Integrand size = 24, antiderivative size = 160 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx=-\frac {163474 \sqrt {1-2 x}}{36015 (2+3 x)^4}-\frac {163474 \sqrt {1-2 x}}{108045 (2+3 x)^3}-\frac {81737 \sqrt {1-2 x}}{151263 (2+3 x)^2}-\frac {81737 \sqrt {1-2 x}}{352947 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^5}+\frac {2 (55633+83544 x)}{5145 \sqrt {1-2 x} (2+3 x)^5}-\frac {163474 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{352947 \sqrt {21}} \]
11/21*(3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^5-163474/7411887*arctanh(1/7*21^(1/2 )*(1-2*x)^(1/2))*21^(1/2)+2/5145*(55633+83544*x)/(2+3*x)^5/(1-2*x)^(1/2)-1 63474/36015*(1-2*x)^(1/2)/(2+3*x)^4-163474/108045*(1-2*x)^(1/2)/(2+3*x)^3- 81737/151263*(1-2*x)^(1/2)/(2+3*x)^2-81737/352947*(1-2*x)^(1/2)/(2+3*x)
Time = 0.34 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.50 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx=\frac {2 \left (-\frac {21 \left (-5615203-42553376 x-99751837 x^2-22641149 x^3+232214817 x^4+323678520 x^5+132413940 x^6\right )}{2 (1-2 x)^{3/2} (2+3 x)^5}-408685 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{37059435} \]
(2*((-21*(-5615203 - 42553376*x - 99751837*x^2 - 22641149*x^3 + 232214817* x^4 + 323678520*x^5 + 132413940*x^6))/(2*(1 - 2*x)^(3/2)*(2 + 3*x)^5) - 40 8685*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]))/37059435
Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {109, 27, 161, 52, 52, 52, 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.
| \(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{5/2} (3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}-\frac {1}{21} \int -\frac {2 (5 x+3) (240 x+133)}{(1-2 x)^{3/2} (3 x+2)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{21} \int \frac {(5 x+3) (240 x+133)}{(1-2 x)^{3/2} (3 x+2)^6}dx+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}\) |
| \(\Big \downarrow \) 161 |
| \(\displaystyle \frac {2}{21} \left (\frac {326948}{245} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^5}dx+\frac {83544 x+55633}{245 \sqrt {1-2 x} (3 x+2)^5}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2}{21} \left (\frac {326948}{245} \left (\frac {1}{4} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {\sqrt {1-2 x}}{28 (3 x+2)^4}\right )+\frac {83544 x+55633}{245 \sqrt {1-2 x} (3 x+2)^5}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2}{21} \left (\frac {326948}{245} \left (\frac {1}{4} \left (\frac {5}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{28 (3 x+2)^4}\right )+\frac {83544 x+55633}{245 \sqrt {1-2 x} (3 x+2)^5}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2}{21} \left (\frac {326948}{245} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{28 (3 x+2)^4}\right )+\frac {83544 x+55633}{245 \sqrt {1-2 x} (3 x+2)^5}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2}{21} \left (\frac {326948}{245} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{28 (3 x+2)^4}\right )+\frac {83544 x+55633}{245 \sqrt {1-2 x} (3 x+2)^5}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2}{21} \left (\frac {326948}{245} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{28 (3 x+2)^4}\right )+\frac {83544 x+55633}{245 \sqrt {1-2 x} (3 x+2)^5}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2}{21} \left (\frac {326948}{245} \left (\frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{28 (3 x+2)^4}\right )+\frac {83544 x+55633}{245 \sqrt {1-2 x} (3 x+2)^5}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}\) |
(11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^5) + (2*((55633 + 83544*x)/ (245*Sqrt[1 - 2*x]*(2 + 3*x)^5) + (326948*(-1/28*Sqrt[1 - 2*x]/(2 + 3*x)^4 + (-1/21*Sqrt[1 - 2*x]/(2 + 3*x)^3 + (5*(-1/14*Sqrt[1 - 2*x]/(2 + 3*x)^2 + (3*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/ (7*Sqrt[21])))/14))/21)/4))/245))/21
3.22.68.3.1 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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1 ) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*( m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f* h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*( n + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c , d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]
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 = 1.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(\frac {132413940 x^{6}+323678520 x^{5}+232214817 x^{4}-22641149 x^{3}-99751837 x^{2}-42553376 x -5615203}{1764735 \left (2+3 x \right )^{5} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {163474 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7411887}\) | \(73\) |
| pseudoelliptic | \(\frac {\frac {163474 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{5} \sqrt {21}}{7411887}-\frac {8827596 x^{6}}{117649}-\frac {21578568 x^{5}}{117649}-\frac {77404939 x^{4}}{588245}+\frac {22641149 x^{3}}{1764735}+\frac {99751837 x^{2}}{1764735}+\frac {42553376 x}{1764735}+\frac {5615203}{1764735}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{5}}\) | \(84\) |
| derivativedivides | \(\frac {\frac {9020754 \left (1-2 x \right )^{\frac {9}{2}}}{823543}-\frac {12464124 \left (1-2 x \right )^{\frac {7}{2}}}{117649}+\frac {4589672 \left (1-2 x \right )^{\frac {5}{2}}}{12005}-\frac {628196 \left (1-2 x \right )^{\frac {3}{2}}}{1029}+\frac {53534 \sqrt {1-2 x}}{147}}{\left (-4-6 x \right )^{5}}-\frac {163474 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7411887}+\frac {10648}{352947 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {90024}{823543 \sqrt {1-2 x}}\) | \(93\) |
| default | \(\frac {\frac {9020754 \left (1-2 x \right )^{\frac {9}{2}}}{823543}-\frac {12464124 \left (1-2 x \right )^{\frac {7}{2}}}{117649}+\frac {4589672 \left (1-2 x \right )^{\frac {5}{2}}}{12005}-\frac {628196 \left (1-2 x \right )^{\frac {3}{2}}}{1029}+\frac {53534 \sqrt {1-2 x}}{147}}{\left (-4-6 x \right )^{5}}-\frac {163474 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7411887}+\frac {10648}{352947 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {90024}{823543 \sqrt {1-2 x}}\) | \(93\) |
| trager | \(-\frac {\left (132413940 x^{6}+323678520 x^{5}+232214817 x^{4}-22641149 x^{3}-99751837 x^{2}-42553376 x -5615203\right ) \sqrt {1-2 x}}{1764735 \left (2+3 x \right )^{5} \left (-1+2 x \right )^{2}}-\frac {81737 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{7411887}\) | \(99\) |
1/1764735*(132413940*x^6+323678520*x^5+232214817*x^4-22641149*x^3-99751837 *x^2-42553376*x-5615203)/(2+3*x)^5/(1-2*x)^(1/2)/(-1+2*x)-163474/7411887*a rctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.90 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx=\frac {408685 \, \sqrt {21} {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (132413940 \, x^{6} + 323678520 \, x^{5} + 232214817 \, x^{4} - 22641149 \, x^{3} - 99751837 \, x^{2} - 42553376 \, x - 5615203\right )} \sqrt {-2 \, x + 1}}{37059435 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} \]
1/37059435*(408685*sqrt(21)*(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840 *x^3 - 112*x^2 + 112*x + 32)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(132413940*x^6 + 323678520*x^5 + 232214817*x^4 - 22641149*x^3 - 99751837*x^2 - 42553376*x - 5615203)*sqrt(-2*x + 1))/(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)
Timed out. \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.91 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx=\frac {81737}{7411887} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (33103485 \, {\left (2 \, x - 1\right )}^{6} + 360460170 \, {\left (2 \, x - 1\right )}^{5} + 1537963392 \, {\left (2 \, x - 1\right )}^{4} + 3164039270 \, {\left (2 \, x - 1\right )}^{3} + 2973379535 \, {\left (2 \, x - 1\right )}^{2} + 1324775760 \, x - 1109790220\right )}}{1764735 \, {\left (243 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 2835 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 13230 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 30870 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 36015 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 16807 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
81737/7411887*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq rt(-2*x + 1))) + 2/1764735*(33103485*(2*x - 1)^6 + 360460170*(2*x - 1)^5 + 1537963392*(2*x - 1)^4 + 3164039270*(2*x - 1)^3 + 2973379535*(2*x - 1)^2 + 1324775760*x - 1109790220)/(243*(-2*x + 1)^(13/2) - 2835*(-2*x + 1)^(11/ 2) + 13230*(-2*x + 1)^(9/2) - 30870*(-2*x + 1)^(7/2) + 36015*(-2*x + 1)^(5 /2) - 16807*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.86 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx=\frac {81737}{7411887} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1936 \, {\left (279 \, x - 178\right )}}{2470629 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {67655655 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 654366510 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 2361386244 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 3770746490 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2249364845 \, \sqrt {-2 \, x + 1}}{197650320 \, {\left (3 \, x + 2\right )}^{5}} \]
81737/7411887*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) + 1936/2470629*(279*x - 178)/((2*x - 1)*sqrt(-2*x + 1)) - 1/197650320*(67655655*(2*x - 1)^4*sqrt(-2*x + 1) + 654366510*(2*x - 1)^3*sqrt(-2*x + 1) + 2361386244*(2*x - 1)^2*sqrt(-2*x + 1) - 3770746490 *(-2*x + 1)^(3/2) + 2249364845*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 1.43 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.80 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx=-\frac {163474\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7411887}-\frac {\frac {73568\,x}{11907}+\frac {3467498\,{\left (2\,x-1\right )}^2}{250047}+\frac {25828892\,{\left (2\,x-1\right )}^3}{1750329}+\frac {20924672\,{\left (2\,x-1\right )}^4}{2917215}+\frac {326948\,{\left (2\,x-1\right )}^5}{194481}+\frac {163474\,{\left (2\,x-1\right )}^6}{1058841}-\frac {184888}{35721}}{\frac {16807\,{\left (1-2\,x\right )}^{3/2}}{243}-\frac {12005\,{\left (1-2\,x\right )}^{5/2}}{81}+\frac {3430\,{\left (1-2\,x\right )}^{7/2}}{27}-\frac {490\,{\left (1-2\,x\right )}^{9/2}}{9}+\frac {35\,{\left (1-2\,x\right )}^{11/2}}{3}-{\left (1-2\,x\right )}^{13/2}} \]
- (163474*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/7411887 - ((73568* x)/11907 + (3467498*(2*x - 1)^2)/250047 + (25828892*(2*x - 1)^3)/1750329 + (20924672*(2*x - 1)^4)/2917215 + (326948*(2*x - 1)^5)/194481 + (163474*(2 *x - 1)^6)/1058841 - 184888/35721)/((16807*(1 - 2*x)^(3/2))/243 - (12005*( 1 - 2*x)^(5/2))/81 + (3430*(1 - 2*x)^(7/2))/27 - (490*(1 - 2*x)^(9/2))/9 + (35*(1 - 2*x)^(11/2))/3 - (1 - 2*x)^(13/2))