\(\int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\) [777]

Optimal result

Integrand size = 24, antiderivative size = 92 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {18}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:

78/847/(1-2*x)^(1/2)-5/11/(1-2*x)^(1/2)/(3+5*x)-18/49*21^(1/2)*arctanh(1/7 
*21^(1/2)*(1-2*x)^(1/2))+300/1331*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^( 
1/2))
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {-151+390 x}{847 \sqrt {1-2 x} (3+5 x)}-\frac {18}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:

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

Output:

(-151 + 390*x)/(847*Sqrt[1 - 2*x]*(3 + 5*x)) - (18*Sqrt[3/7]*ArcTanh[Sqrt[ 
3/7]*Sqrt[1 - 2*x]])/7 + (300*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]] 
)/121
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {114, 27, 169, 27, 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.

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

\(\Big \downarrow \) 114

\(\displaystyle -\frac {1}{11} \int \frac {3 (1-15 x)}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{11} \int \frac {1-15 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {3}{11} \left (-\frac {2}{77} \int -\frac {233-195 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{11} \left (\frac {1}{77} \int \frac {233-195 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {3}{11} \left (\frac {1}{77} \left (1750 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-1089 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {3}{11} \left (\frac {1}{77} \left (1089 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-1750 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {3}{11} \left (\frac {1}{77} \left (726 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-700 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\)

Input:

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

Output:

-5/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (3*(-26/(77*Sqrt[1 - 2*x]) + (726*Sqrt[3 
/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 700*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*S 
qrt[1 - 2*x]])/77))/11
 

Defintions of rubi rules used

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 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 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.64

method	result	size
risch	\(\frac {-151+390 x}{847 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {18 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}+\frac {300 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}\)	\(59\)
derivativedivides	\(-\frac {18 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}+\frac {10 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {300 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}+\frac {8}{847 \sqrt {1-2 x}}\)	\(63\)
default	\(-\frac {18 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}+\frac {10 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {300 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}+\frac {8}{847 \sqrt {1-2 x}}\)	\(63\)
pseudoelliptic	\(\frac {-\frac {151}{847}-\frac {18 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right ) \sqrt {21}}{49}+\frac {300 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}}{1331}+\frac {390 x}{847}}{\sqrt {1-2 x}\, \left (3+5 x \right )}\)	\(81\)
trager	\(-\frac {\left (-151+390 x \right ) \sqrt {1-2 x}}{847 \left (10 x^{2}+x -3\right )}-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )-21 \sqrt {1-2 x}}{2+3 x}\right )}{49}-\frac {150 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\)	\(115\)

Input:

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

Output:

1/847*(-151+390*x)/(3+5*x)/(1-2*x)^(1/2)-18/49*21^(1/2)*arctanh(1/7*21^(1/ 
2)*(1-2*x)^(1/2))+300/1331*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {1050 \, \sqrt {\frac {5}{11}} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {5 \, x - 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 1089 \, \sqrt {\frac {3}{7}} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {3 \, x + 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - {\left (390 \, x - 151\right )} \sqrt {-2 \, x + 1}}{847 \, {\left (10 \, x^{2} + x - 3\right )}} \] Input:

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

Output:

1/847*(1050*sqrt(5/11)*(10*x^2 + x - 3)*log((5*x - 11*sqrt(5/11)*sqrt(-2*x 
 + 1) - 8)/(5*x + 3)) + 1089*sqrt(3/7)*(10*x^2 + x - 3)*log((3*x + 7*sqrt( 
3/7)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - (390*x - 151)*sqrt(-2*x + 1))/(10*x^ 
2 + x - 3)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.49 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=- \frac {30030 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {3388 \sqrt {2} i \sqrt {x - \frac {1}{2}}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {147000 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {239580 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {73500 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{2}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {119790 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{2}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {161700 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {263538 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {80850 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {131769 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} \] Input:

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

Output:

-30030*sqrt(2)*I*(x - 1/2)**(3/2)/(717409*x + 652190*(x - 1/2)**2 - 717409 
/2) - 3388*sqrt(2)*I*sqrt(x - 1/2)/(717409*x + 652190*(x - 1/2)**2 - 71740 
9/2) + 147000*sqrt(55)*I*(x - 1/2)**2*atan(sqrt(110)*sqrt(x - 1/2)/11)/(71 
7409*x + 652190*(x - 1/2)**2 - 717409/2) - 239580*sqrt(21)*I*(x - 1/2)**2* 
atan(sqrt(42)*sqrt(x - 1/2)/7)/(717409*x + 652190*(x - 1/2)**2 - 717409/2) 
 - 73500*sqrt(55)*I*pi*(x - 1/2)**2/(717409*x + 652190*(x - 1/2)**2 - 7174 
09/2) + 119790*sqrt(21)*I*pi*(x - 1/2)**2/(717409*x + 652190*(x - 1/2)**2 
- 717409/2) + 161700*sqrt(55)*I*(x - 1/2)*atan(sqrt(110)*sqrt(x - 1/2)/11) 
/(717409*x + 652190*(x - 1/2)**2 - 717409/2) - 263538*sqrt(21)*I*(x - 1/2) 
*atan(sqrt(42)*sqrt(x - 1/2)/7)/(717409*x + 652190*(x - 1/2)**2 - 717409/2 
) - 80850*sqrt(55)*I*pi*(x - 1/2)/(717409*x + 652190*(x - 1/2)**2 - 717409 
/2) + 131769*sqrt(21)*I*pi*(x - 1/2)/(717409*x + 652190*(x - 1/2)**2 - 717 
409/2)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {150}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {9}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \] Input:

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

Output:

-150/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(- 
2*x + 1))) + 9/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 
3*sqrt(-2*x + 1))) - 2/847*(390*x - 151)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2* 
x + 1))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {150}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {9}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \] Input:

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

Output:

-150/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) 
+ 5*sqrt(-2*x + 1))) + 9/49*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x 
 + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/847*(390*x - 151)/(5*(-2*x + 1)^ 
(3/2) - 11*sqrt(-2*x + 1))
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {\frac {156\,x}{847}-\frac {302}{4235}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {18\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}+\frac {300\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \] Input:

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

Output:

((156*x)/847 - 302/4235)/((11*(1 - 2*x)^(1/2))/5 - (1 - 2*x)^(3/2)) - (18* 
21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 + (300*55^(1/2)*atanh((55 
^(1/2)*(1 - 2*x)^(1/2))/11))/1331
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.28 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {-36750 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x -22050 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )+36750 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x +22050 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )+59895 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +35937 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-59895 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -35937 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )+30030 x -11627}{65219 \sqrt {-2 x +1}\, \left (5 x +3\right )} \] Input:

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

Output:

( - 36750*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x - 
 22050*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) + 3675 
0*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x + 22050*s 
qrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) + 59895*sqrt( 
- 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 35937*sqrt( - 2 
*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 59895*sqrt( - 2*x + 
1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 35937*sqrt( - 2*x + 1)* 
sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) + 30030*x - 11627)/(65219*sqrt 
( - 2*x + 1)*(5*x + 3))