Integrand size = 17, antiderivative size = 68 \[ \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx=-\frac {\sqrt {1-2 x}}{22 (3+5 x)^2}-\frac {3 \sqrt {1-2 x}}{242 (3+5 x)}-\frac {3 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \]
-3/6655*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-1/22*(1-2*x)^(1/2)/( 3+5*x)^2-3/242*(1-2*x)^(1/2)/(3+5*x)
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx=-\frac {5 \sqrt {1-2 x} (4+3 x)}{242 (3+5 x)^2}-\frac {3 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{121 \sqrt {55}} \]
(-5*Sqrt[1 - 2*x]*(4 + 3*x))/(242*(3 + 5*x)^2) - (3*ArcTanh[Sqrt[5/11]*Sqr t[1 - 2*x]])/(121*Sqrt[55])
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {52, 52, 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.
\(\displaystyle \int \frac {1}{\sqrt {1-2 x} (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {3}{22} \int \frac {1}{\sqrt {1-2 x} (5 x+3)^2}dx-\frac {\sqrt {1-2 x}}{22 (5 x+3)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {3}{22} \left (\frac {1}{11} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {\sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {\sqrt {1-2 x}}{22 (5 x+3)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3}{22} \left (-\frac {1}{11} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {\sqrt {1-2 x}}{22 (5 x+3)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{22} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{11 \sqrt {55}}-\frac {\sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {\sqrt {1-2 x}}{22 (5 x+3)^2}\) |
-1/22*Sqrt[1 - 2*x]/(3 + 5*x)^2 + (3*(-1/11*Sqrt[1 - 2*x]/(3 + 5*x) - (2*A rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(11*Sqrt[55])))/22
3.21.64.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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_)^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 = 1.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {-\frac {10}{121}+\frac {25}{242} x +\frac {15}{121} x^{2}}{\left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6655}\) | \(46\) |
pseudoelliptic | \(\frac {-6 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}-275 \sqrt {1-2 x}\, \left (4+3 x \right )}{13310 \left (3+5 x \right )^{2}}\) | \(50\) |
derivativedivides | \(-\frac {2 \sqrt {1-2 x}}{11 \left (-6-10 x \right )^{2}}+\frac {3 \sqrt {1-2 x}}{121 \left (-6-10 x \right )}-\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6655}\) | \(52\) |
default | \(-\frac {2 \sqrt {1-2 x}}{11 \left (-6-10 x \right )^{2}}+\frac {3 \sqrt {1-2 x}}{121 \left (-6-10 x \right )}-\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{6655}\) | \(52\) |
trager | \(-\frac {5 \left (4+3 x \right ) \sqrt {1-2 x}}{242 \left (3+5 x \right )^{2}}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{13310}\) | \(67\) |
5/242*(6*x^2+5*x-4)/(3+5*x)^2/(1-2*x)^(1/2)-3/6655*arctanh(1/11*55^(1/2)*( 1-2*x)^(1/2))*55^(1/2)
Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {3 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 275 \, {\left (3 \, x + 4\right )} \sqrt {-2 \, x + 1}}{13310 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
1/13310*(3*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 275*(3*x + 4)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
Result contains complex when optimal does not.
Time = 2.27 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.40 \[ \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\begin {cases} - \frac {3 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{6655} + \frac {3 \sqrt {2}}{1210 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {\sqrt {2}}{1100 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} - \frac {\sqrt {2}}{500 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\\frac {3 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{6655} - \frac {3 \sqrt {2} i}{1210 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {\sqrt {2} i}{1100 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} + \frac {\sqrt {2} i}{500 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((-3*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/6655 + 3*sqrt(2 )/(1210*sqrt(-1 + 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - sqrt(2)/(1100*sqrt(- 1 + 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) - sqrt(2)/(500*sqrt(-1 + 11/(10*( x + 3/5)))*(x + 3/5)**(5/2)), 1/Abs(x + 3/5) > 10/11), (3*sqrt(55)*I*asin( sqrt(110)/(10*sqrt(x + 3/5)))/6655 - 3*sqrt(2)*I/(1210*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5)) + sqrt(2)*I/(1100*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) + sqrt(2)*I/(500*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(5/2 )), True))
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {3}{13310} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {5 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}}{121 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
3/13310*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* x + 1))) + 5/121*(3*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)
Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx=\frac {3}{13310} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {5 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}}{484 \, {\left (5 \, x + 3\right )}^{2}} \]
3/13310*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 5/484*(3*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx=-\frac {3\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{6655}-\frac {\frac {\sqrt {1-2\,x}}{55}-\frac {3\,{\left (1-2\,x\right )}^{3/2}}{605}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \]