\(\int \frac {(4+3 x)^2}{(5-2 x)^{3/2} (2+3 x+x^2)^2} \, dx\) [465]

Optimal result

Integrand size = 27, antiderivative size = 85 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{3/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {890}{1323 \sqrt {5-2 x}}-\frac {46+37 x}{63 \sqrt {5-2 x} \left (2+3 x+x^2\right )}+\frac {16}{81} \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )-\frac {62 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{49 \sqrt {7}} \] Output:

890/1323/(5-2*x)^(1/2)-1/63*(46+37*x)/(5-2*x)^(1/2)/(x^2+3*x+2)+16/81*arct 
anh(1/3*(5-2*x)^(1/2))-62/343*arctanh(1/7*(5-2*x)^(1/2)*7^(1/2))*7^(1/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{3/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {814+1893 x+890 x^2}{1323 \sqrt {5-2 x} (1+x) (2+x)}+\frac {16}{81} \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )-\frac {62 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{49 \sqrt {7}} \] Input:

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

Output:

(814 + 1893*x + 890*x^2)/(1323*Sqrt[5 - 2*x]*(1 + x)*(2 + x)) + (16*ArcTan 
h[Sqrt[5 - 2*x]/3])/81 - (62*ArcTanh[Sqrt[5 - 2*x]/Sqrt[7]])/(49*Sqrt[7])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1265, 27, 1198, 1197, 25, 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.

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

\(\Big \downarrow \) 1265

\(\displaystyle -\frac {1}{63} \int -\frac {3 (37 x+130)}{(5-2 x)^{3/2} \left (x^2+3 x+2\right )}dx-\frac {37 x+46}{63 \sqrt {5-2 x} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \int \frac {37 x+130}{(5-2 x)^{3/2} \left (x^2+3 x+2\right )}dx-\frac {37 x+46}{63 \sqrt {5-2 x} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1198

\(\displaystyle \frac {1}{21} \left (\frac {1}{63} \int \frac {445 x+1282}{\sqrt {5-2 x} \left (x^2+3 x+2\right )}dx+\frac {890}{63 \sqrt {5-2 x}}\right )-\frac {37 x+46}{63 \sqrt {5-2 x} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {1}{21} \left (\frac {2}{63} \int -\frac {4789-445 (5-2 x)}{(5-2 x)^2-16 (5-2 x)+63}d\sqrt {5-2 x}+\frac {890}{63 \sqrt {5-2 x}}\right )-\frac {37 x+46}{63 \sqrt {5-2 x} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{21} \left (\frac {890}{63 \sqrt {5-2 x}}-\frac {2}{63} \int \frac {4789-445 (5-2 x)}{(5-2 x)^2-16 (5-2 x)+63}d\sqrt {5-2 x}\right )-\frac {37 x+46}{63 \sqrt {5-2 x} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {1}{21} \left (\frac {2}{63} \left (837 \int \frac {1}{-2 x-2}d\sqrt {5-2 x}-392 \int \frac {1}{-2 x-4}d\sqrt {5-2 x}\right )+\frac {890}{63 \sqrt {5-2 x}}\right )-\frac {37 x+46}{63 \sqrt {5-2 x} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{21} \left (\frac {2}{63} \left (\frac {392}{3} \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )-\frac {837 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{\sqrt {7}}\right )+\frac {890}{63 \sqrt {5-2 x}}\right )-\frac {37 x+46}{63 \sqrt {5-2 x} \left (x^2+3 x+2\right )}\)

Input:

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

Output:

-1/63*(46 + 37*x)/(Sqrt[5 - 2*x]*(2 + 3*x + x^2)) + (890/(63*Sqrt[5 - 2*x] 
) + (2*((392*ArcTanh[Sqrt[5 - 2*x]/3])/3 - (837*ArcTanh[Sqrt[5 - 2*x]/Sqrt 
[7]])/Sqrt[7]))/63)/21
 

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 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 1198

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

rule 1265

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x) 
^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(f + g*x)^n, a + b*x 
 + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(f + g*x)^n, a + b*x + c 
*x^2, x], x, 1]}, Simp[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)*((R*(b*c 
*d - b^2*e + 2*a*c*e) - a*S*(2*c*d - b*e) + c*(R*(2*c*d - b*e) - S*(b*d - 2 
*a*e))*x)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p 
 + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m*(a + b*x + c 
*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Q + 
 R*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a 
*c*e^2*(m + 2*p + 3)) - S*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d - b*e + 
 2*c*d*p - b*e*p)) + c*e*(S*(b*d - 2*a*e) - R*(2*c*d - b*e))*(m + 2*p + 4)* 
x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && IGtQ[n, 1] && LtQ[p 
, -1] && NeQ[c*d^2 - b*d*e + a*e^2, 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]
 

Maple [A] (verified)

Time = 1.97 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86

method	result	size
risch	\(\frac {890 x^{2}+1893 x +814}{1323 \left (x^{2}+3 x +2\right ) \sqrt {5-2 x}}+\frac {8 \ln \left (\sqrt {5-2 x}+3\right )}{81}-\frac {62 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{343}-\frac {8 \ln \left (\sqrt {5-2 x}-3\right )}{81}\)	\(73\)
derivativedivides	\(\frac {2 \sqrt {5-2 x}}{49 \left (-2 x -2\right )}-\frac {62 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{343}+\frac {2116}{3969 \sqrt {5-2 x}}+\frac {4}{81 \left (\sqrt {5-2 x}-3\right )}-\frac {8 \ln \left (\sqrt {5-2 x}-3\right )}{81}+\frac {4}{81 \left (\sqrt {5-2 x}+3\right )}+\frac {8 \ln \left (\sqrt {5-2 x}+3\right )}{81}\)	\(95\)
default	\(\frac {2 \sqrt {5-2 x}}{49 \left (-2 x -2\right )}-\frac {62 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{343}+\frac {2116}{3969 \sqrt {5-2 x}}+\frac {4}{81 \left (\sqrt {5-2 x}-3\right )}-\frac {8 \ln \left (\sqrt {5-2 x}-3\right )}{81}+\frac {4}{81 \left (\sqrt {5-2 x}+3\right )}+\frac {8 \ln \left (\sqrt {5-2 x}+3\right )}{81}\)	\(95\)
trager	\(-\frac {\left (890 x^{2}+1893 x +814\right ) \sqrt {5-2 x}}{1323 \left (2 x^{3}+x^{2}-11 x -10\right )}-\frac {8 \ln \left (\frac {-7+x +3 \sqrt {5-2 x}}{2+x}\right )}{81}-\frac {31 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x +7 \sqrt {5-2 x}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )}{x +1}\right )}{343}\)	\(99\)
pseudoelliptic	\(-\frac {62 \left (\sqrt {7}\, \sqrt {5-2 x}\, \left (2+x \right ) \left (x +1\right ) \operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right )+\frac {1372 \left (2+x \right ) \left (x +1\right ) \sqrt {5-2 x}\, \ln \left (\sqrt {5-2 x}-3\right )}{2511}-\frac {1372 \left (2+x \right ) \left (x +1\right ) \sqrt {5-2 x}\, \ln \left (\sqrt {5-2 x}+3\right )}{2511}-\frac {3115 x^{2}}{837}-\frac {4417 x}{558}-\frac {2849}{837}\right )}{343 \sqrt {5-2 x}\, \left (2+x \right ) \left (x +1\right )}\)	\(110\)

Input:

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

Output:

1/1323*(890*x^2+1893*x+814)/(x^2+3*x+2)/(5-2*x)^(1/2)+8/81*ln((5-2*x)^(1/2 
)+3)-62/343*arctanh(1/7*(5-2*x)^(1/2)*7^(1/2))*7^(1/2)-8/81*ln((5-2*x)^(1/ 
2)-3)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.48 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{3/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {2511 \, \sqrt {7} {\left (2 \, x^{3} + x^{2} - 11 \, x - 10\right )} \log \left (\frac {x + \sqrt {7} \sqrt {-2 \, x + 5} - 6}{x + 1}\right ) + 2744 \, {\left (2 \, x^{3} + x^{2} - 11 \, x - 10\right )} \log \left (\sqrt {-2 \, x + 5} + 3\right ) - 2744 \, {\left (2 \, x^{3} + x^{2} - 11 \, x - 10\right )} \log \left (\sqrt {-2 \, x + 5} - 3\right ) - 21 \, {\left (890 \, x^{2} + 1893 \, x + 814\right )} \sqrt {-2 \, x + 5}}{27783 \, {\left (2 \, x^{3} + x^{2} - 11 \, x - 10\right )}} \] Input:

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

Output:

1/27783*(2511*sqrt(7)*(2*x^3 + x^2 - 11*x - 10)*log((x + sqrt(7)*sqrt(-2*x 
 + 5) - 6)/(x + 1)) + 2744*(2*x^3 + x^2 - 11*x - 10)*log(sqrt(-2*x + 5) + 
3) - 2744*(2*x^3 + x^2 - 11*x - 10)*log(sqrt(-2*x + 5) - 3) - 21*(890*x^2 
+ 1893*x + 814)*sqrt(-2*x + 5))/(2*x^3 + x^2 - 11*x - 10)
 

Sympy [A] (verification not implemented)

Time = 62.41 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.64 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{3/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {30 \sqrt {7} \left (\log {\left (\sqrt {5 - 2 x} - \sqrt {7} \right )} - \log {\left (\sqrt {5 - 2 x} + \sqrt {7} \right )}\right )}{343} - \frac {4 \left (\begin {cases} \frac {\sqrt {7} \left (- \frac {\log {\left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} - 1\right )}\right )}{49} & \text {for}\: \sqrt {5 - 2 x} > - \sqrt {7} \wedge \sqrt {5 - 2 x} < \sqrt {7} \end {cases}\right )}{7} - \frac {8 \log {\left (\sqrt {5 - 2 x} - 3 \right )}}{81} + \frac {8 \log {\left (\sqrt {5 - 2 x} + 3 \right )}}{81} + \frac {4}{81 \left (\sqrt {5 - 2 x} + 3\right )} + \frac {4}{81 \left (\sqrt {5 - 2 x} - 3\right )} + \frac {2116}{3969 \sqrt {5 - 2 x}} \] Input:

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

Output:

30*sqrt(7)*(log(sqrt(5 - 2*x) - sqrt(7)) - log(sqrt(5 - 2*x) + sqrt(7)))/3 
43 - 4*Piecewise((sqrt(7)*(-log(sqrt(7)*sqrt(5 - 2*x)/7 - 1)/4 + log(sqrt( 
7)*sqrt(5 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(7)*sqrt(5 - 2*x)/7 + 1)) - 1/(4*(sq 
rt(7)*sqrt(5 - 2*x)/7 - 1)))/49, (sqrt(5 - 2*x) < sqrt(7)) & (sqrt(5 - 2*x 
) > -sqrt(7))))/7 - 8*log(sqrt(5 - 2*x) - 3)/81 + 8*log(sqrt(5 - 2*x) + 3) 
/81 + 4/(81*(sqrt(5 - 2*x) + 3)) + 4/(81*(sqrt(5 - 2*x) - 3)) + 2116/(3969 
*sqrt(5 - 2*x))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.21 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{3/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {31}{343} \, \sqrt {7} \log \left (-\frac {\sqrt {7} - \sqrt {-2 \, x + 5}}{\sqrt {7} + \sqrt {-2 \, x + 5}}\right ) + \frac {2 \, {\left (445 \, {\left (2 \, x - 5\right )}^{2} + 12686 \, x - 9497\right )}}{1323 \, {\left ({\left (-2 \, x + 5\right )}^{\frac {5}{2}} - 16 \, {\left (-2 \, x + 5\right )}^{\frac {3}{2}} + 63 \, \sqrt {-2 \, x + 5}\right )}} + \frac {8}{81} \, \log \left (\sqrt {-2 \, x + 5} + 3\right ) - \frac {8}{81} \, \log \left (\sqrt {-2 \, x + 5} - 3\right ) \] Input:

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

Output:

31/343*sqrt(7)*log(-(sqrt(7) - sqrt(-2*x + 5))/(sqrt(7) + sqrt(-2*x + 5))) 
 + 2/1323*(445*(2*x - 5)^2 + 12686*x - 9497)/((-2*x + 5)^(5/2) - 16*(-2*x 
+ 5)^(3/2) + 63*sqrt(-2*x + 5)) + 8/81*log(sqrt(-2*x + 5) + 3) - 8/81*log( 
sqrt(-2*x + 5) - 3)
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.35 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{3/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {31}{343} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + 2 \, \sqrt {-2 \, x + 5} \right |}}{2 \, {\left (\sqrt {7} + \sqrt {-2 \, x + 5}\right )}}\right ) + \frac {2 \, {\left (445 \, {\left (2 \, x - 5\right )}^{2} + 12686 \, x - 9497\right )}}{1323 \, {\left ({\left (2 \, x - 5\right )}^{2} \sqrt {-2 \, x + 5} - 16 \, {\left (-2 \, x + 5\right )}^{\frac {3}{2}} + 63 \, \sqrt {-2 \, x + 5}\right )}} + \frac {8}{81} \, \log \left (\sqrt {-2 \, x + 5} + 3\right ) - \frac {8}{81} \, \log \left ({\left | \sqrt {-2 \, x + 5} - 3 \right |}\right ) \] Input:

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

Output:

31/343*sqrt(7)*log(1/2*abs(-2*sqrt(7) + 2*sqrt(-2*x + 5))/(sqrt(7) + sqrt( 
-2*x + 5))) + 2/1323*(445*(2*x - 5)^2 + 12686*x - 9497)/((2*x - 5)^2*sqrt( 
-2*x + 5) - 16*(-2*x + 5)^(3/2) + 63*sqrt(-2*x + 5)) + 8/81*log(sqrt(-2*x 
+ 5) + 3) - 8/81*log(abs(sqrt(-2*x + 5) - 3))
 

Mupad [B] (verification not implemented)

Time = 13.87 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.87 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{3/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {16\,\mathrm {atanh}\left (\frac {\sqrt {5-2\,x}}{3}\right )}{81}-\frac {62\,\sqrt {7}\,\mathrm {atanh}\left (\frac {\sqrt {7}\,\sqrt {5-2\,x}}{7}\right )}{343}+\frac {\frac {25372\,x}{1323}+\frac {890\,{\left (2\,x-5\right )}^2}{1323}-\frac {18994}{1323}}{63\,\sqrt {5-2\,x}-16\,{\left (5-2\,x\right )}^{3/2}+{\left (5-2\,x\right )}^{5/2}} \] Input:

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

Output:

(16*atanh((5 - 2*x)^(1/2)/3))/81 - (62*7^(1/2)*atanh((7^(1/2)*(5 - 2*x)^(1 
/2))/7))/343 + ((25372*x)/1323 + (890*(2*x - 5)^2)/1323 - 18994/1323)/(63* 
(5 - 2*x)^(1/2) - 16*(5 - 2*x)^(3/2) + (5 - 2*x)^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.22 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{3/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {2511 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right ) x^{2}+7533 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right ) x +5022 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right )-2511 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right ) x^{2}-7533 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right ) x -5022 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right )-2744 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right ) x^{2}-8232 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right ) x -5488 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right )+2744 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right ) x^{2}+8232 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right ) x +5488 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right )+18690 x^{2}+39753 x +17094}{27783 \sqrt {-2 x +5}\, \left (x^{2}+3 x +2\right )} \] Input:

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

Output:

(2511*sqrt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x**2 + 7533 
*sqrt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x + 5022*sqrt( - 
 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7)) - 2511*sqrt( - 2*x + 5)* 
sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x**2 - 7533*sqrt( - 2*x + 5)*sqrt( 
7)*log(sqrt( - 2*x + 5) + sqrt(7))*x - 5022*sqrt( - 2*x + 5)*sqrt(7)*log(s 
qrt( - 2*x + 5) + sqrt(7)) - 2744*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) - 
3)*x**2 - 8232*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) - 3)*x - 5488*sqrt( - 
 2*x + 5)*log(sqrt( - 2*x + 5) - 3) + 2744*sqrt( - 2*x + 5)*log(sqrt( - 2* 
x + 5) + 3)*x**2 + 8232*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) + 3)*x + 548 
8*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) + 3) + 18690*x**2 + 39753*x + 1709 
4)/(27783*sqrt( - 2*x + 5)*(x**2 + 3*x + 2))