\(\int \frac {5-x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx\) [1011]

Optimal result

Integrand size = 27, antiderivative size = 134 \[ \int \frac {5-x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {41 \sqrt {2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac {681 \sqrt {2+5 x+3 x^2}}{250 (3+2 x)}+\frac {5771 \text {arctanh}\left (\frac {\sqrt {5} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{1000 \sqrt {5}} \] Output:

-13/20*(3*x^2+5*x+2)^(1/2)/(3+2*x)^4-86/75*(3*x^2+5*x+2)^(1/2)/(3+2*x)^3-4 
1/24*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2-681*(3*x^2+5*x+2)^(1/2)/(750+500*x)+577 
1/5000*5^(1/2)*arctanh(5^(1/2)*(1+x)/(3*x^2+5*x+2)^(1/2))
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.54 \[ \int \frac {5-x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (279039+509668 x+314692 x^2+65376 x^3\right )}{(3+2 x)^4}+17313 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{15000} \] Input:

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

Output:

((-5*Sqrt[2 + 5*x + 3*x^2]*(279039 + 509668*x + 314692*x^2 + 65376*x^3))/( 
3 + 2*x)^4 + 17313*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/150 
00
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1237, 27, 1237, 25, 1237, 27, 1228, 1154, 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.

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

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {1}{20} \int \frac {234 x+7}{2 (2 x+3)^4 \sqrt {3 x^2+5 x+2}}dx-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{40} \int \frac {234 x+7}{(2 x+3)^4 \sqrt {3 x^2+5 x+2}}dx-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{40} \left (\frac {1}{15} \int -\frac {4128 x+1067}{(2 x+3)^3 \sqrt {3 x^2+5 x+2}}dx-\frac {688 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{40} \left (-\frac {1}{15} \int \frac {4128 x+1067}{(2 x+3)^3 \sqrt {3 x^2+5 x+2}}dx-\frac {688 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{40} \left (\frac {1}{15} \left (\frac {1}{10} \int -\frac {15 (2050 x+351)}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx-\frac {1025 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )-\frac {688 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{40} \left (\frac {1}{15} \left (-\frac {3}{2} \int \frac {2050 x+351}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx-\frac {1025 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )-\frac {688 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {1}{40} \left (\frac {1}{15} \left (-\frac {3}{2} \left (\frac {5448 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}-\frac {5771}{5} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1025 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )-\frac {688 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{40} \left (\frac {1}{15} \left (-\frac {3}{2} \left (\frac {11542}{5} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {5448 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )-\frac {1025 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )-\frac {688 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{40} \left (\frac {1}{15} \left (-\frac {3}{2} \left (\frac {5448 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}-\frac {5771 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{5 \sqrt {5}}\right )-\frac {1025 \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}\right )-\frac {688 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^3}\right )-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}\)

Input:

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

Output:

(-13*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^4) + ((-688*Sqrt[2 + 5*x + 3*x^2 
])/(15*(3 + 2*x)^3) + ((-1025*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^2 - (3*((54 
48*Sqrt[2 + 5*x + 3*x^2])/(5*(3 + 2*x)) - (5771*ArcTanh[(7 + 8*x)/(2*Sqrt[ 
5]*Sqrt[2 + 5*x + 3*x^2])])/(5*Sqrt[5])))/2)/15)/40
 

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 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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1228

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

rule 1237

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

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.58

method	result	size
risch	\(-\frac {196128 x^{5}+1270956 x^{4}+3233216 x^{3}+4014841 x^{2}+2414531 x +558078}{3000 \left (2 x +3\right )^{4} \sqrt {3 x^{2}+5 x +2}}-\frac {5771 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{10000}\)	\(78\)
trager	\(-\frac {\left (65376 x^{3}+314692 x^{2}+509668 x +279039\right ) \sqrt {3 x^{2}+5 x +2}}{3000 \left (2 x +3\right )^{4}}-\frac {5771 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )-10 \sqrt {3 x^{2}+5 x +2}}{2 x +3}\right )}{10000}\)	\(88\)
default	\(-\frac {13 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{320 \left (x +\frac {3}{2}\right )^{4}}-\frac {43 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{300 \left (x +\frac {3}{2}\right )^{3}}-\frac {41 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{96 \left (x +\frac {3}{2}\right )^{2}}-\frac {681 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{500 \left (x +\frac {3}{2}\right )}-\frac {5771 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{10000}\)	\(116\)

Input:

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

Output:

-1/3000*(196128*x^5+1270956*x^4+3233216*x^3+4014841*x^2+2414531*x+558078)/ 
(2*x+3)^4/(3*x^2+5*x+2)^(1/2)-5771/10000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^ 
(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.93 \[ \int \frac {5-x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx=\frac {17313 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (65376 \, x^{3} + 314692 \, x^{2} + 509668 \, x + 279039\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{60000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \] Input:

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

Output:

1/60000*(17313*sqrt(5)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log((4*sqr 
t(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x 
 + 9)) - 20*(65376*x^3 + 314692*x^2 + 509668*x + 279039)*sqrt(3*x^2 + 5*x 
+ 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)
 

Sympy [F]

\[ \int \frac {5-x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx=- \int \frac {x}{32 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 810 x \sqrt {3 x^{2} + 5 x + 2} + 243 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{32 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 810 x \sqrt {3 x^{2} + 5 x + 2} + 243 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \] Input:

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

Output:

-Integral(x/(32*x**5*sqrt(3*x**2 + 5*x + 2) + 240*x**4*sqrt(3*x**2 + 5*x + 
 2) + 720*x**3*sqrt(3*x**2 + 5*x + 2) + 1080*x**2*sqrt(3*x**2 + 5*x + 2) + 
 810*x*sqrt(3*x**2 + 5*x + 2) + 243*sqrt(3*x**2 + 5*x + 2)), x) - Integral 
(-5/(32*x**5*sqrt(3*x**2 + 5*x + 2) + 240*x**4*sqrt(3*x**2 + 5*x + 2) + 72 
0*x**3*sqrt(3*x**2 + 5*x + 2) + 1080*x**2*sqrt(3*x**2 + 5*x + 2) + 810*x*s 
qrt(3*x**2 + 5*x + 2) + 243*sqrt(3*x**2 + 5*x + 2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.17 \[ \int \frac {5-x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx=-\frac {5771}{10000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {86 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{75 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {41 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{24 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {681 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{250 \, {\left (2 \, x + 3\right )}} \] Input:

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

Output:

-5771/10000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/a 
bs(2*x + 3) - 2) - 13/20*sqrt(3*x^2 + 5*x + 2)/(16*x^4 + 96*x^3 + 216*x^2 
+ 216*x + 81) - 86/75*sqrt(3*x^2 + 5*x + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 
 41/24*sqrt(3*x^2 + 5*x + 2)/(4*x^2 + 12*x + 9) - 681/250*sqrt(3*x^2 + 5*x 
 + 2)/(2*x + 3)
 

Giac [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.44 \[ \int \frac {5-x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx=\frac {1}{10000} \, \sqrt {5} {\left (2724 \, \sqrt {5} \sqrt {3} + 5771 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{6000} \, {\left (\frac {5 \, {\left (\frac {2 \, {\left (\frac {344}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {195}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {1025}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {8172}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {5771 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{10000 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \] Input:

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

Output:

1/10000*sqrt(5)*(2724*sqrt(5)*sqrt(3) + 5771*log(-sqrt(5)*sqrt(3) + 4))*sg 
n(1/(2*x + 3)) - 1/6000*(5*(2*(344/sgn(1/(2*x + 3)) + 195/((2*x + 3)*sgn(1 
/(2*x + 3))))/(2*x + 3) + 1025/sgn(1/(2*x + 3)))/(2*x + 3) + 8172/sgn(1/(2 
*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) - 5771/10000*sqrt(5)*log( 
abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 
 4))/sgn(1/(2*x + 3))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.08 \[ \int \frac {5-x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx=\frac {-20266560 \sqrt {3 x^{2}+5 x +2}\, x^{3}-97554520 \sqrt {3 x^{2}+5 x +2}\, x^{2}-157997080 \sqrt {3 x^{2}+5 x +2}\, x -86502090 \sqrt {3 x^{2}+5 x +2}+8587248 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right ) x^{4}[phone] \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right ) x^{3}[phone] \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right ) x^{2}[phone] \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right ) x [phone] \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}-\sqrt {15}+6 x +9\right )-8587248 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right ) x^{4}-51523488 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right ) x^{3}-115927848 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right ) x^{2}-115927848 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right ) x -43472943 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+\sqrt {15}+6 x +9\right )[phone] \sqrt {3}\, x^{4}[phone] \sqrt {3}\, x^{3}[phone] \sqrt {3}\, x^{2}[phone] \sqrt {3}\, x [phone] \sqrt {3}}{14880000 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

( - 20266560*sqrt(3*x**2 + 5*x + 2)*x**3 - 97554520*sqrt(3*x**2 + 5*x + 2) 
*x**2 - 157997080*sqrt(3*x**2 + 5*x + 2)*x - 86502090*sqrt(3*x**2 + 5*x + 
2) + 8587248*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt(15) + 6*x 
 + 9)*x**4 + 51523488*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) - sqrt( 
15) + 6*x + 9)*x**3 + 115927848*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt( 
3) - sqrt(15) + 6*x + 9)*x**2 + 115927848*sqrt(5)*log(2*sqrt(3*x**2 + 5*x 
+ 2)*sqrt(3) - sqrt(15) + 6*x + 9)*x + 43472943*sqrt(5)*log(2*sqrt(3*x**2 
+ 5*x + 2)*sqrt(3) - sqrt(15) + 6*x + 9) - 8587248*sqrt(5)*log(2*sqrt(3*x* 
*2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x + 9)*x**4 - 51523488*sqrt(5)*log(2* 
sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x + 9)*x**3 - 115927848*sqrt 
(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x + 9)*x**2 - 1159 
27848*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x + 9)*x 
 - 43472943*sqrt(5)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + sqrt(15) + 6*x 
+ 9) + 18881520*sqrt(3)*x**4 + 113289120*sqrt(3)*x**3 + 254900520*sqrt(3)* 
x**2 + 254900520*sqrt(3)*x + 95587695*sqrt(3))/(930000*(16*x**4 + 96*x**3 
+ 216*x**2 + 216*x + 81))