\(\int \frac {(5-x) (2+3 x^2)^{5/2}}{3+2 x} \, dx\) [225]

Optimal result

Integrand size = 24, antiderivative size = 112 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx=\frac {7}{64} (2275-691 x) \sqrt {2+3 x^2}+\frac {7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac {1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}-\frac {162673 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{128 \sqrt {3}}-\frac {15925}{128} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \] Output:

7/64*(2275-691*x)*(3*x^2+2)^(1/2)+7/96*(130-53*x)*(3*x^2+2)^(3/2)+1/60*(39 
-5*x)*(3*x^2+2)^(5/2)-162673/384*arcsinh(1/2*x*6^(1/2))*3^(1/2)-15925/128* 
35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx=\frac {-2 \sqrt {2+3 x^2} \left (-259571+80295 x-34788 x^2+12090 x^3-5616 x^4+720 x^5\right )+477750 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )+813365 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{1920} \] Input:

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

Output:

(-2*Sqrt[2 + 3*x^2]*(-259571 + 80295*x - 34788*x^2 + 12090*x^3 - 5616*x^4 
+ 720*x^5) + 477750*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 
 3*x^2])/Sqrt[35]] + 813365*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/1 
920
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {682, 27, 682, 27, 682, 27, 719, 222, 488, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 682

\(\displaystyle \frac {1}{72} \int \frac {42 (18-53 x) \left (3 x^2+2\right )^{3/2}}{2 x+3}dx+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{12} \int \frac {(18-53 x) \left (3 x^2+2\right )^{3/2}}{2 x+3}dx+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {7}{12} \left (\frac {1}{48} \int \frac {36 (101-691 x) \sqrt {3 x^2+2}}{2 x+3}dx+\frac {1}{8} (130-53 x) \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{12} \left (\frac {3}{4} \int \frac {(101-691 x) \sqrt {3 x^2+2}}{2 x+3}dx+\frac {1}{8} (130-53 x) \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {7}{12} \left (\frac {3}{4} \left (\frac {1}{24} \int \frac {6 (4954-23239 x)}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (2275-691 x)\right )+\frac {1}{8} (130-53 x) \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{12} \left (\frac {3}{4} \left (\frac {1}{4} \int \frac {4954-23239 x}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (2275-691 x)\right )+\frac {1}{8} (130-53 x) \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {7}{12} \left (\frac {3}{4} \left (\frac {1}{4} \left (\frac {79625}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {23239}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )+\frac {1}{4} \sqrt {3 x^2+2} (2275-691 x)\right )+\frac {1}{8} (130-53 x) \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {7}{12} \left (\frac {3}{4} \left (\frac {1}{4} \left (\frac {79625}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {23239 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {1}{4} \sqrt {3 x^2+2} (2275-691 x)\right )+\frac {1}{8} (130-53 x) \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {7}{12} \left (\frac {3}{4} \left (\frac {1}{4} \left (-\frac {79625}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {23239 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {1}{4} \sqrt {3 x^2+2} (2275-691 x)\right )+\frac {1}{8} (130-53 x) \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7}{12} \left (\frac {3}{4} \left (\frac {1}{4} \left (-\frac {23239 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {2275}{2} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (2275-691 x)\right )+\frac {1}{8} (130-53 x) \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}\)

Input:

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

Output:

((39 - 5*x)*(2 + 3*x^2)^(5/2))/60 + (7*(((130 - 53*x)*(2 + 3*x^2)^(3/2))/8 
 + (3*(((2275 - 691*x)*Sqrt[2 + 3*x^2])/4 + ((-23239*ArcSinh[Sqrt[3/2]*x]) 
/(2*Sqrt[3]) - (2275*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2]) 
])/2)/4))/4))/12
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {\left (720 x^{5}-5616 x^{4}+12090 x^{3}-34788 x^{2}+80295 x -259571\right ) \sqrt {3 x^{2}+2}}{960}-\frac {162673 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{384}-\frac {15925 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{128}\)	\(80\)
trager	\(\left (-\frac {3}{4} x^{5}+\frac {117}{20} x^{4}-\frac {403}{32} x^{3}+\frac {2899}{80} x^{2}-\frac {5353}{64} x +\frac {259571}{960}\right ) \sqrt {3 x^{2}+2}-\frac {162673 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{384}-\frac {15925 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{128}\)	\(113\)
default	\(-\frac {x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{12}-\frac {5 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{24}-\frac {5 x \sqrt {3 x^{2}+2}}{8}-\frac {162673 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{384}+\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{20}-\frac {117 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{32}-\frac {4797 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{64}+\frac {455 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{48}+\frac {15925 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{128}-\frac {15925 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{128}\)	\(162\)

Input:

int((5-x)*(3*x^2+2)^(5/2)/(2*x+3),x,method=_RETURNVERBOSE)
 

Output:

-1/960*(720*x^5-5616*x^4+12090*x^3-34788*x^2+80295*x-259571)*(3*x^2+2)^(1/ 
2)-162673/384*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-15925/128*35^(1/2)*arctanh(2/ 
35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx=-\frac {1}{960} \, {\left (720 \, x^{5} - 5616 \, x^{4} + 12090 \, x^{3} - 34788 \, x^{2} + 80295 \, x - 259571\right )} \sqrt {3 \, x^{2} + 2} + \frac {162673}{768} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac {15925}{256} \, \sqrt {35} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) \] Input:

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

Output:

-1/960*(720*x^5 - 5616*x^4 + 12090*x^3 - 34788*x^2 + 80295*x - 259571)*sqr 
t(3*x^2 + 2) + 162673/768*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 
1) + 15925/256*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))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.04 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx=-\frac {1}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {13}{20} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {371}{96} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {455}{48} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {4837}{64} \, \sqrt {3 \, x^{2} + 2} x - \frac {162673}{384} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {15925}{128} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {15925}{64} \, \sqrt {3 \, x^{2} + 2} \] Input:

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

Output:

-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)*arcsinh(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)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.12 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx=-\frac {1}{960} \, {\left (3 \, {\left (2 \, {\left ({\left (24 \, {\left (5 \, x - 39\right )} x + 2015\right )} x - 5798\right )} x + 26765\right )} x - 259571\right )} \sqrt {3 \, x^{2} + 2} + \frac {162673}{384} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {15925}{128} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) \] Input:

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

Output:

-1/960*(3*(2*((24*(5*x - 39)*x + 2015)*x - 5798)*x + 26765)*x - 259571)*sq 
rt(3*x^2 + 2) + 162673/384*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 159 
25/128*sqrt(35)*log(-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)))
 

Mupad [B] (verification not implemented)

Time = 5.87 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx=\frac {\sqrt {35}\,\left (1114750\,\ln \left (x+\frac {3}{2}\right )-1114750\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{8960}-\frac {162673\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{384}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {9\,x^5}{4}-\frac {351\,x^4}{20}+\frac {1209\,x^3}{32}-\frac {8697\,x^2}{80}+\frac {16059\,x}{64}-\frac {259571}{320}\right )}{3} \] Input:

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

Output:

(35^(1/2)*(1114750*log(x + 3/2) - 1114750*log(x - (3^(1/2)*35^(1/2)*(x^2 + 
 2/3)^(1/2))/9 - 4/9)))/8960 - (162673*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2 
))/384 - (3^(1/2)*(x^2 + 2/3)^(1/2)*((16059*x)/64 - (8697*x^2)/80 + (1209* 
x^3)/32 - (351*x^4)/20 + (9*x^5)/4 - 259571/320))/3
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.71 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx=\frac {15925 \sqrt {35}\, \mathit {atan} \left (\frac {2 \sqrt {3 x^{2}+2}\, i +2 \sqrt {3}\, i x}{\sqrt {35}-3 \sqrt {3}}\right ) i}{128}-\frac {3 \sqrt {3 x^{2}+2}\, x^{5}}{4}+\frac {117 \sqrt {3 x^{2}+2}\, x^{4}}{20}-\frac {403 \sqrt {3 x^{2}+2}\, x^{3}}{32}+\frac {2899 \sqrt {3 x^{2}+2}\, x^{2}}{80}-\frac {5353 \sqrt {3 x^{2}+2}\, x}{64}+\frac {259571 \sqrt {3 x^{2}+2}}{960}+\frac {15925 \sqrt {35}\, \mathrm {log}\left (4 \sqrt {3 x^{2}+2}\, \sqrt {3}\, x +3 \sqrt {105}+12 x^{2}-27\right )}{256}-\frac {15925 \sqrt {35}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}+2}+\sqrt {35}+2 \sqrt {3}\, x +3 \sqrt {3}}{\sqrt {2}}\right )}{128}-\frac {162673 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{384} \] Input:

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

Output:

(477750*sqrt(35)*atan((2*sqrt(3*x**2 + 2)*i + 2*sqrt(3)*i*x)/(sqrt(35) - 3 
*sqrt(3)))*i - 2880*sqrt(3*x**2 + 2)*x**5 + 22464*sqrt(3*x**2 + 2)*x**4 - 
48360*sqrt(3*x**2 + 2)*x**3 + 139152*sqrt(3*x**2 + 2)*x**2 - 321180*sqrt(3 
*x**2 + 2)*x + 1038284*sqrt(3*x**2 + 2) + 238875*sqrt(35)*log(4*sqrt(3*x** 
2 + 2)*sqrt(3)*x + 3*sqrt(105) + 12*x**2 - 27) - 477750*sqrt(35)*log((2*sq 
rt(3*x**2 + 2) + sqrt(35) + 2*sqrt(3)*x + 3*sqrt(3))/sqrt(2)) - 1626730*sq 
rt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)))/3840