\(\int \frac {(5-2 x)^{5/2} \sqrt {4+3 x}}{2+3 x+x^2} \, dx\) [346]

Optimal result

Integrand size = 29, antiderivative size = 135 \[ \int \frac {(5-2 x)^{5/2} \sqrt {4+3 x}}{2+3 x+x^2} \, dx=-\frac {169}{6} \sqrt {5-2 x} \sqrt {4+3 x}-(5-2 x)^{3/2} \sqrt {4+3 x}-\frac {8951 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )}{6 \sqrt {6}}+486 \sqrt {2} \arctan \left (\frac {3 \sqrt {4+3 x}}{\sqrt {2} \sqrt {5-2 x}}\right )-98 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {7} \sqrt {4+3 x}}{\sqrt {5-2 x}}\right ) \] Output:

-169/6*(5-2*x)^(1/2)*(4+3*x)^(1/2)-(5-2*x)^(3/2)*(4+3*x)^(1/2)-8951/36*arc 
sin(1/23*46^(1/2)*(4+3*x)^(1/2))*6^(1/2)+486*2^(1/2)*arctan(3/2*(4+3*x)^(1 
/2)*2^(1/2)/(5-2*x)^(1/2))-98*7^(1/2)*arctanh(7^(1/2)*(4+3*x)^(1/2)/(5-2*x 
)^(1/2))
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90 \[ \int \frac {(5-2 x)^{5/2} \sqrt {4+3 x}}{2+3 x+x^2} \, dx=\frac {1}{6} \sqrt {5-2 x} \sqrt {4+3 x} (-199+12 x)-486 \sqrt {2} \arctan \left (\frac {\sqrt {10-4 x}}{3 \sqrt {4+3 x}}\right )+\frac {8951 \arctan \left (\frac {\sqrt {\frac {15}{2}-3 x}}{\sqrt {4+3 x}}\right )}{6 \sqrt {6}}-98 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7} \sqrt {4+3 x}}\right ) \] Input:

Integrate[((5 - 2*x)^(5/2)*Sqrt[4 + 3*x])/(2 + 3*x + x^2),x]
 

Output:

(Sqrt[5 - 2*x]*Sqrt[4 + 3*x]*(-199 + 12*x))/6 - 486*Sqrt[2]*ArcTan[Sqrt[10 
 - 4*x]/(3*Sqrt[4 + 3*x])] + (8951*ArcTan[Sqrt[15/2 - 3*x]/Sqrt[4 + 3*x]]) 
/(6*Sqrt[6]) - 98*Sqrt[7]*ArcTanh[Sqrt[5 - 2*x]/(Sqrt[7]*Sqrt[4 + 3*x])]
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.46, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1201, 27, 90, 60, 64, 223, 2153, 2009}

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.

\(\displaystyle \int \frac {(5-2 x)^{5/2} \sqrt {3 x+4}}{x^2+3 x+2} \, dx\)

\(\Big \downarrow \) 1201

\(\displaystyle \int \frac {\sqrt {5-2 x} (211 x+260)}{\sqrt {3 x+4} \left (x^2+3 x+2\right )}dx-2 \int \frac {2 (20-3 x) \sqrt {5-2 x}}{\sqrt {3 x+4}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {\sqrt {5-2 x} (211 x+260)}{\sqrt {3 x+4} \left (x^2+3 x+2\right )}dx-4 \int \frac {(20-3 x) \sqrt {5-2 x}}{\sqrt {3 x+4}}dx\)

\(\Big \downarrow \) 90

\(\displaystyle \int \frac {\sqrt {5-2 x} (211 x+260)}{\sqrt {3 x+4} \left (x^2+3 x+2\right )}dx-4 \left (\frac {169}{8} \int \frac {\sqrt {5-2 x}}{\sqrt {3 x+4}}dx+\frac {1}{4} \sqrt {3 x+4} (5-2 x)^{3/2}\right )\)

\(\Big \downarrow \) 60

\(\displaystyle \int \frac {\sqrt {5-2 x} (211 x+260)}{\sqrt {3 x+4} \left (x^2+3 x+2\right )}dx-4 \left (\frac {169}{8} \left (\frac {23}{6} \int \frac {1}{\sqrt {5-2 x} \sqrt {3 x+4}}dx+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{4} \sqrt {3 x+4} (5-2 x)^{3/2}\right )\)

\(\Big \downarrow \) 64

\(\displaystyle \int \frac {\sqrt {5-2 x} (211 x+260)}{\sqrt {3 x+4} \left (x^2+3 x+2\right )}dx-4 \left (\frac {169}{8} \left (\frac {23}{9} \int \frac {1}{\sqrt {\frac {23}{3}-\frac {2}{3} (3 x+4)}}d\sqrt {3 x+4}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{4} \sqrt {3 x+4} (5-2 x)^{3/2}\right )\)

\(\Big \downarrow \) 223

\(\displaystyle \int \frac {\sqrt {5-2 x} (211 x+260)}{\sqrt {3 x+4} \left (x^2+3 x+2\right )}dx-4 \left (\frac {169}{8} \left (\frac {23 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{3 \sqrt {6}}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{4} \sqrt {3 x+4} (5-2 x)^{3/2}\right )\)

\(\Big \downarrow \) 2153

\(\displaystyle \int \left (\frac {98 \sqrt {5-2 x}}{(2 x+2) \sqrt {3 x+4}}+\frac {324 \sqrt {5-2 x}}{(2 x+4) \sqrt {3 x+4}}\right )dx-4 \left (\frac {169}{8} \left (\frac {23 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{3 \sqrt {6}}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{4} \sqrt {3 x+4} (5-2 x)^{3/2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -108 \sqrt {6} \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )-98 \sqrt {\frac {2}{3}} \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )-4 \left (\frac {169}{8} \left (\frac {23 \arcsin \left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{3 \sqrt {6}}+\frac {1}{3} \sqrt {5-2 x} \sqrt {3 x+4}\right )+\frac {1}{4} \sqrt {3 x+4} (5-2 x)^{3/2}\right )+486 \sqrt {2} \arctan \left (\frac {3 \sqrt {3 x+4}}{\sqrt {2} \sqrt {5-2 x}}\right )-98 \sqrt {7} \text {arctanh}\left (\frac {\sqrt {7} \sqrt {3 x+4}}{\sqrt {5-2 x}}\right )\)

Input:

Int[((5 - 2*x)^(5/2)*Sqrt[4 + 3*x])/(2 + 3*x + x^2),x]
 

Output:

-98*Sqrt[2/3]*ArcSin[Sqrt[2/23]*Sqrt[4 + 3*x]] - 108*Sqrt[6]*ArcSin[Sqrt[2 
/23]*Sqrt[4 + 3*x]] - 4*(((5 - 2*x)^(3/2)*Sqrt[4 + 3*x])/4 + (169*((Sqrt[5 
 - 2*x]*Sqrt[4 + 3*x])/3 + (23*ArcSin[Sqrt[2/23]*Sqrt[4 + 3*x]])/(3*Sqrt[6 
])))/8) + 486*Sqrt[2]*ArcTan[(3*Sqrt[4 + 3*x])/(Sqrt[2]*Sqrt[5 - 2*x])] - 
98*Sqrt[7]*ArcTanh[(Sqrt[7]*Sqrt[4 + 3*x])/Sqrt[5 - 2*x]]
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 64

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp 
[2/b   Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] 
 /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] 
 || PosQ[b])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 223

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

rule 1201

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2153

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* 
x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ 
[m] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] && IGtQ[n, 0])
 

Maple [A] (verified)

Time = 1.92 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {\sqrt {5-2 x}\, \sqrt {3 x +4}\, \left (8951 \sqrt {3}\, \sqrt {2}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right )-17496 \sqrt {2}\, \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 x^{2}+7 x +20}}\right )+3528 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 x^{2}+7 x +20}}\right )-144 x \sqrt {-6 x^{2}+7 x +20}+2388 \sqrt {-6 x^{2}+7 x +20}\right )}{72 \sqrt {-6 x^{2}+7 x +20}}\)	\(129\)
risch	\(-\frac {\left (-199+12 x \right ) \sqrt {3 x +4}\, \left (-5+2 x \right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{6 \sqrt {-\left (3 x +4\right ) \left (-5+2 x \right )}\, \sqrt {5-2 x}}-\frac {\left (\frac {8951 \sqrt {6}\, \arcsin \left (\frac {12 x}{23}-\frac {7}{23}\right )}{72}+49 \sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (33+19 x \right ) \sqrt {7}}{14 \sqrt {-6 \left (x +1\right )^{2}+19 x +26}}\right )-243 \sqrt {2}\, \arctan \left (\frac {\left (26+31 x \right ) \sqrt {2}}{12 \sqrt {-6 \left (2+x \right )^{2}+44+31 x}}\right )\right ) \sqrt {\left (5-2 x \right ) \left (3 x +4\right )}}{\sqrt {5-2 x}\, \sqrt {3 x +4}}\)	\(156\)

Input:

int((5-2*x)^(5/2)*(3*x+4)^(1/2)/(x^2+3*x+2),x,method=_RETURNVERBOSE)
 

Output:

-1/72*(5-2*x)^(1/2)*(3*x+4)^(1/2)*(8951*3^(1/2)*2^(1/2)*arcsin(12/23*x-7/2 
3)-17496*2^(1/2)*arctan(1/12*(26+31*x)*2^(1/2)/(-6*x^2+7*x+20)^(1/2))+3528 
*7^(1/2)*arctanh(1/14*(33+19*x)*7^(1/2)/(-6*x^2+7*x+20)^(1/2))-144*x*(-6*x 
^2+7*x+20)^(1/2)+2388*(-6*x^2+7*x+20)^(1/2))/(-6*x^2+7*x+20)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.17 \[ \int \frac {(5-2 x)^{5/2} \sqrt {4+3 x}}{2+3 x+x^2} \, dx=\frac {1}{6} \, {\left (12 \, x - 199\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} - 243 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (31 \, x + 26\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{12 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) + \frac {8951}{72} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} {\left (12 \, x - 7\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5}}{12 \, {\left (6 \, x^{2} - 7 \, x - 20\right )}}\right ) + \frac {49}{2} \, \sqrt {7} \log \left (-\frac {4 \, \sqrt {7} {\left (19 \, x + 33\right )} \sqrt {3 \, x + 4} \sqrt {-2 \, x + 5} - 193 \, x^{2} - 1450 \, x - 1649}{x^{2} + 2 \, x + 1}\right ) \] Input:

integrate((5-2*x)^(5/2)*(4+3*x)^(1/2)/(x^2+3*x+2),x, algorithm="fricas")
 

Output:

1/6*(12*x - 199)*sqrt(3*x + 4)*sqrt(-2*x + 5) - 243*sqrt(2)*arctan(1/12*sq 
rt(2)*(31*x + 26)*sqrt(3*x + 4)*sqrt(-2*x + 5)/(6*x^2 - 7*x - 20)) + 8951/ 
72*sqrt(6)*arctan(1/12*sqrt(6)*(12*x - 7)*sqrt(3*x + 4)*sqrt(-2*x + 5)/(6* 
x^2 - 7*x - 20)) + 49/2*sqrt(7)*log(-(4*sqrt(7)*(19*x + 33)*sqrt(3*x + 4)* 
sqrt(-2*x + 5) - 193*x^2 - 1450*x - 1649)/(x^2 + 2*x + 1))
 

Sympy [F]

\[ \int \frac {(5-2 x)^{5/2} \sqrt {4+3 x}}{2+3 x+x^2} \, dx=\int \frac {\left (5 - 2 x\right )^{\frac {5}{2}} \sqrt {3 x + 4}}{\left (x + 1\right ) \left (x + 2\right )}\, dx \] Input:

integrate((5-2*x)**(5/2)*(4+3*x)**(1/2)/(x**2+3*x+2),x)
 

Output:

Integral((5 - 2*x)**(5/2)*sqrt(3*x + 4)/((x + 1)*(x + 2)), x)
 

Maxima [F]

\[ \int \frac {(5-2 x)^{5/2} \sqrt {4+3 x}}{2+3 x+x^2} \, dx=\int { \frac {\sqrt {3 \, x + 4} {\left (-2 \, x + 5\right )}^{\frac {5}{2}}}{x^{2} + 3 \, x + 2} \,d x } \] Input:

integrate((5-2*x)^(5/2)*(4+3*x)^(1/2)/(x^2+3*x+2),x, algorithm="maxima")
 

Output:

integrate(sqrt(3*x + 4)*(-2*x + 5)^(5/2)/(x^2 + 3*x + 2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (100) = 200\).

Time = 0.52 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.26 \[ \int \frac {(5-2 x)^{5/2} \sqrt {4+3 x}}{2+3 x+x^2} \, dx=81 \, \sqrt {6} \sqrt {3} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {3} \sqrt {3 \, x + 4} {\left (\frac {{\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{2}}{3 \, x + 4} - 4\right )}}{18 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}\right )\right )} + \frac {1}{18} \, {\left (4 \, \sqrt {3} {\left (3 \, x + 4\right )} - 215 \, \sqrt {3}\right )} \sqrt {3 \, x + 4} \sqrt {-6 \, x + 15} + \frac {49}{6} \, \sqrt {42} \sqrt {6} \log \left (\frac {{\left | -2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}{{\left | 2 \, \sqrt {42} + \frac {\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}}{\sqrt {3 \, x + 4}} - \frac {4 \, \sqrt {3 \, x + 4}}{\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}} \right |}}\right ) - \frac {8951}{72} \, \sqrt {6} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {3 \, x + 4} {\left (\frac {{\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}^{2}}{3 \, x + 4} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-6 \, x + 15} - \sqrt {46}\right )}}\right )\right )} \] Input:

integrate((5-2*x)^(5/2)*(4+3*x)^(1/2)/(x^2+3*x+2),x, algorithm="giac")
 

Output:

81*sqrt(6)*sqrt(3)*(pi + 2*arctan(-1/18*sqrt(3)*sqrt(3*x + 4)*((sqrt(2)*sq 
rt(-6*x + 15) - sqrt(46))^2/(3*x + 4) - 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt 
(46)))) + 1/18*(4*sqrt(3)*(3*x + 4) - 215*sqrt(3))*sqrt(3*x + 4)*sqrt(-6*x 
 + 15) + 49/6*sqrt(42)*sqrt(6)*log(abs(-2*sqrt(42) + (sqrt(2)*sqrt(-6*x + 
15) - sqrt(46))/sqrt(3*x + 4) - 4*sqrt(3*x + 4)/(sqrt(2)*sqrt(-6*x + 15) - 
 sqrt(46)))/abs(2*sqrt(42) + (sqrt(2)*sqrt(-6*x + 15) - sqrt(46))/sqrt(3*x 
 + 4) - 4*sqrt(3*x + 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46)))) - 8951/72*s 
qrt(6)*(pi + 2*arctan(-1/4*sqrt(3*x + 4)*((sqrt(2)*sqrt(-6*x + 15) - sqrt( 
46))^2/(3*x + 4) - 4)/(sqrt(2)*sqrt(-6*x + 15) - sqrt(46))))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-2 x)^{5/2} \sqrt {4+3 x}}{2+3 x+x^2} \, dx=\int \frac {{\left (5-2\,x\right )}^{5/2}\,\sqrt {3\,x+4}}{x^2+3\,x+2} \,d x \] Input:

int(((5 - 2*x)^(5/2)*(3*x + 4)^(1/2))/(3*x + x^2 + 2),x)
 

Output:

int(((5 - 2*x)^(5/2)*(3*x + 4)^(1/2))/(3*x + x^2 + 2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.76 \[ \int \frac {(5-2 x)^{5/2} \sqrt {4+3 x}}{2+3 x+x^2} \, dx=\frac {8951 \sqrt {6}\, \mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{36}+486 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {23}}{2}-\frac {3 \sqrt {3}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )}{2}\right )-486 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {23}}{2}+\frac {3 \sqrt {3}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )}{2}\right )+2 \sqrt {3 x +4}\, \sqrt {-2 x +5}\, x -\frac {199 \sqrt {3 x +4}\, \sqrt {-2 x +5}}{6}-49 \sqrt {7}\, \mathrm {log}\left (-\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )-\sqrt {2}\right )+49 \sqrt {7}\, \mathrm {log}\left (-\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )+\sqrt {2}\right )-49 \sqrt {7}\, \mathrm {log}\left (\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )-\sqrt {2}\right )+49 \sqrt {7}\, \mathrm {log}\left (\sqrt {23}+\sqrt {21}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +5}\, \sqrt {3}}{\sqrt {23}}\right )}{2}\right )+\sqrt {2}\right ) \] Input:

int((5-2*x)^(5/2)*(4+3*x)^(1/2)/(x^2+3*x+2),x)
 

Output:

(8951*sqrt(6)*asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23)) + 17496*sqrt(2)*at 
an((sqrt(23) - 3*sqrt(3)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2)) 
/2) - 17496*sqrt(2)*atan((sqrt(23) + 3*sqrt(3)*tan(asin((sqrt( - 2*x + 5)* 
sqrt(3))/sqrt(23))/2))/2) + 72*sqrt(3*x + 4)*sqrt( - 2*x + 5)*x - 1194*sqr 
t(3*x + 4)*sqrt( - 2*x + 5) - 1764*sqrt(7)*log( - sqrt(23) + sqrt(21)*tan( 
asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2) - sqrt(2)) + 1764*sqrt(7)*log 
( - sqrt(23) + sqrt(21)*tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2) + 
 sqrt(2)) - 1764*sqrt(7)*log(sqrt(23) + sqrt(21)*tan(asin((sqrt( - 2*x + 5 
)*sqrt(3))/sqrt(23))/2) - sqrt(2)) + 1764*sqrt(7)*log(sqrt(23) + sqrt(21)* 
tan(asin((sqrt( - 2*x + 5)*sqrt(3))/sqrt(23))/2) + sqrt(2)))/36