\(\int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx\) [543]

Optimal result

Integrand size = 24, antiderivative size = 113 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3}+\frac {52 \sqrt {1-2 x}}{21 (2+3 x)^2}+\frac {1207 \sqrt {1-2 x}}{49 (2+3 x)}+\frac {83264 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}-50 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:

1/3*(1-2*x)^(1/2)/(2+3*x)^3+52/21*(1-2*x)^(1/2)/(2+3*x)^2+1207*(1-2*x)^(1/ 
2)/(98+147*x)+83264/1029*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-50*5 
5^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (5087+14848 x+10863 x^2\right )}{49 (2+3 x)^3}+\frac {83264 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}-50 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:

Integrate[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)),x]
 

Output:

(Sqrt[1 - 2*x]*(5087 + 14848*x + 10863*x^2))/(49*(2 + 3*x)^3) + (83264*Arc 
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[21]) - 50*Sqrt[55]*ArcTanh[Sqrt[5/ 
11]*Sqrt[1 - 2*x]]
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {110, 25, 168, 27, 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.

\(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^4 (5 x+3)} \, dx\)

\(\Big \downarrow \) 110

\(\displaystyle \frac {\sqrt {1-2 x}}{3 (3 x+2)^3}-\frac {1}{3} \int -\frac {18-25 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \int \frac {18-25 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{3} \left (\frac {1}{14} \int \frac {6 (229-260 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {3}{7} \int \frac {229-260 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{3} \left (\frac {3}{7} \left (\frac {1}{7} \int \frac {9854-6035 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {1207 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{3} \left (\frac {3}{7} \left (\frac {1}{7} \left (67375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-41632 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {1207 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{3} \left (\frac {3}{7} \left (\frac {1}{7} \left (41632 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-67375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {1207 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {3}{7} \left (\frac {1}{7} \left (\frac {83264 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-2450 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {1207 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {52 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3}\)

Input:

Int[Sqrt[1 - 2*x]/((2 + 3*x)^4*(3 + 5*x)),x]
 

Output:

Sqrt[1 - 2*x]/(3*(2 + 3*x)^3) + ((52*Sqrt[1 - 2*x])/(7*(2 + 3*x)^2) + (3*( 
(1207*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + ((83264*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2* 
x]])/Sqrt[21] - 2450*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/7))/7)/3
 

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 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 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (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 = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.61

method	result	size
risch	\(-\frac {21726 x^{3}+18833 x^{2}-4674 x -5087}{49 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {83264 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{1029}-50 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )\)	\(69\)
derivativedivides	\(-50 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )-\frac {54 \left (\frac {1207 \left (1-2 x \right )^{\frac {5}{2}}}{147}-\frac {7346 \left (1-2 x \right )^{\frac {3}{2}}}{189}+\frac {1243 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {83264 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{1029}\)	\(75\)
default	\(-50 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )-\frac {54 \left (\frac {1207 \left (1-2 x \right )^{\frac {5}{2}}}{147}-\frac {7346 \left (1-2 x \right )^{\frac {3}{2}}}{189}+\frac {1243 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {83264 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{1029}\)	\(75\)
pseudoelliptic	\(\frac {83264 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}-51450 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \sqrt {55}+21 \sqrt {1-2 x}\, \left (10863 x^{2}+14848 x +5087\right )}{1029 \left (2+3 x \right )^{3}}\)	\(80\)
trager	\(\frac {\left (10863 x^{2}+14848 x +5087\right ) \sqrt {1-2 x}}{49 \left (2+3 x \right )^{3}}+25 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )-\frac {41632 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1029}\)	\(116\)

Input:

int((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x,method=_RETURNVERBOSE)
 

Output:

-1/49*(21726*x^3+18833*x^2-4674*x-5087)/(2+3*x)^3/(1-2*x)^(1/2)+83264/1029 
*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-50*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 = 130, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {25725 \, \sqrt {55} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 41632 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (10863 \, x^{2} + 14848 \, x + 5087\right )} \sqrt {-2 \, x + 1}}{1029 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x, algorithm="fricas")
 

Output:

1/1029*(25725*sqrt(55)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x + sqrt(55)*sq 
rt(-2*x + 1) - 8)/(5*x + 3)) + 41632*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8) 
*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(10863*x^2 + 1484 
8*x + 5087)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
 

Sympy [A] (verification not implemented)

Time = 88.16 (sec) , antiderivative size = 578, normalized size of antiderivative = 5.12 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx =\text {Too large to display} \] Input:

integrate((1-2*x)**(1/2)/(2+3*x)**4/(3+5*x),x)
 

Output:

-275*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 
21)/3))/7 + 25*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2* 
x) + sqrt(55)/5)) + 660*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 
 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2* 
x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > - 
sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3))) - 264*Piecewise((sqrt(21)*(3* 
log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1) 
/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x 
)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqr 
t(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x 
) < sqrt(21)/3))) + 112*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqrt(1 - 2*x) 
/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqrt(21)*sqrt 
(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 1/(48*(sqrt 
(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) + 1/ 
(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 - 
 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3 
)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=25 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {41632}{1029} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (10863 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 51422 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 60907 \, \sqrt {-2 \, x + 1}\right )}}{49 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \] Input:

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x, algorithm="maxima")
 

Output:

25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1 
))) - 41632/1029*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3 
*sqrt(-2*x + 1))) + 2/49*(10863*(-2*x + 1)^(5/2) - 51422*(-2*x + 1)^(3/2) 
+ 60907*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=25 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {41632}{1029} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {10863 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 51422 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 60907 \, \sqrt {-2 \, x + 1}}{196 \, {\left (3 \, x + 2\right )}^{3}} \] Input:

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x, algorithm="giac")
 

Output:

25*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr 
t(-2*x + 1))) - 41632/1029*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x 
+ 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/196*(10863*(2*x - 1)^2*sqrt(-2*x 
+ 1) - 51422*(-2*x + 1)^(3/2) + 60907*sqrt(-2*x + 1))/(3*x + 2)^3
 

Mupad [B] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {83264\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1029}-50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {2486\,\sqrt {1-2\,x}}{27}-\frac {14692\,{\left (1-2\,x\right )}^{3/2}}{189}+\frac {2414\,{\left (1-2\,x\right )}^{5/2}}{147}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \] Input:

int((1 - 2*x)^(1/2)/((3*x + 2)^4*(5*x + 3)),x)
 

Output:

(83264*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1029 - 50*55^(1/2)*at 
anh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((2486*(1 - 2*x)^(1/2))/27 - (14692*( 
1 - 2*x)^(3/2))/189 + (2414*(1 - 2*x)^(5/2))/147)/((98*x)/3 + 7*(2*x - 1)^ 
2 + (2*x - 1)^3 - 98/27)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.07 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {228123 \sqrt {-2 x +1}\, x^{2}+311808 \sqrt {-2 x +1}\, x +106827 \sqrt {-2 x +1}+694575 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{3}+1389150 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+926100 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +205800 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-694575 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{3}-1389150 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-926100 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -205800 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-1124064 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}-2248128 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-1498752 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -333056 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+1124064 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}+2248128 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}+1498752 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +333056 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{27783 x^{3}+55566 x^{2}+37044 x +8232} \] Input:

int((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x)
 

Output:

(228123*sqrt( - 2*x + 1)*x**2 + 311808*sqrt( - 2*x + 1)*x + 106827*sqrt( - 
 2*x + 1) + 694575*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**3 + 1389 
150*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**2 + 926100*sqrt(55)*log 
(5*sqrt( - 2*x + 1) - sqrt(55))*x + 205800*sqrt(55)*log(5*sqrt( - 2*x + 1) 
 - sqrt(55)) - 694575*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**3 - 1 
389150*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 - 926100*sqrt(55)* 
log(5*sqrt( - 2*x + 1) + sqrt(55))*x - 205800*sqrt(55)*log(5*sqrt( - 2*x + 
 1) + sqrt(55)) - 1124064*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**3 
 - 2248128*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 - 1498752*sqrt 
(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x - 333056*sqrt(21)*log(3*sqrt( - 
2*x + 1) - sqrt(21)) + 1124064*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) 
*x**3 + 2248128*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 + 1498752 
*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x + 333056*sqrt(21)*log(3*sqr 
t( - 2*x + 1) + sqrt(21)))/(1029*(27*x**3 + 54*x**2 + 36*x + 8))