Integrand size = 24, antiderivative size = 133 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {\sqrt {1-2 x}}{4 (2+3 x)^4}+\frac {139 \sqrt {1-2 x}}{84 (2+3 x)^3}+\frac {14555 \sqrt {1-2 x}}{1176 (2+3 x)^2}+\frac {337955 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {11656955 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
11656955/28812*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-250*arctanh(1/ 11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+1/4*(1-2*x)^(1/2)/(2+3*x)^4+139/84*(1- 2*x)^(1/2)/(2+3*x)^3+14555/1176*(1-2*x)^(1/2)/(2+3*x)^2+337955/2744*(1-2*x )^(1/2)/(2+3*x)
Time = 0.44 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {\sqrt {1-2 x} \left (2849254+12587542 x+18555225 x^2+9124785 x^3\right )}{2744 (2+3 x)^4}+\frac {11656955 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372 \sqrt {21}}-250 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(2849254 + 12587542*x + 18555225*x^2 + 9124785*x^3))/(2744* (2 + 3*x)^4) + (11656955*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 250*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {110, 25, 168, 27, 168, 27, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^5 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x}}{4 (3 x+2)^4}-\frac {1}{4} \int -\frac {23-35 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \int \frac {23-35 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)}dx+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{21} \int \frac {5 (507-695 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {139 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {5}{21} \int \frac {507-695 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {139 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (\frac {5}{21} \left (\frac {1}{14} \int \frac {3 (12827-14555 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {2911 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {139 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \int \frac {12827-14555 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {2911 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {139 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {551827-337955 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {67591 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {2911 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {139 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \left (3773000 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-2331391 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {67591 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {2911 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {139 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \left (2331391 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-3773000 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {67591 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {2911 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {139 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \left (\frac {4662782 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-137200 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {67591 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {2911 \sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {139 \sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{4 (3 x+2)^4}\) |
Sqrt[1 - 2*x]/(4*(2 + 3*x)^4) + ((139*Sqrt[1 - 2*x])/(21*(2 + 3*x)^3) + (5 *((2911*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2) + (3*((67591*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + ((4662782*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 137200*Sq rt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/7))/14))/21)/4
3.19.37.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (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 + 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 = 1.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {18249570 x^{4}+27985665 x^{3}+6619859 x^{2}-6889034 x -2849254}{2744 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {11656955 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{28812}-250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\) | \(74\) |
| derivativedivides | \(-250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {337955 \left (1-2 x \right )^{\frac {7}{2}}}{8232}-\frac {3070705 \left (1-2 x \right )^{\frac {5}{2}}}{10584}+\frac {3100927 \left (1-2 x \right )^{\frac {3}{2}}}{4536}-\frac {116015 \sqrt {1-2 x}}{216}\right )}{\left (-4-6 x \right )^{4}}+\frac {11656955 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{28812}\) | \(84\) |
| default | \(-250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}-\frac {162 \left (\frac {337955 \left (1-2 x \right )^{\frac {7}{2}}}{8232}-\frac {3070705 \left (1-2 x \right )^{\frac {5}{2}}}{10584}+\frac {3100927 \left (1-2 x \right )^{\frac {3}{2}}}{4536}-\frac {116015 \sqrt {1-2 x}}{216}\right )}{\left (-4-6 x \right )^{4}}+\frac {11656955 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{28812}\) | \(84\) |
| pseudoelliptic | \(\frac {23313910 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-14406000 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{4} \sqrt {55}+21 \sqrt {1-2 x}\, \left (9124785 x^{3}+18555225 x^{2}+12587542 x +2849254\right )}{57624 \left (2+3 x \right )^{4}}\) | \(85\) |
| trager | \(\frac {\left (9124785 x^{3}+18555225 x^{2}+12587542 x +2849254\right ) \sqrt {1-2 x}}{2744 \left (2+3 x \right )^{4}}+125 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )+\frac {11656955 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{57624}\) | \(121\) |
-1/2744*(18249570*x^4+27985665*x^3+6619859*x^2-6889034*x-2849254)/(2+3*x)^ 4/(1-2*x)^(1/2)+11656955/28812*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2 )-250*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {7203000 \, \sqrt {55} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11656955 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (9124785 \, x^{3} + 18555225 \, x^{2} + 12587542 \, x + 2849254\right )} \sqrt {-2 \, x + 1}}{57624 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/57624*(7203000*sqrt(55)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((5* x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 11656955*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3* x + 2)) + 21*(9124785*x^3 + 18555225*x^2 + 12587542*x + 2849254)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Time = 130.80 (sec) , antiderivative size = 833, normalized size of antiderivative = 6.26 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx=\text {Too large to display} \]
-1375*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3))/7 + 125*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5)) + 3300*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x )/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3))) - 1320*Piecewise((sqrt(21) *(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21) *sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3))) + 528*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqrt(21)* sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 1/(48*( sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x) /7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(2 1)/3))) - 224*Piecewise((sqrt(21)*(35*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/25 6 - 35*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/256 + 35/(256*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 15/(256*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 5/(192*(sqrt(2 1)*sqrt(1 - 2*x)/7 + 1)**3) + 1/(128*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**4)...
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx=125 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {11656955}{57624} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9124785 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 64484805 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 151945423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 119379435 \, \sqrt {-2 \, x + 1}}{1372 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 11656955/57624*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) - 1/1372*(9124785*(-2*x + 1)^(7/2) - 64484805*(-2*x + 1)^(5/2) + 151945423*(-2*x + 1)^(3/2) - 119379435*sqrt(-2*x + 1))/(81*( 2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx=125 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {11656955}{57624} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {9124785 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 64484805 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 151945423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 119379435 \, \sqrt {-2 \, x + 1}}{21952 \, {\left (3 \, x + 2\right )}^{4}} \]
125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sq rt(-2*x + 1))) - 11656955/57624*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt( -2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/21952*(9124785*(2*x - 1)^3*s qrt(-2*x + 1) + 64484805*(2*x - 1)^2*sqrt(-2*x + 1) - 151945423*(-2*x + 1) ^(3/2) + 119379435*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.55 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx=\frac {11656955\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{28812}-250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {116015\,\sqrt {1-2\,x}}{108}-\frac {3100927\,{\left (1-2\,x\right )}^{3/2}}{2268}+\frac {3070705\,{\left (1-2\,x\right )}^{5/2}}{5292}-\frac {337955\,{\left (1-2\,x\right )}^{7/2}}{4116}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]
(11656955*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/28812 - 250*55^(1/ 2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((116015*(1 - 2*x)^(1/2))/108 - (3100927*(1 - 2*x)^(3/2))/2268 + (3070705*(1 - 2*x)^(5/2))/5292 - (337955* (1 - 2*x)^(7/2))/4116)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3 )/3 + (2*x - 1)^4 - 1715/81)