Integrand size = 24, antiderivative size = 180 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx=-\frac {182335 \sqrt {1-2 x}}{294 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)^2}+\frac {29 \sqrt {1-2 x}}{7 (2+3 x)^2 (3+5 x)^2}+\frac {4042 \sqrt {1-2 x}}{49 (2+3 x) (3+5 x)^2}+\frac {4031135 \sqrt {1-2 x}}{1078 (3+5 x)}+\frac {2528082}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {551075}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
2528082/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-551075/121*arctan h(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-182335/294*(1-2*x)^(1/2)/(3+5*x)^2 +1/3*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^2+29/7*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x )^2+4042/49*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+4031135/1078*(1-2*x)^(1/2)/(3+ 5*x)
Time = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (91763734+573620246 x+1343346156 x^2+1396877220 x^3+544203225 x^4\right )}{1078 (2+3 x)^3 (3+5 x)^2}+\frac {2528082}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {551075}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(91763734 + 573620246*x + 1343346156*x^2 + 1396877220*x^3 + 544203225*x^4))/(1078*(2 + 3*x)^3*(3 + 5*x)^2) + (2528082*Sqrt[3/7]*ArcTa nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (551075*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sq rt[1 - 2*x]])/11
Time = 0.27 (sec) , antiderivative size = 201, 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 = {110, 25, 168, 27, 168, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^4 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}-\frac {1}{3} \int -\frac {28-45 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \int \frac {28-45 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^3}dx+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{14} \int \frac {2 (2012-3045 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {87 \sqrt {1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \int \frac {2012-3045 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {87 \sqrt {1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{7} \int \frac {219247-303150 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {12126 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {87 \sqrt {1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{7} \left (-\frac {1}{22} \int \frac {33 (478024-547005 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {182335 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {12126 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {87 \sqrt {1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{7} \left (-\frac {3}{2} \int \frac {478024-547005 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {182335 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {12126 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {87 \sqrt {1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{7} \left (-\frac {3}{2} \left (-\frac {1}{11} \int \frac {19746632-12093405 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {4031135 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {182335 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {12126 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {87 \sqrt {1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{7} \left (-\frac {3}{2} \left (\frac {1}{11} \left (83426706 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-135013375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {4031135 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {182335 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {12126 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {87 \sqrt {1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{7} \left (-\frac {3}{2} \left (\frac {1}{11} \left (135013375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-83426706 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {4031135 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {182335 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {12126 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {87 \sqrt {1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{7} \left (-\frac {3}{2} \left (\frac {1}{11} \left (54005350 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-55617804 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {4031135 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {182335 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {12126 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}\right )+\frac {87 \sqrt {1-2 x}}{7 (3 x+2)^2 (5 x+3)^2}\right )+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^2}\) |
Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*x)^2) + ((87*Sqrt[1 - 2*x])/(7*(2 + 3* x)^2*(3 + 5*x)^2) + ((12126*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^2) + ((- 182335*Sqrt[1 - 2*x])/(2*(3 + 5*x)^2) - (3*((-4031135*Sqrt[1 - 2*x])/(11*( 3 + 5*x)) + (-55617804*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 540053 50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11))/2)/7)/7)/3
3.19.56.3.1 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 = 1.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(-\frac {1088406450 x^{5}+2249551215 x^{4}+1289815092 x^{3}-196105664 x^{2}-390092778 x -91763734}{1078 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}+\frac {2528082 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}-\frac {551075 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\) | \(86\) |
| derivativedivides | \(\frac {-\frac {165625 \left (1-2 x \right )^{\frac {3}{2}}}{11}+32875 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {551075 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {972 \left (\frac {7297 \left (1-2 x \right )^{\frac {5}{2}}}{294}-\frac {22048 \left (1-2 x \right )^{\frac {3}{2}}}{189}+\frac {7403 \sqrt {1-2 x}}{54}\right )}{\left (-4-6 x \right )^{3}}+\frac {2528082 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(103\) |
| default | \(\frac {-\frac {165625 \left (1-2 x \right )^{\frac {3}{2}}}{11}+32875 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {551075 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {972 \left (\frac {7297 \left (1-2 x \right )^{\frac {5}{2}}}{294}-\frac {22048 \left (1-2 x \right )^{\frac {3}{2}}}{189}+\frac {7403 \sqrt {1-2 x}}{54}\right )}{\left (-4-6 x \right )^{3}}+\frac {2528082 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(103\) |
| pseudoelliptic | \(\frac {611795844 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right )^{2} \sqrt {21}-378037450 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right )^{2} \sqrt {55}+77 \sqrt {1-2 x}\, \left (544203225 x^{4}+1396877220 x^{3}+1343346156 x^{2}+573620246 x +91763734\right )}{83006 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}\) | \(111\) |
| trager | \(\frac {\left (544203225 x^{4}+1396877220 x^{3}+1343346156 x^{2}+573620246 x +91763734\right ) \sqrt {1-2 x}}{1078 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}-\frac {551075 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{242}-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-414244353621\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-414244353621\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-414244353621\right )+2949429 \sqrt {1-2 x}}{2+3 x}\right )}{343}\) | \(133\) |
-1/1078*(1088406450*x^5+2249551215*x^4+1289815092*x^3-196105664*x^2-390092 778*x-91763734)/(2+3*x)^3/(1-2*x)^(1/2)/(3+5*x)^2+2528082/343*arctanh(1/7* 21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-551075/121*arctanh(1/11*55^(1/2)*(1-2*x)^ (1/2))*55^(1/2)
Time = 0.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {189018725 \, \sqrt {11} \sqrt {5} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 305897922 \, \sqrt {7} \sqrt {3} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (544203225 \, x^{4} + 1396877220 \, x^{3} + 1343346156 \, x^{2} + 573620246 \, x + 91763734\right )} \sqrt {-2 \, x + 1}}{83006 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
1/83006*(189018725*sqrt(11)*sqrt(5)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766* x^2 + 564*x + 72)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3 )) + 305897922*sqrt(7)*sqrt(3)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(544203225*x^4 + 1396877220*x^3 + 1343346156*x^2 + 573620246*x + 917637 34)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72 )
Time = 112.69 (sec) , antiderivative size = 865, normalized size of antiderivative = 4.81 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx=\text {Too large to display} \]
-25350*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqr t(21)/3))/7 + 25350*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/11 + 36720*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)*s qrt(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))) - 7416*Piecewise((s qrt(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)*s qrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(s qrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (s qrt(1 - 2*x) < sqrt(21)/3))) + 1008*Piecewise((sqrt(21)*(-5*log(sqrt(21)*s qrt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(s qrt(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(21)/3))) + 67000*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/1 1 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5))) + 11000*Piecewise((sqr...
Time = 0.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {551075}{242} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1264041}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {544203225 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 4970567340 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 17019867294 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 25893807436 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 14768524001 \, \sqrt {-2 \, x + 1}}{539 \, {\left (675 \, {\left (2 \, x - 1\right )}^{5} + 7695 \, {\left (2 \, x - 1\right )}^{4} + 35082 \, {\left (2 \, x - 1\right )}^{3} + 79954 \, {\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \]
551075/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( -2*x + 1))) - 1264041/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqr t(21) + 3*sqrt(-2*x + 1))) + 1/539*(544203225*(-2*x + 1)^(9/2) - 497056734 0*(-2*x + 1)^(7/2) + 17019867294*(-2*x + 1)^(5/2) - 25893807436*(-2*x + 1) ^(3/2) + 14768524001*sqrt(-2*x + 1))/(675*(2*x - 1)^5 + 7695*(2*x - 1)^4 + 35082*(2*x - 1)^3 + 79954*(2*x - 1)^2 + 182182*x - 49588)
Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {551075}{242} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {1264041}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {125 \, {\left (1325 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2893 \, \sqrt {-2 \, x + 1}\right )}}{44 \, {\left (5 \, x + 3\right )}^{2}} + \frac {9 \, {\left (65673 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 308672 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 362747 \, \sqrt {-2 \, x + 1}\right )}}{196 \, {\left (3 \, x + 2\right )}^{3}} \]
551075/242*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1264041/343*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*s qrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/44*(1325*(-2*x + 1)^(3 /2) - 2893*sqrt(-2*x + 1))/(5*x + 3)^2 + 9/196*(65673*(2*x - 1)^2*sqrt(-2* x + 1) - 308672*(-2*x + 1)^(3/2) + 362747*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 1.46 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {2528082\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {551075\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}+\frac {\frac {27399859\,\sqrt {1-2\,x}}{675}-\frac {3699115348\,{\left (1-2\,x\right )}^{3/2}}{51975}+\frac {1891096366\,{\left (1-2\,x\right )}^{5/2}}{40425}-\frac {110457052\,{\left (1-2\,x\right )}^{7/2}}{8085}+\frac {806227\,{\left (1-2\,x\right )}^{9/2}}{539}}{\frac {182182\,x}{675}+\frac {79954\,{\left (2\,x-1\right )}^2}{675}+\frac {3898\,{\left (2\,x-1\right )}^3}{75}+\frac {57\,{\left (2\,x-1\right )}^4}{5}+{\left (2\,x-1\right )}^5-\frac {49588}{675}} \]
(2528082*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - (551075*55^(1 /2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 + ((27399859*(1 - 2*x)^(1/2) )/675 - (3699115348*(1 - 2*x)^(3/2))/51975 + (1891096366*(1 - 2*x)^(5/2))/ 40425 - (110457052*(1 - 2*x)^(7/2))/8085 + (806227*(1 - 2*x)^(9/2))/539)/( (182182*x)/675 + (79954*(2*x - 1)^2)/675 + (3898*(2*x - 1)^3)/75 + (57*(2* x - 1)^4)/5 + (2*x - 1)^5 - 49588/675)