Integrand size = 24, antiderivative size = 92 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {18}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-18/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+300/1331*arctanh(1/11* 55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+78/847/(1-2*x)^(1/2)-5/11/(3+5*x)/(1-2*x) ^(1/2)
Time = 0.22 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {-151+390 x}{847 \sqrt {1-2 x} (3+5 x)}-\frac {18}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(-151 + 390*x)/(847*Sqrt[1 - 2*x]*(3 + 5*x)) - (18*Sqrt[3/7]*ArcTanh[Sqrt[ 3/7]*Sqrt[1 - 2*x]])/7 + (300*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]] )/121
Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {114, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {1}{11} \int \frac {3 (1-15 x)}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{11} \int \frac {1-15 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {3}{11} \left (-\frac {2}{77} \int -\frac {233-195 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{11} \left (\frac {1}{77} \int \frac {233-195 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {3}{11} \left (\frac {1}{77} \left (1750 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-1089 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {3}{11} \left (\frac {1}{77} \left (1089 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-1750 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {3}{11} \left (\frac {1}{77} \left (726 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-700 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {26}{77 \sqrt {1-2 x}}\right )-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}\) |
-5/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (3*(-26/(77*Sqrt[1 - 2*x]) + (726*Sqrt[3 /7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 700*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*S qrt[1 - 2*x]])/77))/11
3.22.22.3.1 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 = 1.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {390 x -151}{847 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}+\frac {300 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}\) | \(59\) |
| derivativedivides | \(\frac {10 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {300 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}+\frac {8}{847 \sqrt {1-2 x}}\) | \(63\) |
| default | \(\frac {10 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {300 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}+\frac {8}{847 \sqrt {1-2 x}}\) | \(63\) |
| pseudoelliptic | \(-\frac {119790 \left (\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right ) \sqrt {21}}{5}-\frac {490 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}}{3993}-\frac {91 x}{363}+\frac {1057}{10890}\right )}{\sqrt {1-2 x}\, \left (195657+326095 x \right )}\) | \(82\) |
| trager | \(-\frac {\left (390 x -151\right ) \sqrt {1-2 x}}{847 \left (10 x^{2}+x -3\right )}-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{49}-\frac {150 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\) | \(114\) |
1/847*(390*x-151)/(3+5*x)/(1-2*x)^(1/2)-18/49*arctanh(1/7*21^(1/2)*(1-2*x) ^(1/2))*21^(1/2)+300/1331*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 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {7350 \, \sqrt {11} \sqrt {5} {\left (10 \, x^{2} + x - 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 11979 \, \sqrt {7} \sqrt {3} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (390 \, x - 151\right )} \sqrt {-2 \, x + 1}}{65219 \, {\left (10 \, x^{2} + x - 3\right )}} \]
1/65219*(7350*sqrt(11)*sqrt(5)*(10*x^2 + x - 3)*log(-(sqrt(11)*sqrt(5)*sqr t(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 11979*sqrt(7)*sqrt(3)*(10*x^2 + x - 3) *log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(390*x - 1 51)*sqrt(-2*x + 1))/(10*x^2 + x - 3)
Result contains complex when optimal does not.
Time = 5.92 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=- \frac {30030 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {3388 \sqrt {2} i \sqrt {x - \frac {1}{2}}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {147000 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {239580 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {73500 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{2}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {119790 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{2}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {161700 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {263538 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} - \frac {80850 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} + \frac {131769 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )}{717409 x + 652190 \left (x - \frac {1}{2}\right )^{2} - \frac {717409}{2}} \]
-30030*sqrt(2)*I*(x - 1/2)**(3/2)/(717409*x + 652190*(x - 1/2)**2 - 717409 /2) - 3388*sqrt(2)*I*sqrt(x - 1/2)/(717409*x + 652190*(x - 1/2)**2 - 71740 9/2) + 147000*sqrt(55)*I*(x - 1/2)**2*atan(sqrt(110)*sqrt(x - 1/2)/11)/(71 7409*x + 652190*(x - 1/2)**2 - 717409/2) - 239580*sqrt(21)*I*(x - 1/2)**2* atan(sqrt(42)*sqrt(x - 1/2)/7)/(717409*x + 652190*(x - 1/2)**2 - 717409/2) - 73500*sqrt(55)*I*pi*(x - 1/2)**2/(717409*x + 652190*(x - 1/2)**2 - 7174 09/2) + 119790*sqrt(21)*I*pi*(x - 1/2)**2/(717409*x + 652190*(x - 1/2)**2 - 717409/2) + 161700*sqrt(55)*I*(x - 1/2)*atan(sqrt(110)*sqrt(x - 1/2)/11) /(717409*x + 652190*(x - 1/2)**2 - 717409/2) - 263538*sqrt(21)*I*(x - 1/2) *atan(sqrt(42)*sqrt(x - 1/2)/7)/(717409*x + 652190*(x - 1/2)**2 - 717409/2 ) - 80850*sqrt(55)*I*pi*(x - 1/2)/(717409*x + 652190*(x - 1/2)**2 - 717409 /2) + 131769*sqrt(21)*I*pi*(x - 1/2)/(717409*x + 652190*(x - 1/2)**2 - 717 409/2)
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {150}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {9}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
-150/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(- 2*x + 1))) + 9/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/847*(390*x - 151)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2* x + 1))
Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=-\frac {150}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {9}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
-150/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 9/49*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/847*(390*x - 151)/(5*(-2*x + 1)^ (3/2) - 11*sqrt(-2*x + 1))
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx=\frac {\frac {156\,x}{847}-\frac {302}{4235}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {18\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}+\frac {300\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]