Integrand size = 26, antiderivative size = 173 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {181304825 \sqrt {1-2 x}}{12096 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {2051 \sqrt {1-2 x}}{216 (2+3 x)^3 \sqrt {3+5 x}}+\frac {22957 \sqrt {1-2 x}}{288 (2+3 x)^2 \sqrt {3+5 x}}+\frac {3997345 \sqrt {1-2 x}}{4032 (2+3 x) \sqrt {3+5 x}}+\frac {46095555 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}} \]
46095555/3136*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+7/12 *(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(1/2)-181304825/12096*(1-2*x)^(1/2)/(3+5* x)^(1/2)+2051/216*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^(1/2)+22957/288*(1-2*x)^ (1/2)/(2+3*x)^2/(3+5*x)^(1/2)+3997345/4032*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^( 1/2)
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.49 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\frac {-\frac {7 \sqrt {1-2 x} \left (103735088+628209228 x+1426133132 x^2+1438446565 x^3+543914475 x^4\right )}{(2+3 x)^4 \sqrt {3+5 x}}+46095555 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136} \]
((-7*Sqrt[1 - 2*x]*(103735088 + 628209228*x + 1426133132*x^2 + 1438446565* x^3 + 543914475*x^4))/((2 + 3*x)^4*Sqrt[3 + 5*x]) + 46095555*Sqrt[7]*ArcTa n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3136
Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {109, 27, 166, 27, 168, 27, 168, 27, 169, 27, 104, 217}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^5 (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{12} \int \frac {(425-388 x) \sqrt {1-2 x}}{2 (3 x+2)^4 (5 x+3)^{3/2}}dx+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \int \frac {(425-388 x) \sqrt {1-2 x}}{(3 x+2)^4 (5 x+3)^{3/2}}dx+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{24} \left (\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}-\frac {1}{9} \int -\frac {11 (7433-10764 x)}{2 \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{3/2}}dx\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {11}{18} \int \frac {7433-10764 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{24} \left (\frac {11}{18} \left (\frac {1}{14} \int \frac {35 (39287-50088 x)}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {6261 \sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {11}{18} \left (\frac {5}{4} \int \frac {39287-50088 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {6261 \sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{24} \left (\frac {11}{18} \left (\frac {5}{4} \left (\frac {1}{7} \int \frac {4635749-4360740 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {218037 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {6261 \sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {11}{18} \left (\frac {5}{4} \left (\frac {1}{14} \int \frac {4635749-4360740 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {218037 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {6261 \sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{24} \left (\frac {11}{18} \left (\frac {5}{4} \left (\frac {1}{14} \left (-\frac {2}{11} \int \frac {248915997}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {72521930 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {218037 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {6261 \sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{24} \left (\frac {11}{18} \left (\frac {5}{4} \left (\frac {1}{14} \left (-22628727 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {72521930 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {218037 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {6261 \sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{24} \left (\frac {11}{18} \left (\frac {5}{4} \left (\frac {1}{14} \left (-45257454 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {72521930 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {218037 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {6261 \sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{24} \left (\frac {11}{18} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {45257454 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}-\frac {72521930 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )+\frac {218037 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {6261 \sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {2051 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt {5 x+3}}\) |
(7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*Sqrt[3 + 5*x]) + ((2051*Sqrt[1 - 2*x]) /(9*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (11*((6261*Sqrt[1 - 2*x])/(2*(2 + 3*x)^2* Sqrt[3 + 5*x]) + (5*((218037*Sqrt[1 - 2*x])/(7*(2 + 3*x)*Sqrt[3 + 5*x]) + ((-72521930*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) + (45257454*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/14))/4))/18)/24
3.25.45.3.1 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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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[((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] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(134)=268\).
Time = 1.19 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.72
| method | result | size |
| default | \(-\frac {\left (18668699775 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+60984419265 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+79653119040 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+7614802650 x^{4} \sqrt {-10 x^{2}-x +3}+51995786040 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+20138251910 x^{3} \sqrt {-10 x^{2}-x +3}+16963164240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +19965863848 x^{2} \sqrt {-10 x^{2}-x +3}+2212586640 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8794929192 x \sqrt {-10 x^{2}-x +3}+1452291232 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{6272 \left (2+3 x \right )^{4} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(298\) |
-1/6272*(18668699775*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^( 1/2))*x^5+60984419265*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^ (1/2))*x^4+79653119040*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3) ^(1/2))*x^3+7614802650*x^4*(-10*x^2-x+3)^(1/2)+51995786040*7^(1/2)*arctan( 1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+20138251910*x^3*(-10*x^2-x +3)^(1/2)+16963164240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^ (1/2))*x+19965863848*x^2*(-10*x^2-x+3)^(1/2)+2212586640*7^(1/2)*arctan(1/1 4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+8794929192*x*(-10*x^2-x+3)^(1/2)+ 1452291232*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^4/(-10*x^2-x+3)^(1/2 )/(3+5*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\frac {46095555 \, \sqrt {7} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (543914475 \, x^{4} + 1438446565 \, x^{3} + 1426133132 \, x^{2} + 628209228 \, x + 103735088\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6272 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]
1/6272*(46095555*sqrt(7)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^ 2 + x - 3)) - 14*(543914475*x^4 + 1438446565*x^3 + 1426133132*x^2 + 628209 228*x + 103735088)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(405*x^5 + 1323*x^4 + 172 8*x^3 + 1128*x^2 + 368*x + 48)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{5} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (134) = 268\).
Time = 0.30 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.71 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {46095555}{6272} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {181304825 \, x}{6048 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {189299515}{12096 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{108 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {13181}{648 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {466361}{2592 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {1301839}{576 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
-46095555/6272*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 181304825/6048*x/sqrt(-10*x^2 - x + 3) - 189299515/12096/sqrt(-10*x^2 - x + 3) + 343/108/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)* x^3 + 216*sqrt(-10*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt (-10*x^2 - x + 3)) + 13181/648/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10 *x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 466361/2592/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 1301839/576/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt (-10*x^2 - x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (134) = 268\).
Time = 0.54 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.47 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {9219111}{12544} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {605}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {605 \, \sqrt {10} {\left (77025 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 51138136 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 12067876800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {984130112000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {3936520448000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{224 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
-9219111/12544*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(- 10*x + 5) - sqrt(22)))) - 605/2*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt( 22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 605/224*sqrt(10)*(77025*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 51138136* ((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr t(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 12067876800*((sqrt(2)*sqrt(-10*x + 5 ) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - s qrt(22)))^3 + 984130112000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 3936520448000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(( (sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt (2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^5\,{\left (5\,x+3\right )}^{3/2}} \,d x \]