Integrand size = 24, antiderivative size = 85 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx=\frac {4}{231 (1-2 x)^{3/2}}+\frac {272}{5929 \sqrt {1-2 x}}+\frac {18}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
4/231/(1-2*x)^(3/2)+18/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50 /1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+272/5929/(1-2*x)^(1/2)
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx=-\frac {4 (-281+408 x)}{17787 (1-2 x)^{3/2}}+\frac {18}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(-4*(-281 + 408*x))/(17787*(1 - 2*x)^(3/2)) + (18*Sqrt[3/7]*ArcTanh[Sqrt[3 /7]*Sqrt[1 - 2*x]])/49 - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]) /121
Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {96, 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.
\(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2) (5 x+3)} \, dx\) |
\(\Big \downarrow \) 96 |
\(\displaystyle \frac {1}{77} \int \frac {30 x+53}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {4}{231 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{77} \left (\frac {272}{77 \sqrt {1-2 x}}-\frac {2}{77} \int -\frac {2040 x+2449}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx\right )+\frac {4}{231 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{77} \left (\frac {1}{77} \int \frac {2040 x+2449}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {272}{77 \sqrt {1-2 x}}\right )+\frac {4}{231 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (6125 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-3267 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {272}{77 \sqrt {1-2 x}}\right )+\frac {4}{231 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (3267 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-6125 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {272}{77 \sqrt {1-2 x}}\right )+\frac {4}{231 (1-2 x)^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (2178 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-2450 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {272}{77 \sqrt {1-2 x}}\right )+\frac {4}{231 (1-2 x)^{3/2}}\) |
4/(231*(1 - 2*x)^(3/2)) + (272/(77*Sqrt[1 - 2*x]) + (2178*Sqrt[3/7]*ArcTan h[Sqrt[3/7]*Sqrt[1 - 2*x]] - 2450*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2 *x]])/77)/77
3.22.76.3.1 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[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S imp[1/((b*e - a*f)*(d*e - c*f)) Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*((e + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]
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 = 3.49 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {4}{231 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}-\frac {50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {272}{5929 \sqrt {1-2 x}}\) | \(56\) |
default | \(\frac {4}{231 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}-\frac {50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {272}{5929 \sqrt {1-2 x}}\) | \(56\) |
pseudoelliptic | \(-\frac {36 \left (\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \sqrt {21}}{2}-\frac {8575 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \sqrt {55}}{23958}+\frac {952 x}{1089}-\frac {1967}{3267}\right )}{343 \left (1-2 x \right )^{\frac {3}{2}}}\) | \(75\) |
trager | \(-\frac {4 \left (408 x -281\right ) \sqrt {1-2 x}}{17787 \left (-1+2 x \right )^{2}}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{343}-\frac {25 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\) | \(111\) |
4/231/(1-2*x)^(3/2)+18/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50 /1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+272/5929/(1-2*x)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (55) = 110\).
Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.44 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx=\frac {25725 \, \sqrt {11} \sqrt {5} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 35937 \, \sqrt {7} \sqrt {3} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - 308 \, {\left (408 \, x - 281\right )} \sqrt {-2 \, x + 1}}{1369599 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
1/1369599*(25725*sqrt(11)*sqrt(5)*(4*x^2 - 4*x + 1)*log((sqrt(11)*sqrt(5)* sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 35937*sqrt(7)*sqrt(3)*(4*x^2 - 4*x + 1)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) - 308*(408 *x - 281)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)
Result contains complex when optimal does not.
Time = 4.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx=- \frac {50 \sqrt {55} i \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{1331} + \frac {18 \sqrt {21} i \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{343} - \frac {136 \sqrt {2} i}{5929 \sqrt {x - \frac {1}{2}}} + \frac {\sqrt {2} i}{231 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} + \frac {\sqrt {2} i}{20 \left (x - \frac {1}{2}\right )^{\frac {5}{2}}} \]
-50*sqrt(55)*I*atan(sqrt(110)*sqrt(x - 1/2)/11)/1331 + 18*sqrt(21)*I*atan( sqrt(42)*sqrt(x - 1/2)/7)/343 - 136*sqrt(2)*I/(5929*sqrt(x - 1/2)) + sqrt( 2)*I/(231*(x - 1/2)**(3/2)) + sqrt(2)*I/(20*(x - 1/2)**(5/2))
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx=\frac {25}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4 \, {\left (408 \, x - 281\right )}}{17787 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]
25/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* x + 1))) - 9/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) - 4/17787*(408*x - 281)/(-2*x + 1)^(3/2)
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx=\frac {25}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {9}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (408 \, x - 281\right )}}{17787 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]
25/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/343*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/17787*(408*x - 281)/((2*x - 1)*sq rt(-2*x + 1))
Time = 1.56 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx=\frac {18\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {544\,x}{5929}-\frac {1124}{17787}}{{\left (1-2\,x\right )}^{3/2}}-\frac {50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]