Integrand size = 24, antiderivative size = 160 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {1676975 \sqrt {1-2 x}}{7546 (3+5 x)}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac {145 \sqrt {1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac {7585 \sqrt {1-2 x}}{343 (2+3 x) (3+5 x)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-1051695/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+32750/121*arcta nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-1676975/7546*(1-2*x)^(1/2)/(3+5*x )+1/7*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)+145/98*(1-2*x)^(1/2)/(2+3*x)^2/(3+5* x)+7585/343*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (12724912+58335165 x+89054820 x^2+45278325 x^3\right )}{7546 (2+3 x)^3 (3+5 x)}-\frac {1051695}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {32750}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-1/7546*(Sqrt[1 - 2*x]*(12724912 + 58335165*x + 89054820*x^2 + 45278325*x^ 3))/((2 + 3*x)^3*(3 + 5*x)) - (1051695*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (32750*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11
Time = 0.25 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {114, 27, 168, 168, 168, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{21} \int \frac {15 (5-7 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \int \frac {5-7 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2}dx+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{14} \int \frac {528-725 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx+\frac {29 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{14} \left (\frac {1}{7} \int \frac {39773-45510 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {3034 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {29 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{14} \left (\frac {1}{7} \left (-\frac {1}{11} \int \frac {1642939-1006185 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {335395 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {3034 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {29 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (6941187 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-11233250 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {335395 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {3034 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {29 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (11233250 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-6941187 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {335395 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {3034 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {29 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (4493300 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-4627458 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {335395 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {3034 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {29 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 (5 x+3)}\) |
Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*(3 + 5*x)) + (5*((29*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)) + ((3034*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) + ((-335 395*Sqrt[1 - 2*x])/(11*(3 + 5*x)) + (-4627458*Sqrt[3/7]*ArcTanh[Sqrt[3/7]* Sqrt[1 - 2*x]] + 4493300*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11) /7)/14))/7
3.21.57.3.1 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[(((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 = 1.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(\frac {90556650 x^{4}+132831315 x^{3}+27615510 x^{2}-32885341 x -12724912}{7546 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (3+5 x \right )}-\frac {1051695 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {32750 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\) | \(81\) |
| derivativedivides | \(\frac {250 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {32750 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {\frac {612765 \left (1-2 x \right )^{\frac {5}{2}}}{343}-\frac {412380 \left (1-2 x \right )^{\frac {3}{2}}}{49}+\frac {69399 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{3}}-\frac {1051695 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(91\) |
| default | \(\frac {250 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {32750 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {\frac {612765 \left (1-2 x \right )^{\frac {5}{2}}}{343}-\frac {412380 \left (1-2 x \right )^{\frac {3}{2}}}{49}+\frac {69399 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{3}}-\frac {1051695 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(91\) |
| pseudoelliptic | \(\frac {-254510190 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right ) \sqrt {21}+157265500 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right ) \sqrt {55}-77 \sqrt {1-2 x}\, \left (45278325 x^{3}+89054820 x^{2}+58335165 x +12724912\right )}{581042 \left (2+3 x \right )^{3} \left (3+5 x \right )}\) | \(102\) |
| trager | \(-\frac {\left (45278325 x^{3}+89054820 x^{2}+58335165 x +12724912\right ) \sqrt {1-2 x}}{7546 \left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {45 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11470276461\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11470276461\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-11470276461\right )-490791 \sqrt {1-2 x}}{2+3 x}\right )}{4802}-\frac {125 \operatorname {RootOf}\left (\textit {\_Z}^{2}-943855\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-943855\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-943855\right )+7205 \sqrt {1-2 x}}{3+5 x}\right )}{121}\) | \(129\) |
1/7546*(90556650*x^4+132831315*x^3+27615510*x^2-32885341*x-12724912)/(2+3* x)^3/(1-2*x)^(1/2)/(3+5*x)-1051695/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2) )*21^(1/2)+32750/121*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 = 1.01 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx=\frac {78632750 \, \sqrt {11} \sqrt {5} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 127255095 \, \sqrt {7} \sqrt {3} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (45278325 \, x^{3} + 89054820 \, x^{2} + 58335165 \, x + 12724912\right )} \sqrt {-2 \, x + 1}}{581042 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
1/581042*(78632750*sqrt(11)*sqrt(5)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 1272550 95*sqrt(7)*sqrt(3)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((sqrt(7) *sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(45278325*x^3 + 8905482 0*x^2 + 58335165*x + 12724912)*sqrt(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^ 2 + 148*x + 24)
Result contains complex when optimal does not.
Time = 16.65 (sec) , antiderivative size = 10610, normalized size of antiderivative = 66.31 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx=\text {Too large to display} \]
-7807820443315200*sqrt(2)*I*(x - 1/2)**(37/2)/(175666939176960*(x - 1/2)** 19 + 2242681256825856*(x - 1/2)**18 + 13013992410693120*(x - 1/2)**17 + 45 309871214657280*(x - 1/2)**16 + 105165127647740160*(x - 1/2)**15 + 1708584 63353424000*(x - 1/2)**14 + 198271754584795584*(x - 1/2)**13 + 16434052938 1052640*(x - 1/2)**12 + 95348514688063560*(x - 1/2)**11 + 3687900244408842 0*(x - 1/2)**10 + 8558206289250310*(x - 1/2)**9 + 902714909962019*(x - 1/2 )**8) - 90832262075089920*sqrt(2)*I*(x - 1/2)**(35/2)/(175666939176960*(x - 1/2)**19 + 2242681256825856*(x - 1/2)**18 + 13013992410693120*(x - 1/2)* *17 + 45309871214657280*(x - 1/2)**16 + 105165127647740160*(x - 1/2)**15 + 170858463353424000*(x - 1/2)**14 + 198271754584795584*(x - 1/2)**13 + 164 340529381052640*(x - 1/2)**12 + 95348514688063560*(x - 1/2)**11 + 36879002 444088420*(x - 1/2)**10 + 8558206289250310*(x - 1/2)**9 + 902714909962019* (x - 1/2)**8) - 475503973738295040*sqrt(2)*I*(x - 1/2)**(33/2)/(1756669391 76960*(x - 1/2)**19 + 2242681256825856*(x - 1/2)**18 + 13013992410693120*( x - 1/2)**17 + 45309871214657280*(x - 1/2)**16 + 105165127647740160*(x - 1 /2)**15 + 170858463353424000*(x - 1/2)**14 + 198271754584795584*(x - 1/2)* *13 + 164340529381052640*(x - 1/2)**12 + 95348514688063560*(x - 1/2)**11 + 36879002444088420*(x - 1/2)**10 + 8558206289250310*(x - 1/2)**9 + 9027149 09962019*(x - 1/2)**8) - 1475079622907692032*sqrt(2)*I*(x - 1/2)**(31/2)/( 175666939176960*(x - 1/2)**19 + 2242681256825856*(x - 1/2)**18 + 130139...
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {16375}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1051695}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {45278325 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 313944615 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 725394915 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 558527921 \, \sqrt {-2 \, x + 1}}{3773 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \]
-16375/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( -2*x + 1))) + 1051695/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sq rt(21) + 3*sqrt(-2*x + 1))) + 1/3773*(45278325*(-2*x + 1)^(7/2) - 31394461 5*(-2*x + 1)^(5/2) + 725394915*(-2*x + 1)^(3/2) - 558527921*sqrt(-2*x + 1) )/(135*(2*x - 1)^4 + 1242*(2*x - 1)^3 + 4284*(2*x - 1)^2 + 13132*x - 2793)
Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx=-\frac {16375}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1051695}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {625 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} - \frac {9 \, {\left (68085 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 320740 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 377839 \, \sqrt {-2 \, x + 1}\right )}}{2744 \, {\left (3 \, x + 2\right )}^{3}} \]
-16375/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1051695/4802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6* sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 625/11*sqrt(-2*x + 1)/(5* x + 3) - 9/2744*(68085*(2*x - 1)^2*sqrt(-2*x + 1) - 320740*(-2*x + 1)^(3/2 ) + 377839*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx=\frac {32750\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {1051695\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {\frac {11398529\,\sqrt {1-2\,x}}{10395}-\frac {2302841\,{\left (1-2\,x\right )}^{3/2}}{1617}+\frac {6976547\,{\left (1-2\,x\right )}^{5/2}}{11319}-\frac {335395\,{\left (1-2\,x\right )}^{7/2}}{3773}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \]
(32750*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 - (1051695*21^(1 /2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - ((11398529*(1 - 2*x)^(1/2) )/10395 - (2302841*(1 - 2*x)^(3/2))/1617 + (6976547*(1 - 2*x)^(5/2))/11319 - (335395*(1 - 2*x)^(7/2))/3773)/((13132*x)/135 + (476*(2*x - 1)^2)/15 + (46*(2*x - 1)^3)/5 + (2*x - 1)^4 - 931/45)