Integrand size = 17, antiderivative size = 69 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {242}{125} \sqrt {1-2 x}+\frac {22}{75} (1-2 x)^{3/2}+\frac {2}{25} (1-2 x)^{5/2}-\frac {242}{125} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:
242/125*(1-2*x)^(1/2)+22/75*(1-2*x)^(3/2)+2/25*(1-2*x)^(5/2)-242/625*55^(1 /2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \left (433-170 x+60 x^2\right )-726 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1875} \] Input:
Integrate[(1 - 2*x)^(5/2)/(3 + 5*x),x]
Output:
(10*Sqrt[1 - 2*x]*(433 - 170*x + 60*x^2) - 726*Sqrt[55]*ArcTanh[Sqrt[5/11] *Sqrt[1 - 2*x]])/1875
Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {60, 60, 60, 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.
\(\displaystyle \int \frac {(1-2 x)^{5/2}}{5 x+3} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {11}{5} \int \frac {(1-2 x)^{3/2}}{5 x+3}dx+\frac {2}{25} (1-2 x)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {11}{5} \left (\frac {11}{5} \int \frac {\sqrt {1-2 x}}{5 x+3}dx+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {11}{5} \left (\frac {11}{5} \left (\frac {11}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {2}{5} \sqrt {1-2 x}\right )+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {11}{5} \left (\frac {11}{5} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {11}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {11}{5} \left (\frac {11}{5} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {2}{5} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\) |
Input:
Int[(1 - 2*x)^(5/2)/(3 + 5*x),x]
Output:
(2*(1 - 2*x)^(5/2))/25 + (11*((2*(1 - 2*x)^(3/2))/15 + (11*((2*Sqrt[1 - 2* x])/5 - (2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5))/5))/5
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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 = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
pseudoelliptic | \(-\frac {242 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{625}+\frac {2 \sqrt {1-2 x}\, \left (60 x^{2}-170 x +433\right )}{375}\) | \(39\) |
risch | \(-\frac {2 \left (60 x^{2}-170 x +433\right ) \left (-1+2 x \right )}{375 \sqrt {1-2 x}}-\frac {242 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{625}\) | \(44\) |
derivativedivides | \(\frac {242 \sqrt {1-2 x}}{125}+\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{75}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{25}-\frac {242 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{625}\) | \(47\) |
default | \(\frac {242 \sqrt {1-2 x}}{125}+\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{75}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{25}-\frac {242 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{625}\) | \(47\) |
trager | \(\left (\frac {8}{25} x^{2}-\frac {68}{75} x +\frac {866}{375}\right ) \sqrt {1-2 x}-\frac {121 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{625}\) | \(64\) |
Input:
int((1-2*x)^(5/2)/(3+5*x),x,method=_RETURNVERBOSE)
Output:
-242/625*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))+2/375*(1-2*x)^(1/2) *(60*x^2-170*x+433)
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {2}{375} \, {\left (60 \, x^{2} - 170 \, x + 433\right )} \sqrt {-2 \, x + 1} + \frac {121}{125} \, \sqrt {\frac {11}{5}} \log \left (\frac {5 \, x + 5 \, \sqrt {\frac {11}{5}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \] Input:
integrate((1-2*x)^(5/2)/(3+5*x),x, algorithm="fricas")
Output:
2/375*(60*x^2 - 170*x + 433)*sqrt(-2*x + 1) + 121/125*sqrt(11/5)*log((5*x + 5*sqrt(11/5)*sqrt(-2*x + 1) - 8)/(5*x + 3))
Result contains complex when optimal does not.
Time = 2.03 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.93 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\begin {cases} \frac {8 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{125} - \frac {484 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{1875} + \frac {5566 \sqrt {5} i \sqrt {10 x - 5}}{9375} + \frac {242 \sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{625} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {8 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{125} - \frac {484 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{1875} + \frac {5566 \sqrt {5} \sqrt {5 - 10 x}}{9375} + \frac {121 \sqrt {55} \log {\left (x + \frac {3}{5} \right )}}{625} - \frac {242 \sqrt {55} \log {\left (\sqrt {\frac {5}{11} - \frac {10 x}{11}} + 1 \right )}}{625} & \text {otherwise} \end {cases} \] Input:
integrate((1-2*x)**(5/2)/(3+5*x),x)
Output:
Piecewise((8*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/125 - 484*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/1875 + 5566*sqrt(5)*I*sqrt(10*x - 5)/9375 + 242*sqrt (55)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/625, Abs(x + 3/5) > 11/10), (8*s qrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/125 - 484*sqrt(5)*sqrt(5 - 10*x)*(x + 3 /5)/1875 + 5566*sqrt(5)*sqrt(5 - 10*x)/9375 + 121*sqrt(55)*log(x + 3/5)/62 5 - 242*sqrt(55)*log(sqrt(5/11 - 10*x/11) + 1)/625, True))
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {2}{25} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{125} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)/(3+5*x),x, algorithm="maxima")
Output:
2/25*(-2*x + 1)^(5/2) + 22/75*(-2*x + 1)^(3/2) + 121/625*sqrt(55)*log(-(sq rt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/125*sqrt(- 2*x + 1)
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {2}{25} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{75} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {242}{125} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)/(3+5*x),x, algorithm="giac")
Output:
2/25*(2*x - 1)^2*sqrt(-2*x + 1) + 22/75*(-2*x + 1)^(3/2) + 121/625*sqrt(55 )*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1 ))) + 242/125*sqrt(-2*x + 1)
Time = 1.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {242\,\sqrt {1-2\,x}}{125}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{75}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{25}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{625} \] Input:
int((1 - 2*x)^(5/2)/(5*x + 3),x)
Output:
(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*242i)/625 + (242*(1 - 2*x )^(1/2))/125 + (22*(1 - 2*x)^(3/2))/75 + (2*(1 - 2*x)^(5/2))/25
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx=\frac {8 \sqrt {-2 x +1}\, x^{2}}{25}-\frac {68 \sqrt {-2 x +1}\, x}{75}+\frac {866 \sqrt {-2 x +1}}{375}+\frac {121 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )}{625}-\frac {121 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )}{625} \] Input:
int((1-2*x)^(5/2)/(3+5*x),x)
Output:
(600*sqrt( - 2*x + 1)*x**2 - 1700*sqrt( - 2*x + 1)*x + 4330*sqrt( - 2*x + 1) + 363*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 363*sqrt(55)*log(5* sqrt( - 2*x + 1) + sqrt(55)))/1875