Integrand size = 24, antiderivative size = 77 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=-\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}-6 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-6/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+64/121*arctanh(1/11*55^( 1/2)*(1-2*x)^(1/2))*55^(1/2)-5/11*(1-2*x)^(1/2)/(3+5*x)
Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=-\frac {5 \sqrt {1-2 x}}{33+55 x}-6 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(-5*Sqrt[1 - 2*x])/(33 + 55*x) - 6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2* x]] + (64*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11
Time = 0.18 (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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {1}{11} \int \frac {23-15 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {5 \sqrt {1-2 x}}{11 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{11} \left (99 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-160 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {5 \sqrt {1-2 x}}{11 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{11} \left (160 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-99 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {5 \sqrt {1-2 x}}{11 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{11} \left (64 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-66 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {5 \sqrt {1-2 x}}{11 (5 x+3)}\) |
(-5*Sqrt[1 - 2*x])/(11*(3 + 5*x)) + (-66*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[ 1 - 2*x]] + 64*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11
3.21.54.3.1 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 | \(\frac {2 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {64 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(54\) |
| default | \(\frac {2 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {64 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(54\) |
| risch | \(\frac {-\frac {5}{11}+\frac {10 x}{11}}{\left (3+5 x \right ) \sqrt {1-2 x}}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}+\frac {64 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\) | \(59\) |
| pseudoelliptic | \(\frac {-726 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right ) \sqrt {21}+448 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-385 \sqrt {1-2 x}}{2541+4235 x}\) | \(65\) |
| trager | \(-\frac {5 \sqrt {1-2 x}}{11 \left (3+5 x \right )}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{7}-\frac {32 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}\) | \(106\) |
2/11*(1-2*x)^(1/2)/(-6/5-2*x)+64/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))* 55^(1/2)-6/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=\frac {224 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 363 \, \sqrt {7} \sqrt {3} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 385 \, \sqrt {-2 \, x + 1}}{847 \, {\left (5 \, x + 3\right )}} \]
1/847*(224*sqrt(11)*sqrt(5)*(5*x + 3)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1 ) - 5*x + 8)/(5*x + 3)) + 363*sqrt(7)*sqrt(3)*(5*x + 3)*log((sqrt(7)*sqrt( 3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 385*sqrt(-2*x + 1))/(5*x + 3)
Result contains complex when optimal does not.
Time = 5.67 (sec) , antiderivative size = 515, normalized size of antiderivative = 6.69 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=- \frac {140 \sqrt {55} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {4340 \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 )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {7260 \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 )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {2170 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {3630 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {154 \sqrt {55} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {4774 \sqrt {55} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {7986 \sqrt {21} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {2387 \sqrt {55} i \pi \sqrt {x - \frac {1}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {3993 \sqrt {21} i \pi \sqrt {x - \frac {1}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {770 \sqrt {2} i \left (x - \frac {1}{2}\right )}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} \]
-140*sqrt(55)*I*(x - 1/2)**(3/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(8470* (x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) + 4340*sqrt(55)*I*(x - 1/2)**(3/2)* atan(sqrt(110)*sqrt(x - 1/2)/11)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/ 2)) - 7260*sqrt(21)*I*(x - 1/2)**(3/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(847 0*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) - 2170*sqrt(55)*I*pi*(x - 1/2)**( 3/2)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) + 3630*sqrt(21)*I*pi*(x - 1/2)**(3/2)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) - 154*sqrt(55)* I*sqrt(x - 1/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) + 4774*sqrt(55)*I*sqrt(x - 1/2)*atan(sqrt(110)*sqrt( x - 1/2)/11)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) - 7986*sqrt(21)* I*sqrt(x - 1/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(8470*(x - 1/2)**(3/2) + 93 17*sqrt(x - 1/2)) - 2387*sqrt(55)*I*pi*sqrt(x - 1/2)/(8470*(x - 1/2)**(3/2 ) + 9317*sqrt(x - 1/2)) + 3993*sqrt(21)*I*pi*sqrt(x - 1/2)/(8470*(x - 1/2) **(3/2) + 9317*sqrt(x - 1/2)) - 770*sqrt(2)*I*(x - 1/2)/(8470*(x - 1/2)**( 3/2) + 9317*sqrt(x - 1/2))
Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=-\frac {32}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {3}{7} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {5 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} \]
-32/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* x + 1))) + 3/7*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*s qrt(-2*x + 1))) - 5/11*sqrt(-2*x + 1)/(5*x + 3)
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=-\frac {32}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {3}{7} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {5 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} \]
-32/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 3/7*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 5/11*sqrt(-2*x + 1)/(5*x + 3)
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx=\frac {64\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {6\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {2\,\sqrt {1-2\,x}}{11\,\left (2\,x+\frac {6}{5}\right )} \]