Integrand size = 24, antiderivative size = 172 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=-\frac {169975 \sqrt {1-2 x}}{54 (3+5 x)^2}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {581 \sqrt {1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac {1256 \sqrt {1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac {113875 \sqrt {1-2 x}}{6 (3+5 x)}+\frac {785570 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
7/9*(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x)^2+785570/21*arctanh(1/7*21^(1/2)*(1-2* x)^(1/2))*21^(1/2)-23115*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-169 975/54*(1-2*x)^(1/2)/(3+5*x)^2+581/27*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2+12 56/3*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+113875/6*(1-2*x)^(1/2)/(3+5*x)
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (864074+5401374 x+12649336 x^2+13153400 x^3+5124375 x^4\right )}{2 (2+3 x)^3 (3+5 x)^2}+\frac {785570 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-23115 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(864074 + 5401374*x + 12649336*x^2 + 13153400*x^3 + 5124375 *x^4))/(2*(2 + 3*x)^3*(3 + 5*x)^2) + (785570*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2* x]])/Sqrt[21] - 23115*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {109, 166, 27, 168, 27, 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.
\(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^4 (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {1}{9} \int \frac {(232-233 x) \sqrt {1-2 x}}{(3 x+2)^3 (5 x+3)^3}dx+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 166 |
\(\displaystyle \frac {1}{9} \left (\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}-\frac {1}{6} \int -\frac {2 (13130-19869 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \int \frac {13130-19869 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (\frac {1}{7} \int \frac {35 (40877-56520 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {11304 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (5 \int \frac {40877-56520 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {11304 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (5 \left (-\frac {1}{22} \int \frac {99 (29708-33995 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {33995 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11304 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (5 \left (-\frac {9}{2} \int \frac {29708-33995 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {33995 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11304 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (5 \left (-\frac {9}{2} \left (-\frac {1}{11} \int \frac {33 (37188-22775 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {22775 \sqrt {1-2 x}}{5 x+3}\right )-\frac {33995 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11304 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (5 \left (-\frac {9}{2} \left (-3 \int \frac {37188-22775 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {22775 \sqrt {1-2 x}}{5 x+3}\right )-\frac {33995 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11304 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (5 \left (-\frac {9}{2} \left (-3 \left (254265 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-157114 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {22775 \sqrt {1-2 x}}{5 x+3}\right )-\frac {33995 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11304 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (5 \left (-\frac {9}{2} \left (-3 \left (157114 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-254265 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {22775 \sqrt {1-2 x}}{5 x+3}\right )-\frac {33995 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11304 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (5 \left (-\frac {9}{2} \left (-3 \left (\frac {314228 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-9246 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {22775 \sqrt {1-2 x}}{5 x+3}\right )-\frac {33995 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11304 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^2}\right )+\frac {581 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}\right )+\frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}\) |
(7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 + 5*x)^2) + ((581*Sqrt[1 - 2*x])/(3* (2 + 3*x)^2*(3 + 5*x)^2) + ((11304*Sqrt[1 - 2*x])/((2 + 3*x)*(3 + 5*x)^2) + 5*((-33995*Sqrt[1 - 2*x])/(2*(3 + 5*x)^2) - (9*((-22775*Sqrt[1 - 2*x])/( 3 + 5*x) - 3*((314228*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 9246*Sq rt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])))/2))/3)/9
3.20.99.3.1 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 = 1.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.50
method | result | size |
risch | \(-\frac {10248750 x^{5}+21182425 x^{4}+12145272 x^{3}-1846588 x^{2}-3673226 x -864074}{2 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}+\frac {785570 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21}-23115 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(86\) |
derivativedivides | \(\frac {-75075 \left (1-2 x \right )^{\frac {3}{2}}+163955 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-23115 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {108 \left (\frac {6883 \left (1-2 x \right )^{\frac {5}{2}}}{6}-\frac {145600 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {342265 \sqrt {1-2 x}}{54}\right )}{\left (-4-6 x \right )^{3}}+\frac {785570 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21}\) | \(103\) |
default | \(\frac {-75075 \left (1-2 x \right )^{\frac {3}{2}}+163955 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-23115 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {108 \left (\frac {6883 \left (1-2 x \right )^{\frac {5}{2}}}{6}-\frac {145600 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {342265 \sqrt {1-2 x}}{54}\right )}{\left (-4-6 x \right )^{3}}+\frac {785570 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21}\) | \(103\) |
pseudoelliptic | \(\frac {1571140 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right )^{2} \sqrt {21}-970830 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right )^{2} \sqrt {55}+21 \sqrt {1-2 x}\, \left (5124375 x^{4}+13153400 x^{3}+12649336 x^{2}+5401374 x +864074\right )}{42 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}\) | \(111\) |
trager | \(\frac {\left (5124375 x^{4}+13153400 x^{3}+12649336 x^{2}+5401374 x +864074\right ) \sqrt {1-2 x}}{2 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}+\frac {23115 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{2}-\frac {392785 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{21}\) | \(133\) |
-1/2*(10248750*x^5+21182425*x^4+12145272*x^3-1846588*x^2-3673226*x-864074) /(2+3*x)^3/(1-2*x)^(1/2)/(3+5*x)^2+785570/21*arctanh(1/7*21^(1/2)*(1-2*x)^ (1/2))*21^(1/2)-23115*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {485415 \, \sqrt {55} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 785570 \, \sqrt {21} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (5124375 \, x^{4} + 13153400 \, x^{3} + 12649336 \, x^{2} + 5401374 \, x + 864074\right )} \sqrt {-2 \, x + 1}}{42 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
1/42*(485415*sqrt(55)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 785570*sqrt(21)*( 675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((3*x - sqrt(21) *sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(5124375*x^4 + 13153400*x^3 + 1264933 6*x^2 + 5401374*x + 864074)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)
Time = 112.73 (sec) , antiderivative size = 864, normalized size of antiderivative = 5.02 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\text {Too large to display} \]
-128634*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sq rt(21)/3))/7 + 11694*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt( 1 - 2*x) + sqrt(55)/5)) + 188496*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)*sq rt(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))) - 38808*Piecewise((s qrt(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)*s qrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(s qrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (s qrt(1 - 2*x) < sqrt(21)/3))) + 5488*Piecewise((sqrt(21)*(-5*log(sqrt(21)*s qrt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(s qrt(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))) + 333960*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))) + 53240*Piecewise((sq...
Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {23115}{2} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {392785}{21} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {5124375 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 46804300 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 160263994 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 243823580 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 139064695 \, \sqrt {-2 \, x + 1}}{675 \, {\left (2 \, x - 1\right )}^{5} + 7695 \, {\left (2 \, x - 1\right )}^{4} + 35082 \, {\left (2 \, x - 1\right )}^{3} + 79954 \, {\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588} \]
23115/2*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* x + 1))) - 392785/21*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + (5124375*(-2*x + 1)^(9/2) - 46804300*(-2*x + 1)^(7 /2) + 160263994*(-2*x + 1)^(5/2) - 243823580*(-2*x + 1)^(3/2) + 139064695* sqrt(-2*x + 1))/(675*(2*x - 1)^5 + 7695*(2*x - 1)^4 + 35082*(2*x - 1)^3 + 79954*(2*x - 1)^2 + 182182*x - 49588)
Time = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {23115}{2} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {392785}{21} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {55 \, {\left (1365 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2981 \, \sqrt {-2 \, x + 1}\right )}}{4 \, {\left (5 \, x + 3\right )}^{2}} + \frac {61947 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 291200 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 342265 \, \sqrt {-2 \, x + 1}}{4 \, {\left (3 \, x + 2\right )}^{3}} \]
23115/2*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 392785/21*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(- 2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 55/4*(1365*(-2*x + 1)^(3/2) - 2 981*sqrt(-2*x + 1))/(5*x + 3)^2 + 1/4*(61947*(2*x - 1)^2*sqrt(-2*x + 1) - 291200*(-2*x + 1)^(3/2) + 342265*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {785570\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21}-23115\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {27812939\,\sqrt {1-2\,x}}{135}-\frac {48764716\,{\left (1-2\,x\right )}^{3/2}}{135}+\frac {160263994\,{\left (1-2\,x\right )}^{5/2}}{675}-\frac {1872172\,{\left (1-2\,x\right )}^{7/2}}{27}+\frac {22775\,{\left (1-2\,x\right )}^{9/2}}{3}}{\frac {182182\,x}{675}+\frac {79954\,{\left (2\,x-1\right )}^2}{675}+\frac {3898\,{\left (2\,x-1\right )}^3}{75}+\frac {57\,{\left (2\,x-1\right )}^4}{5}+{\left (2\,x-1\right )}^5-\frac {49588}{675}} \]
(785570*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/21 - 23115*55^(1/2)* atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((27812939*(1 - 2*x)^(1/2))/135 - ( 48764716*(1 - 2*x)^(3/2))/135 + (160263994*(1 - 2*x)^(5/2))/675 - (1872172 *(1 - 2*x)^(7/2))/27 + (22775*(1 - 2*x)^(9/2))/3)/((182182*x)/675 + (79954 *(2*x - 1)^2)/675 + (3898*(2*x - 1)^3)/75 + (57*(2*x - 1)^4)/5 + (2*x - 1) ^5 - 49588/675)