Integrand size = 24, antiderivative size = 137 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx=\frac {3 \sqrt {1-2 x}}{28 (2+3 x)^4}+\frac {3 \sqrt {1-2 x}}{4 (2+3 x)^3}+\frac {45 \sqrt {1-2 x}}{8 (2+3 x)^2}+\frac {3135 \sqrt {1-2 x}}{56 (2+3 x)}+\frac {36045}{28} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-1250 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
36045/196*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1250/11*arctanh(1/1 1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+3/28*(1-2*x)^(1/2)/(2+3*x)^4+3/4*(1-2*x )^(1/2)/(2+3*x)^3+45/8*(1-2*x)^(1/2)/(2+3*x)^2+3135/56*(1-2*x)^(1/2)/(2+3* x)
Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx=\frac {3 \sqrt {1-2 x} \left (8810+38922 x+57375 x^2+28215 x^3\right )}{56 (2+3 x)^4}+\frac {36045}{28} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-1250 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(3*Sqrt[1 - 2*x]*(8810 + 38922*x + 57375*x^2 + 28215*x^3))/(56*(2 + 3*x)^4 ) + (36045*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/28 - 1250*Sqrt[5/11 ]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {114, 27, 168, 27, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^5 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{28} \int \frac {7 (11-15 x)}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {11-15 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{21} \int \frac {105 (11-15 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {3 \sqrt {1-2 x}}{(3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (5 \int \frac {11-15 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {3 \sqrt {1-2 x}}{(3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (5 \left (\frac {1}{14} \int \frac {7 (119-135 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {9 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {3 \sqrt {1-2 x}}{(3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (5 \left (\frac {1}{2} \int \frac {119-135 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {9 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {3 \sqrt {1-2 x}}{(3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (5 \left (\frac {1}{2} \left (\frac {1}{7} \int \frac {5119-3135 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {627 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {9 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {3 \sqrt {1-2 x}}{(3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{4} \left (5 \left (\frac {1}{2} \left (\frac {1}{7} \left (35000 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-21627 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {627 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {9 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {3 \sqrt {1-2 x}}{(3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (5 \left (\frac {1}{2} \left (\frac {1}{7} \left (21627 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-35000 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {627 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {9 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {3 \sqrt {1-2 x}}{(3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (5 \left (\frac {1}{2} \left (\frac {1}{7} \left (14418 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-14000 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {627 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {9 \sqrt {1-2 x}}{2 (3 x+2)^2}\right )+\frac {3 \sqrt {1-2 x}}{(3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}\) |
(3*Sqrt[1 - 2*x])/(28*(2 + 3*x)^4) + ((3*Sqrt[1 - 2*x])/(2 + 3*x)^3 + 5*(( 9*Sqrt[1 - 2*x])/(2*(2 + 3*x)^2) + ((627*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + (1 4418*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 14000*Sqrt[5/11]*ArcTanh [Sqrt[5/11]*Sqrt[1 - 2*x]])/7)/2))/4
3.21.46.3.1 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[(((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 = 3.40 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(-\frac {3 \left (56430 x^{4}+86535 x^{3}+20469 x^{2}-21302 x -8810\right )}{56 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {36045 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{196}-\frac {1250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}\) | \(74\) |
| derivativedivides | \(-\frac {1250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {486 \left (\frac {1045 \left (1-2 x \right )^{\frac {7}{2}}}{168}-\frac {1055 \left (1-2 x \right )^{\frac {5}{2}}}{24}+\frac {22373 \left (1-2 x \right )^{\frac {3}{2}}}{216}-\frac {369133 \sqrt {1-2 x}}{4536}\right )}{\left (-4-6 x \right )^{4}}+\frac {36045 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{196}\) | \(84\) |
| default | \(-\frac {1250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {486 \left (\frac {1045 \left (1-2 x \right )^{\frac {7}{2}}}{168}-\frac {1055 \left (1-2 x \right )^{\frac {5}{2}}}{24}+\frac {22373 \left (1-2 x \right )^{\frac {3}{2}}}{216}-\frac {369133 \sqrt {1-2 x}}{4536}\right )}{\left (-4-6 x \right )^{4}}+\frac {36045 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{196}\) | \(84\) |
| pseudoelliptic | \(\frac {792990 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-490000 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{4} \sqrt {55}+231 \sqrt {1-2 x}\, \left (28215 x^{3}+57375 x^{2}+38922 x +8810\right )}{4312 \left (2+3 x \right )^{4}}\) | \(85\) |
| trager | \(\frac {3 \left (28215 x^{3}+57375 x^{2}+38922 x +8810\right ) \sqrt {1-2 x}}{56 \left (2+3 x \right )^{4}}-\frac {625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{11}-\frac {36045 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{392}\) | \(121\) |
-3/56*(56430*x^4+86535*x^3+20469*x^2-21302*x-8810)/(2+3*x)^4/(1-2*x)^(1/2) +36045/196*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1250/11*arctanh(1/ 11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx=\frac {245000 \, \sqrt {11} \sqrt {5} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 396495 \, \sqrt {7} \sqrt {3} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 231 \, {\left (28215 \, x^{3} + 57375 \, x^{2} + 38922 \, x + 8810\right )} \sqrt {-2 \, x + 1}}{4312 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/4312*(245000*sqrt(11)*sqrt(5)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*l og((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 396495*sqrt(7) *sqrt(3)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(-(sqrt(7)*sqrt(3)*sq rt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 231*(28215*x^3 + 57375*x^2 + 38922*x + 8810)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Result contains complex when optimal does not.
Time = 30.39 (sec) , antiderivative size = 25400, normalized size of antiderivative = 185.40 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx=\text {Too large to display} \]
50454063292862952898560*sqrt(2)*I*(x - 1/2)**(83/2)/(270375841569409125580 8*(x - 1/2)**42 + 63087696366195462635520*(x - 1/2)**41 + 6992219680586663 77543680*(x - 1/2)**40 + 4894553776410664642805760*(x - 1/2)**39 + 2426882 9141369545520578560*(x - 1/2)**38 + 90603628794446303276826624*(x - 1/2)** 37 + 264260583983801717890744320*(x - 1/2)**36 + 6166080292955373417450700 80*(x - 1/2)**35 + 1168986055539456210391695360*(x - 1/2)**34 + 1818422753 061376327275970560*(x - 1/2)**33 + 2333642533095432953337495552*(x - 1/2)* *32 + 2475075413889095556570071040*(x - 1/2)**31 + 21656909871529586119988 12160*(x - 1/2)**30 + 1554855067699560029127352320*(x - 1/2)**29 + 9069987 89491410016990955520*(x - 1/2)**28 + 423266101762658007929112576*(x - 1/2) **27 + 154315766267635732057488960*(x - 1/2)**26 + 42361190740135298996173 440*(x - 1/2)**25 + 8236898199470752582589280*(x - 1/2)**24 + 101154890168 9390668037280*(x - 1/2)**23 + 59007019265214455635508*(x - 1/2)**22) + 112 0088252001604262952960*sqrt(2)*I*(x - 1/2)**(81/2)/(2703758415694091255808 *(x - 1/2)**42 + 63087696366195462635520*(x - 1/2)**41 + 69922196805866637 7543680*(x - 1/2)**40 + 4894553776410664642805760*(x - 1/2)**39 + 24268829 141369545520578560*(x - 1/2)**38 + 90603628794446303276826624*(x - 1/2)**3 7 + 264260583983801717890744320*(x - 1/2)**36 + 61660802929553734174507008 0*(x - 1/2)**35 + 1168986055539456210391695360*(x - 1/2)**34 + 18184227530 61376327275970560*(x - 1/2)**33 + 2333642533095432953337495552*(x - 1/2...
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx=\frac {625}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {36045}{392} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3 \, {\left (28215 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 199395 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 469833 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 369133 \, \sqrt {-2 \, x + 1}\right )}}{28 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
625/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 36045/392*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 3/28*(28215*(-2*x + 1)^(7/2) - 199395*(-2*x + 1)^(5 /2) + 469833*(-2*x + 1)^(3/2) - 369133*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 7 56*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx=\frac {625}{11} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {36045}{392} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {3 \, {\left (28215 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 199395 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 469833 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 369133 \, \sqrt {-2 \, x + 1}\right )}}{448 \, {\left (3 \, x + 2\right )}^{4}} \]
625/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5 *sqrt(-2*x + 1))) - 36045/392*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2 *x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 3/448*(28215*(2*x - 1)^3*sqrt(-2 *x + 1) + 199395*(2*x - 1)^2*sqrt(-2*x + 1) - 469833*(-2*x + 1)^(3/2) + 36 9133*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.38 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx=\frac {36045\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{196}-\frac {1250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {369133\,\sqrt {1-2\,x}}{756}-\frac {22373\,{\left (1-2\,x\right )}^{3/2}}{36}+\frac {1055\,{\left (1-2\,x\right )}^{5/2}}{4}-\frac {1045\,{\left (1-2\,x\right )}^{7/2}}{28}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]
(36045*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/196 - (1250*55^(1/2)* atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/11 + ((369133*(1 - 2*x)^(1/2))/756 - (22373*(1 - 2*x)^(3/2))/36 + (1055*(1 - 2*x)^(5/2))/4 - (1045*(1 - 2*x)^( 7/2))/28)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)