Integrand size = 24, antiderivative size = 108 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx=\frac {242 \sqrt {1-2 x}}{15625}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {2 (1-2 x)^{5/2}}{3125}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {18}{25} (1-2 x)^{9/2}-\frac {27}{220} (1-2 x)^{11/2}-\frac {242 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]
22/9375*(1-2*x)^(3/2)+2/3125*(1-2*x)^(5/2)-3897/3500*(1-2*x)^(7/2)+18/25*( 1-2*x)^(9/2)-27/220*(1-2*x)^(11/2)-242/78125*arctanh(1/11*55^(1/2)*(1-2*x) ^(1/2))*55^(1/2)+242/15625*(1-2*x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx=\frac {5 \sqrt {1-2 x} \left (-1796318+7726195 x-3564885 x^2-15572250 x^3+6142500 x^4+14175000 x^5\right )-55902 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{18046875} \]
(5*Sqrt[1 - 2*x]*(-1796318 + 7726195*x - 3564885*x^2 - 15572250*x^3 + 6142 500*x^4 + 14175000*x^5) - 55902*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]] )/18046875
Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {99, 2009}
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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^3}{5 x+3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {27}{20} (1-2 x)^{9/2}-\frac {162}{25} (1-2 x)^{7/2}+\frac {(1-2 x)^{5/2}}{125 (5 x+3)}+\frac {3897}{500} (1-2 x)^{5/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {242 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625}-\frac {27}{220} (1-2 x)^{11/2}+\frac {18}{25} (1-2 x)^{9/2}-\frac {3897 (1-2 x)^{7/2}}{3500}+\frac {2 (1-2 x)^{5/2}}{3125}+\frac {22 (1-2 x)^{3/2}}{9375}+\frac {242 \sqrt {1-2 x}}{15625}\) |
(242*Sqrt[1 - 2*x])/15625 + (22*(1 - 2*x)^(3/2))/9375 + (2*(1 - 2*x)^(5/2) )/3125 - (3897*(1 - 2*x)^(7/2))/3500 + (18*(1 - 2*x)^(9/2))/25 - (27*(1 - 2*x)^(11/2))/220 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1562 5
3.20.69.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 1.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50
method | result | size |
pseudoelliptic | \(-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}+\frac {\sqrt {1-2 x}\, \left (14175000 x^{5}+6142500 x^{4}-15572250 x^{3}-3564885 x^{2}+7726195 x -1796318\right )}{3609375}\) | \(54\) |
risch | \(-\frac {\left (14175000 x^{5}+6142500 x^{4}-15572250 x^{3}-3564885 x^{2}+7726195 x -1796318\right ) \left (-1+2 x \right )}{3609375 \sqrt {1-2 x}}-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}\) | \(59\) |
derivativedivides | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{9375}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{3125}-\frac {3897 \left (1-2 x \right )^{\frac {7}{2}}}{3500}+\frac {18 \left (1-2 x \right )^{\frac {9}{2}}}{25}-\frac {27 \left (1-2 x \right )^{\frac {11}{2}}}{220}-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}+\frac {242 \sqrt {1-2 x}}{15625}\) | \(74\) |
default | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{9375}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{3125}-\frac {3897 \left (1-2 x \right )^{\frac {7}{2}}}{3500}+\frac {18 \left (1-2 x \right )^{\frac {9}{2}}}{25}-\frac {27 \left (1-2 x \right )^{\frac {11}{2}}}{220}-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}+\frac {242 \sqrt {1-2 x}}{15625}\) | \(74\) |
trager | \(\left (\frac {216}{55} x^{5}+\frac {468}{275} x^{4}-\frac {41526}{9625} x^{3}-\frac {237659}{240625} x^{2}+\frac {1545239}{721875} x -\frac {1796318}{3609375}\right ) \sqrt {1-2 x}+\frac {121 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{78125}\) | \(79\) |
-242/78125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+1/3609375*(1-2*x) ^(1/2)*(14175000*x^5+6142500*x^4-15572250*x^3-3564885*x^2+7726195*x-179631 8)
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx=\frac {121}{78125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{3609375} \, {\left (14175000 \, x^{5} + 6142500 \, x^{4} - 15572250 \, x^{3} - 3564885 \, x^{2} + 7726195 \, x - 1796318\right )} \sqrt {-2 \, x + 1} \]
121/78125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8) /(5*x + 3)) + 1/3609375*(14175000*x^5 + 6142500*x^4 - 15572250*x^3 - 35648 85*x^2 + 7726195*x - 1796318)*sqrt(-2*x + 1)
Time = 2.00 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx=- \frac {27 \left (1 - 2 x\right )^{\frac {11}{2}}}{220} + \frac {18 \left (1 - 2 x\right )^{\frac {9}{2}}}{25} - \frac {3897 \left (1 - 2 x\right )^{\frac {7}{2}}}{3500} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{3125} + \frac {22 \left (1 - 2 x\right )^{\frac {3}{2}}}{9375} + \frac {242 \sqrt {1 - 2 x}}{15625} + \frac {121 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{78125} \]
-27*(1 - 2*x)**(11/2)/220 + 18*(1 - 2*x)**(9/2)/25 - 3897*(1 - 2*x)**(7/2) /3500 + 2*(1 - 2*x)**(5/2)/3125 + 22*(1 - 2*x)**(3/2)/9375 + 242*sqrt(1 - 2*x)/15625 + 121*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/78125
Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx=-\frac {27}{220} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {18}{25} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {3897}{3500} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{15625} \, \sqrt {-2 \, x + 1} \]
-27/220*(-2*x + 1)^(11/2) + 18/25*(-2*x + 1)^(9/2) - 3897/3500*(-2*x + 1)^ (7/2) + 2/3125*(-2*x + 1)^(5/2) + 22/9375*(-2*x + 1)^(3/2) + 121/78125*sqr t(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/15625*sqrt(-2*x + 1)
Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx=\frac {27}{220} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {18}{25} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {3897}{3500} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{3125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{78125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {242}{15625} \, \sqrt {-2 \, x + 1} \]
27/220*(2*x - 1)^5*sqrt(-2*x + 1) + 18/25*(2*x - 1)^4*sqrt(-2*x + 1) + 389 7/3500*(2*x - 1)^3*sqrt(-2*x + 1) + 2/3125*(2*x - 1)^2*sqrt(-2*x + 1) + 22 /9375*(-2*x + 1)^(3/2) + 121/78125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*s qrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/15625*sqrt(-2*x + 1)
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx=\frac {242\,\sqrt {1-2\,x}}{15625}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{9375}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{3125}-\frac {3897\,{\left (1-2\,x\right )}^{7/2}}{3500}+\frac {18\,{\left (1-2\,x\right )}^{9/2}}{25}-\frac {27\,{\left (1-2\,x\right )}^{11/2}}{220}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{78125} \]