\(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\) [731]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx=-\frac {505 \sqrt {1-2 x}}{154 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {33465 \sqrt {1-2 x}}{1694 (3+5 x)}+\frac {1908}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {32025}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:

-505/154*(1-2*x)^(1/2)/(3+5*x)^2+3/7*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+33465 
*(1-2*x)^(1/2)/(5082+8470*x)+1908/49*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x) 
^(1/2))-32025/1331*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {1908}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {\frac {11 \sqrt {1-2 x} \left (190406+619170 x+501975 x^2\right )}{(2+3 x) (3+5 x)^2}-448350 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{18634} \] Input:

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

Output:

(1908*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + ((11*Sqrt[1 - 2*x]*( 
190406 + 619170*x + 501975*x^2))/((2 + 3*x)*(3 + 5*x)^2) - 448350*Sqrt[55] 
*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/18634
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {114, 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.

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{7} \int \frac {56-75 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{7} \left (-\frac {1}{22} \int \frac {3 (1322-1515 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {505 \sqrt {1-2 x}}{22 (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (-\frac {3}{22} \int \frac {1322-1515 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {505 \sqrt {1-2 x}}{22 (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{7} \left (-\frac {3}{22} \left (-\frac {1}{11} \int \frac {54646-33465 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {11155 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {505 \sqrt {1-2 x}}{22 (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{7} \left (-\frac {3}{22} \left (\frac {1}{11} \left (230868 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-373625 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {11155 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {505 \sqrt {1-2 x}}{22 (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{7} \left (-\frac {3}{22} \left (\frac {1}{11} \left (373625 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-230868 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {11155 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {505 \sqrt {1-2 x}}{22 (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{7} \left (-\frac {3}{22} \left (\frac {1}{11} \left (149450 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-153912 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {11155 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {505 \sqrt {1-2 x}}{22 (5 x+3)^2}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\)

Input:

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

Output:

(3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^2) + ((-505*Sqrt[1 - 2*x])/(22*(3 
 + 5*x)^2) - (3*((-11155*Sqrt[1 - 2*x])/(11*(3 + 5*x)) + (-153912*Sqrt[3/7 
]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 149450*Sqrt[5/11]*ArcTanh[Sqrt[5/11]* 
Sqrt[1 - 2*x]])/11))/22)/7
 

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 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 = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.60

method	result	size
risch	\(-\frac {1003950 x^{3}+736365 x^{2}-238358 x -190406}{1694 \left (2+3 x \right ) \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}+\frac {1908 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}-\frac {32025 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}\)	\(76\)
derivativedivides	\(\frac {-\frac {16125 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {3175 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {32025 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}-\frac {18 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}+\frac {1908 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}\)	\(82\)
default	\(\frac {-\frac {16125 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {3175 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {32025 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}-\frac {18 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}+\frac {1908 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}\)	\(82\)
pseudoelliptic	\(\frac {5079096 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {21}-3138450 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {55}+77 \sqrt {1-2 x}\, \left (501975 x^{2}+619170 x +190406\right )}{130438 \left (2+3 x \right ) \left (3+5 x \right )^{2}}\)	\(97\)
trager	\(\frac {\left (501975 x^{2}+619170 x +190406\right ) \sqrt {1-2 x}}{1694 \left (3+5 x \right )^{2} \left (2+3 x \right )}+\frac {954 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )-21 \sqrt {1-2 x}}{2+3 x}\right )}{49}-\frac {32025 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{2662}\)	\(124\)

Input:

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

Output:

-1/1694*(1003950*x^3+736365*x^2-238358*x-190406)/(2+3*x)/(1-2*x)^(1/2)/(3+ 
5*x)^2+1908/49*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-32025/1331*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 = 130, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {224175 \, \sqrt {\frac {5}{11}} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac {5 \, x + 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 230868 \, \sqrt {\frac {3}{7}} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac {3 \, x - 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + {\left (501975 \, x^{2} + 619170 \, x + 190406\right )} \sqrt {-2 \, x + 1}}{1694 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \] Input:

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

Output:

1/1694*(224175*sqrt(5/11)*(75*x^3 + 140*x^2 + 87*x + 18)*log((5*x + 11*sqr 
t(5/11)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 230868*sqrt(3/7)*(75*x^3 + 140*x^ 
2 + 87*x + 18)*log((3*x - 7*sqrt(3/7)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + (50 
1975*x^2 + 619170*x + 190406)*sqrt(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 1 
8)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.15 (sec) , antiderivative size = 3624, normalized size of antiderivative = 28.76 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1866900000*sqrt(55)*I*(x - 1/2)**(15/2)*atan(sqrt(110)/(10*sqrt(x - 1/2))) 
/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 484968484 
00*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2 
)**(7/2) + 6684099653*(x - 1/2)**(5/2)) - 92286600000*sqrt(55)*I*(x - 1/2) 
**(15/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(3913140000*(x - 1/2)**(15/2) + 
21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 5397785316 
0*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)** 
(5/2)) + 1437480000*sqrt(21)*I*(x - 1/2)**(15/2)*atan(sqrt(42)/(6*sqrt(x - 
 1/2)))/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48 
496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*( 
x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 153810360000*sqrt(21)*I*( 
x - 1/2)**(15/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(3913140000*(x - 1/2)**(15 
/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 5397 
7853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 
1/2)**(5/2)) - 76905180000*sqrt(21)*I*pi*(x - 1/2)**(15/2)/(3913140000*(x 
- 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(1 
1/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 66840 
99653*(x - 1/2)**(5/2)) + 46143300000*sqrt(55)*I*pi*(x - 1/2)**(15/2)/(391 
3140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x 
 - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)*...
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {32025}{2662} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {954}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {501975 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 2242290 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2501939 \, \sqrt {-2 \, x + 1}}{847 \, {\left (75 \, {\left (2 \, x - 1\right )}^{3} + 505 \, {\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \] Input:

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

Output:

32025/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( 
-2*x + 1))) - 954/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) 
 + 3*sqrt(-2*x + 1))) + 1/847*(501975*(-2*x + 1)^(5/2) - 2242290*(-2*x + 1 
)^(3/2) + 2501939*sqrt(-2*x + 1))/(75*(2*x - 1)^3 + 505*(2*x - 1)^2 + 2266 
*x - 286)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {32025}{2662} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {954}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {27 \, \sqrt {-2 \, x + 1}}{7 \, {\left (3 \, x + 2\right )}} - \frac {25 \, {\left (645 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1397 \, \sqrt {-2 \, x + 1}\right )}}{484 \, {\left (5 \, x + 3\right )}^{2}} \] Input:

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

Output:

32025/2662*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) 
 + 5*sqrt(-2*x + 1))) - 954/49*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(- 
2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 27/7*sqrt(-2*x + 1)/(3*x + 2) - 
 25/484*(645*(-2*x + 1)^(3/2) - 1397*sqrt(-2*x + 1))/(5*x + 3)^2
 

Mupad [B] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {1908\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {32025\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {227449\,\sqrt {1-2\,x}}{5775}-\frac {149486\,{\left (1-2\,x\right )}^{3/2}}{4235}+\frac {6693\,{\left (1-2\,x\right )}^{5/2}}{847}}{\frac {2266\,x}{75}+\frac {101\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {286}{75}} \] Input:

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

Output:

(1908*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - (32025*55^(1/2)*a 
tanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331 + ((227449*(1 - 2*x)^(1/2))/5775 
 - (149486*(1 - 2*x)^(3/2))/4235 + (6693*(1 - 2*x)^(5/2))/847)/((2266*x)/7 
5 + (101*(2*x - 1)^2)/15 + (2*x - 1)^3 - 286/75)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.75 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx=\frac {38652075 \sqrt {-2 x +1}\, x^{2}[phone] \sqrt {-2 x +1}\, x [phone] \sqrt {-2 x +1}[phone] \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{3}[phone] \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}[phone] \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x [phone] \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-117691875 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{3}-219691500 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-136522575 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -28246050 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-190466100 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}-355536720 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-220940676 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -45711864 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )[phone] \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}[phone] \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}[phone] \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x [phone] \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{9782850 x^{3}[phone] x^{2}[phone] x +2347884} \] Input:

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

Output:

(38652075*sqrt( - 2*x + 1)*x**2 + 47676090*sqrt( - 2*x + 1)*x + 14661262*s 
qrt( - 2*x + 1) + 117691875*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x* 
*3 + 219691500*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**2 + 13652257 
5*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x + 28246050*sqrt(55)*log(5* 
sqrt( - 2*x + 1) - sqrt(55)) - 117691875*sqrt(55)*log(5*sqrt( - 2*x + 1) + 
 sqrt(55))*x**3 - 219691500*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x* 
*2 - 136522575*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x - 28246050*sq 
rt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) - 190466100*sqrt(21)*log(3*sqrt( 
 - 2*x + 1) - sqrt(21))*x**3 - 355536720*sqrt(21)*log(3*sqrt( - 2*x + 1) - 
 sqrt(21))*x**2 - 220940676*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x 
- 45711864*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 190466100*sqrt(21 
)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**3 + 355536720*sqrt(21)*log(3*sqrt( 
 - 2*x + 1) + sqrt(21))*x**2 + 220940676*sqrt(21)*log(3*sqrt( - 2*x + 1) + 
 sqrt(21))*x + 45711864*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(1304 
38*(75*x**3 + 140*x**2 + 87*x + 18))