Integrand size = 24, antiderivative size = 179 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx=-\frac {6845810}{195657 (1-2 x)^{3/2}}-\frac {77527480}{5021863 \sqrt {1-2 x}}-\frac {5165}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2}+\frac {9}{2 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {167960}{847 (1-2 x)^{3/2} (3+5 x)}+\frac {182655}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {7570625 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]
-6845810/195657/(1-2*x)^(3/2)-5165/154/(1-2*x)^(3/2)/(3+5*x)^2+3/14/(1-2*x )^(3/2)/(2+3*x)^2/(3+5*x)^2+9/2/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^2+167960/847 /(1-2*x)^(3/2)/(3+5*x)+182655/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^ (1/2)-7570625/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-7752748 0/5021863/(1-2*x)^(1/2)
Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {13236365823+9944654283 x-99160158305 x^2-93885376440 x^3+188418548700 x^4+209324196000 x^5}{30131178 (1-2 x)^{3/2} \left (6+19 x+15 x^2\right )^2}+\frac {182655}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {7570625 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]
(13236365823 + 9944654283*x - 99160158305*x^2 - 93885376440*x^3 + 18841854 8700*x^4 + 209324196000*x^5)/(30131178*(1 - 2*x)^(3/2)*(6 + 19*x + 15*x^2) ^2) + (182655*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (7570625*S qrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641
Time = 0.27 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {114, 168, 27, 168, 27, 168, 25, 169, 27, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^3 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{14} \int \frac {37-165 x}{(1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^3}dx+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \int \frac {35 (73-567 x)}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^3}dx+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (5 \int \frac {73-567 x}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^3}dx+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (5 \left (-\frac {1}{22} \int \frac {2 (421-21693 x)}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2}dx-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (5 \left (-\frac {1}{11} \int \frac {421-21693 x}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2}dx-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (5 \left (\frac {1}{11} \left (\frac {1}{11} \int -\frac {1007760 x+180701}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx+\frac {67184}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{14} \left (5 \left (\frac {1}{11} \left (\frac {67184}{11 (1-2 x)^{3/2} (5 x+3)}-\frac {1}{11} \int \frac {1007760 x+180701}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx\right )-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (5 \left (\frac {1}{11} \left (\frac {1}{11} \left (\frac {2}{231} \int \frac {3 (2515967-20537430 x)}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {2738324}{231 (1-2 x)^{3/2}}\right )+\frac {67184}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (5 \left (\frac {1}{11} \left (\frac {1}{11} \left (\frac {1}{77} \int \frac {2515967-20537430 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {2738324}{231 (1-2 x)^{3/2}}\right )+\frac {67184}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (5 \left (\frac {1}{11} \left (\frac {1}{11} \left (\frac {1}{77} \left (-\frac {2}{77} \int -\frac {379795411-232582440 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {31010992}{77 \sqrt {1-2 x}}\right )-\frac {2738324}{231 (1-2 x)^{3/2}}\right )+\frac {67184}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (5 \left (\frac {1}{11} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \int \frac {379795411-232582440 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {31010992}{77 \sqrt {1-2 x}}\right )-\frac {2738324}{231 (1-2 x)^{3/2}}\right )+\frac {67184}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{14} \left (5 \left (\frac {1}{11} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (2596724375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-1604551113 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {31010992}{77 \sqrt {1-2 x}}\right )-\frac {2738324}{231 (1-2 x)^{3/2}}\right )+\frac {67184}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{14} \left (5 \left (\frac {1}{11} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (1604551113 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-2596724375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {31010992}{77 \sqrt {1-2 x}}\right )-\frac {2738324}{231 (1-2 x)^{3/2}}\right )+\frac {67184}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{14} \left (5 \left (\frac {1}{11} \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (1069700742 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-1038689750 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {31010992}{77 \sqrt {1-2 x}}\right )-\frac {2738324}{231 (1-2 x)^{3/2}}\right )+\frac {67184}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1033}{11 (1-2 x)^{3/2} (5 x+3)^2}\right )+\frac {63}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}\) |
3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^2) + (63/((1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^2) + 5*(-1033/(11*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + (67184/(1 1*(1 - 2*x)^(3/2)*(3 + 5*x)) + (-2738324/(231*(1 - 2*x)^(3/2)) + (-3101099 2/(77*Sqrt[1 - 2*x]) + (1069700742*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2* x]] - 1038689750*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77)/77)/11) /11))/14
3.22.99.3.1 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*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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[(((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 = 1.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(-\frac {209324196000 x^{5}+188418548700 x^{4}-93885376440 x^{3}-99160158305 x^{2}+9944654283 x +13236365823}{30131178 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right )}+\frac {182655 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}-\frac {7570625 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}\) | \(91\) |
| derivativedivides | \(\frac {-\frac {265625 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {578125 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {7570625 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}-\frac {26244 \left (\frac {221 \left (1-2 x \right )^{\frac {3}{2}}}{36}-\frac {1561 \sqrt {1-2 x}}{108}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {182655 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {64}{1369599 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {13056}{35153041 \sqrt {1-2 x}}\) | \(112\) |
| default | \(\frac {-\frac {265625 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {578125 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {7570625 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}-\frac {26244 \left (\frac {221 \left (1-2 x \right )^{\frac {3}{2}}}{36}-\frac {1561 \sqrt {1-2 x}}{108}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {182655 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {64}{1369599 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {13056}{35153041 \sqrt {1-2 x}}\) | \(112\) |
| pseudoelliptic | \(\frac {\frac {4412121941}{10043726}-\frac {182655 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {21}}{2401}+\frac {7570625 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \left (15 x^{2}+19 x +6\right )^{2} \sqrt {55}}{161051}+\frac {34887366000 x^{5}}{5021863}+\frac {31403091450 x^{4}}{5021863}-\frac {15647562740 x^{3}}{5021863}-\frac {99160158305 x^{2}}{30131178}+\frac {3314884761 x}{10043726}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}\) | \(133\) |
| trager | \(\frac {\left (209324196000 x^{5}+188418548700 x^{4}-93885376440 x^{3}-99160158305 x^{2}+9944654283 x +13236365823\right ) \sqrt {1-2 x}}{30131178 \left (30 x^{3}+23 x^{2}-7 x -6\right )^{2}}+\frac {625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-8069862295\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-8069862295\right ) x +666215 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-8069862295\right )}{3+5 x}\right )}{322102}-\frac {182655 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{4802}\) | \(141\) |
-1/30131178*(209324196000*x^5+188418548700*x^4-93885376440*x^3-99160158305 *x^2+9944654283*x+13236365823)/(15*x^2+19*x+6)^2/(1-2*x)^(1/2)/(-1+2*x)+18 2655/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-7570625/161051*arct anh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.24 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {54531211875 \, \sqrt {11} \sqrt {5} {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 88250311215 \, \sqrt {7} \sqrt {3} {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (209324196000 \, x^{5} + 188418548700 \, x^{4} - 93885376440 \, x^{3} - 99160158305 \, x^{2} + 9944654283 \, x + 13236365823\right )} \sqrt {-2 \, x + 1}}{2320100706 \, {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )}} \]
1/2320100706*(54531211875*sqrt(11)*sqrt(5)*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 88250311215*sqrt(7)*sqrt(3)*(900*x^6 + 1380*x^5 + 109*x ^4 - 682*x^3 - 227*x^2 + 84*x + 36)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(209324196000*x^5 + 188418548700*x^4 - 938853764 40*x^3 - 99160158305*x^2 + 9944654283*x + 13236365823)*sqrt(-2*x + 1))/(90 0*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)
Result contains complex when optimal does not.
Time = 15.82 (sec) , antiderivative size = 2966, normalized size of antiderivative = 16.57 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx=\text {Too large to display} \]
3868311142080000*sqrt(2)*I*(x - 1/2)**(17/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 1458270 5295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 561239785103659 2*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 21920924557224000*sqrt(2 )*I*(x - 1/2)**(15/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 105920 2535612298*(x - 1/2)**3) + 49676964263942400*sqrt(2)*I*(x - 1/2)**(13/2)/( 501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 96536606255 81280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3 ) + 56275446111672480*sqrt(2)*I*(x - 1/2)**(11/2)/(501141752496000*(x - 1/ 2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14 582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851 036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 31867497856150880*s qrt(2)*I*(x - 1/2)**(9/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800 *(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2 )**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 10 59202535612298*(x - 1/2)**3) + 7216395978913044*sqrt(2)*I*(x - 1/2)**(7/2) /(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 965366...
Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {7570625}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {182655}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (26165524500 \, {\left (2 \, x - 1\right )}^{5} + 177932259675 \, {\left (2 \, x - 1\right )}^{4} + 403131105480 \, {\left (2 \, x - 1\right )}^{3} + 304294845085 \, {\left (2 \, x - 1\right )}^{2} - 25803008 \, x + 14988512\right )}}{15065589 \, {\left (225 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2040 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 6934 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 10472 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 5929 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
7570625/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*s qrt(-2*x + 1))) - 182655/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/ (sqrt(21) + 3*sqrt(-2*x + 1))) + 2/15065589*(26165524500*(2*x - 1)^5 + 177 932259675*(2*x - 1)^4 + 403131105480*(2*x - 1)^3 + 304294845085*(2*x - 1)^ 2 - 25803008*x + 14988512)/(225*(-2*x + 1)^(11/2) - 2040*(-2*x + 1)^(9/2) + 6934*(-2*x + 1)^(7/2) - 10472*(-2*x + 1)^(5/2) + 5929*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {7570625}{322102} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {182655}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {64 \, {\left (1224 \, x - 689\right )}}{105459123 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {2 \, {\left (5550396300 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 37744400445 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 85516621432 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 64553088299 \, \sqrt {-2 \, x + 1}\right )}}{3195731 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]
7570625/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt (55) + 5*sqrt(-2*x + 1))) - 182655/4802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/105459123*(1224*x - 689)/((2*x - 1)*sqrt(-2*x + 1)) + 2/3195731*(5550396300*(2*x - 1)^3*sqrt( -2*x + 1) + 37744400445*(2*x - 1)^2*sqrt(-2*x + 1) - 85516621432*(-2*x + 1 )^(3/2) + 64553088299*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2
Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^3} \, dx=\frac {182655\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {7570625\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051}+\frac {\frac {1580752442\,{\left (2\,x-1\right )}^2}{8804565}-\frac {8704\,x}{571725}+\frac {53750814064\,{\left (2\,x-1\right )}^3}{225983835}+\frac {1581620086\,{\left (2\,x-1\right )}^4}{15065589}+\frac {77527480\,{\left (2\,x-1\right )}^5}{5021863}+\frac {5056}{571725}}{\frac {5929\,{\left (1-2\,x\right )}^{3/2}}{225}-\frac {10472\,{\left (1-2\,x\right )}^{5/2}}{225}+\frac {6934\,{\left (1-2\,x\right )}^{7/2}}{225}-\frac {136\,{\left (1-2\,x\right )}^{9/2}}{15}+{\left (1-2\,x\right )}^{11/2}} \]
(182655*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (7570625*55^( 1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/161051 + ((1580752442*(2*x - 1) ^2)/8804565 - (8704*x)/571725 + (53750814064*(2*x - 1)^3)/225983835 + (158 1620086*(2*x - 1)^4)/15065589 + (77527480*(2*x - 1)^5)/5021863 + 5056/5717 25)/((5929*(1 - 2*x)^(3/2))/225 - (10472*(1 - 2*x)^(5/2))/225 + (6934*(1 - 2*x)^(7/2))/225 - (136*(1 - 2*x)^(9/2))/15 + (1 - 2*x)^(11/2))