Integrand size = 17, antiderivative size = 63 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {6}{121 \sqrt {1-2 x}}-\frac {1}{11 \sqrt {1-2 x} (3+5 x)}-\frac {6}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-6/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+6/121/(1-2*x)^(1/2)- 1/11/(3+5*x)/(1-2*x)^(1/2)
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {2 (-22+15 (1-2 x))}{121 (-11+5 (1-2 x)) \sqrt {1-2 x}}-\frac {6}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(2*(-22 + 15*(1 - 2*x)))/(121*(-11 + 5*(1 - 2*x))*Sqrt[1 - 2*x]) - (6*Sqrt [5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {52, 61, 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.
\(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (5 x+3)^2} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {3}{11} \int \frac {1}{(1-2 x)^{3/2} (5 x+3)}dx-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {3}{11} \left (\frac {5}{11} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {2}{11 \sqrt {1-2 x}}\right )-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3}{11} \left (\frac {2}{11 \sqrt {1-2 x}}-\frac {5}{11} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{11} \left (\frac {2}{11 \sqrt {1-2 x}}-\frac {2}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}\) |
-1/11*1/(Sqrt[1 - 2*x]*(3 + 5*x)) + (3*(2/(11*Sqrt[1 - 2*x]) - (2*Sqrt[5/1 1]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11))/11
3.22.21.3.1 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] :> 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_)^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 = 1.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {30 x +7}{121 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}\) | \(41\) |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {4}{121 \sqrt {1-2 x}}\) | \(45\) |
default | \(\frac {2 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}-\frac {6 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {4}{121 \sqrt {1-2 x}}\) | \(45\) |
pseudoelliptic | \(-\frac {30 \left (\sqrt {55}\, \left (x +\frac {3}{5}\right ) \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )-11 x -\frac {77}{30}\right )}{\sqrt {1-2 x}\, \left (3993+6655 x \right )}\) | \(49\) |
trager | \(-\frac {\left (30 x +7\right ) \sqrt {1-2 x}}{121 \left (10 x^{2}+x -3\right )}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\) | \(70\) |
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {3 \, \sqrt {11} \sqrt {5} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (30 \, x + 7\right )} \sqrt {-2 \, x + 1}}{1331 \, {\left (10 \, x^{2} + x - 3\right )}} \]
1/1331*(3*sqrt(11)*sqrt(5)*(10*x^2 + x - 3)*log((sqrt(11)*sqrt(5)*sqrt(-2* x + 1) + 5*x - 8)/(5*x + 3)) - 11*(30*x + 7)*sqrt(-2*x + 1))/(10*x^2 + x - 3)
Result contains complex when optimal does not.
Time = 1.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.78 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\begin {cases} - \frac {6 \sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{1331} + \frac {3 \sqrt {2}}{121 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {\sqrt {2}}{110 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\\frac {6 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{1331} - \frac {3 \sqrt {2} i}{121 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {\sqrt {2} i}{110 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((-6*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/1331 + 3*sqrt(2 )/(121*sqrt(-1 + 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - sqrt(2)/(110*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)), 1/Abs(x + 3/5) > 10/11), (6*sqrt(5 5)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/1331 - 3*sqrt(2)*I/(121*sqrt(1 - 1 1/(10*(x + 3/5)))*sqrt(x + 3/5)) + sqrt(2)*I/(110*sqrt(1 - 11/(10*(x + 3/5 )))*(x + 3/5)**(3/2)), True))
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {3}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (30 \, x + 7\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
3/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2/121*(30*x + 7)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {3}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2 \, {\left (30 \, x + 7\right )}}{121 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
3/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5 *sqrt(-2*x + 1))) - 2/121*(30*x + 7)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))
Time = 1.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {\frac {12\,x}{121}+\frac {14}{605}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {6\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]