Integrand size = 24, antiderivative size = 146 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {245865}{41503 \sqrt {1-2 x}}-\frac {36175}{1078 \sqrt {1-2 x} (3+5 x)}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}+\frac {165}{49 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {70065}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {24000}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-70065/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+24000/1331*arctan h(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+245865/41503/(1-2*x)^(1/2)-36175/1 078/(3+5*x)/(1-2*x)^(1/2)+3/14/(2+3*x)^2/(3+5*x)/(1-2*x)^(1/2)+165/49/(2+3 *x)/(3+5*x)/(1-2*x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {-4664333-5050290 x+17711235 x^2+22127850 x^3}{83006 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}-\frac {70065}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {24000}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(-4664333 - 5050290*x + 17711235*x^2 + 22127850*x^3)/(83006*Sqrt[1 - 2*x]* (2 + 3*x)^2*(3 + 5*x)) - (70065*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] )/343 + (24000*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121
Time = 0.24 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {114, 27, 168, 168, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{14} \int \frac {5 (8-21 x)}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}dx+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \int \frac {8-21 x}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2}dx+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \int \frac {457-1650 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}dx+\frac {66}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (-\frac {1}{11} \int \frac {3 (2657-21705 x)}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {7235}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {66}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \int \frac {2657-21705 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {7235}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {66}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \left (-\frac {2}{77} \int -\frac {401281-245865 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {32782}{77 \sqrt {1-2 x}}\right )-\frac {7235}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {66}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \left (\frac {1}{77} \int \frac {401281-245865 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {32782}{77 \sqrt {1-2 x}}\right )-\frac {7235}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {66}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \left (\frac {1}{77} \left (2744000 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-1695573 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {32782}{77 \sqrt {1-2 x}}\right )-\frac {7235}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {66}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \left (\frac {1}{77} \left (1695573 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-2744000 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {32782}{77 \sqrt {1-2 x}}\right )-\frac {7235}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {66}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (-\frac {3}{11} \left (\frac {1}{77} \left (1130382 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-1097600 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {32782}{77 \sqrt {1-2 x}}\right )-\frac {7235}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {66}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}\) |
3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)) + (5*(66/(7*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)) + (-7235/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (3*(-32782/(77*Sqr t[1 - 2*x]) + (1130382*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 109760 0*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77))/11)/7))/14
3.22.24.3.1 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 = 3.52 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {22127850 x^{3}+17711235 x^{2}-5050290 x -4664333}{83006 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}-\frac {70065 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {24000 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}\) | \(76\) |
| derivativedivides | \(\frac {250 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {24000 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {\frac {243 \left (1-2 x \right )^{\frac {3}{2}}}{7}-\frac {4023 \sqrt {1-2 x}}{49}}{\left (-4-6 x \right )^{2}}-\frac {70065 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {32}{41503 \sqrt {1-2 x}}\) | \(91\) |
| default | \(\frac {250 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {24000 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {\frac {243 \left (1-2 x \right )^{\frac {3}{2}}}{7}-\frac {4023 \sqrt {1-2 x}}{49}}{\left (-4-6 x \right )^{2}}-\frac {70065 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {32}{41503 \sqrt {1-2 x}}\) | \(91\) |
| pseudoelliptic | \(\frac {-\frac {4664333}{83006}-\frac {70065 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {21}}{2401}+\frac {24000 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {55}}{1331}+\frac {11063925 x^{3}}{41503}+\frac {17711235 x^{2}}{83006}-\frac {360735 x}{5929}}{\left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {1-2 x}}\) | \(113\) |
| trager | \(-\frac {\left (22127850 x^{3}+17711235 x^{2}-5050290 x -4664333\right ) \sqrt {1-2 x}}{83006 \left (2+3 x \right )^{2} \left (10 x^{2}+x -3\right )}+\frac {12000 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}+\frac {405 \operatorname {RootOf}\left (\textit {\_Z}^{2}-628509\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-628509\right ) x +3633 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-628509\right )}{2+3 x}\right )}{4802}\) | \(131\) |
1/83006*(22127850*x^3+17711235*x^2-5050290*x-4664333)/(3+5*x)/(1-2*x)^(1/2 )/(2+3*x)^2-70065/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+24000/ 1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {57624000 \, \sqrt {11} \sqrt {5} {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 93256515 \, \sqrt {7} \sqrt {3} {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (22127850 \, x^{3} + 17711235 \, x^{2} - 5050290 \, x - 4664333\right )} \sqrt {-2 \, x + 1}}{6391462 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )}} \]
1/6391462*(57624000*sqrt(11)*sqrt(5)*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 1 2)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 93256515* sqrt(7)*sqrt(3)*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*log((sqrt(7)*sqrt( 3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(22127850*x^3 + 17711235*x^2 - 5050290*x - 4664333)*sqrt(-2*x + 1))/(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)
Result contains complex when optimal does not.
Time = 11.65 (sec) , antiderivative size = 2222, normalized size of antiderivative = 15.22 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx=\text {Too large to display} \]
-245353600800*sqrt(2)*I*(x - 1/2)**(13/2)/(82833347520*(x - 1/2)**7 + 4776 72304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/ 2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 113690431 0080*sqrt(2)*I*(x - 1/2)**(11/2)/(82833347520*(x - 1/2)**7 + 477672304032* (x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 7 32218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 1975315945680*sqrt (2)*I*(x - 1/2)**(9/2)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)* *6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 73221866964 4*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 1525208808816*sqrt(2)*I*(x - 1/2)**(7/2)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 11016 83522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2 )**3 + 168804902882*(x - 1/2)**2) - 441655676154*sqrt(2)*I*(x - 1/2)**(5/2 )/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 1688 04902882*(x - 1/2)**2) - 65076704*sqrt(2)*I*(x - 1/2)**(3/2)/(82833347520* (x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 12 70264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1 /2)**2) + 1493614080000*sqrt(55)*I*(x - 1/2)**7*atan(sqrt(110)*sqrt(x - 1/ 2)/11)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 11016835220 16*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**...
Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {12000}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {70065}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {11063925 \, {\left (2 \, x - 1\right )}^{3} + 50903010 \, {\left (2 \, x - 1\right )}^{2} + 117027330 \, x - 58496417}{41503 \, {\left (45 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 309 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 707 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 539 \, \sqrt {-2 \, x + 1}\right )}} \]
-12000/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt (-2*x + 1))) + 70065/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqr t(21) + 3*sqrt(-2*x + 1))) - 1/41503*(11063925*(2*x - 1)^3 + 50903010*(2*x - 1)^2 + 117027330*x - 58496417)/(45*(-2*x + 1)^(7/2) - 309*(-2*x + 1)^(5 /2) + 707*(-2*x + 1)^(3/2) - 539*sqrt(-2*x + 1))
Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {12000}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {70065}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (428910 \, x - 214279\right )}}{41503 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} + \frac {27 \, {\left (63 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 149 \, \sqrt {-2 \, x + 1}\right )}}{196 \, {\left (3 \, x + 2\right )}^{2}} \]
-12000/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55 ) + 5*sqrt(-2*x + 1))) + 70065/4802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*s qrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/41503*(428910*x - 214279 )/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1)) + 27/196*(63*(-2*x + 1)^(3/2) - 149*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {\frac {1114546\,x}{17787}+\frac {1131178\,{\left (2\,x-1\right )}^2}{41503}+\frac {245865\,{\left (2\,x-1\right )}^3}{41503}-\frac {8356631}{266805}}{\frac {539\,\sqrt {1-2\,x}}{45}-\frac {707\,{\left (1-2\,x\right )}^{3/2}}{45}+\frac {103\,{\left (1-2\,x\right )}^{5/2}}{15}-{\left (1-2\,x\right )}^{7/2}}-\frac {70065\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}+\frac {24000\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]
((1114546*x)/17787 + (1131178*(2*x - 1)^2)/41503 + (245865*(2*x - 1)^3)/41 503 - 8356631/266805)/((539*(1 - 2*x)^(1/2))/45 - (707*(1 - 2*x)^(3/2))/45 + (103*(1 - 2*x)^(5/2))/15 - (1 - 2*x)^(7/2)) - (70065*21^(1/2)*atanh((21 ^(1/2)*(1 - 2*x)^(1/2))/7))/2401 + (24000*55^(1/2)*atanh((55^(1/2)*(1 - 2* x)^(1/2))/11))/1331