Integrand size = 24, antiderivative size = 132 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {13900}{17787 (1-2 x)^{3/2}}+\frac {159800}{456533 \sqrt {1-2 x}}-\frac {340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {4050}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {15250 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
13900/17787/(1-2*x)^(3/2)-340/77/(1-2*x)^(3/2)/(3+5*x)+3/7/(1-2*x)^(3/2)/( 2+3*x)/(3+5*x)-4050/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1525 0/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+159800/456533/(1-2*x )^(1/2)
Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {-2209989+5548760 x+5028300 x^2-14382000 x^3}{1369599 (1-2 x)^{3/2} \left (6+19 x+15 x^2\right )}-\frac {4050}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {15250 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
(-2209989 + 5548760*x + 5028300*x^2 - 14382000*x^3)/(1369599*(1 - 2*x)^(3/ 2)*(6 + 19*x + 15*x^2)) - (4050*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] )/343 + (15250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331
Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {114, 27, 168, 27, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{7} \int \frac {5 (1-21 x)}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2}dx+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \int \frac {1-21 x}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2}dx+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{7} \left (-\frac {1}{11} \int -\frac {5 (204 x+37)}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx-\frac {68}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \left (\frac {5}{11} \int \frac {204 x+37}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx-\frac {68}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{7} \left (\frac {5}{11} \left (\frac {556}{231 (1-2 x)^{3/2}}-\frac {2}{231} \int \frac {3 (487-4170 x)}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )-\frac {68}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \left (\frac {5}{11} \left (\frac {556}{231 (1-2 x)^{3/2}}-\frac {1}{77} \int \frac {487-4170 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )-\frac {68}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{7} \left (\frac {5}{11} \left (\frac {1}{77} \left (\frac {2}{77} \int -\frac {75851-47940 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {6392}{77 \sqrt {1-2 x}}\right )+\frac {556}{231 (1-2 x)^{3/2}}\right )-\frac {68}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \left (\frac {5}{11} \left (\frac {1}{77} \left (\frac {6392}{77 \sqrt {1-2 x}}-\frac {1}{77} \int \frac {75851-47940 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx\right )+\frac {556}{231 (1-2 x)^{3/2}}\right )-\frac {68}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {5}{7} \left (\frac {5}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (323433 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-523075 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {6392}{77 \sqrt {1-2 x}}\right )+\frac {556}{231 (1-2 x)^{3/2}}\right )-\frac {68}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5}{7} \left (\frac {5}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (523075 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-323433 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {6392}{77 \sqrt {1-2 x}}\right )+\frac {556}{231 (1-2 x)^{3/2}}\right )-\frac {68}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{7} \left (\frac {5}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (209230 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-215622 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {6392}{77 \sqrt {1-2 x}}\right )+\frac {556}{231 (1-2 x)^{3/2}}\right )-\frac {68}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}\) |
3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)) + (5*(-68/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) + (5*(556/(231*(1 - 2*x)^(3/2)) + (6392/(77*Sqrt[1 - 2*x]) + (-21 5622*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 209230*Sqrt[5/11]*ArcTan h[Sqrt[5/11]*Sqrt[1 - 2*x]])/77)/77))/11))/7
3.22.88.3.1 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*(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 = 4.77 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(\frac {14382000 x^{3}-5028300 x^{2}-5548760 x +2209989}{1369599 \left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {4050 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {15250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}\) | \(81\) |
| derivativedivides | \(\frac {250 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {15250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {54 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}-\frac {4050 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {16}{17787 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2176}{456533 \sqrt {1-2 x}}\) | \(88\) |
| default | \(\frac {250 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {15250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {54 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}-\frac {4050 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {16}{17787 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2176}{456533 \sqrt {1-2 x}}\) | \(88\) |
| pseudoelliptic | \(-\frac {91500 \left (\frac {8103293}{31384500}-\frac {395307 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (30 x^{3}+23 x^{2}-7 x -6\right ) \sqrt {21}}{1464610}+\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (30 x^{3}+23 x^{2}-7 x -6\right ) \sqrt {55}}{6}+\frac {35156 x^{3}}{20923}-\frac {61457 x^{2}}{104615}-\frac {62282 x}{96075}\right )}{14641 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right ) \left (3+5 x \right )}\) | \(119\) |
| trager | \(-\frac {\left (14382000 x^{3}-5028300 x^{2}-5548760 x +2209989\right ) \sqrt {1-2 x}}{1369599 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}-\frac {7625 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{14641}+\frac {2025 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{2401}\) | \(133\) |
1/1369599*(14382000*x^3-5028300*x^2-5548760*x+2209989)/(15*x^2+19*x+6)/(1- 2*x)^(1/2)/(-1+2*x)-4050/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2) +15250/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {54922875 \, \sqrt {11} \sqrt {5} {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 88944075 \, \sqrt {7} \sqrt {3} {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (14382000 \, x^{3} - 5028300 \, x^{2} - 5548760 \, x + 2209989\right )} \sqrt {-2 \, x + 1}}{105459123 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} \]
1/105459123*(54922875*sqrt(11)*sqrt(5)*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6 )*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 88944075*s qrt(7)*sqrt(3)*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log((sqrt(7)*sqrt(3)*s qrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(14382000*x^3 - 5028300*x^2 - 554 8760*x + 2209989)*sqrt(-2*x + 1))/(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)
Result contains complex when optimal does not.
Time = 9.51 (sec) , antiderivative size = 2966, normalized size of antiderivative = 22.47 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx=\text {Too large to display} \]
-3986690400000*sqrt(2)*I*(x - 1/2)**(17/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 662850240685 248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1/2 )**4 + 48145569800559*(x - 1/2)**3) - 22659187320000*sqrt(2)*I*(x - 1/2)** (15/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 43880 2755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908*( x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) - 51564023280000*sqrt(2)*I*(x - 1/2)**(13/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 6628502406 85248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1 /2)**4 + 48145569800559*(x - 1/2)**3) - 58784347960800*sqrt(2)*I*(x - 1/2) **(11/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438 802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908 *(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3 ) - 33664789429040*sqrt(2)*I*(x - 1/2)**(9/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 662850240 685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) - 7840313302668*sqrt(2)*I*(x - 1/2) **(7/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 4388 02755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 5631302031589...
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {7625}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2025}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4 \, {\left (1797750 \, {\left (2 \, x - 1\right )}^{3} + 4136175 \, {\left (2 \, x - 1\right )}^{2} + 209440 \, x - 128436\right )}}{1369599 \, {\left (15 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 68 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 77 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
-7625/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt (-2*x + 1))) + 2025/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt (21) + 3*sqrt(-2*x + 1))) - 4/1369599*(1797750*(2*x - 1)^3 + 4136175*(2*x - 1)^2 + 209440*x - 128436)/(15*(-2*x + 1)^(7/2) - 68*(-2*x + 1)^(5/2) + 7 7*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {7625}{14641} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2025}{2401} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (591090 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1343273 \, \sqrt {-2 \, x + 1}\right )}}{456533 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} + \frac {16 \, {\left (816 \, x - 485\right )}}{1369599 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]
-7625/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55 ) + 5*sqrt(-2*x + 1))) + 2025/2401*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sq rt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/456533*(591090*(-2*x + 1) ^(3/2) - 1343273*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9) + 16/1369599 *(816*x - 485)/((2*x - 1)*sqrt(-2*x + 1))
Time = 1.87 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {15250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641}-\frac {4050\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {\frac {2176\,x}{53361}+\frac {367660\,{\left (2\,x-1\right )}^2}{456533}+\frac {159800\,{\left (2\,x-1\right )}^3}{456533}-\frac {2224}{88935}}{\frac {77\,{\left (1-2\,x\right )}^{3/2}}{15}-\frac {68\,{\left (1-2\,x\right )}^{5/2}}{15}+{\left (1-2\,x\right )}^{7/2}} \]