Integrand size = 24, antiderivative size = 95 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx=\frac {2 \sqrt {1-2 x}}{3125}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {9}{40} (1-2 x)^{9/2}-\frac {2 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]
-45473/5000*(1-2*x)^(3/2)+34371/5000*(1-2*x)^(5/2)-2889/1400*(1-2*x)^(7/2) +9/40*(1-2*x)^(9/2)-2/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+ 2/3125*(1-2*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx=\frac {5 \sqrt {1-2 x} \left (-88776+27865 x+177930 x^2+203625 x^3+78750 x^4\right )-14 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{109375} \]
(5*Sqrt[1 - 2*x]*(-88776 + 27865*x + 177930*x^2 + 203625*x^3 + 78750*x^4) - 14*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/109375
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.
\(\displaystyle \int \frac {\sqrt {1-2 x} (3 x+2)^4}{5 x+3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {81}{40} (1-2 x)^{7/2}+\frac {2889}{200} (1-2 x)^{5/2}-\frac {34371 (1-2 x)^{3/2}}{1000}+\frac {\sqrt {1-2 x}}{625 (5 x+3)}+\frac {136419 \sqrt {1-2 x}}{5000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}+\frac {9}{40} (1-2 x)^{9/2}-\frac {2889 (1-2 x)^{7/2}}{1400}+\frac {34371 (1-2 x)^{5/2}}{5000}-\frac {45473 (1-2 x)^{3/2}}{5000}+\frac {2 \sqrt {1-2 x}}{3125}\) |
(2*Sqrt[1 - 2*x])/3125 - (45473*(1 - 2*x)^(3/2))/5000 + (34371*(1 - 2*x)^( 5/2))/5000 - (2889*(1 - 2*x)^(7/2))/1400 + (9*(1 - 2*x)^(9/2))/40 - (2*Sqr t[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125
3.19.28.3.1 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 = 0.97 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.52
method | result | size |
pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {\sqrt {1-2 x}\, \left (78750 x^{4}+203625 x^{3}+177930 x^{2}+27865 x -88776\right )}{21875}\) | \(49\) |
risch | \(-\frac {\left (78750 x^{4}+203625 x^{3}+177930 x^{2}+27865 x -88776\right ) \left (-1+2 x \right )}{21875 \sqrt {1-2 x}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}\) | \(54\) |
derivativedivides | \(-\frac {45473 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {34371 \left (1-2 x \right )^{\frac {5}{2}}}{5000}-\frac {2889 \left (1-2 x \right )^{\frac {7}{2}}}{1400}+\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{40}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {2 \sqrt {1-2 x}}{3125}\) | \(65\) |
default | \(-\frac {45473 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {34371 \left (1-2 x \right )^{\frac {5}{2}}}{5000}-\frac {2889 \left (1-2 x \right )^{\frac {7}{2}}}{1400}+\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{40}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{15625}+\frac {2 \sqrt {1-2 x}}{3125}\) | \(65\) |
trager | \(\left (\frac {18}{5} x^{4}+\frac {1629}{175} x^{3}+\frac {35586}{4375} x^{2}+\frac {5573}{4375} x -\frac {88776}{21875}\right ) \sqrt {1-2 x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{15625}\) | \(74\) |
-2/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+1/21875*(1-2*x)^(1/ 2)*(78750*x^4+203625*x^3+177930*x^2+27865*x-88776)
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx=\frac {1}{15625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac {1}{21875} \, {\left (78750 \, x^{4} + 203625 \, x^{3} + 177930 \, x^{2} + 27865 \, x - 88776\right )} \sqrt {-2 \, x + 1} \]
1/15625*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/( 5*x + 3)) + 1/21875*(78750*x^4 + 203625*x^3 + 177930*x^2 + 27865*x - 88776 )*sqrt(-2*x + 1)
Time = 2.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx=\frac {9 \left (1 - 2 x\right )^{\frac {9}{2}}}{40} - \frac {2889 \left (1 - 2 x\right )^{\frac {7}{2}}}{1400} + \frac {34371 \left (1 - 2 x\right )^{\frac {5}{2}}}{5000} - \frac {45473 \left (1 - 2 x\right )^{\frac {3}{2}}}{5000} + \frac {2 \sqrt {1 - 2 x}}{3125} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{15625} \]
9*(1 - 2*x)**(9/2)/40 - 2889*(1 - 2*x)**(7/2)/1400 + 34371*(1 - 2*x)**(5/2 )/5000 - 45473*(1 - 2*x)**(3/2)/5000 + 2*sqrt(1 - 2*x)/3125 + sqrt(55)*(lo g(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/15625
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx=\frac {9}{40} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {2889}{1400} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {34371}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {45473}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2}{3125} \, \sqrt {-2 \, x + 1} \]
9/40*(-2*x + 1)^(9/2) - 2889/1400*(-2*x + 1)^(7/2) + 34371/5000*(-2*x + 1) ^(5/2) - 45473/5000*(-2*x + 1)^(3/2) + 1/15625*sqrt(55)*log(-(sqrt(55) - 5 *sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/3125*sqrt(-2*x + 1)
Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx=\frac {9}{40} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {2889}{1400} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {34371}{5000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {45473}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{15625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2}{3125} \, \sqrt {-2 \, x + 1} \]
9/40*(2*x - 1)^4*sqrt(-2*x + 1) + 2889/1400*(2*x - 1)^3*sqrt(-2*x + 1) + 3 4371/5000*(2*x - 1)^2*sqrt(-2*x + 1) - 45473/5000*(-2*x + 1)^(3/2) + 1/156 25*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr t(-2*x + 1))) + 2/3125*sqrt(-2*x + 1)
Time = 1.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^4}{3+5 x} \, dx=\frac {2\,\sqrt {1-2\,x}}{3125}-\frac {45473\,{\left (1-2\,x\right )}^{3/2}}{5000}+\frac {34371\,{\left (1-2\,x\right )}^{5/2}}{5000}-\frac {2889\,{\left (1-2\,x\right )}^{7/2}}{1400}+\frac {9\,{\left (1-2\,x\right )}^{9/2}}{40}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{15625} \]