3*x^2+2)^(3/2)-3/8*x*(3*x^2+2)^(1/2)-1529/32*arcsinh(1/2*x*6^(1/2))*3^(1
/2)+13/12*(3*(x+3/2)^2-9*x-19/4)^(3/2)-117/16*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+455
/32*(12*(x+3/2)^2-36*x-19)^(1/2)-455/32*35^(1/2)*arctanh(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 = 0.787574, size = 126, normalized size = 1.37 \[ -\frac{1}{8} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{13}{12} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{123}{16} \, \sqrt{3 \, x^{2} + 2} x - \frac{1529}{32} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{455}{32} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{455}{16} \, \sqrt{3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(3*x^2 + 2)^(3/2)*x + 13/12*(3*x^2 + 2)^(3/2) - 123/16*sqrt(3*x^2 + 2)*x -
1529/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 455/32*sqrt(35)*arcsinh(3/2*sqrt(6)*x/a
bs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 455/16*sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.301139, size = 138, normalized size = 1.5 \[ -\frac{1}{48} \,{\left (18 \, x^{3} - 156 \, x^{2} + 381 \, x - 1469\right )} \sqrt{3 \, x^{2} + 2} + \frac{1529}{64} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac{455}{64} \, \sqrt{35} \log \left (-\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(18*x^3 - 156*x^2 + 381*x - 1469)*sqrt(3*x^2 + 2) + 1529/64*sqrt(3)*log(sq
rt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 455/64*sqrt(35)*log(-(sqrt(35)*sqrt(3*x^2
+ 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.297023, size = 157, normalized size = 1.71 \[ -\frac{1}{48} \,{\left (3 \,{\left (2 \,{\left (3 \, x - 26\right )} x + 127\right )} x - 1469\right )} \sqrt{3 \, x^{2} + 2} + \frac{1529}{32} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{455}{32} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(3*(2*(3*x - 26)*x + 127)*x - 1469)*sqrt(3*x^2 + 2) + 1529/32*sqrt(3)*ln(-
sqrt(3)*x + sqrt(3*x^2 + 2)) + 455/32*sqrt(35)*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)))