Integrand size = 24, antiderivative size = 105 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx=\frac {218}{2541 (1-2 x)^{3/2}}+\frac {3274}{65219 \sqrt {1-2 x}}-\frac {5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac {54}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1400 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
218/2541/(1-2*x)^(3/2)-5/11/(1-2*x)^(3/2)/(3+5*x)-54/343*arctanh(1/7*21^(1 /2)*(1-2*x)^(1/2))*21^(1/2)+1400/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2) )*55^(1/2)+3274/65219/(1-2*x)^(1/2)
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {9111-74108 x+98220 x^2}{195657 (1-2 x)^{3/2} (3+5 x)}-\frac {54}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1400 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
-1/195657*(9111 - 74108*x + 98220*x^2)/((1 - 2*x)^(3/2)*(3 + 5*x)) - (54*S qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (1400*Sqrt[5/11]*ArcTanh[S qrt[5/11]*Sqrt[1 - 2*x]])/1331
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {114, 25, 169, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {1}{11} \int -\frac {75 x+17}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{11} \int \frac {75 x+17}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{11} \left (\frac {218}{231 (1-2 x)^{3/2}}-\frac {2}{231} \int -\frac {3 (1635 x+1)}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{77} \int \frac {1635 x+1}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {218}{231 (1-2 x)^{3/2}}\right )-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{77} \left (\frac {3274}{77 \sqrt {1-2 x}}-\frac {2}{77} \int \frac {19567-24555 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx\right )+\frac {218}{231 (1-2 x)^{3/2}}\right )-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{77} \left (\frac {3274}{77 \sqrt {1-2 x}}-\frac {1}{77} \int \frac {19567-24555 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx\right )+\frac {218}{231 (1-2 x)^{3/2}}\right )-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (107811 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-171500 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {3274}{77 \sqrt {1-2 x}}\right )+\frac {218}{231 (1-2 x)^{3/2}}\right )-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (171500 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-107811 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {3274}{77 \sqrt {1-2 x}}\right )+\frac {218}{231 (1-2 x)^{3/2}}\right )-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (68600 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-71874 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {3274}{77 \sqrt {1-2 x}}\right )+\frac {218}{231 (1-2 x)^{3/2}}\right )-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}\) |
-5/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) + (218/(231*(1 - 2*x)^(3/2)) + (3274/(77 *Sqrt[1 - 2*x]) + (-71874*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 686 00*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77)/77)/11
3.22.87.3.1 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] && 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 = 1.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.69
| method | result | size |
| derivativedivides | \(\frac {50 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {1400 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}-\frac {54 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {8}{2541 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {824}{65219 \sqrt {1-2 x}}\) | \(72\) |
| default | \(\frac {50 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {1400 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}-\frac {54 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {8}{2541 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {824}{65219 \sqrt {1-2 x}}\) | \(72\) |
| pseudoelliptic | \(\frac {2371842 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (10 x^{2}+x -3\right ) \sqrt {21}-1440600 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (10 x^{2}+x -3\right ) \sqrt {55}-7562940 x^{2}+5706316 x -701547}{\left (1-2 x \right )^{\frac {3}{2}} \left (45196767+75327945 x \right )}\) | \(93\) |
| trager | \(-\frac {\left (98220 x^{2}-74108 x +9111\right ) \sqrt {1-2 x}}{195657 \left (-1+2 x \right )^{2} \left (3+5 x \right )}+\frac {27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{343}-\frac {700 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{14641}\) | \(123\) |
50/1331*(1-2*x)^(1/2)/(-6/5-2*x)+1400/14641*arctanh(1/11*55^(1/2)*(1-2*x)^ (1/2))*55^(1/2)-54/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+8/2541 /(1-2*x)^(3/2)+824/65219/(1-2*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx=\frac {720300 \, \sqrt {11} \sqrt {5} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 1185921 \, \sqrt {7} \sqrt {3} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (98220 \, x^{2} - 74108 \, x + 9111\right )} \sqrt {-2 \, x + 1}}{15065589 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
1/15065589*(720300*sqrt(11)*sqrt(5)*(20*x^3 - 8*x^2 - 7*x + 3)*log(-(sqrt( 11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 1185921*sqrt(7)*sqrt(3) *(20*x^3 - 8*x^2 - 7*x + 3)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5) /(3*x + 2)) - 77*(98220*x^2 - 74108*x + 9111)*sqrt(-2*x + 1))/(20*x^3 - 8* x^2 - 7*x + 3)
Result contains complex when optimal does not.
Time = 6.85 (sec) , antiderivative size = 1352, normalized size of antiderivative = 12.88 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx=\text {Too large to display} \]
1440600000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/( 15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070 *(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 2371842000*sqrt(21)*I* (x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(15065589000*(x - 1/2)**( 11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 2005 2298959*(x - 1/2)**(5/2)) - 720300000*sqrt(55)*I*pi*(x - 1/2)**(11/2)/(150 65589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) + 1185921000*sqrt(21)*I*pi* (x - 1/2)**(11/2)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)** (9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) + 475 3980000*sqrt(55)*I*(x - 1/2)**(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(1506 5589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 7827078600*sqrt(21)*I*(x - 1/2)**(9/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 200522989 59*(x - 1/2)**(5/2)) - 2376990000*sqrt(55)*I*pi*(x - 1/2)**(9/2)/(15065589 000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/ 2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) + 3913539300*sqrt(21)*I*pi*(x - 1/2)**(9/2)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) + 522937...
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {700}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {27}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (24555 \, {\left (2 \, x - 1\right )}^{2} + 24112 \, x - 15444\right )}}{195657 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
-700/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( -2*x + 1))) + 27/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/195657*(24555*(2*x - 1)^2 + 24112*x - 15444)/(5* (-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {700}{14641} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {27}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (309 \, x - 193\right )}}{195657 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {125 \, \sqrt {-2 \, x + 1}}{1331 \, {\left (5 \, x + 3\right )}} \]
-700/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 27/343*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(- 2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/195657*(309*x - 193)/((2*x - 1)*sqrt(-2*x + 1)) - 125/1331*sqrt(-2*x + 1)/(5*x + 3)
Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx=\frac {1400\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641}-\frac {54\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {4384\,x}{88935}+\frac {3274\,{\left (2\,x-1\right )}^2}{65219}-\frac {936}{29645}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}} \]