3.1374 \(\int \frac{(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac{(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}-\frac{1}{8} (193-63 x) \sqrt{3 x^2+2}+\frac{193}{16} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )+\frac{663}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

-((193 - 63*x)*Sqrt[2 + 3*x^2])/8 - ((21 + x)*(2 + 3*x^2)^(3/2))/(6*(3 + 2*x)) +
(663*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/16 + (193*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[3
5]*Sqrt[2 + 3*x^2])])/16

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Rubi [A]  time = 0.180545, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(x+21) \left (3 x^2+2\right )^{3/2}}{6 (2 x+3)}-\frac{1}{8} (193-63 x) \sqrt{3 x^2+2}+\frac{193}{16} \sqrt{35} \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )+\frac{663}{16} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((193 - 63*x)*Sqrt[2 + 3*x^2])/8 - ((21 + x)*(2 + 3*x^2)^(3/2))/(6*(3 + 2*x)) +
(663*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/16 + (193*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[3
5]*Sqrt[2 + 3*x^2])])/16

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Rubi in Sympy [A]  time = 18.4764, size = 87, normalized size = 0.9 \[ - \frac{\left (- 1512 x + 4632\right ) \sqrt{3 x^{2} + 2}}{192} + \frac{663 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{16} + \frac{193 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{16} - \frac{\left (2 x + 42\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{12 \left (2 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-1512*x + 4632)*sqrt(3*x**2 + 2)/192 + 663*sqrt(3)*asinh(sqrt(6)*x/2)/16 + 193
*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/16 - (2*x + 42)*(3*x*
*2 + 2)**(3/2)/(12*(2*x + 3))

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Mathematica [A]  time = 0.220962, size = 102, normalized size = 1.05 \[ \frac{1}{16} \left (193 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{2 \sqrt{3 x^2+2} \left (12 x^3-126 x^2+599 x+1905\right )}{6 x+9}-193 \sqrt{35} \log (2 x+3)+663 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[2 + 3*x^2]*(1905 + 599*x - 126*x^2 + 12*x^3))/(9 + 6*x) + 663*Sqrt[3]*
ArcSinh[Sqrt[3/2]*x] - 193*Sqrt[35]*Log[3 + 2*x] + 193*Sqrt[35]*Log[2*(4 - 9*x +
Sqrt[35]*Sqrt[2 + 3*x^2])])/16

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Maple [A]  time = 0.016, size = 131, normalized size = 1.4 \[ -{\frac{13}{70} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{193}{210} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{63\,x}{8}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}+{\frac{663\,\sqrt{3}}{16}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{193}{16}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{193\,\sqrt{35}}{16}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{39\,x}{70} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/70/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(5/2)-193/210*(3*(x+3/2)^2-9*x-19/4)^(3/2)
+63/8*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)+663/16*arcsinh(1/2*x*6^(1/2))*3^(1/2)-193/1
6*(12*(x+3/2)^2-36*x-19)^(1/2)+193/16*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12
*(x+3/2)^2-36*x-19)^(1/2))+39/70*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)

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Maxima [A]  time = 0.778592, size = 134, normalized size = 1.38 \[ -\frac{1}{12} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} + \frac{63}{8} \, \sqrt{3 \, x^{2} + 2} x + \frac{663}{16} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) - \frac{193}{16} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{193}{8} \, \sqrt{3 \, x^{2} + 2} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{4 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(3*x^2 + 2)^(3/2) + 63/8*sqrt(3*x^2 + 2)*x + 663/16*sqrt(3)*arcsinh(1/2*sq
rt(6)*x) - 193/16*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(
2*x + 3)) - 193/8*sqrt(3*x^2 + 2) - 13/4*(3*x^2 + 2)^(3/2)/(2*x + 3)

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Fricas [A]  time = 0.301625, size = 163, normalized size = 1.68 \[ \frac{1989 \, \sqrt{3}{\left (2 \, x + 3\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 579 \, \sqrt{35}{\left (2 \, x + 3\right )} \log \left (\frac{\sqrt{35} \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \,{\left (12 \, x^{3} - 126 \, x^{2} + 599 \, x + 1905\right )} \sqrt{3 \, x^{2} + 2}}{96 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96*(1989*sqrt(3)*(2*x + 3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 579*s
qrt(35)*(2*x + 3)*log((sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/
(4*x^2 + 12*x + 9)) - 4*(12*x^3 - 126*x^2 + 599*x + 1905)*sqrt(3*x^2 + 2))/(2*x
+ 3)

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.544572, size = 641, normalized size = 6.61 \[ \frac{193}{16} \, \sqrt{35}{\rm ln}\left (\sqrt{35}{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )} - 9\right ){\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - \frac{663}{16} \, \sqrt{3}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{3} + 2 \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{2 \, \sqrt{35}}{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{3} + \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}}\right ){\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - \frac{455}{32} \, \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) + \frac{3 \,{\left (704 \,{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{5}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - 323 \, \sqrt{35}{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{4}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - 1944 \,{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{3}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) + 1158 \, \sqrt{35}{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{2}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) + 1872 \,{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - 1263 \, \sqrt{35}{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{8 \,{\left ({\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )}^{2} - 3\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

193/16*sqrt(35)*ln(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)
/(2*x + 3)) - 9)*sign(1/(2*x + 3)) - 663/16*sqrt(3)*ln(1/2*abs(-2*sqrt(3) + 2*sq
rt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35)/(2*x + 3))/(sqrt(3) + sqrt(-
18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)))*sign(1/(2*x + 3)) - 45
5/32*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)*sign(1/(2*x + 3)) + 3/8*(704*(sqrt
(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^5*sign(1/(2*x + 3)) -
323*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^4*
sign(1/(2*x + 3)) - 1944*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2
*x + 3))^3*sign(1/(2*x + 3)) + 1158*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^
2 + 3) + sqrt(35)/(2*x + 3))^2*sign(1/(2*x + 3)) + 1872*(sqrt(-18/(2*x + 3) + 35
/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))*sign(1/(2*x + 3)) - 1263*sqrt(35)*sign(1
/(2*x + 3)))/((sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^2
- 3)^3