Integrand size = 24, antiderivative size = 133 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {35845 \sqrt {1-2 x}}{1078 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac {162 \sqrt {1-2 x}}{49 (2+3 x) (3+5 x)}-\frac {22479}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {4900}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-22479/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+4900/121*arctanh(1 /11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-35845/1078*(1-2*x)^(1/2)/(3+5*x)+3/14 *(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)+162/49*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (136021+419448 x+322605 x^2\right )}{1078 (2+3 x)^2 (3+5 x)}-\frac {22479}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {4900}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-1/1078*(Sqrt[1 - 2*x]*(136021 + 419448*x + 322605*x^2))/((2 + 3*x)^2*(3 + 5*x)) - (22479*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (4900*Sqr t[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11
Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {114, 168, 168, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{14} \int \frac {58-75 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^2}dx+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \int \frac {4253-4860 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx+\frac {324 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (-\frac {1}{11} \int \frac {175579-107535 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {35845 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {324 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (741807 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-1200500 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {35845 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {324 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (1200500 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-741807 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {35845 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {324 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (\frac {1}{11} \left (480200 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-494538 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {35845 \sqrt {1-2 x}}{11 (5 x+3)}\right )+\frac {324 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)}\) |
(3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)) + ((324*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) + ((-35845*Sqrt[1 - 2*x])/(11*(3 + 5*x)) + (-494538*Sqrt [3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 480200*Sqrt[5/11]*ArcTanh[Sqrt[5/ 11]*Sqrt[1 - 2*x]])/11)/7)/14
3.21.56.3.1 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[((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] && ILtQ[m, -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 = 1.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.57
| method | result | size |
| risch | \(\frac {645210 x^{3}+516291 x^{2}-147406 x -136021}{1078 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}-\frac {22479 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {4900 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\) | \(76\) |
| derivativedivides | \(\frac {50 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {4900 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {\frac {3861 \left (1-2 x \right )^{\frac {3}{2}}}{49}-\frac {1305 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{2}}-\frac {22479 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(82\) |
| default | \(\frac {50 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {4900 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {\frac {3861 \left (1-2 x \right )^{\frac {3}{2}}}{49}-\frac {1305 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{2}}-\frac {22479 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(82\) |
| pseudoelliptic | \(\frac {-5439918 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {21}+3361400 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {55}-77 \sqrt {1-2 x}\, \left (322605 x^{2}+419448 x +136021\right )}{83006 \left (2+3 x \right )^{2} \left (3+5 x \right )}\) | \(97\) |
| trager | \(-\frac {\left (322605 x^{2}+419448 x +136021\right ) \sqrt {1-2 x}}{1078 \left (2+3 x \right )^{2} \left (3+5 x \right )}-\frac {2450 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}+\frac {177 \operatorname {RootOf}\left (\textit {\_Z}^{2}-338709\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-338709\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-338709\right )+2667 \sqrt {1-2 x}}{2+3 x}\right )}{686}\) | \(123\) |
1/1078*(645210*x^3+516291*x^2-147406*x-136021)/(3+5*x)/(1-2*x)^(1/2)/(2+3* x)^2-22479/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+4900/121*arcta nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {1680700 \, \sqrt {11} \sqrt {5} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 2719959 \, \sqrt {7} \sqrt {3} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (322605 \, x^{2} + 419448 \, x + 136021\right )} \sqrt {-2 \, x + 1}}{83006 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]
1/83006*(1680700*sqrt(11)*sqrt(5)*(45*x^3 + 87*x^2 + 56*x + 12)*log(-(sqrt (11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 2719959*sqrt(7)*sqrt(3 )*(45*x^3 + 87*x^2 + 56*x + 12)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(322605*x^2 + 419448*x + 136021)*sqrt(-2*x + 1))/(45* x^3 + 87*x^2 + 56*x + 12)
Result contains complex when optimal does not.
Time = 11.38 (sec) , antiderivative size = 3624, normalized size of antiderivative = 27.25 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=\text {Too large to display} \]
-222264000*sqrt(55)*I*(x - 1/2)**(15/2)*atan(sqrt(110)/(10*sqrt(x - 1/2))) /(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104* (x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7 /2) + 1096135733*(x - 1/2)**(5/2)) + 21559608000*sqrt(55)*I*(x - 1/2)**(15 /2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(537878880*(x - 1/2)**(15/2) + 310176 8208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/ 2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 6 82149600*sqrt(21)*I*(x - 1/2)**(15/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(53 7878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 35932818240*sqrt(21)*I*(x - 1/2)**(15/2)* atan(sqrt(42)*sqrt(x - 1/2)/7)/(537878880*(x - 1/2)**(15/2) + 3101768208*( x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9 /2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 1077980 4000*sqrt(55)*I*pi*(x - 1/2)**(15/2)/(537878880*(x - 1/2)**(15/2) + 310176 8208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/ 2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 1 7966409120*sqrt(21)*I*pi*(x - 1/2)**(15/2)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*( x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(...
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {2450}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22479}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {322605 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 1484106 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1705585 \, \sqrt {-2 \, x + 1}}{539 \, {\left (45 \, {\left (2 \, x - 1\right )}^{3} + 309 \, {\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \]
-2450/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(- 2*x + 1))) + 22479/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(2 1) + 3*sqrt(-2*x + 1))) - 1/539*(322605*(-2*x + 1)^(5/2) - 1484106*(-2*x + 1)^(3/2) + 1705585*sqrt(-2*x + 1))/(45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 14 14*x - 168)
Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {2450}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {22479}{686} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {125 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} + \frac {9 \, {\left (429 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1015 \, \sqrt {-2 \, x + 1}\right )}}{196 \, {\left (3 \, x + 2\right )}^{2}} \]
-2450/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22479/686*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt (-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/11*sqrt(-2*x + 1)/(5*x + 3) + 9/196*(429*(-2*x + 1)^(3/2) - 1015*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 1.55 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {4900\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {22479\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {48731\,\sqrt {1-2\,x}}{693}-\frac {494702\,{\left (1-2\,x\right )}^{3/2}}{8085}+\frac {7169\,{\left (1-2\,x\right )}^{5/2}}{539}}{\frac {1414\,x}{45}+\frac {103\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {56}{15}} \]