Integrand size = 24, antiderivative size = 125 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx=-\frac {1115}{1617 (1-2 x)^{3/2}}-\frac {12295}{41503 \sqrt {1-2 x}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {33}{14 (1-2 x)^{3/2} (2+3 x)}+\frac {3645}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-1115/1617/(1-2*x)^(3/2)+3/14/(1-2*x)^(3/2)/(2+3*x)^2+33/14/(1-2*x)^(3/2)/ (2+3*x)+3645/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1250/1331*a rctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-12295/41503/(1-2*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {245383-594687 x-438840 x^2+1327860 x^3}{249018 (1-2 x)^{3/2} (2+3 x)^2}+\frac {3645}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {1250}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(245383 - 594687*x - 438840*x^2 + 1327860*x^3)/(249018*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (3645*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1250* Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121
Time = 0.24 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {114, 27, 168, 27, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^3 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{14} \int \frac {7 (1-15 x)}{(1-2 x)^{5/2} (3 x+2)^2 (5 x+3)}dx+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {1-15 x}{(1-2 x)^{5/2} (3 x+2)^2 (5 x+3)}dx+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \int -\frac {5 (165 x+29)}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx+\frac {33}{7 (1-2 x)^{3/2} (3 x+2)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {33}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \int \frac {165 x+29}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{2} \left (\frac {33}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {446}{231 (1-2 x)^{3/2}}-\frac {2}{231} \int \frac {3 (443-3345 x)}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {33}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {446}{231 (1-2 x)^{3/2}}-\frac {1}{77} \int \frac {443-3345 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{2} \left (\frac {33}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {2}{77} \int -\frac {63619-36885 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {4918}{77 \sqrt {1-2 x}}\right )+\frac {446}{231 (1-2 x)^{3/2}}\right )\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {33}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {4918}{77 \sqrt {1-2 x}}-\frac {1}{77} \int \frac {63619-36885 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx\right )+\frac {446}{231 (1-2 x)^{3/2}}\right )\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{2} \left (\frac {33}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {1}{77} \left (264627 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-428750 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {4918}{77 \sqrt {1-2 x}}\right )+\frac {446}{231 (1-2 x)^{3/2}}\right )\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {33}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {1}{77} \left (428750 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-264627 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {4918}{77 \sqrt {1-2 x}}\right )+\frac {446}{231 (1-2 x)^{3/2}}\right )\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {33}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {1}{77} \left (171500 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-176418 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {4918}{77 \sqrt {1-2 x}}\right )+\frac {446}{231 (1-2 x)^{3/2}}\right )\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2}\) |
3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (33/(7*(1 - 2*x)^(3/2)*(2 + 3*x)) - ( 5*(446/(231*(1 - 2*x)^(3/2)) + (4918/(77*Sqrt[1 - 2*x]) + (-176418*Sqrt[3/ 7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 171500*Sqrt[5/11]*ArcTanh[Sqrt[5/11] *Sqrt[1 - 2*x]])/77)/77))/7)/2
3.22.78.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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]
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]
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]
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])
Time = 3.58 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.67
| method | result | size |
| derivativedivides | \(-\frac {1250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {486 \left (\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{2}-\frac {581 \sqrt {1-2 x}}{18}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {3645 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {16}{11319 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2144}{290521 \sqrt {1-2 x}}\) | \(84\) |
| default | \(-\frac {1250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {486 \left (\frac {27 \left (1-2 x \right )^{\frac {3}{2}}}{2}-\frac {581 \sqrt {1-2 x}}{18}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {3645 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {16}{11319 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2144}{290521 \sqrt {1-2 x}}\) | \(84\) |
| pseudoelliptic | \(\frac {-\frac {3645 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{2} \sqrt {21}}{2401}+\frac {1250 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{2} \sqrt {55}}{1331}+\frac {221310 x^{3}}{41503}-\frac {73140 x^{2}}{41503}-\frac {198229 x}{83006}+\frac {245383}{249018}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{2}}\) | \(106\) |
| trager | \(\frac {\left (1327860 x^{3}-438840 x^{2}-594687 x +245383\right ) \sqrt {1-2 x}}{249018 \left (6 x^{2}+x -2\right )^{2}}-\frac {625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}-\frac {3645 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{4802}\) | \(124\) |
-1250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-486/2401*(27/2*(1 -2*x)^(3/2)-581/18*(1-2*x)^(1/2))/(-4-6*x)^2+3645/2401*arctanh(1/7*21^(1/2 )*(1-2*x)^(1/2))*21^(1/2)+16/11319/(1-2*x)^(3/2)+2144/290521/(1-2*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {9003750 \, \sqrt {11} \sqrt {5} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 14554485 \, \sqrt {7} \sqrt {3} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (1327860 \, x^{3} - 438840 \, x^{2} - 594687 \, x + 245383\right )} \sqrt {-2 \, x + 1}}{19174386 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
1/19174386*(9003750*sqrt(11)*sqrt(5)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)* log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 14554485*sqrt (7)*sqrt(3)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(-(sqrt(7)*sqrt(3)*sqr t(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(1327860*x^3 - 438840*x^2 - 594687* x + 245383)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)
Result contains complex when optimal does not.
Time = 9.79 (sec) , antiderivative size = 1352, normalized size of antiderivative = 10.82 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx=\text {Too large to display} \]
-3889620000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/ (4141667376*(x - 1/2)**(11/2) + 14495835816*(x - 1/2)**(9/2) + 16911808452 *(x - 1/2)**(7/2) + 6576814398*(x - 1/2)**(5/2)) + 6287537520*sqrt(21)*I*( x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(4141667376*(x - 1/2)**(11 /2) + 14495835816*(x - 1/2)**(9/2) + 16911808452*(x - 1/2)**(7/2) + 657681 4398*(x - 1/2)**(5/2)) - 3143768760*sqrt(21)*I*pi*(x - 1/2)**(11/2)/(41416 67376*(x - 1/2)**(11/2) + 14495835816*(x - 1/2)**(9/2) + 16911808452*(x - 1/2)**(7/2) + 6576814398*(x - 1/2)**(5/2)) + 1944810000*sqrt(55)*I*pi*(x - 1/2)**(11/2)/(4141667376*(x - 1/2)**(11/2) + 14495835816*(x - 1/2)**(9/2) + 16911808452*(x - 1/2)**(7/2) + 6576814398*(x - 1/2)**(5/2)) - 136136700 00*sqrt(55)*I*(x - 1/2)**(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(414166737 6*(x - 1/2)**(11/2) + 14495835816*(x - 1/2)**(9/2) + 16911808452*(x - 1/2) **(7/2) + 6576814398*(x - 1/2)**(5/2)) + 22006381320*sqrt(21)*I*(x - 1/2)* *(9/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(4141667376*(x - 1/2)**(11/2) + 1449 5835816*(x - 1/2)**(9/2) + 16911808452*(x - 1/2)**(7/2) + 6576814398*(x - 1/2)**(5/2)) - 11003190660*sqrt(21)*I*pi*(x - 1/2)**(9/2)/(4141667376*(x - 1/2)**(11/2) + 14495835816*(x - 1/2)**(9/2) + 16911808452*(x - 1/2)**(7/2 ) + 6576814398*(x - 1/2)**(5/2)) + 6806835000*sqrt(55)*I*pi*(x - 1/2)**(9/ 2)/(4141667376*(x - 1/2)**(11/2) + 14495835816*(x - 1/2)**(9/2) + 16911808 452*(x - 1/2)**(7/2) + 6576814398*(x - 1/2)**(5/2)) - 15882615000*sqrt(...
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {625}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3645}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {331965 \, {\left (2 \, x - 1\right )}^{3} + 776475 \, {\left (2 \, x - 1\right )}^{2} - 75264 \, x + 46256}{124509 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
625/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 *x + 1))) - 3645/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) + 1/124509*(331965*(2*x - 1)^3 + 776475*(2*x - 1)^2 - 75264*x + 46256)/(9*(-2*x + 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {625}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3645}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (804 \, x - 479\right )}}{871563 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {27 \, {\left (243 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 581 \, \sqrt {-2 \, x + 1}\right )}}{9604 \, {\left (3 \, x + 2\right )}^{2}} \]
625/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3645/4802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt( -2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/871563*(804*x - 479)/((2*x - 1)*sqrt(-2*x + 1)) - 27/9604*(243*(-2*x + 1)^(3/2) - 581*sqrt(-2*x + 1)) /(3*x + 2)^2
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {3645\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {1250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {12325\,{\left (2\,x-1\right )}^2}{17787}-\frac {512\,x}{7623}+\frac {12295\,{\left (2\,x-1\right )}^3}{41503}+\frac {944}{22869}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \]