Integrand size = 24, antiderivative size = 178 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}+\frac {813716}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
813716/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-112875/11*arctanh(1 /11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-176065/126*(1-2*x)^(1/2)/(3+5*x)^2+7/ 9*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^2+28/3*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2 +1301/7*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+117955/14*(1-2*x)^(1/2)/(3+5*x)
Time = 0.35 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (2685098+16784696 x+39307638 x^2+40874010 x^3+15923925 x^4\right )}{14 (2+3 x)^3 (3+5 x)^2}+\frac {813716}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(2685098 + 16784696*x + 39307638*x^2 + 40874010*x^3 + 15923 925*x^4))/(14*(2 + 3*x)^3*(3 + 5*x)^2) + (813716*Sqrt[3/7]*ArcTanh[Sqrt[3/ 7]*Sqrt[1 - 2*x]])/7 - 112875*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {109, 168, 27, 168, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^4 (5 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{9} \int \frac {190-303 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^3}dx+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{14} \int \frac {14 (1943-2940 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {1943-2940 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^3}dx+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {211708-292725 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3}dx+\frac {11709 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (-\frac {1}{22} \int \frac {99 (153862-176065 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {176065 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11709 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (-\frac {9}{2} \int \frac {153862-176065 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {176065 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11709 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (-\frac {9}{2} \left (-\frac {1}{11} \int \frac {33 (192602-117955 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {117955 \sqrt {1-2 x}}{5 x+3}\right )-\frac {176065 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11709 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (-\frac {9}{2} \left (-3 \int \frac {192602-117955 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {117955 \sqrt {1-2 x}}{5 x+3}\right )-\frac {176065 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11709 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (-\frac {9}{2} \left (-3 \left (1316875 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-813716 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {117955 \sqrt {1-2 x}}{5 x+3}\right )-\frac {176065 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11709 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (-\frac {9}{2} \left (-3 \left (813716 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-1316875 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {117955 \sqrt {1-2 x}}{5 x+3}\right )-\frac {176065 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11709 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (-\frac {9}{2} \left (-3 \left (\frac {1627432 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{\sqrt {21}}-526750 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {117955 \sqrt {1-2 x}}{5 x+3}\right )-\frac {176065 \sqrt {1-2 x}}{2 (5 x+3)^2}\right )+\frac {11709 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {84 \sqrt {1-2 x}}{(3 x+2)^2 (5 x+3)^2}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}\) |
(7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*x)^2) + ((84*Sqrt[1 - 2*x])/((2 + 3*x)^2*(3 + 5*x)^2) + (11709*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^2) + (( -176065*Sqrt[1 - 2*x])/(2*(3 + 5*x)^2) - (9*((-117955*Sqrt[1 - 2*x])/(3 + 5*x) - 3*((1627432*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 526750*Sqr t[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])))/2)/7)/9
3.20.27.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 = 1.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(-\frac {31847850 x^{5}+65824095 x^{4}+37741266 x^{3}-5738246 x^{2}-11414500 x -2685098}{14 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}+\frac {813716 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}-\frac {112875 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}\) | \(86\) |
| derivativedivides | \(\frac {-33625 \left (1-2 x \right )^{\frac {3}{2}}+73425 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {112875 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {324 \left (\frac {3544 \left (1-2 x \right )^{\frac {5}{2}}}{21}-\frac {21418 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {25172 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {813716 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(103\) |
| default | \(\frac {-33625 \left (1-2 x \right )^{\frac {3}{2}}+73425 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {112875 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {324 \left (\frac {3544 \left (1-2 x \right )^{\frac {5}{2}}}{21}-\frac {21418 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {25172 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {813716 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(103\) |
| pseudoelliptic | \(\frac {17901752 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right )^{2} \sqrt {21}-11061750 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \left (3+5 x \right )^{2} \sqrt {55}+77 \sqrt {1-2 x}\, \left (15923925 x^{4}+40874010 x^{3}+39307638 x^{2}+16784696 x +2685098\right )}{1078 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}\) | \(111\) |
| trager | \(\frac {\left (15923925 x^{4}+40874010 x^{3}+39307638 x^{2}+16784696 x +2685098\right ) \sqrt {1-2 x}}{14 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}+\frac {112875 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{22}-\frac {406858 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{49}\) | \(133\) |
-1/14*(31847850*x^5+65824095*x^4+37741266*x^3-5738246*x^2-11414500*x-26850 98)/(2+3*x)^3/(1-2*x)^(1/2)/(3+5*x)^2+813716/49*arctanh(1/7*21^(1/2)*(1-2* x)^(1/2))*21^(1/2)-112875/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {5530875 \, \sqrt {11} \sqrt {5} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 8950876 \, \sqrt {7} \sqrt {3} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (15923925 \, x^{4} + 40874010 \, x^{3} + 39307638 \, x^{2} + 16784696 \, x + 2685098\right )} \sqrt {-2 \, x + 1}}{1078 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
1/1078*(5530875*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)) + 8950876*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*(1 5923925*x^4 + 40874010*x^3 + 39307638*x^2 + 16784696*x + 2685098)*sqrt(-2* x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)
Time = 118.30 (sec) , antiderivative size = 865, normalized size of antiderivative = 4.86 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\text {Too large to display} \]
-57110*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqr t(21)/3))/7 + 57110*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/11 + 83208*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))) - 16968*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))) + 2352*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(21)/3))) + 149600*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x) /11 - 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))) + 24200*Piecewise((s...
Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {112875}{22} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {406858}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {15923925 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 145443720 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 498018162 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 757678432 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 432141633 \, \sqrt {-2 \, x + 1}}{7 \, {\left (675 \, {\left (2 \, x - 1\right )}^{5} + 7695 \, {\left (2 \, x - 1\right )}^{4} + 35082 \, {\left (2 \, x - 1\right )}^{3} + 79954 \, {\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \]
112875/22*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(- 2*x + 1))) - 406858/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) + 1/7*(15923925*(-2*x + 1)^(9/2) - 145443720*(-2*x + 1)^(7/2) + 498018162*(-2*x + 1)^(5/2) - 757678432*(-2*x + 1)^(3/2) + 43 2141633*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 = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {112875}{22} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {406858}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {25 \, {\left (1345 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2937 \, \sqrt {-2 \, x + 1}\right )}}{4 \, {\left (5 \, x + 3\right )}^{2}} + \frac {3 \, {\left (15948 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 74963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 88102 \, \sqrt {-2 \, x + 1}\right )}}{7 \, {\left (3 \, x + 2\right )}^{3}} \]
112875/22*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 406858/49*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt (-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 25/4*(1345*(-2*x + 1)^(3/2) - 2937*sqrt(-2*x + 1))/(5*x + 3)^2 + 3/7*(15948*(2*x - 1)^2*sqrt(-2*x + 1) - 74963*(-2*x + 1)^(3/2) + 88102*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 1.41 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx=\frac {813716\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {112875\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {6859391\,\sqrt {1-2\,x}}{75}-\frac {108239776\,{\left (1-2\,x\right )}^{3/2}}{675}+\frac {166006054\,{\left (1-2\,x\right )}^{5/2}}{1575}-\frac {9696248\,{\left (1-2\,x\right )}^{7/2}}{315}+\frac {23591\,{\left (1-2\,x\right )}^{9/2}}{7}}{\frac {182182\,x}{675}+\frac {79954\,{\left (2\,x-1\right )}^2}{675}+\frac {3898\,{\left (2\,x-1\right )}^3}{75}+\frac {57\,{\left (2\,x-1\right )}^4}{5}+{\left (2\,x-1\right )}^5-\frac {49588}{675}} \]
(813716*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - (112875*55^(1/2 )*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/11 + ((6859391*(1 - 2*x)^(1/2))/75 - (108239776*(1 - 2*x)^(3/2))/675 + (166006054*(1 - 2*x)^(5/2))/1575 - (9 696248*(1 - 2*x)^(7/2))/315 + (23591*(1 - 2*x)^(9/2))/7)/((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)