Integrand size = 24, antiderivative size = 154 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx=-\frac {15313 \sqrt {1-2 x}}{444528 (2+3 x)}-\frac {653 \sqrt {1-2 x} (3+5 x)^2}{2520 (2+3 x)^4}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{5 (2+3 x)^5}-\frac {\sqrt {1-2 x} (413424+664915 x)}{317520 (2+3 x)^3}-\frac {15313 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \]
-1/18*(1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^6-15313/4667544*arctanh(1/7*21^(1/2) *(1-2*x)^(1/2))*21^(1/2)-15313/444528*(1-2*x)^(1/2)/(2+3*x)-653/2520*(3+5* x)^2*(1-2*x)^(1/2)/(2+3*x)^4+2/5*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^5-1/31752 0*(413424+664915*x)*(1-2*x)^(1/2)/(2+3*x)^3
Time = 0.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.49 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (1660816-10947400 x-75153042 x^2-122053374 x^3-46991565 x^4+18605295 x^5\right )}{2 (2+3 x)^6}-76565 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{23337720} \]
((-21*Sqrt[1 - 2*x]*(1660816 - 10947400*x - 75153042*x^2 - 122053374*x^3 - 46991565*x^4 + 18605295*x^5))/(2*(2 + 3*x)^6) - 76565*Sqrt[21]*ArcTanh[Sq rt[3/7]*Sqrt[1 - 2*x]])/23337720
Time = 0.24 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {108, 27, 166, 27, 166, 162, 52, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^3}{(3 x+2)^7} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{18} \int \frac {3 (2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^6}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {(2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^6}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {12 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}-\frac {1}{15} \int -\frac {3 (131-130 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^5}dx\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {(131-130 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^5}dx+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{84} \int \frac {(9323-8405 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {653 \sqrt {1-2 x} (5 x+3)^2}{84 (3 x+2)^4}\right )+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{84} \left (\frac {76565}{126} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (664915 x+413424)}{126 (3 x+2)^3}\right )-\frac {653 \sqrt {1-2 x} (5 x+3)^2}{84 (3 x+2)^4}\right )+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{84} \left (\frac {76565}{126} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (664915 x+413424)}{126 (3 x+2)^3}\right )-\frac {653 \sqrt {1-2 x} (5 x+3)^2}{84 (3 x+2)^4}\right )+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{84} \left (\frac {76565}{126} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (664915 x+413424)}{126 (3 x+2)^3}\right )-\frac {653 \sqrt {1-2 x} (5 x+3)^2}{84 (3 x+2)^4}\right )+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{84} \left (\frac {76565}{126} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (664915 x+413424)}{126 (3 x+2)^3}\right )-\frac {653 \sqrt {1-2 x} (5 x+3)^2}{84 (3 x+2)^4}\right )+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{5 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}\) |
-1/18*((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^6 + ((12*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(5*(2 + 3*x)^5) + ((-653*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(84*(2 + 3*x)^ 4) + (-1/126*(Sqrt[1 - 2*x]*(413424 + 664915*x))/(2 + 3*x)^3 + (76565*(-1/ 7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[2 1])))/126)/84)/5)/6
3.19.93.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
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*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
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 = 3.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43
| method | result | size |
| risch | \(\frac {37210590 x^{6}-112588425 x^{5}-197115183 x^{4}-28252710 x^{3}+53258242 x^{2}+14269032 x -1660816}{2222640 \left (2+3 x \right )^{6} \sqrt {1-2 x}}-\frac {15313 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\) | \(66\) |
| pseudoelliptic | \(\frac {-153130 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}-21 \sqrt {1-2 x}\, \left (18605295 x^{5}-46991565 x^{4}-122053374 x^{3}-75153042 x^{2}-10947400 x +1660816\right )}{46675440 \left (2+3 x \right )^{6}}\) | \(70\) |
| derivativedivides | \(-\frac {11664 \left (-\frac {15313 \left (1-2 x \right )^{\frac {11}{2}}}{10668672}-\frac {3037 \left (1-2 x \right )^{\frac {9}{2}}}{41150592}+\frac {256271 \left (1-2 x \right )^{\frac {7}{2}}}{4898880}-\frac {923549 \left (1-2 x \right )^{\frac {5}{2}}}{4898880}+\frac {1822247 \left (1-2 x \right )^{\frac {3}{2}}}{7558272}-\frac {750337 \sqrt {1-2 x}}{7558272}\right )}{\left (-4-6 x \right )^{6}}-\frac {15313 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\) | \(84\) |
| default | \(-\frac {11664 \left (-\frac {15313 \left (1-2 x \right )^{\frac {11}{2}}}{10668672}-\frac {3037 \left (1-2 x \right )^{\frac {9}{2}}}{41150592}+\frac {256271 \left (1-2 x \right )^{\frac {7}{2}}}{4898880}-\frac {923549 \left (1-2 x \right )^{\frac {5}{2}}}{4898880}+\frac {1822247 \left (1-2 x \right )^{\frac {3}{2}}}{7558272}-\frac {750337 \sqrt {1-2 x}}{7558272}\right )}{\left (-4-6 x \right )^{6}}-\frac {15313 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\) | \(84\) |
| trager | \(-\frac {\left (18605295 x^{5}-46991565 x^{4}-122053374 x^{3}-75153042 x^{2}-10947400 x +1660816\right ) \sqrt {1-2 x}}{2222640 \left (2+3 x \right )^{6}}+\frac {15313 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{9335088}\) | \(87\) |
1/2222640*(37210590*x^6-112588425*x^5-197115183*x^4-28252710*x^3+53258242* x^2+14269032*x-1660816)/(2+3*x)^6/(1-2*x)^(1/2)-15313/4667544*arctanh(1/7* 21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {76565 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (18605295 \, x^{5} - 46991565 \, x^{4} - 122053374 \, x^{3} - 75153042 \, x^{2} - 10947400 \, x + 1660816\right )} \sqrt {-2 \, x + 1}}{46675440 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
1/46675440*(76565*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 216 0*x^2 + 576*x + 64)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 2 1*(18605295*x^5 - 46991565*x^4 - 122053374*x^3 - 75153042*x^2 - 10947400*x + 1660816)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 21 60*x^2 + 576*x + 64)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {15313}{9335088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {18605295 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 956655 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 678093066 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 2443710654 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 3125153605 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1286827955 \, \sqrt {-2 \, x + 1}}{1111320 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]
15313/9335088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq rt(-2*x + 1))) + 1/1111320*(18605295*(-2*x + 1)^(11/2) + 956655*(-2*x + 1) ^(9/2) - 678093066*(-2*x + 1)^(7/2) + 2443710654*(-2*x + 1)^(5/2) - 312515 3605*(-2*x + 1)^(3/2) + 1286827955*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 1020 6*(2*x - 1)^5 + 59535*(2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^ 2 + 605052*x - 184877)
Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {15313}{9335088} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {18605295 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - 956655 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 678093066 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 2443710654 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 3125153605 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1286827955 \, \sqrt {-2 \, x + 1}}{71124480 \, {\left (3 \, x + 2\right )}^{6}} \]
15313/9335088*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) - 1/71124480*(18605295*(2*x - 1)^5*sqrt(-2*x + 1) - 956655*(2*x - 1)^4*sqrt(-2*x + 1) - 678093066*(2*x - 1)^3*sqrt(-2*x + 1) - 2443710654*(2*x - 1)^2*sqrt(-2*x + 1) + 3125153605*(-2*x + 1)^(3/2) - 1 286827955*sqrt(-2*x + 1))/(3*x + 2)^6
Time = 1.53 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {\frac {750337\,\sqrt {1-2\,x}}{472392}-\frac {1822247\,{\left (1-2\,x\right )}^{3/2}}{472392}+\frac {923549\,{\left (1-2\,x\right )}^{5/2}}{306180}-\frac {256271\,{\left (1-2\,x\right )}^{7/2}}{306180}+\frac {3037\,{\left (1-2\,x\right )}^{9/2}}{2571912}+\frac {15313\,{\left (1-2\,x\right )}^{11/2}}{666792}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}}-\frac {15313\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4667544} \]
((750337*(1 - 2*x)^(1/2))/472392 - (1822247*(1 - 2*x)^(3/2))/472392 + (923 549*(1 - 2*x)^(5/2))/306180 - (256271*(1 - 2*x)^(7/2))/306180 + (3037*(1 - 2*x)^(9/2))/2571912 + (15313*(1 - 2*x)^(11/2))/666792)/((67228*x)/81 + (1 2005*(2*x - 1)^2)/27 + (6860*(2*x - 1)^3)/27 + (245*(2*x - 1)^4)/3 + 14*(2 *x - 1)^5 + (2*x - 1)^6 - 184877/729) - (15313*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/4667544