3.26.58 \(\int \frac {(5-x) (3+2 x)^{7/2}}{(2+5 x+3 x^2)^2} \, dx\) [2558]

3.26.58.1 Optimal result

Integrand size = 27, antiderivative size = 98 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=\frac {1358}{27} \sqrt {3+2 x}+\frac {826}{27} (3+2 x)^{3/2}-\frac {(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-154 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {2800}{27} \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]

output

826/27*(3+2*x)^(3/2)-1/3*(3+2*x)^(5/2)*(121+139*x)/(3*x^2+5*x+2)-154*arcta 
nh((3+2*x)^(1/2))+2800/81*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)+135 
8/27*(3+2*x)^(1/2)
 

3.26.58.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.83 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=-154 \text {arctanh}\left (\sqrt {3+2 x}\right )+\frac {1}{81} \left (-\frac {3 \sqrt {3+2 x} \left (2129+1843 x-400 x^2+48 x^3\right )}{2+5 x+3 x^2}+2800 \sqrt {15} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\right ) \]

input

Integrate[((5 - x)*(3 + 2*x)^(7/2))/(2 + 5*x + 3*x^2)^2,x]
 

output

-154*ArcTanh[Sqrt[3 + 2*x]] + ((-3*Sqrt[3 + 2*x]*(2129 + 1843*x - 400*x^2 
+ 48*x^3))/(2 + 5*x + 3*x^2) + 2800*Sqrt[15]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2* 
x]])/81
 

3.26.58.3 Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1233, 27, 1196, 25, 1196, 1197, 1480, 220}

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.

\(\displaystyle \int \frac {(5-x) (2 x+3)^{7/2}}{\left (3 x^2+5 x+2\right )^2} \, dx\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {1}{3} \int \frac {7 (2 x+3)^{3/2} (59 x+26)}{3 x^2+5 x+2}dx-\frac {(2 x+3)^{5/2} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{3} \int \frac {(2 x+3)^{3/2} (59 x+26)}{3 x^2+5 x+2}dx-\frac {(2 x+3)^{5/2} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}\)

\(\Big \downarrow \) 1196

\(\displaystyle \frac {7}{3} \left (\frac {1}{3} \int -\frac {(2-97 x) \sqrt {2 x+3}}{3 x^2+5 x+2}dx+\frac {118}{9} (2 x+3)^{3/2}\right )-\frac {(2 x+3)^{5/2} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {7}{3} \left (\frac {118}{9} (2 x+3)^{3/2}-\frac {1}{3} \int \frac {(2-97 x) \sqrt {2 x+3}}{3 x^2+5 x+2}dx\right )-\frac {(2 x+3)^{5/2} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}\)

\(\Big \downarrow \) 1196

\(\displaystyle \frac {7}{3} \left (\frac {1}{3} \left (\frac {194}{3} \sqrt {2 x+3}-\frac {1}{3} \int \frac {109 x+406}{\sqrt {2 x+3} \left (3 x^2+5 x+2\right )}dx\right )+\frac {118}{9} (2 x+3)^{3/2}\right )-\frac {(2 x+3)^{5/2} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {7}{3} \left (\frac {1}{3} \left (\frac {194}{3} \sqrt {2 x+3}-\frac {2}{3} \int \frac {109 (2 x+3)+485}{3 (2 x+3)^2-8 (2 x+3)+5}d\sqrt {2 x+3}\right )+\frac {118}{9} (2 x+3)^{3/2}\right )-\frac {(2 x+3)^{5/2} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {7}{3} \left (\frac {1}{3} \left (\frac {194}{3} \sqrt {2 x+3}-\frac {2}{3} \left (1000 \int \frac {1}{3 (2 x+3)-5}d\sqrt {2 x+3}-891 \int \frac {1}{3 (2 x+3)-3}d\sqrt {2 x+3}\right )\right )+\frac {118}{9} (2 x+3)^{3/2}\right )-\frac {(2 x+3)^{5/2} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {7}{3} \left (\frac {1}{3} \left (\frac {194}{3} \sqrt {2 x+3}-\frac {2}{3} \left (297 \text {arctanh}\left (\sqrt {2 x+3}\right )-200 \sqrt {\frac {5}{3}} \text {arctanh}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )\right )\right )+\frac {118}{9} (2 x+3)^{3/2}\right )-\frac {(2 x+3)^{5/2} (139 x+121)}{3 \left (3 x^2+5 x+2\right )}\)

input

Int[((5 - x)*(3 + 2*x)^(7/2))/(2 + 5*x + 3*x^2)^2,x]
 

output

-1/3*((3 + 2*x)^(5/2)*(121 + 139*x))/(2 + 5*x + 3*x^2) + (7*((118*(3 + 2*x 
)^(3/2))/9 + ((194*Sqrt[3 + 2*x])/3 - (2*(297*ArcTanh[Sqrt[3 + 2*x]] - 200 
*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]]))/3)/3))/3
 

3.26.58.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 220

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

rule 1196

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

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

3.26.58.4 Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82

method	result	size
risch	\(-\frac {\left (48 x^{3}-400 x^{2}+1843 x +2129\right ) \sqrt {3+2 x}}{27 \left (3 x^{2}+5 x +2\right )}-77 \ln \left (\sqrt {3+2 x}+1\right )+\frac {2800 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{81}+77 \ln \left (\sqrt {3+2 x}-1\right )\)	\(80\)
trager	\(-\frac {\left (48 x^{3}-400 x^{2}+1843 x +2129\right ) \sqrt {3+2 x}}{27 \left (3 x^{2}+5 x +2\right )}+77 \ln \left (\frac {-2-x +\sqrt {3+2 x}}{1+x}\right )+\frac {1400 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right ) x +15 \sqrt {3+2 x}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-15\right )}{2+3 x}\right )}{81}\)	\(103\)
pseudoelliptic	\(\frac {8400 \sqrt {15}\, \left (x +\frac {2}{3}\right ) \left (1+x \right ) \operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right )+\left (18711 x^{2}+31185 x +12474\right ) \ln \left (\sqrt {3+2 x}-1\right )+\left (-18711 x^{2}-31185 x -12474\right ) \ln \left (\sqrt {3+2 x}+1\right )-144 \left (x^{3}-\frac {25}{3} x^{2}+\frac {1843}{48} x +\frac {2129}{48}\right ) \sqrt {3+2 x}}{243 x^{2}+405 x +162}\)	\(103\)
derivativedivides	\(-\frac {8 \left (3+2 x \right )^{\frac {3}{2}}}{27}+\frac {184 \sqrt {3+2 x}}{27}-\frac {6}{\sqrt {3+2 x}-1}+77 \ln \left (\sqrt {3+2 x}-1\right )-\frac {4250 \sqrt {3+2 x}}{81 \left (\frac {4}{3}+2 x \right )}+\frac {2800 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{81}-\frac {6}{\sqrt {3+2 x}+1}-77 \ln \left (\sqrt {3+2 x}+1\right )\)	\(104\)
default	\(-\frac {8 \left (3+2 x \right )^{\frac {3}{2}}}{27}+\frac {184 \sqrt {3+2 x}}{27}-\frac {6}{\sqrt {3+2 x}-1}+77 \ln \left (\sqrt {3+2 x}-1\right )-\frac {4250 \sqrt {3+2 x}}{81 \left (\frac {4}{3}+2 x \right )}+\frac {2800 \,\operatorname {arctanh}\left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{81}-\frac {6}{\sqrt {3+2 x}+1}-77 \ln \left (\sqrt {3+2 x}+1\right )\)	\(104\)

input

int((5-x)*(3+2*x)^(7/2)/(3*x^2+5*x+2)^2,x,method=_RETURNVERBOSE)
 

output

-1/27*(48*x^3-400*x^2+1843*x+2129)/(3*x^2+5*x+2)*(3+2*x)^(1/2)-77*ln((3+2* 
x)^(1/2)+1)+2800/81*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)+77*ln((3+ 
2*x)^(1/2)-1)
 

3.26.58.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.32 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=\frac {1400 \, \sqrt {5} \sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 6237 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 6237 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 3 \, {\left (48 \, x^{3} - 400 \, x^{2} + 1843 \, x + 2129\right )} \sqrt {2 \, x + 3}}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

input

integrate((5-x)*(3+2*x)^(7/2)/(3*x^2+5*x+2)^2,x, algorithm="fricas")
 

output

1/81*(1400*sqrt(5)*sqrt(3)*(3*x^2 + 5*x + 2)*log((sqrt(5)*sqrt(3)*sqrt(2*x 
 + 3) + 3*x + 7)/(3*x + 2)) - 6237*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) + 1 
) + 6237*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) - 1) - 3*(48*x^3 - 400*x^2 + 
1843*x + 2129)*sqrt(2*x + 3))/(3*x^2 + 5*x + 2)
 

3.26.58.6 Sympy [A] (verification not implemented)

Time = 41.59 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.41 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=- \frac {8 \left (2 x + 3\right )^{\frac {3}{2}}}{27} + \frac {184 \sqrt {2 x + 3}}{27} - \frac {325 \sqrt {15} \left (\log {\left (\sqrt {2 x + 3} - \frac {\sqrt {15}}{3} \right )} - \log {\left (\sqrt {2 x + 3} + \frac {\sqrt {15}}{3} \right )}\right )}{27} + \frac {42500 \left (\begin {cases} \frac {\sqrt {15} \left (- \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {15} \sqrt {2 x + 3}}{5} - 1\right )}\right )}{75} & \text {for}\: \sqrt {2 x + 3} > - \frac {\sqrt {15}}{3} \wedge \sqrt {2 x + 3} < \frac {\sqrt {15}}{3} \end {cases}\right )}{27} + 77 \log {\left (\sqrt {2 x + 3} - 1 \right )} - 77 \log {\left (\sqrt {2 x + 3} + 1 \right )} - \frac {6}{\sqrt {2 x + 3} + 1} - \frac {6}{\sqrt {2 x + 3} - 1} \]

input

integrate((5-x)*(3+2*x)**(7/2)/(3*x**2+5*x+2)**2,x)
 

output

-8*(2*x + 3)**(3/2)/27 + 184*sqrt(2*x + 3)/27 - 325*sqrt(15)*(log(sqrt(2*x 
 + 3) - sqrt(15)/3) - log(sqrt(2*x + 3) + sqrt(15)/3))/27 + 42500*Piecewis 
e((sqrt(15)*(-log(sqrt(15)*sqrt(2*x + 3)/5 - 1)/4 + log(sqrt(15)*sqrt(2*x 
+ 3)/5 + 1)/4 - 1/(4*(sqrt(15)*sqrt(2*x + 3)/5 + 1)) - 1/(4*(sqrt(15)*sqrt 
(2*x + 3)/5 - 1)))/75, (sqrt(2*x + 3) > -sqrt(15)/3) & (sqrt(2*x + 3) < sq 
rt(15)/3)))/27 + 77*log(sqrt(2*x + 3) - 1) - 77*log(sqrt(2*x + 3) + 1) - 6 
/(sqrt(2*x + 3) + 1) - 6/(sqrt(2*x + 3) - 1)
 

3.26.58.7 Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {8}{27} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1400}{81} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {184}{27} \, \sqrt {2 \, x + 3} - \frac {2 \, {\left (2611 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 2935 \, \sqrt {2 \, x + 3}\right )}}{27 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 77 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 77 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \]

input

integrate((5-x)*(3+2*x)^(7/2)/(3*x^2+5*x+2)^2,x, algorithm="maxima")
 

output

-8/27*(2*x + 3)^(3/2) - 1400/81*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3)) 
/(sqrt(15) + 3*sqrt(2*x + 3))) + 184/27*sqrt(2*x + 3) - 2/27*(2611*(2*x + 
3)^(3/2) - 2935*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 77*log(sqrt(2 
*x + 3) + 1) + 77*log(sqrt(2*x + 3) - 1)
 

3.26.58.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.22 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=-\frac {8}{27} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1400}{81} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) + \frac {184}{27} \, \sqrt {2 \, x + 3} - \frac {2 \, {\left (2611 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 2935 \, \sqrt {2 \, x + 3}\right )}}{27 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 77 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 77 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \]

input

integrate((5-x)*(3+2*x)^(7/2)/(3*x^2+5*x+2)^2,x, algorithm="giac")
 

output

-8/27*(2*x + 3)^(3/2) - 1400/81*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt( 
2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) + 184/27*sqrt(2*x + 3) - 2/27*(261 
1*(2*x + 3)^(3/2) - 2935*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 77*l 
og(sqrt(2*x + 3) + 1) + 77*log(abs(sqrt(2*x + 3) - 1))
 

3.26.58.9 Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92 \[ \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx=\frac {184\,\sqrt {2\,x+3}}{27}-\frac {\frac {5870\,\sqrt {2\,x+3}}{81}-\frac {5222\,{\left (2\,x+3\right )}^{3/2}}{81}}{\frac {16\,x}{3}-{\left (2\,x+3\right )}^2+\frac {19}{3}}-\frac {8\,{\left (2\,x+3\right )}^{3/2}}{27}+\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,154{}\mathrm {i}-\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,2800{}\mathrm {i}}{81} \]

input

int(-((2*x + 3)^(7/2)*(x - 5))/(5*x + 3*x^2 + 2)^2,x)
 

output

atan((2*x + 3)^(1/2)*1i)*154i - ((5870*(2*x + 3)^(1/2))/81 - (5222*(2*x + 
3)^(3/2))/81)/((16*x)/3 - (2*x + 3)^2 + 19/3) - (15^(1/2)*atan((15^(1/2)*( 
2*x + 3)^(1/2)*1i)/5)*2800i)/81 + (184*(2*x + 3)^(1/2))/27 - (8*(2*x + 3)^ 
(3/2))/27