\(\int \frac {(4+3 x)^2}{(5-2 x)^{5/2} (2+3 x+x^2)^2} \, dx\) [477]

Optimal result

Integrand size = 27, antiderivative size = 98 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {3778}{11907 (5-2 x)^{3/2}}+\frac {37138}{250047 \sqrt {5-2 x}}-\frac {46+37 x}{63 (5-2 x)^{3/2} \left (2+3 x+x^2\right )}+\frac {32 \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )}{2187}-\frac {66 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{343 \sqrt {7}} \] Output:

3778/11907/(5-2*x)^(3/2)+37138/250047/(5-2*x)^(1/2)-1/63*(46+37*x)/(5-2*x) 
^(3/2)/(x^2+3*x+2)+32/2187*arctanh(1/3*(5-2*x)^(1/2))-66/2401*arctanh(1/7* 
(5-2*x)^(1/2)*7^(1/2))*7^(1/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {347482+499679 x+42200 x^2-74276 x^3}{250047 (5-2 x)^{3/2} \left (2+3 x+x^2\right )}+\frac {32 \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )}{2187}-\frac {66 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{343 \sqrt {7}} \] Input:

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

Output:

(347482 + 499679*x + 42200*x^2 - 74276*x^3)/(250047*(5 - 2*x)^(3/2)*(2 + 3 
*x + x^2)) + (32*ArcTanh[Sqrt[5 - 2*x]/3])/2187 - (66*ArcTanh[Sqrt[5 - 2*x 
]/Sqrt[7]])/(343*Sqrt[7])
 

Rubi [A] (verified)

Time = 0.48 (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 = {1265, 25, 1198, 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.

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

\(\Big \downarrow \) 1265

\(\displaystyle -\frac {1}{63} \int -\frac {185 x+482}{(5-2 x)^{5/2} \left (x^2+3 x+2\right )}dx-\frac {37 x+46}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{63} \int \frac {185 x+482}{(5-2 x)^{5/2} \left (x^2+3 x+2\right )}dx-\frac {37 x+46}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1198

\(\displaystyle \frac {1}{63} \left (\frac {1}{63} \int \frac {1889 x+4562}{(5-2 x)^{3/2} \left (x^2+3 x+2\right )}dx+\frac {3778}{189 (5-2 x)^{3/2}}\right )-\frac {37 x+46}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1198

\(\displaystyle \frac {1}{63} \left (\frac {1}{63} \left (\frac {1}{63} \int \frac {18569 x+42626}{\sqrt {5-2 x} \left (x^2+3 x+2\right )}dx+\frac {37138}{63 \sqrt {5-2 x}}\right )+\frac {3778}{189 (5-2 x)^{3/2}}\right )-\frac {37 x+46}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {1}{63} \left (\frac {1}{63} \left (\frac {2}{63} \int -\frac {178097-18569 (5-2 x)}{(5-2 x)^2-16 (5-2 x)+63}d\sqrt {5-2 x}+\frac {37138}{63 \sqrt {5-2 x}}\right )+\frac {3778}{189 (5-2 x)^{3/2}}\right )-\frac {37 x+46}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{63} \left (\frac {1}{63} \left (\frac {37138}{63 \sqrt {5-2 x}}-\frac {2}{63} \int \frac {178097-18569 (5-2 x)}{(5-2 x)^2-16 (5-2 x)+63}d\sqrt {5-2 x}\right )+\frac {3778}{189 (5-2 x)^{3/2}}\right )-\frac {37 x+46}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {1}{63} \left (\frac {1}{63} \left (\frac {2}{63} \left (24057 \int \frac {1}{-2 x-2}d\sqrt {5-2 x}-5488 \int \frac {1}{-2 x-4}d\sqrt {5-2 x}\right )+\frac {37138}{63 \sqrt {5-2 x}}\right )+\frac {3778}{189 (5-2 x)^{3/2}}\right )-\frac {37 x+46}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{63} \left (\frac {1}{63} \left (\frac {2}{63} \left (\frac {5488}{3} \text {arctanh}\left (\frac {1}{3} \sqrt {5-2 x}\right )-\frac {24057 \text {arctanh}\left (\frac {\sqrt {5-2 x}}{\sqrt {7}}\right )}{\sqrt {7}}\right )+\frac {37138}{63 \sqrt {5-2 x}}\right )+\frac {3778}{189 (5-2 x)^{3/2}}\right )-\frac {37 x+46}{63 (5-2 x)^{3/2} \left (x^2+3 x+2\right )}\)

Input:

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

Output:

-1/63*(46 + 37*x)/((5 - 2*x)^(3/2)*(2 + 3*x + x^2)) + (3778/(189*(5 - 2*x) 
^(3/2)) + (37138/(63*Sqrt[5 - 2*x]) + (2*((5488*ArcTanh[Sqrt[5 - 2*x]/3])/ 
3 - (24057*ArcTanh[Sqrt[5 - 2*x]/Sqrt[7]])/Sqrt[7]))/63)/63)/63
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 = 1.98 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87

method	result	size
risch	\(\frac {74276 x^{3}-42200 x^{2}-499679 x -347482}{250047 \left (x^{2}+3 x +2\right ) \sqrt {5-2 x}\, \left (-5+2 x \right )}+\frac {16 \ln \left (\sqrt {5-2 x}+3\right )}{2187}-\frac {66 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{2401}-\frac {16 \ln \left (\sqrt {5-2 x}-3\right )}{2187}\)	\(85\)
derivativedivides	\(\frac {4}{729 \left (\sqrt {5-2 x}+3\right )}+\frac {16 \ln \left (\sqrt {5-2 x}+3\right )}{2187}+\frac {2116}{11907 \left (5-2 x \right )^{\frac {3}{2}}}+\frac {32936}{250047 \sqrt {5-2 x}}+\frac {2 \sqrt {5-2 x}}{343 \left (-2 x -2\right )}-\frac {66 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{2401}+\frac {4}{729 \left (\sqrt {5-2 x}-3\right )}-\frac {16 \ln \left (\sqrt {5-2 x}-3\right )}{2187}\)	\(104\)
default	\(\frac {4}{729 \left (\sqrt {5-2 x}+3\right )}+\frac {16 \ln \left (\sqrt {5-2 x}+3\right )}{2187}+\frac {2116}{11907 \left (5-2 x \right )^{\frac {3}{2}}}+\frac {32936}{250047 \sqrt {5-2 x}}+\frac {2 \sqrt {5-2 x}}{343 \left (-2 x -2\right )}-\frac {66 \,\operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right ) \sqrt {7}}{2401}+\frac {4}{729 \left (\sqrt {5-2 x}-3\right )}-\frac {16 \ln \left (\sqrt {5-2 x}-3\right )}{2187}\)	\(104\)
trager	\(-\frac {\left (74276 x^{3}-42200 x^{2}-499679 x -347482\right ) \sqrt {5-2 x}}{250047 \left (-5+2 x \right )^{2} \left (x^{2}+3 x +2\right )}+\frac {33 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x -6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )+7 \sqrt {5-2 x}}{x +1}\right )}{2401}+\frac {16 \ln \left (\frac {3 \sqrt {5-2 x}+7-x}{2+x}\right )}{2187}\)	\(107\)
pseudoelliptic	\(\frac {\frac {132 \sqrt {5-2 x}\, \left (2+x \right ) \sqrt {7}\, \left (x +1\right ) \left (x -\frac {5}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {5-2 x}\, \sqrt {7}}{7}\right )}{2401}+\frac {32 \sqrt {5-2 x}\, \left (2+x \right ) \left (x +1\right ) \left (x -\frac {5}{2}\right ) \ln \left (\sqrt {5-2 x}-3\right )}{2187}-\frac {32 \sqrt {5-2 x}\, \left (2+x \right ) \left (x +1\right ) \left (x -\frac {5}{2}\right ) \ln \left (\sqrt {5-2 x}+3\right )}{2187}-\frac {74276 x^{3}}{250047}+\frac {42200 x^{2}}{250047}+\frac {499679 x}{250047}+\frac {347482}{250047}}{\left (5-2 x \right )^{\frac {3}{2}} \left (x +1\right ) \left (2+x \right )}\)	\(124\)

Input:

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

Output:

1/250047*(74276*x^3-42200*x^2-499679*x-347482)/(x^2+3*x+2)/(5-2*x)^(1/2)/( 
-5+2*x)+16/2187*ln((5-2*x)^(1/2)+3)-66/2401*arctanh(1/7*(5-2*x)^(1/2)*7^(1 
/2))*7^(1/2)-16/2187*ln((5-2*x)^(1/2)-3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (73) = 146\).

Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.62 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {72171 \, \sqrt {7} {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )} \log \left (\frac {x + \sqrt {7} \sqrt {-2 \, x + 5} - 6}{x + 1}\right ) + 38416 \, {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )} \log \left (\sqrt {-2 \, x + 5} + 3\right ) - 38416 \, {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )} \log \left (\sqrt {-2 \, x + 5} - 3\right ) - 21 \, {\left (74276 \, x^{3} - 42200 \, x^{2} - 499679 \, x - 347482\right )} \sqrt {-2 \, x + 5}}{5250987 \, {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )}} \] Input:

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

Output:

1/5250987*(72171*sqrt(7)*(4*x^4 - 8*x^3 - 27*x^2 + 35*x + 50)*log((x + sqr 
t(7)*sqrt(-2*x + 5) - 6)/(x + 1)) + 38416*(4*x^4 - 8*x^3 - 27*x^2 + 35*x + 
 50)*log(sqrt(-2*x + 5) + 3) - 38416*(4*x^4 - 8*x^3 - 27*x^2 + 35*x + 50)* 
log(sqrt(-2*x + 5) - 3) - 21*(74276*x^3 - 42200*x^2 - 499679*x - 347482)*s 
qrt(-2*x + 5))/(4*x^4 - 8*x^3 - 27*x^2 + 35*x + 50)
 

Sympy [A] (verification not implemented)

Time = 60.58 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.41 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {32 \sqrt {7} \left (\log {\left (\sqrt {5 - 2 x} - \sqrt {7} \right )} - \log {\left (\sqrt {5 - 2 x} + \sqrt {7} \right )}\right )}{2401} - \frac {4 \left (\begin {cases} \frac {\sqrt {7} \left (- \frac {\log {\left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {7} \sqrt {5 - 2 x}}{7} - 1\right )}\right )}{49} & \text {for}\: \sqrt {5 - 2 x} > - \sqrt {7} \wedge \sqrt {5 - 2 x} < \sqrt {7} \end {cases}\right )}{49} - \frac {16 \log {\left (\sqrt {5 - 2 x} - 3 \right )}}{2187} + \frac {16 \log {\left (\sqrt {5 - 2 x} + 3 \right )}}{2187} + \frac {4}{729 \left (\sqrt {5 - 2 x} + 3\right )} + \frac {4}{729 \left (\sqrt {5 - 2 x} - 3\right )} + \frac {32936}{250047 \sqrt {5 - 2 x}} + \frac {2116}{11907 \left (5 - 2 x\right )^{\frac {3}{2}}} \] Input:

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

Output:

32*sqrt(7)*(log(sqrt(5 - 2*x) - sqrt(7)) - log(sqrt(5 - 2*x) + sqrt(7)))/2 
401 - 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*(s 
qrt(7)*sqrt(5 - 2*x)/7 - 1)))/49, (sqrt(5 - 2*x) < sqrt(7)) & (sqrt(5 - 2* 
x) > -sqrt(7))))/49 - 16*log(sqrt(5 - 2*x) - 3)/2187 + 16*log(sqrt(5 - 2*x 
) + 3)/2187 + 4/(729*(sqrt(5 - 2*x) + 3)) + 4/(729*(sqrt(5 - 2*x) - 3)) + 
32936/(250047*sqrt(5 - 2*x)) + 2116/(11907*(5 - 2*x)**(3/2))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {33}{2401} \, \sqrt {7} \log \left (-\frac {\sqrt {7} - \sqrt {-2 \, x + 5}}{\sqrt {7} + \sqrt {-2 \, x + 5}}\right ) - \frac {2 \, {\left (18569 \, {\left (2 \, x - 5\right )}^{3} + 257435 \, {\left (2 \, x - 5\right )}^{2} + 1363992 \, x - 4809714\right )}}{250047 \, {\left ({\left (-2 \, x + 5\right )}^{\frac {7}{2}} - 16 \, {\left (-2 \, x + 5\right )}^{\frac {5}{2}} + 63 \, {\left (-2 \, x + 5\right )}^{\frac {3}{2}}\right )}} + \frac {16}{2187} \, \log \left (\sqrt {-2 \, x + 5} + 3\right ) - \frac {16}{2187} \, \log \left (\sqrt {-2 \, x + 5} - 3\right ) \] Input:

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

Output:

33/2401*sqrt(7)*log(-(sqrt(7) - sqrt(-2*x + 5))/(sqrt(7) + sqrt(-2*x + 5)) 
) - 2/250047*(18569*(2*x - 5)^3 + 257435*(2*x - 5)^2 + 1363992*x - 4809714 
)/((-2*x + 5)^(7/2) - 16*(-2*x + 5)^(5/2) + 63*(-2*x + 5)^(3/2)) + 16/2187 
*log(sqrt(-2*x + 5) + 3) - 16/2187*log(sqrt(-2*x + 5) - 3)
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.21 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {33}{2401} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + 2 \, \sqrt {-2 \, x + 5} \right |}}{2 \, {\left (\sqrt {7} + \sqrt {-2 \, x + 5}\right )}}\right ) + \frac {2 \, {\left (2101 \, {\left (-2 \, x + 5\right )}^{\frac {3}{2}} - 16165 \, \sqrt {-2 \, x + 5}\right )}}{250047 \, {\left ({\left (2 \, x - 5\right )}^{2} + 32 \, x - 17\right )}} + \frac {92 \, {\left (716 \, x - 2273\right )}}{250047 \, {\left (2 \, x - 5\right )} \sqrt {-2 \, x + 5}} + \frac {16}{2187} \, \log \left (\sqrt {-2 \, x + 5} + 3\right ) - \frac {16}{2187} \, \log \left ({\left | \sqrt {-2 \, x + 5} - 3 \right |}\right ) \] Input:

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

Output:

33/2401*sqrt(7)*log(1/2*abs(-2*sqrt(7) + 2*sqrt(-2*x + 5))/(sqrt(7) + sqrt 
(-2*x + 5))) + 2/250047*(2101*(-2*x + 5)^(3/2) - 16165*sqrt(-2*x + 5))/((2 
*x - 5)^2 + 32*x - 17) + 92/250047*(716*x - 2273)/((2*x - 5)*sqrt(-2*x + 5 
)) + 16/2187*log(sqrt(-2*x + 5) + 3) - 16/2187*log(abs(sqrt(-2*x + 5) - 3) 
)
 

Mupad [B] (verification not implemented)

Time = 13.58 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.86 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {32\,\mathrm {atanh}\left (\frac {\sqrt {5-2\,x}}{3}\right )}{2187}-\frac {66\,\sqrt {7}\,\mathrm {atanh}\left (\frac {\sqrt {7}\,\sqrt {5-2\,x}}{7}\right )}{2401}-\frac {\frac {129904\,x}{11907}+\frac {514870\,{\left (2\,x-5\right )}^2}{250047}+\frac {37138\,{\left (2\,x-5\right )}^3}{250047}-\frac {458068}{11907}}{63\,{\left (5-2\,x\right )}^{3/2}-16\,{\left (5-2\,x\right )}^{5/2}+{\left (5-2\,x\right )}^{7/2}} \] Input:

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

Output:

(32*atanh((5 - 2*x)^(1/2)/3))/2187 - (66*7^(1/2)*atanh((7^(1/2)*(5 - 2*x)^ 
(1/2))/7))/2401 - ((129904*x)/11907 + (514870*(2*x - 5)^2)/250047 + (37138 
*(2*x - 5)^3)/250047 - 458068/11907)/(63*(5 - 2*x)^(3/2) - 16*(5 - 2*x)^(5 
/2) + (5 - 2*x)^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.80 \[ \int \frac {(4+3 x)^2}{(5-2 x)^{5/2} \left (2+3 x+x^2\right )^2} \, dx=\frac {144342 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right ) x^{3}+72171 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right ) x^{2}-793881 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right ) x -721710 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}-\sqrt {7}\right )-144342 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right ) x^{3}-72171 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right ) x^{2}+793881 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right ) x +721710 \sqrt {-2 x +5}\, \sqrt {7}\, \mathrm {log}\left (\sqrt {-2 x +5}+\sqrt {7}\right )-76832 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right ) x^{3}-38416 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right ) x^{2}+422576 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right ) x +384160 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}-3\right )+76832 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right ) x^{3}+38416 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right ) x^{2}-422576 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right ) x -384160 \sqrt {-2 x +5}\, \mathrm {log}\left (\sqrt {-2 x +5}+3\right )+1559796 x^{3}-886200 x^{2}-10493259 x -7297122}{5250987 \sqrt {-2 x +5}\, \left (2 x^{3}+x^{2}-11 x -10\right )} \] Input:

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

Output:

(144342*sqrt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x**3 + 72 
171*sqrt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x**2 - 793881 
*sqrt( - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7))*x - 721710*sqrt( 
 - 2*x + 5)*sqrt(7)*log(sqrt( - 2*x + 5) - sqrt(7)) - 144342*sqrt( - 2*x + 
 5)*sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x**3 - 72171*sqrt( - 2*x + 5)* 
sqrt(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x**2 + 793881*sqrt( - 2*x + 5)*sqr 
t(7)*log(sqrt( - 2*x + 5) + sqrt(7))*x + 721710*sqrt( - 2*x + 5)*sqrt(7)*l 
og(sqrt( - 2*x + 5) + sqrt(7)) - 76832*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 
5) - 3)*x**3 - 38416*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) - 3)*x**2 + 422 
576*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) - 3)*x + 384160*sqrt( - 2*x + 5) 
*log(sqrt( - 2*x + 5) - 3) + 76832*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) + 
 3)*x**3 + 38416*sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) + 3)*x**2 - 422576* 
sqrt( - 2*x + 5)*log(sqrt( - 2*x + 5) + 3)*x - 384160*sqrt( - 2*x + 5)*log 
(sqrt( - 2*x + 5) + 3) + 1559796*x**3 - 886200*x**2 - 10493259*x - 7297122 
)/(5250987*sqrt( - 2*x + 5)*(2*x**3 + x**2 - 11*x - 10))