Integrand size = 24, antiderivative size = 95 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx=\frac {242 \sqrt {1-2 x}}{3125}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {2}{625} (1-2 x)^{5/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {1}{10} (1-2 x)^{9/2}-\frac {242 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
22/1875*(1-2*x)^(3/2)+2/625*(1-2*x)^(5/2)-111/350*(1-2*x)^(7/2)+1/10*(1-2* x)^(9/2)-242/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+242/3125* (1-2*x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx=\frac {5 \sqrt {1-2 x} \left (-8188+69995 x-91410 x^2-43500 x^3+105000 x^4\right )-5082 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{328125} \]
(5*Sqrt[1 - 2*x]*(-8188 + 69995*x - 91410*x^2 - 43500*x^3 + 105000*x^4) - 5082*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/328125
Time = 0.21 (sec) , antiderivative size = 95, 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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^2}{5 x+3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {9}{10} (1-2 x)^{7/2}+\frac {(1-2 x)^{5/2}}{25 (5 x+3)}+\frac {111}{50} (1-2 x)^{5/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {242 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}+\frac {1}{10} (1-2 x)^{9/2}-\frac {111}{350} (1-2 x)^{7/2}+\frac {2}{625} (1-2 x)^{5/2}+\frac {22 (1-2 x)^{3/2}}{1875}+\frac {242 \sqrt {1-2 x}}{3125}\) |
(242*Sqrt[1 - 2*x])/3125 + (22*(1 - 2*x)^(3/2))/1875 + (2*(1 - 2*x)^(5/2)) /625 - (111*(1 - 2*x)^(7/2))/350 + (1 - 2*x)^(9/2)/10 - (242*Sqrt[11/5]*Ar cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125
3.20.70.3.1 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 = 1.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.52
method | result | size |
pseudoelliptic | \(-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {\sqrt {1-2 x}\, \left (105000 x^{4}-43500 x^{3}-91410 x^{2}+69995 x -8188\right )}{65625}\) | \(49\) |
risch | \(-\frac {\left (105000 x^{4}-43500 x^{3}-91410 x^{2}+69995 x -8188\right ) \left (-1+2 x \right )}{65625 \sqrt {1-2 x}}-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(54\) |
derivativedivides | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{625}-\frac {111 \left (1-2 x \right )^{\frac {7}{2}}}{350}+\frac {\left (1-2 x \right )^{\frac {9}{2}}}{10}-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {242 \sqrt {1-2 x}}{3125}\) | \(65\) |
default | \(\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{1875}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{625}-\frac {111 \left (1-2 x \right )^{\frac {7}{2}}}{350}+\frac {\left (1-2 x \right )^{\frac {9}{2}}}{10}-\frac {242 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {242 \sqrt {1-2 x}}{3125}\) | \(65\) |
trager | \(\left (\frac {8}{5} x^{4}-\frac {116}{175} x^{3}-\frac {6094}{4375} x^{2}+\frac {13999}{13125} x -\frac {8188}{65625}\right ) \sqrt {1-2 x}-\frac {121 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{15625}\) | \(74\) |
-242/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+1/65625*(1-2*x)^( 1/2)*(105000*x^4-43500*x^3-91410*x^2+69995*x-8188)
Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx=\frac {121}{15625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{65625} \, {\left (105000 \, x^{4} - 43500 \, x^{3} - 91410 \, x^{2} + 69995 \, x - 8188\right )} \sqrt {-2 \, x + 1} \]
121/15625*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8) /(5*x + 3)) + 1/65625*(105000*x^4 - 43500*x^3 - 91410*x^2 + 69995*x - 8188 )*sqrt(-2*x + 1)
Time = 1.75 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx=\frac {\left (1 - 2 x\right )^{\frac {9}{2}}}{10} - \frac {111 \left (1 - 2 x\right )^{\frac {7}{2}}}{350} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{625} + \frac {22 \left (1 - 2 x\right )^{\frac {3}{2}}}{1875} + \frac {242 \sqrt {1 - 2 x}}{3125} + \frac {121 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{15625} \]
(1 - 2*x)**(9/2)/10 - 111*(1 - 2*x)**(7/2)/350 + 2*(1 - 2*x)**(5/2)/625 + 22*(1 - 2*x)**(3/2)/1875 + 242*sqrt(1 - 2*x)/3125 + 121*sqrt(55)*(log(sqrt (1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/15625
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx=\frac {1}{10} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {111}{350} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{3125} \, \sqrt {-2 \, x + 1} \]
1/10*(-2*x + 1)^(9/2) - 111/350*(-2*x + 1)^(7/2) + 2/625*(-2*x + 1)^(5/2) + 22/1875*(-2*x + 1)^(3/2) + 121/15625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2 *x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/3125*sqrt(-2*x + 1)
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.12 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx=\frac {1}{10} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {111}{350} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{15625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {242}{3125} \, \sqrt {-2 \, x + 1} \]
1/10*(2*x - 1)^4*sqrt(-2*x + 1) + 111/350*(2*x - 1)^3*sqrt(-2*x + 1) + 2/6 25*(2*x - 1)^2*sqrt(-2*x + 1) + 22/1875*(-2*x + 1)^(3/2) + 121/15625*sqrt( 55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/3125*sqrt(-2*x + 1)
Time = 1.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{3+5 x} \, dx=\frac {242\,\sqrt {1-2\,x}}{3125}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{1875}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {111\,{\left (1-2\,x\right )}^{7/2}}{350}+\frac {{\left (1-2\,x\right )}^{9/2}}{10}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{15625} \]