Integrand size = 24, antiderivative size = 72 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx=\frac {4}{77 \sqrt {1-2 x}}+\frac {6}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {10}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
6/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-10/121*arctanh(1/11*55^( 1/2)*(1-2*x)^(1/2))*55^(1/2)+4/77/(1-2*x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx=\frac {4}{77 \sqrt {1-2 x}}+\frac {6}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {10}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
4/(77*Sqrt[1 - 2*x]) + (6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (10*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {96, 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.
\(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2) (5 x+3)} \, dx\) |
\(\Big \downarrow \) 96 |
\(\displaystyle \frac {1}{77} \int \frac {30 x+53}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {4}{77 \sqrt {1-2 x}}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{77} \left (175 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-99 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {4}{77 \sqrt {1-2 x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{77} \left (99 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-175 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {4}{77 \sqrt {1-2 x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{77} \left (66 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-70 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {4}{77 \sqrt {1-2 x}}\) |
4/(77*Sqrt[1 - 2*x]) + (66*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 70 *Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77
3.22.11.3.1 Defintions of rubi rules used
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[(((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 = 47, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {4}{77 \sqrt {1-2 x}}\) | \(47\) |
default | \(\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {4}{77 \sqrt {1-2 x}}\) | \(47\) |
risch | \(\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {4}{77 \sqrt {1-2 x}}\) | \(47\) |
pseudoelliptic | \(\frac {726 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}\, \sqrt {1-2 x}-490 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\, \sqrt {1-2 x}+308}{5929 \sqrt {1-2 x}}\) | \(62\) |
trager | \(-\frac {4 \sqrt {1-2 x}}{77 \left (-1+2 x \right )}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{49}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}\) | \(106\) |
6/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-10/121*arctanh(1/11*55^( 1/2)*(1-2*x)^(1/2))*55^(1/2)+4/77/(1-2*x)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (46) = 92\).
Time = 0.23 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx=\frac {245 \, \sqrt {11} \sqrt {5} {\left (2 \, x - 1\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 363 \, \sqrt {7} \sqrt {3} {\left (2 \, x - 1\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - 308 \, \sqrt {-2 \, x + 1}}{5929 \, {\left (2 \, x - 1\right )}} \]
1/5929*(245*sqrt(11)*sqrt(5)*(2*x - 1)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1 ) + 5*x - 8)/(5*x + 3)) + 363*sqrt(7)*sqrt(3)*(2*x - 1)*log(-(sqrt(7)*sqrt (3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) - 308*sqrt(-2*x + 1))/(2*x - 1)
Time = 3.32 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx=- \frac {3 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{49} + \frac {5 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{121} + \frac {4}{77 \sqrt {1 - 2 x}} \]
-3*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21 )/3))/49 + 5*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/121 + 4/(77*sqrt(1 - 2*x))
Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx=\frac {5}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4}{77 \, \sqrt {-2 \, x + 1}} \]
5/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq rt(-2*x + 1))) + 4/77/sqrt(-2*x + 1)
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx=\frac {5}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {3}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4}{77 \, \sqrt {-2 \, x + 1}} \]
5/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5* sqrt(-2*x + 1))) - 3/49*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1 ))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/77/sqrt(-2*x + 1)
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx=\frac {6\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {10\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}+\frac {4}{77\,\sqrt {1-2\,x}} \]