Integrand size = 24, antiderivative size = 77 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx=\frac {3 \sqrt {1-2 x}}{7 (2+3 x)}+\frac {72}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-10 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
72/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-10/11*arctanh(1/11*55^( 1/2)*(1-2*x)^(1/2))*55^(1/2)+3/7*(1-2*x)^(1/2)/(2+3*x)
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx=\frac {3 \sqrt {1-2 x}}{14+21 x}+\frac {72}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-10 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(3*Sqrt[1 - 2*x])/(14 + 21*x) + (72*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2 *x]])/7 - 10*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {114, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{7} \int \frac {26-15 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {3 \sqrt {1-2 x}}{7 (3 x+2)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{7} \left (175 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-108 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{7} \left (108 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-175 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{7} \left (72 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-70 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2)}\) |
(3*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + (72*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 70*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/7
3.21.43.3.1 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[(((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 = 1.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {2 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}+\frac {72 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(54\) |
| default | \(-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {2 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}+\frac {72 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(54\) |
| risch | \(-\frac {3 \left (-1+2 x \right )}{7 \left (2+3 x \right ) \sqrt {1-2 x}}+\frac {72 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}\) | \(59\) |
| pseudoelliptic | \(\frac {792 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}-490 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right ) \sqrt {55}+231 \sqrt {1-2 x}}{1078+1617 x}\) | \(65\) |
| trager | \(\frac {3 \sqrt {1-2 x}}{7 \left (2+3 x \right )}-\frac {36 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{49}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{11}\) | \(106\) |
-10/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-2/7*(1-2*x)^(1/2)/(-4 /3-2*x)+72/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx=\frac {245 \, \sqrt {11} \sqrt {5} {\left (3 \, x + 2\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 396 \, \sqrt {7} \sqrt {3} {\left (3 \, x + 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 231 \, \sqrt {-2 \, x + 1}}{539 \, {\left (3 \, x + 2\right )}} \]
1/539*(245*sqrt(11)*sqrt(5)*(3*x + 2)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 396*sqrt(7)*sqrt(3)*(3*x + 2)*log(-(sqrt(7)*sqrt( 3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 231*sqrt(-2*x + 1))/(3*x + 2)
Result contains complex when optimal does not.
Time = 5.94 (sec) , antiderivative size = 515, normalized size of antiderivative = 6.69 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx=- \frac {2940 \sqrt {55} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} + \frac {132 \sqrt {21} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {42}}{6 \sqrt {x - \frac {1}{2}}} \right )}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} + \frac {4884 \sqrt {21} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} - \frac {2442 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} + \frac {1470 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} - \frac {3430 \sqrt {55} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} + \frac {154 \sqrt {21} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {42}}{6 \sqrt {x - \frac {1}{2}}} \right )}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} + \frac {5698 \sqrt {21} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} - \frac {2849 \sqrt {21} i \pi \sqrt {x - \frac {1}{2}}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} + \frac {1715 \sqrt {55} i \pi \sqrt {x - \frac {1}{2}}}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} + \frac {462 \sqrt {2} i \left (x - \frac {1}{2}\right )}{3234 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 3773 \sqrt {x - \frac {1}{2}}} \]
-2940*sqrt(55)*I*(x - 1/2)**(3/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(3234*( x - 1/2)**(3/2) + 3773*sqrt(x - 1/2)) + 132*sqrt(21)*I*(x - 1/2)**(3/2)*at an(sqrt(42)/(6*sqrt(x - 1/2)))/(3234*(x - 1/2)**(3/2) + 3773*sqrt(x - 1/2) ) + 4884*sqrt(21)*I*(x - 1/2)**(3/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(3234* (x - 1/2)**(3/2) + 3773*sqrt(x - 1/2)) - 2442*sqrt(21)*I*pi*(x - 1/2)**(3/ 2)/(3234*(x - 1/2)**(3/2) + 3773*sqrt(x - 1/2)) + 1470*sqrt(55)*I*pi*(x - 1/2)**(3/2)/(3234*(x - 1/2)**(3/2) + 3773*sqrt(x - 1/2)) - 3430*sqrt(55)*I *sqrt(x - 1/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(3234*(x - 1/2)**(3/2) + 3 773*sqrt(x - 1/2)) + 154*sqrt(21)*I*sqrt(x - 1/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(3234*(x - 1/2)**(3/2) + 3773*sqrt(x - 1/2)) + 5698*sqrt(21)*I*sq rt(x - 1/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(3234*(x - 1/2)**(3/2) + 3773*s qrt(x - 1/2)) - 2849*sqrt(21)*I*pi*sqrt(x - 1/2)/(3234*(x - 1/2)**(3/2) + 3773*sqrt(x - 1/2)) + 1715*sqrt(55)*I*pi*sqrt(x - 1/2)/(3234*(x - 1/2)**(3 /2) + 3773*sqrt(x - 1/2)) + 462*sqrt(2)*I*(x - 1/2)/(3234*(x - 1/2)**(3/2) + 3773*sqrt(x - 1/2))
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx=\frac {5}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {36}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {3 \, \sqrt {-2 \, x + 1}}{7 \, {\left (3 \, x + 2\right )}} \]
5/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 36/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq rt(-2*x + 1))) + 3/7*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx=\frac {5}{11} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {36}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {3 \, \sqrt {-2 \, x + 1}}{7 \, {\left (3 \, x + 2\right )}} \]
5/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s qrt(-2*x + 1))) - 36/49*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1 ))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 3/7*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx=\frac {72\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {10\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {2\,\sqrt {1-2\,x}}{7\,\left (2\,x+\frac {4}{3}\right )} \]