\(\int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx\) [615]

Optimal result

Integrand size = 24, antiderivative size = 93 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx=-\frac {11 \sqrt {1-2 x}}{10 (3+5 x)^2}+\frac {71 \sqrt {1-2 x}}{10 (3+5 x)}+14 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2379 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}} \] Output:

-11/10*(1-2*x)^(1/2)/(3+5*x)^2+71*(1-2*x)^(1/2)/(30+50*x)+14*21^(1/2)*arct 
anh(1/7*21^(1/2)*(1-2*x)^(1/2))-2379/275*55^(1/2)*arctanh(1/11*55^(1/2)*(1 
-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} (202+355 x)}{10 (3+5 x)^2}+14 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2379 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}} \] Input:

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

Output:

(Sqrt[1 - 2*x]*(202 + 355*x))/(10*(3 + 5*x)^2) + 14*Sqrt[21]*ArcTanh[Sqrt[ 
3/7]*Sqrt[1 - 2*x]] - (2379*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55] 
)
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {109, 168, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {1}{10} \int \frac {92-107 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {11 \sqrt {1-2 x}}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} \left (\frac {1}{11} \int \frac {33 (116-71 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {71 \sqrt {1-2 x}}{5 x+3}\right )-\frac {11 \sqrt {1-2 x}}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (3 \int \frac {116-71 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {71 \sqrt {1-2 x}}{5 x+3}\right )-\frac {11 \sqrt {1-2 x}}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{10} \left (3 \left (793 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-490 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {71 \sqrt {1-2 x}}{5 x+3}\right )-\frac {11 \sqrt {1-2 x}}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{10} \left (3 \left (490 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-793 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {71 \sqrt {1-2 x}}{5 x+3}\right )-\frac {11 \sqrt {1-2 x}}{10 (5 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{10} \left (3 \left (140 \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1586 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{\sqrt {55}}\right )+\frac {71 \sqrt {1-2 x}}{5 x+3}\right )-\frac {11 \sqrt {1-2 x}}{10 (5 x+3)^2}\)

Input:

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

Output:

(-11*Sqrt[1 - 2*x])/(10*(3 + 5*x)^2) + ((71*Sqrt[1 - 2*x])/(3 + 5*x) + 3*( 
140*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (1586*ArcTanh[Sqrt[5/11]* 
Sqrt[1 - 2*x]])/Sqrt[55]))/10
 

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 109

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

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 = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.69

method	result	size
risch	\(-\frac {710 x^{2}+49 x -202}{10 \left (3+5 x \right )^{2} \sqrt {1-2 x}}+14 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )-\frac {2379 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{275}\)	\(64\)
derivativedivides	\(\frac {-71 \left (1-2 x \right )^{\frac {3}{2}}+\frac {759 \sqrt {1-2 x}}{5}}{\left (-6-10 x \right )^{2}}-\frac {2379 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{275}+14 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )\)	\(66\)
default	\(\frac {-71 \left (1-2 x \right )^{\frac {3}{2}}+\frac {759 \sqrt {1-2 x}}{5}}{\left (-6-10 x \right )^{2}}-\frac {2379 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{275}+14 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )\)	\(66\)
pseudoelliptic	\(\frac {7700 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right )^{2} \sqrt {21}-4758 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}+55 \sqrt {1-2 x}\, \left (355 x +202\right )}{550 \left (3+5 x \right )^{2}}\)	\(75\)
trager	\(\frac {\left (355 x +202\right ) \sqrt {1-2 x}}{10 \left (3+5 x \right )^{2}}+\frac {2379 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{550}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )\)	\(111\)

Input:

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

Output:

-1/10*(710*x^2+49*x-202)/(3+5*x)^2/(1-2*x)^(1/2)+14*21^(1/2)*arctanh(1/7*2 
1^(1/2)*(1-2*x)^(1/2))-2379/275*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 = 110, normalized size of antiderivative = 1.18 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx=\frac {2379 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 3850 \, \sqrt {21} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 55 \, {\left (355 \, x + 202\right )} \sqrt {-2 \, x + 1}}{550 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:

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

Output:

1/550*(2379*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1 
) - 8)/(5*x + 3)) + 3850*sqrt(21)*(25*x^2 + 30*x + 9)*log((3*x - sqrt(21)* 
sqrt(-2*x + 1) - 5)/(3*x + 2)) + 55*(355*x + 202)*sqrt(-2*x + 1))/(25*x^2 
+ 30*x + 9)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (82) = 164\).

Time = 44.60 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.99 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx=- 7 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right ) + \frac {49 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{11} + \frac {1628 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{5} + \frac {968 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{5} \] Input:

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

Output:

-7*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21 
)/3)) + 49*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + 
 sqrt(55)/5))/11 + 1628*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/1 
1 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 
2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) 
 > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/5 + 968*Piecewise((sqrt(5 
5)*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x) 
/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqrt(55)*sqr 
t(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 1/(16*(s 
qrt(55)*sqrt(1 - 2*x)/11 - 1)**2))/6655, (sqrt(1 - 2*x) > -sqrt(55)/5) & ( 
sqrt(1 - 2*x) < sqrt(55)/5)))/5
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.18 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx=\frac {2379}{550} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - 7 \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {355 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 759 \, \sqrt {-2 \, x + 1}}{5 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \] Input:

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

Output:

2379/550*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 
*x + 1))) - 7*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq 
rt(-2*x + 1))) - 1/5*(355*(-2*x + 1)^(3/2) - 759*sqrt(-2*x + 1))/(25*(2*x 
- 1)^2 + 220*x + 11)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.15 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx=\frac {2379}{550} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - 7 \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {355 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 759 \, \sqrt {-2 \, x + 1}}{20 \, {\left (5 \, x + 3\right )}^{2}} \] Input:

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

Output:

2379/550*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 
 5*sqrt(-2*x + 1))) - 7*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1 
))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/20*(355*(-2*x + 1)^(3/2) - 759*sqrt( 
-2*x + 1))/(5*x + 3)^2
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx=14\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )-\frac {2379\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{275}+\frac {\frac {759\,\sqrt {1-2\,x}}{125}-\frac {71\,{\left (1-2\,x\right )}^{3/2}}{25}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \] Input:

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

Output:

14*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7) - (2379*55^(1/2)*atanh((55 
^(1/2)*(1 - 2*x)^(1/2))/11))/275 + ((759*(1 - 2*x)^(1/2))/125 - (71*(1 - 2 
*x)^(3/2))/25)/((44*x)/5 + (2*x - 1)^2 + 11/25)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.70 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx=\frac {19525 \sqrt {-2 x +1}\, x +11110 \sqrt {-2 x +1}+59475 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+71370 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +21411 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-59475 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-71370 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -21411 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-96250 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-115500 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -34650 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+96250 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}+115500 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +34650 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{13750 x^{2}+16500 x +4950} \] Input:

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

Output:

(19525*sqrt( - 2*x + 1)*x + 11110*sqrt( - 2*x + 1) + 59475*sqrt(55)*log(5* 
sqrt( - 2*x + 1) - sqrt(55))*x**2 + 71370*sqrt(55)*log(5*sqrt( - 2*x + 1) 
- sqrt(55))*x + 21411*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 59475* 
sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 - 71370*sqrt(55)*log(5*sq 
rt( - 2*x + 1) + sqrt(55))*x - 21411*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqr 
t(55)) - 96250*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 - 115500*s 
qrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x - 34650*sqrt(21)*log(3*sqrt( 
- 2*x + 1) - sqrt(21)) + 96250*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) 
*x**2 + 115500*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x + 34650*sqrt( 
21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(550*(25*x**2 + 30*x + 9))