Integrand size = 22, antiderivative size = 116 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {160}{3087 (1-2 x)^{3/2}}+\frac {160}{2401 \sqrt {1-2 x}}+\frac {1}{63 (1-2 x)^{3/2} (2+3 x)^3}-\frac {16}{147 (1-2 x)^{3/2} (2+3 x)^2}-\frac {16}{147 (1-2 x)^{3/2} (2+3 x)}-\frac {160 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401} \]
160/3087/(1-2*x)^(3/2)+1/63/(1-2*x)^(3/2)/(2+3*x)^3-16/147/(1-2*x)^(3/2)/( 2+3*x)^2-16/147/(1-2*x)^(3/2)/(2+3*x)-160/16807*arctanh(1/7*21^(1/2)*(1-2* x)^(1/2))*21^(1/2)+160/2401/(1-2*x)^(1/2)
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.60 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {8 \left (-\frac {7 \left (-2237-11280 x-4464 x^2+28800 x^3+25920 x^4\right )}{8 (1-2 x)^{3/2} (2+3 x)^3}-60 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )}{50421} \]
(8*((-7*(-2237 - 11280*x - 4464*x^2 + 28800*x^3 + 25920*x^4))/(8*(1 - 2*x) ^(3/2)*(2 + 3*x)^3) - 60*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]))/50421
Time = 0.19 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {87, 52, 52, 61, 61, 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.
| \(\displaystyle \int \frac {5 x+3}{(1-2 x)^{5/2} (3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {32}{21} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^3}dx+\frac {1}{63 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {32}{21} \left (\frac {1}{2} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^2}dx-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{63 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {32}{21} \left (\frac {1}{2} \left (\frac {5}{7} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)}dx-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{63 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {32}{21} \left (\frac {1}{2} \left (\frac {5}{7} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{63 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {32}{21} \left (\frac {1}{2} \left (\frac {5}{7} \left (\frac {3}{7} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{63 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {32}{21} \left (\frac {1}{2} \left (\frac {5}{7} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{63 (1-2 x)^{3/2} (3 x+2)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {32}{21} \left (\frac {1}{2} \left (\frac {5}{7} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )-\frac {1}{14 (1-2 x)^{3/2} (3 x+2)^2}\right )+\frac {1}{63 (1-2 x)^{3/2} (3 x+2)^3}\) |
1/(63*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (32*(-1/14*1/((1 - 2*x)^(3/2)*(2 + 3* x)^2) + (-1/7*1/((1 - 2*x)^(3/2)*(2 + 3*x)) + (5*(2/(21*(1 - 2*x)^(3/2)) + (3*(2/(7*Sqrt[1 - 2*x]) - (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/ 7))/7))/7)/2))/21
3.22.44.3.1 Defintions of rubi rules used
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] :> 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 1.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(\frac {25920 x^{4}+28800 x^{3}-4464 x^{2}-11280 x -2237}{7203 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {160 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{16807}\) | \(63\) |
| pseudoelliptic | \(\frac {\frac {160 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{3} \sqrt {21}}{16807}-\frac {8640 x^{4}}{2401}-\frac {9600 x^{3}}{2401}+\frac {1488 x^{2}}{2401}+\frac {3760 x}{2401}+\frac {2237}{7203}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{3}}\) | \(74\) |
| derivativedivides | \(\frac {\frac {9288 \left (1-2 x \right )^{\frac {5}{2}}}{16807}-\frac {960 \left (1-2 x \right )^{\frac {3}{2}}}{343}+\frac {1200 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{3}}-\frac {160 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{16807}+\frac {88}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {776}{16807 \sqrt {1-2 x}}\) | \(75\) |
| default | \(\frac {\frac {9288 \left (1-2 x \right )^{\frac {5}{2}}}{16807}-\frac {960 \left (1-2 x \right )^{\frac {3}{2}}}{343}+\frac {1200 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{3}}-\frac {160 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{16807}+\frac {88}{7203 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {776}{16807 \sqrt {1-2 x}}\) | \(75\) |
| trager | \(-\frac {\left (25920 x^{4}+28800 x^{3}-4464 x^{2}-11280 x -2237\right ) \sqrt {1-2 x}}{7203 \left (2+3 x \right )^{3} \left (-1+2 x \right )^{2}}+\frac {80 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{16807}\) | \(89\) |
1/7203*(25920*x^4+28800*x^3-4464*x^2-11280*x-2237)/(2+3*x)^3/(1-2*x)^(1/2) /(-1+2*x)-160/16807*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {240 \, \sqrt {7} \sqrt {3} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (25920 \, x^{4} + 28800 \, x^{3} - 4464 \, x^{2} - 11280 \, x - 2237\right )} \sqrt {-2 \, x + 1}}{50421 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
1/50421*(240*sqrt(7)*sqrt(3)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 7*(25920*x^ 4 + 28800*x^3 - 4464*x^2 - 11280*x - 2237)*sqrt(-2*x + 1))/(108*x^5 + 108* x^4 - 45*x^3 - 58*x^2 + 4*x + 8)
Time = 171.16 (sec) , antiderivative size = 566, normalized size of antiderivative = 4.88 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {388 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{117649} - \frac {1536 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{2401} + \frac {744 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{343} + \frac {48 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{49} + \frac {776}{16807 \sqrt {1 - 2 x}} + \frac {88}{7203 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
388*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 1)/3))/117649 - 1536*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/ 7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqr t(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/2401 + 744*Piecewise((sqrt(21)*( 3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2 *x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*s qrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2 *x) < sqrt(21)/3)))/343 + 48*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqrt(21) *sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 1/(48* (sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x )/7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt( 21)/3)))/49 + 776/(16807*sqrt(1 - 2*x)) + 88/(7203*(1 - 2*x)**(3/2))
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.95 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {80}{16807} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8 \, {\left (1620 \, {\left (2 \, x - 1\right )}^{4} + 10080 \, {\left (2 \, x - 1\right )}^{3} + 19404 \, {\left (2 \, x - 1\right )}^{2} + 18816 \, x - 13181\right )}}{7203 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 343 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
80/16807*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 *x + 1))) + 8/7203*(1620*(2*x - 1)^4 + 10080*(2*x - 1)^3 + 19404*(2*x - 1) ^2 + 18816*x - 13181)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) + 441*(- 2*x + 1)^(5/2) - 343*(-2*x + 1)^(3/2))
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.82 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=\frac {80}{16807} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {8 \, {\left (1620 \, {\left (2 \, x - 1\right )}^{4} + 10080 \, {\left (2 \, x - 1\right )}^{3} + 19404 \, {\left (2 \, x - 1\right )}^{2} + 18816 \, x - 13181\right )}}{7203 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}^{3}} \]
80/16807*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 8/7203*(1620*(2*x - 1)^4 + 10080*(2*x - 1)^3 + 19404* (2*x - 1)^2 + 18816*x - 13181)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))^3
Time = 1.37 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx=-\frac {\frac {1024\,x}{1323}+\frac {352\,{\left (2\,x-1\right )}^2}{441}+\frac {1280\,{\left (2\,x-1\right )}^3}{3087}+\frac {160\,{\left (2\,x-1\right )}^4}{2401}-\frac {2152}{3969}}{\frac {343\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {49\,{\left (1-2\,x\right )}^{5/2}}{3}+7\,{\left (1-2\,x\right )}^{7/2}-{\left (1-2\,x\right )}^{9/2}}-\frac {160\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{16807} \]