Integrand size = 24, antiderivative size = 113 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3}+\frac {49 \sqrt {1-2 x}}{9 (2+3 x)^2}+\frac {1138 \sqrt {1-2 x}}{21 (2+3 x)}+\frac {78506 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-110 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
78506/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-110*arctanh(1/11*55 ^(1/2)*(1-2*x)^(1/2))*55^(1/2)+7/9*(1-2*x)^(1/2)/(2+3*x)^3+49/9*(1-2*x)^(1 /2)/(2+3*x)^2+1138/21*(1-2*x)^(1/2)/(2+3*x)
Time = 0.37 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (4797+13999 x+10242 x^2\right )}{21 (2+3 x)^3}+\frac {78506 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-110 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(4797 + 13999*x + 10242*x^2))/(21*(2 + 3*x)^3) + (78506*Arc Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21]) - 110*Sqrt[55]*ArcTanh[Sqrt[5 /11]*Sqrt[1 - 2*x]]
Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {109, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^4 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{9} \int \frac {120-163 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{14} \int \frac {42 (216-245 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {49 \sqrt {1-2 x}}{(3 x+2)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (3 \int \frac {216-245 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {49 \sqrt {1-2 x}}{(3 x+2)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (3 \left (\frac {1}{7} \int \frac {9291-5690 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {1138 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {49 \sqrt {1-2 x}}{(3 x+2)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{9} \left (3 \left (\frac {1}{7} \left (63525 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-39253 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {1138 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {49 \sqrt {1-2 x}}{(3 x+2)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{9} \left (3 \left (\frac {1}{7} \left (39253 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-63525 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {1138 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {49 \sqrt {1-2 x}}{(3 x+2)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{9} \left (3 \left (\frac {1}{7} \left (\frac {78506 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-2310 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {1138 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {49 \sqrt {1-2 x}}{(3 x+2)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3}\) |
(7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3) + ((49*Sqrt[1 - 2*x])/(2 + 3*x)^2 + 3*(( 1138*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + ((78506*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x ]])/Sqrt[21] - 2310*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/7))/9
3.20.5.3.1 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*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_) )^(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 = 1.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(-\frac {20484 x^{3}+17756 x^{2}-4405 x -4797}{21 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {78506 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}-110 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(69\) |
| derivativedivides | \(-110 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1138 \left (1-2 x \right )^{\frac {5}{2}}}{63}-\frac {6926 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {8204 \sqrt {1-2 x}}{81}\right )}{\left (-4-6 x \right )^{3}}+\frac {78506 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(75\) |
| default | \(-110 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {54 \left (\frac {1138 \left (1-2 x \right )^{\frac {5}{2}}}{63}-\frac {6926 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {8204 \sqrt {1-2 x}}{81}\right )}{\left (-4-6 x \right )^{3}}+\frac {78506 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{441}\) | \(75\) |
| pseudoelliptic | \(\frac {78506 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}-48510 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \sqrt {55}+21 \sqrt {1-2 x}\, \left (10242 x^{2}+13999 x +4797\right )}{441 \left (2+3 x \right )^{3}}\) | \(80\) |
| trager | \(\frac {\left (10242 x^{2}+13999 x +4797\right ) \sqrt {1-2 x}}{21 \left (2+3 x \right )^{3}}-55 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )-\frac {39253 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{441}\) | \(116\) |
-1/21*(20484*x^3+17756*x^2-4405*x-4797)/(2+3*x)^3/(1-2*x)^(1/2)+78506/441* arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-110*arctanh(1/11*55^(1/2)*(1- 2*x)^(1/2))*55^(1/2)
Time = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.15 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {24255 \, \sqrt {55} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 39253 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (10242 \, x^{2} + 13999 \, x + 4797\right )} \sqrt {-2 \, x + 1}}{441 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/441*(24255*sqrt(55)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x + sqrt(55)*sqr t(-2*x + 1) - 8)/(5*x + 3)) + 39253*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)* log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(10242*x^2 + 13999 *x + 4797)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
Time = 75.37 (sec) , antiderivative size = 581, normalized size of antiderivative = 5.14 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx=- \frac {605 \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 )}{7} + 55 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right ) + 1452 \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 ) - \frac {1736 \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 )}{3} + \frac {784 \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 )}{3} \]
-605*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 21)/3))/7 + 55*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2* x) + sqrt(55)/5)) + 1452*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) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3))) - 1736*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)))/3 + 784*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(2 1)/3)))/3
Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx=55 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {39253}{441} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (5121 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 24241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 28714 \, \sqrt {-2 \, x + 1}\right )}}{21 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
55*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1 ))) - 39253/441*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3* sqrt(-2*x + 1))) + 4/21*(5121*(-2*x + 1)^(5/2) - 24241*(-2*x + 1)^(3/2) + 28714*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx=55 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {39253}{441} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {5121 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 24241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 28714 \, \sqrt {-2 \, x + 1}}{42 \, {\left (3 \, x + 2\right )}^{3}} \]
55*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr t(-2*x + 1))) - 39253/441*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/42*(5121*(2*x - 1)^2*sqrt(-2*x + 1 ) - 24241*(-2*x + 1)^(3/2) + 28714*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 1.41 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx=\frac {78506\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441}-110\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {16408\,\sqrt {1-2\,x}}{81}-\frac {13852\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {2276\,{\left (1-2\,x\right )}^{5/2}}{63}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \]