\(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\) [722]

Optimal result

Integrand size = 24, antiderivative size = 106 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}-\frac {426}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:

-340*(1-2*x)^(1/2)/(231+385*x)+3/7*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)-426/49*21 
^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))+650/121*55^(1/2)*arctanh(1/11*5 
5^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} (647+1020 x)}{77 \left (6+19 x+15 x^2\right )}-\frac {426}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:

Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^2),x]
 

Output:

-1/77*(Sqrt[1 - 2*x]*(647 + 1020*x))/(6 + 19*x + 15*x^2) - (426*Sqrt[3/7]* 
ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + (650*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*S 
qrt[1 - 2*x]])/11
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {114, 168, 174, 73, 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.

\(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2} \, dx\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{7} \int \frac {41-45 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{7} \left (-\frac {1}{11} \int \frac {1663-1020 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {340 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{7} \left (\frac {1}{11} \left (7029 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-11375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {340 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{7} \left (\frac {1}{11} \left (11375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-7029 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {340 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{7} \left (\frac {1}{11} \left (4550 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-4686 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {340 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\)

Input:

Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^2),x]
 

Output:

(3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) + ((-340*Sqrt[1 - 2*x])/(11*(3 + 
 5*x)) + (-4686*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 4550*Sqrt[5/1 
1]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11)/7
 

Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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])
 

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {\left (647+1020 x \right ) \left (-1+2 x \right )}{77 \left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}-\frac {426 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}+\frac {650 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{121}\)	\(69\)
derivativedivides	\(\frac {6 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}-\frac {426 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}+\frac {10 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {650 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{121}\)	\(70\)
default	\(\frac {6 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}-\frac {426 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}+\frac {10 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {650 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{121}\)	\(70\)
pseudoelliptic	\(\frac {-51546 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (15 x^{2}+19 x +6\right ) \sqrt {21}+31850 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (15 x^{2}+19 x +6\right ) \sqrt {55}-77 \sqrt {1-2 x}\, \left (647+1020 x \right )}{88935 x^{2}+112651 x +35574}\)	\(85\)
trager	\(-\frac {\left (647+1020 x \right ) \sqrt {1-2 x}}{77 \left (15 x^{2}+19 x +6\right )}-\frac {213 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )-21 \sqrt {1-2 x}}{2+3 x}\right )}{49}+\frac {325 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}\)	\(117\)

Input:

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

Output:

1/77*(647+1020*x)*(-1+2*x)/(15*x^2+19*x+6)/(1-2*x)^(1/2)-426/49*21^(1/2)*a 
rctanh(1/7*21^(1/2)*(1-2*x)^(1/2))+650/121*55^(1/2)*arctanh(1/11*55^(1/2)* 
(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {2275 \, \sqrt {\frac {5}{11}} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac {5 \, x - 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 2343 \, \sqrt {\frac {3}{7}} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac {3 \, x + 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - {\left (1020 \, x + 647\right )} \sqrt {-2 \, x + 1}}{77 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \] Input:

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

Output:

1/77*(2275*sqrt(5/11)*(15*x^2 + 19*x + 6)*log((5*x - 11*sqrt(5/11)*sqrt(-2 
*x + 1) - 8)/(5*x + 3)) + 2343*sqrt(3/7)*(15*x^2 + 19*x + 6)*log((3*x + 7* 
sqrt(3/7)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - (1020*x + 647)*sqrt(-2*x + 1))/ 
(15*x^2 + 19*x + 6)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.73 (sec) , antiderivative size = 988, normalized size of antiderivative = 9.32 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-314160*sqrt(2)*I*(x - 1/2)**(5/2)/(456533*x + 355740*(x - 1/2)**3 + 80634 
4*(x - 1/2)**2 - 456533/2) - 356356*sqrt(2)*I*(x - 1/2)**(3/2)/(456533*x + 
 355740*(x - 1/2)**3 + 806344*(x - 1/2)**2 - 456533/2) - 29400*sqrt(55)*I* 
(x - 1/2)**3*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(456533*x + 355740*(x - 1/ 
2)**3 + 806344*(x - 1/2)**2 - 456533/2) + 1881600*sqrt(55)*I*(x - 1/2)**3* 
atan(sqrt(110)*sqrt(x - 1/2)/11)/(456533*x + 355740*(x - 1/2)**3 + 806344* 
(x - 1/2)**2 - 456533/2) - 43560*sqrt(21)*I*(x - 1/2)**3*atan(sqrt(42)/(6* 
sqrt(x - 1/2)))/(456533*x + 355740*(x - 1/2)**3 + 806344*(x - 1/2)**2 - 45 
6533/2) - 3136320*sqrt(21)*I*(x - 1/2)**3*atan(sqrt(42)*sqrt(x - 1/2)/7)/( 
456533*x + 355740*(x - 1/2)**3 + 806344*(x - 1/2)**2 - 456533/2) - 940800* 
sqrt(55)*I*pi*(x - 1/2)**3/(456533*x + 355740*(x - 1/2)**3 + 806344*(x - 1 
/2)**2 - 456533/2) + 1568160*sqrt(21)*I*pi*(x - 1/2)**3/(456533*x + 355740 
*(x - 1/2)**3 + 806344*(x - 1/2)**2 - 456533/2) - 66640*sqrt(55)*I*(x - 1/ 
2)**2*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(456533*x + 355740*(x - 1/2)**3 + 
 806344*(x - 1/2)**2 - 456533/2) + 4264960*sqrt(55)*I*(x - 1/2)**2*atan(sq 
rt(110)*sqrt(x - 1/2)/11)/(456533*x + 355740*(x - 1/2)**3 + 806344*(x - 1/ 
2)**2 - 456533/2) - 98736*sqrt(21)*I*(x - 1/2)**2*atan(sqrt(42)/(6*sqrt(x 
- 1/2)))/(456533*x + 355740*(x - 1/2)**3 + 806344*(x - 1/2)**2 - 456533/2) 
 - 7108992*sqrt(21)*I*(x - 1/2)**2*atan(sqrt(42)*sqrt(x - 1/2)/7)/(456533* 
x + 355740*(x - 1/2)**3 + 806344*(x - 1/2)**2 - 456533/2) - 2132480*sqr...
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {325}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {213}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1157 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \] Input:

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

Output:

-325/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 
*x + 1))) + 213/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 
 3*sqrt(-2*x + 1))) + 4/77*(510*(-2*x + 1)^(3/2) - 1157*sqrt(-2*x + 1))/(1 
5*(2*x - 1)^2 + 136*x + 9)
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {325}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {213}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1157 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \] Input:

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

Output:

-325/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 
 5*sqrt(-2*x + 1))) + 213/49*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2* 
x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/77*(510*(-2*x + 1)^(3/2) - 1157 
*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)
 

Mupad [B] (verification not implemented)

Time = 1.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {650\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {426\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {\frac {4628\,\sqrt {1-2\,x}}{1155}-\frac {136\,{\left (1-2\,x\right )}^{3/2}}{77}}{\frac {136\,x}{15}+{\left (2\,x-1\right )}^2+\frac {3}{5}} \] Input:

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

Output:

(650*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 - (426*21^(1/2)*at 
anh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - ((4628*(1 - 2*x)^(1/2))/1155 - (13 
6*(1 - 2*x)^(3/2))/77)/((136*x)/15 + (2*x - 1)^2 + 3/5)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.37 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {-78540 \sqrt {-2 x +1}\, x -49819 \sqrt {-2 x +1}-238875 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}-302575 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x -95550 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )+238875 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}+302575 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x +95550 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )+386595 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+489687 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +154638 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-386595 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-489687 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -154638 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{88935 x^{2}+112651 x +35574} \] Input:

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

Output:

( - 78540*sqrt( - 2*x + 1)*x - 49819*sqrt( - 2*x + 1) - 238875*sqrt(55)*lo 
g(5*sqrt( - 2*x + 1) - sqrt(55))*x**2 - 302575*sqrt(55)*log(5*sqrt( - 2*x 
+ 1) - sqrt(55))*x - 95550*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) + 2 
38875*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 + 302575*sqrt(55)*l 
og(5*sqrt( - 2*x + 1) + sqrt(55))*x + 95550*sqrt(55)*log(5*sqrt( - 2*x + 1 
) + sqrt(55)) + 386595*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 + 
489687*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 154638*sqrt(21)*log 
(3*sqrt( - 2*x + 1) - sqrt(21)) - 386595*sqrt(21)*log(3*sqrt( - 2*x + 1) + 
 sqrt(21))*x**2 - 489687*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 1 
54638*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(5929*(15*x**2 + 19*x + 
 6))