Integrand size = 24, antiderivative size = 92 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx=-\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}+\frac {228}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
228/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50/121*arctanh(1/11*5 5^(1/2)*(1-2*x)^(1/2))*55^(1/2)-58/539/(1-2*x)^(1/2)+3/7/(2+3*x)/(1-2*x)^( 1/2)
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {115-174 x}{539 \sqrt {1-2 x} (2+3 x)}+\frac {228}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(115 - 174*x)/(539*Sqrt[1 - 2*x]*(2 + 3*x)) + (228*Sqrt[3/7]*ArcTanh[Sqrt[ 3/7]*Sqrt[1 - 2*x]])/49 - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]] )/11
Time = 0.19 (sec) , antiderivative size = 98, 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 = {114, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{7} \int \frac {8-45 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {3}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{7} \left (-\frac {2}{77} \int -\frac {964-435 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {58}{77 \sqrt {1-2 x}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{77} \int \frac {964-435 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {58}{77 \sqrt {1-2 x}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{77} \left (6125 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-3762 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {58}{77 \sqrt {1-2 x}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{77} \left (3762 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-6125 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {58}{77 \sqrt {1-2 x}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{77} \left (2508 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-2450 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {58}{77 \sqrt {1-2 x}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2)}\) |
3/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (-58/(77*Sqrt[1 - 2*x]) + (2508*Sqrt[3/7]* ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 2450*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt [1 - 2*x]])/77)/7
3.22.12.3.1 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 = 1.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(-\frac {174 x -115}{539 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {228 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}-\frac {50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\) | \(59\) |
| derivativedivides | \(-\frac {50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}+\frac {228 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {8}{539 \sqrt {1-2 x}}\) | \(63\) |
| default | \(-\frac {50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}+\frac {228 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {8}{539 \sqrt {1-2 x}}\) | \(63\) |
| pseudoelliptic | \(\frac {\frac {115}{539}+\frac {228 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}}{343}-\frac {50 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right ) \sqrt {55}}{121}-\frac {174 x}{539}}{\left (2+3 x \right ) \sqrt {1-2 x}}\) | \(81\) |
| trager | \(\frac {\left (174 x -115\right ) \sqrt {1-2 x}}{3234 x^{2}+539 x -1078}-\frac {114 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{343}+\frac {25 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}\) | \(114\) |
-1/539*(174*x-115)/(2+3*x)/(1-2*x)^(1/2)+228/343*arctanh(1/7*21^(1/2)*(1-2 *x)^(1/2))*21^(1/2)-50/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {8575 \, \sqrt {11} \sqrt {5} {\left (6 \, x^{2} + x - 2\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 13794 \, \sqrt {7} \sqrt {3} {\left (6 \, x^{2} + x - 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (174 \, x - 115\right )} \sqrt {-2 \, x + 1}}{41503 \, {\left (6 \, x^{2} + x - 2\right )}} \]
1/41503*(8575*sqrt(11)*sqrt(5)*(6*x^2 + x - 2)*log((sqrt(11)*sqrt(5)*sqrt( -2*x + 1) + 5*x - 8)/(5*x + 3)) + 13794*sqrt(7)*sqrt(3)*(6*x^2 + x - 2)*lo g(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(174*x - 115 )*sqrt(-2*x + 1))/(6*x^2 + x - 2)
Result contains complex when optimal does not.
Time = 6.14 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {13398 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {2156 \sqrt {2} i \sqrt {x - \frac {1}{2}}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {102900 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} + \frac {165528 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {82764 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{2}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} + \frac {51450 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{2}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {120050 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} + \frac {193116 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} - \frac {96558 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} + \frac {60025 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )}{290521 x + 249018 \left (x - \frac {1}{2}\right )^{2} - \frac {290521}{2}} \]
13398*sqrt(2)*I*(x - 1/2)**(3/2)/(290521*x + 249018*(x - 1/2)**2 - 290521/ 2) - 2156*sqrt(2)*I*sqrt(x - 1/2)/(290521*x + 249018*(x - 1/2)**2 - 290521 /2) - 102900*sqrt(55)*I*(x - 1/2)**2*atan(sqrt(110)*sqrt(x - 1/2)/11)/(290 521*x + 249018*(x - 1/2)**2 - 290521/2) + 165528*sqrt(21)*I*(x - 1/2)**2*a tan(sqrt(42)*sqrt(x - 1/2)/7)/(290521*x + 249018*(x - 1/2)**2 - 290521/2) - 82764*sqrt(21)*I*pi*(x - 1/2)**2/(290521*x + 249018*(x - 1/2)**2 - 29052 1/2) + 51450*sqrt(55)*I*pi*(x - 1/2)**2/(290521*x + 249018*(x - 1/2)**2 - 290521/2) - 120050*sqrt(55)*I*(x - 1/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/( 290521*x + 249018*(x - 1/2)**2 - 290521/2) + 193116*sqrt(21)*I*(x - 1/2)*a tan(sqrt(42)*sqrt(x - 1/2)/7)/(290521*x + 249018*(x - 1/2)**2 - 290521/2) - 96558*sqrt(21)*I*pi*(x - 1/2)/(290521*x + 249018*(x - 1/2)**2 - 290521/2 ) + 60025*sqrt(55)*I*pi*(x - 1/2)/(290521*x + 249018*(x - 1/2)**2 - 290521 /2)
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {25}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {114}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (174 \, x - 115\right )}}{539 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \]
25/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 114/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/539*(174*x - 115)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {25}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {114}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2 \, {\left (174 \, x - 115\right )}}{539 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \]
25/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5 *sqrt(-2*x + 1))) - 114/343*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/539*(174*x - 115)/(3*(-2*x + 1)^ (3/2) - 7*sqrt(-2*x + 1))
Time = 1.70 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {228\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {116\,x}{539}-\frac {230}{1617}}{\frac {7\,\sqrt {1-2\,x}}{3}-{\left (1-2\,x\right )}^{3/2}}-\frac {50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121} \]