Integrand size = 24, antiderivative size = 124 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=-\frac {1133 \sqrt {1-2 x}}{30 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)^2}+\frac {7103 \sqrt {1-2 x}}{30 (3+5 x)}+1400 \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {7209}{5} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/3*(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^2-7209/25*arctanh(1/11*55^(1/2)*(1-2*x)^ (1/2))*55^(1/2)+1400/3*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1133/3 0*(1-2*x)^(1/2)/(3+5*x)^2+7103/30*(1-2*x)^(1/2)/(3+5*x)
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=1400 \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1}{50} \left (\frac {5 \sqrt {1-2 x} \left (13474+43806 x+35515 x^2\right )}{(2+3 x) (3+5 x)^2}-14418 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right ) \]
1400*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + ((5*Sqrt[1 - 2*x]*(13474 + 43806*x + 35515*x^2))/((2 + 3*x)*(3 + 5*x)^2) - 14418*Sqrt[55]*ArcTanh[ Sqrt[5/11]*Sqrt[1 - 2*x]])/50
Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {109, 166, 25, 168, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^2 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{3} \int \frac {(166-101 x) \sqrt {1-2 x}}{(3 x+2) (5 x+3)^3}dx+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{10} \int -\frac {9266-10601 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {1133 \sqrt {1-2 x}}{10 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (-\frac {1}{10} \int \frac {9266-10601 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {1133 \sqrt {1-2 x}}{10 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{10} \left (\frac {1}{11} \int \frac {33 (11598-7103 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {7103 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1133 \sqrt {1-2 x}}{10 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{10} \left (3 \int \frac {11598-7103 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {7103 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1133 \sqrt {1-2 x}}{10 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{10} \left (3 \left (79299 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-49000 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {7103 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1133 \sqrt {1-2 x}}{10 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{10} \left (3 \left (49000 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-79299 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {7103 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1133 \sqrt {1-2 x}}{10 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{10} \left (3 \left (14000 \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-14418 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {7103 \sqrt {1-2 x}}{5 x+3}\right )-\frac {1133 \sqrt {1-2 x}}{10 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)^2}\) |
(7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*(3 + 5*x)^2) + ((-1133*Sqrt[1 - 2*x])/(10 *(3 + 5*x)^2) + ((7103*Sqrt[1 - 2*x])/(3 + 5*x) + 3*(14000*Sqrt[7/3]*ArcTa nh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 14418*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/10)/3
3.20.97.3.1 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*((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_))^(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 = 3.36 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(-\frac {71030 x^{3}+52097 x^{2}-16858 x -13474}{10 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (2+3 x \right )}+\frac {1400 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3}-\frac {7209 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{25}\) | \(76\) |
| derivativedivides | \(\frac {-1551 \left (1-2 x \right )^{\frac {3}{2}}+\frac {16819 \sqrt {1-2 x}}{5}}{\left (-6-10 x \right )^{2}}-\frac {7209 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{25}-\frac {98 \sqrt {1-2 x}}{3 \left (-\frac {4}{3}-2 x \right )}+\frac {1400 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3}\) | \(82\) |
| default | \(\frac {-1551 \left (1-2 x \right )^{\frac {3}{2}}+\frac {16819 \sqrt {1-2 x}}{5}}{\left (-6-10 x \right )^{2}}-\frac {7209 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{25}-\frac {98 \sqrt {1-2 x}}{3 \left (-\frac {4}{3}-2 x \right )}+\frac {1400 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{3}\) | \(82\) |
| pseudoelliptic | \(\frac {70000 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {21}-43254 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right ) \left (3+5 x \right )^{2} \sqrt {55}+15 \sqrt {1-2 x}\, \left (35515 x^{2}+43806 x +13474\right )}{150 \left (2+3 x \right ) \left (3+5 x \right )^{2}}\) | \(97\) |
| trager | \(\frac {\left (35515 x^{2}+43806 x +13474\right ) \sqrt {1-2 x}}{10 \left (3+5 x \right )^{2} \left (2+3 x \right )}-\frac {7209 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{50}-\frac {700 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{3}\) | \(123\) |
-1/10*(71030*x^3+52097*x^2-16858*x-13474)/(3+5*x)^2/(1-2*x)^(1/2)/(2+3*x)+ 1400/3*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-7209/25*arctanh(1/11*5 5^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {21627 \, \sqrt {11} \sqrt {5} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 35000 \, \sqrt {7} \sqrt {3} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 15 \, {\left (35515 \, x^{2} + 43806 \, x + 13474\right )} \sqrt {-2 \, x + 1}}{150 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]
1/150*(21627*sqrt(11)*sqrt(5)*(75*x^3 + 140*x^2 + 87*x + 18)*log((sqrt(11) *sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 35000*sqrt(7)*sqrt(3)*(75* x^3 + 140*x^2 + 87*x + 18)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5) /(3*x + 2)) + 15*(35515*x^2 + 43806*x + 13474)*sqrt(-2*x + 1))/(75*x^3 + 1 40*x^2 + 87*x + 18)
Time = 53.92 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.94 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=- 231 \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 ) + 147 \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 ) + 1372 \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 {34848 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{5} + \frac {10648 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{5} \]
-231*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 21)/3)) + 147*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x ) + sqrt(55)/5)) + 1372*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))) + 34848*Piecewise((sqrt(55)*( -log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1) /4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/ 11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5 )))/5 + 10648*Piecewise((sqrt(55)*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x) /11 + 1)) + 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sq rt(1 - 2*x)/11 - 1)) - 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)**2))/6655, (s qrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/5
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {7209}{50} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {700}{3} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {35515 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 158642 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 177023 \, \sqrt {-2 \, x + 1}}{5 \, {\left (75 \, {\left (2 \, x - 1\right )}^{3} + 505 \, {\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \]
7209/50*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* x + 1))) - 700/3*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) + 1/5*(35515*(-2*x + 1)^(5/2) - 158642*(-2*x + 1)^(3/2) + 177023*sqrt(-2*x + 1))/(75*(2*x - 1)^3 + 505*(2*x - 1)^2 + 2266*x - 286)
Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {7209}{50} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {700}{3} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {49 \, \sqrt {-2 \, x + 1}}{3 \, x + 2} - \frac {11 \, {\left (705 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1529 \, \sqrt {-2 \, x + 1}\right )}}{20 \, {\left (5 \, x + 3\right )}^{2}} \]
7209/50*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 700/3*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 49*sqrt(-2*x + 1)/(3*x + 2) - 11/20 *(705*(-2*x + 1)^(3/2) - 1529*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {1400\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3}-\frac {7209\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{25}+\frac {\frac {177023\,\sqrt {1-2\,x}}{375}-\frac {158642\,{\left (1-2\,x\right )}^{3/2}}{375}+\frac {7103\,{\left (1-2\,x\right )}^{5/2}}{75}}{\frac {2266\,x}{75}+\frac {101\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {286}{75}} \]