Integrand size = 24, antiderivative size = 154 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {44545 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}-\frac {307295 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/9*(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)-307295/63*arctanh(1/7*21^(1/2)*(1-2*x) ^(1/2))*21^(1/2)+3014*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-44545/ 18*(1-2*x)^(1/2)/(3+5*x)+917/54*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)+6649/27*(1 -2*x)^(1/2)/(2+3*x)/(3+5*x)
Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.62 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (112668+516513 x+788512 x^2+400905 x^3\right )}{6 (2+3 x)^3 (3+5 x)}-\frac {307295 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-1/6*(Sqrt[1 - 2*x]*(112668 + 516513*x + 788512*x^2 + 400905*x^3))/((2 + 3 *x)^3*(3 + 5*x)) - (307295*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21]) + 3014*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.26 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {109, 166, 25, 168, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^4 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{9} \int \frac {(197-163 x) \sqrt {1-2 x}}{(3 x+2)^3 (5 x+3)^2}dx+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{9} \left (\frac {917 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}-\frac {1}{6} \int -\frac {16180-22273 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \int \frac {16180-22273 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx+\frac {917 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (\frac {1}{7} \int \frac {105 (11621-13298 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {13298 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {917 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (15 \int \frac {11621-13298 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {13298 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {917 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (15 \left (-\frac {1}{11} \int \frac {33 (14547-8909 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {8909 \sqrt {1-2 x}}{5 x+3}\right )+\frac {13298 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {917 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (15 \left (-3 \int \frac {14547-8909 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {8909 \sqrt {1-2 x}}{5 x+3}\right )+\frac {13298 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {917 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (15 \left (-3 \left (99462 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-61459 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {8909 \sqrt {1-2 x}}{5 x+3}\right )+\frac {13298 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {917 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (15 \left (-3 \left (61459 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-99462 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {8909 \sqrt {1-2 x}}{5 x+3}\right )+\frac {13298 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {917 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (15 \left (-3 \left (\frac {122918 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-18084 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {8909 \sqrt {1-2 x}}{5 x+3}\right )+\frac {13298 \sqrt {1-2 x}}{(3 x+2) (5 x+3)}\right )+\frac {917 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}\) |
(7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 + 5*x)) + ((917*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x)) + ((13298*Sqrt[1 - 2*x])/((2 + 3*x)*(3 + 5*x)) + 15*( (-8909*Sqrt[1 - 2*x])/(3 + 5*x) - 3*((122918*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2* x]])/Sqrt[21] - 18084*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])))/6)/9
3.20.88.3.1 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 = 1.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.53
| method | result | size |
| risch | \(\frac {801810 x^{4}+1176119 x^{3}+244514 x^{2}-291177 x -112668}{6 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (3+5 x \right )}-\frac {307295 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{63}+3014 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(81\) |
| derivativedivides | \(\frac {242 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+3014 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {20193 \left (1-2 x \right )^{\frac {5}{2}}-\frac {285460 \left (1-2 x \right )^{\frac {3}{2}}}{3}+\frac {336385 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{3}}-\frac {307295 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{63}\) | \(91\) |
| default | \(\frac {242 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+3014 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}+\frac {20193 \left (1-2 x \right )^{\frac {5}{2}}-\frac {285460 \left (1-2 x \right )^{\frac {3}{2}}}{3}+\frac {336385 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{3}}-\frac {307295 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{63}\) | \(91\) |
| pseudoelliptic | \(\frac {-614590 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right ) \sqrt {21}+379764 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right ) \sqrt {55}-21 \sqrt {1-2 x}\, \left (400905 x^{3}+788512 x^{2}+516513 x +112668\right )}{126 \left (2+3 x \right )^{3} \left (3+5 x \right )}\) | \(102\) |
| trager | \(-\frac {\left (400905 x^{3}+788512 x^{2}+516513 x +112668\right ) \sqrt {1-2 x}}{6 \left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {307295 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{126}-11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1032295\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1032295\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1032295\right )+7535 \sqrt {1-2 x}}{3+5 x}\right )\) | \(128\) |
1/6*(801810*x^4+1176119*x^3+244514*x^2-291177*x-112668)/(2+3*x)^3/(1-2*x)^ (1/2)/(3+5*x)-307295/63*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+3014* arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx=\frac {189882 \, \sqrt {55} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 307295 \, \sqrt {21} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (400905 \, x^{3} + 788512 \, x^{2} + 516513 \, x + 112668\right )} \sqrt {-2 \, x + 1}}{126 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
1/126*(189882*sqrt(55)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 307295*sqrt(21)*(135*x^4 + 35 1*x^3 + 342*x^2 + 148*x + 24)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(400905*x^3 + 788512*x^2 + 516513*x + 112668)*sqrt(-2*x + 1))/ (135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)
Time = 85.88 (sec) , antiderivative size = 700, normalized size of antiderivative = 4.55 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx=\frac {16698 \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} - 1518 \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 ) - 30492 \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 {25088 \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 {5488 \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} - 26620 \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 ) \]
16698*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3))/7 - 1518*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5)) - 30492*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))) + 25088*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)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt( 1 - 2*x) < sqrt(21)/3)))/3 - 5488*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqr t(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqr t(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)))/3 - 26620*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/1 1 - 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)))
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx=-1507 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {307295}{126} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {400905 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 2779739 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 6422815 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4945325 \, \sqrt {-2 \, x + 1}}{3 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \]
-1507*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 307295/126*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/3*(400905*(-2*x + 1)^(7/2) - 2779739*(-2*x + 1)^( 5/2) + 6422815*(-2*x + 1)^(3/2) - 4945325*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 1242*(2*x - 1)^3 + 4284*(2*x - 1)^2 + 13132*x - 2793)
Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx=-1507 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {307295}{126} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {605 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {60579 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 285460 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 336385 \, \sqrt {-2 \, x + 1}}{24 \, {\left (3 \, x + 2\right )}^{3}} \]
-1507*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5* sqrt(-2*x + 1))) + 307295/126*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2 *x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 605*sqrt(-2*x + 1)/(5*x + 3) - 1 /24*(60579*(2*x - 1)^2*sqrt(-2*x + 1) - 285460*(-2*x + 1)^(3/2) + 336385*s qrt(-2*x + 1))/(3*x + 2)^3
Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx=3014\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {307295\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{63}-\frac {\frac {989065\,\sqrt {1-2\,x}}{81}-\frac {1284563\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {2779739\,{\left (1-2\,x\right )}^{5/2}}{405}-\frac {8909\,{\left (1-2\,x\right )}^{7/2}}{9}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \]
3014*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) - (307295*21^(1/2)*atan h((21^(1/2)*(1 - 2*x)^(1/2))/7))/63 - ((989065*(1 - 2*x)^(1/2))/81 - (1284 563*(1 - 2*x)^(3/2))/81 + (2779739*(1 - 2*x)^(5/2))/405 - (8909*(1 - 2*x)^ (7/2))/9)/((13132*x)/135 + (476*(2*x - 1)^2)/15 + (46*(2*x - 1)^3)/5 + (2* x - 1)^4 - 931/45)