Integrand size = 24, antiderivative size = 93 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {16807}{176 \sqrt {1-2 x}}+\frac {806121 \sqrt {1-2 x}}{5000}-\frac {17019}{500} (1-2 x)^{3/2}+\frac {5751 (1-2 x)^{5/2}}{1000}-\frac {243}{560} (1-2 x)^{7/2}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}} \]
-17019/500*(1-2*x)^(3/2)+5751/1000*(1-2*x)^(5/2)-243/560*(1-2*x)^(7/2)-2/3 78125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+16807/176/(1-2*x)^(1/2 )+806121/5000*(1-2*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.66 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)} \, dx=-\frac {-10972384+10459053 x+3732300 x^2+1545885 x^3+334125 x^4}{48125 \sqrt {1-2 x}}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}} \]
-1/48125*(-10972384 + 10459053*x + 3732300*x^2 + 1545885*x^3 + 334125*x^4) /Sqrt[1 - 2*x] - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(6875*Sqrt[55])
Time = 0.22 (sec) , antiderivative size = 93, 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 = {98, 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.
\(\displaystyle \int \frac {(3 x+2)^5}{(1-2 x)^{3/2} (5 x+3)} \, dx\) |
\(\Big \downarrow \) 98 |
\(\displaystyle \int \left (-\frac {243 x^3}{10 \sqrt {1-2 x}}-\frac {7857 x^2}{100 \sqrt {1-2 x}}-\frac {107433 x}{1000 \sqrt {1-2 x}}-\frac {848277}{10000 \sqrt {1-2 x}}+\frac {1}{6875 \sqrt {1-2 x} (5 x+3)}+\frac {16807}{176 (1-2 x)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{6875 \sqrt {55}}-\frac {243}{560} (1-2 x)^{7/2}+\frac {5751 (1-2 x)^{5/2}}{1000}-\frac {17019}{500} (1-2 x)^{3/2}+\frac {806121 \sqrt {1-2 x}}{5000}+\frac {16807}{176 \sqrt {1-2 x}}\) |
16807/(176*Sqrt[1 - 2*x]) + (806121*Sqrt[1 - 2*x])/5000 - (17019*(1 - 2*x) ^(3/2))/500 + (5751*(1 - 2*x)^(5/2))/1000 - (243*(1 - 2*x)^(7/2))/560 - (2 *ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(6875*Sqrt[55])
3.22.5.3.1 Defintions of rubi rules used
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x _)), x_] :> Int[ExpandIntegrand[(e + f*x)^FractionalPart[p], (c + d*x)^n*(( e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]
Time = 1.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53
method | result | size |
risch | \(-\frac {334125 x^{4}+1545885 x^{3}+3732300 x^{2}+10459053 x -10972384}{48125 \sqrt {1-2 x}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{378125}\) | \(49\) |
pseudoelliptic | \(\frac {-18376875 x^{4}-14 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\, \sqrt {1-2 x}-85023675 x^{3}-205276500 x^{2}-575247915 x +603481120}{2646875 \sqrt {1-2 x}}\) | \(55\) |
derivativedivides | \(-\frac {17019 \left (1-2 x \right )^{\frac {3}{2}}}{500}+\frac {5751 \left (1-2 x \right )^{\frac {5}{2}}}{1000}-\frac {243 \left (1-2 x \right )^{\frac {7}{2}}}{560}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{378125}+\frac {16807}{176 \sqrt {1-2 x}}+\frac {806121 \sqrt {1-2 x}}{5000}\) | \(65\) |
default | \(-\frac {17019 \left (1-2 x \right )^{\frac {3}{2}}}{500}+\frac {5751 \left (1-2 x \right )^{\frac {5}{2}}}{1000}-\frac {243 \left (1-2 x \right )^{\frac {7}{2}}}{560}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{378125}+\frac {16807}{176 \sqrt {1-2 x}}+\frac {806121 \sqrt {1-2 x}}{5000}\) | \(65\) |
trager | \(\frac {\left (334125 x^{4}+1545885 x^{3}+3732300 x^{2}+10459053 x -10972384\right ) \sqrt {1-2 x}}{-48125+96250 x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{378125}\) | \(82\) |
-1/48125*(334125*x^4+1545885*x^3+3732300*x^2+10459053*x-10972384)/(1-2*x)^ (1/2)-2/378125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.80 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {7 \, \sqrt {55} {\left (2 \, x - 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (334125 \, x^{4} + 1545885 \, x^{3} + 3732300 \, x^{2} + 10459053 \, x - 10972384\right )} \sqrt {-2 \, x + 1}}{2646875 \, {\left (2 \, x - 1\right )}} \]
1/2646875*(7*sqrt(55)*(2*x - 1)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5 *x + 3)) + 55*(334125*x^4 + 1545885*x^3 + 3732300*x^2 + 10459053*x - 10972 384)*sqrt(-2*x + 1))/(2*x - 1)
Time = 2.92 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.06 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)} \, dx=- \frac {243 \left (1 - 2 x\right )^{\frac {7}{2}}}{560} + \frac {5751 \left (1 - 2 x\right )^{\frac {5}{2}}}{1000} - \frac {17019 \left (1 - 2 x\right )^{\frac {3}{2}}}{500} + \frac {806121 \sqrt {1 - 2 x}}{5000} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{378125} + \frac {16807}{176 \sqrt {1 - 2 x}} \]
-243*(1 - 2*x)**(7/2)/560 + 5751*(1 - 2*x)**(5/2)/1000 - 17019*(1 - 2*x)** (3/2)/500 + 806121*sqrt(1 - 2*x)/5000 + sqrt(55)*(log(sqrt(1 - 2*x) - sqrt (55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/378125 + 16807/(176*sqrt(1 - 2* x))
Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.88 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)} \, dx=-\frac {243}{560} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {5751}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {17019}{500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{378125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {806121}{5000} \, \sqrt {-2 \, x + 1} + \frac {16807}{176 \, \sqrt {-2 \, x + 1}} \]
-243/560*(-2*x + 1)^(7/2) + 5751/1000*(-2*x + 1)^(5/2) - 17019/500*(-2*x + 1)^(3/2) + 1/378125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 806121/5000*sqrt(-2*x + 1) + 16807/176/sqrt(-2*x + 1)
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.06 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {243}{560} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {5751}{1000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {17019}{500} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{378125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {806121}{5000} \, \sqrt {-2 \, x + 1} + \frac {16807}{176 \, \sqrt {-2 \, x + 1}} \]
243/560*(2*x - 1)^3*sqrt(-2*x + 1) + 5751/1000*(2*x - 1)^2*sqrt(-2*x + 1) - 17019/500*(-2*x + 1)^(3/2) + 1/378125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 806121/5000*sqrt(-2*x + 1) + 16807/176/sqrt(-2*x + 1)
Time = 1.36 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {16807}{176\,\sqrt {1-2\,x}}+\frac {806121\,\sqrt {1-2\,x}}{5000}-\frac {17019\,{\left (1-2\,x\right )}^{3/2}}{500}+\frac {5751\,{\left (1-2\,x\right )}^{5/2}}{1000}-\frac {243\,{\left (1-2\,x\right )}^{7/2}}{560}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{378125} \]