Integrand size = 24, antiderivative size = 106 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {117649}{352 \sqrt {1-2 x}}+\frac {70752609 \sqrt {1-2 x}}{100000}-\frac {1997451 (1-2 x)^{3/2}}{10000}+\frac {507627 (1-2 x)^{5/2}}{10000}-\frac {43011 (1-2 x)^{7/2}}{5600}+\frac {81}{160} (1-2 x)^{9/2}-\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{34375 \sqrt {55}} \]
-1997451/10000*(1-2*x)^(3/2)+507627/10000*(1-2*x)^(5/2)-43011/5600*(1-2*x) ^(7/2)+81/160*(1-2*x)^(9/2)-2/1890625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2)) *55^(1/2)+117649/352/(1-2*x)^(1/2)+70752609/100000*(1-2*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {-\frac {55 \left (-213097384+207964053 x+85159800 x^2+48323385 x^3+19824750 x^4+3898125 x^5\right )}{\sqrt {1-2 x}}-14 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{13234375} \]
((-55*(-213097384 + 207964053*x + 85159800*x^2 + 48323385*x^3 + 19824750*x ^4 + 3898125*x^5))/Sqrt[1 - 2*x] - 14*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/13234375
Time = 0.24 (sec) , antiderivative size = 106, 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.
\(\displaystyle \int \frac {(3 x+2)^6}{(1-2 x)^{3/2} (5 x+3)} \, dx\) |
\(\Big \downarrow \) 98 |
\(\displaystyle \int \left (-\frac {729 x^4}{10 \sqrt {1-2 x}}-\frac {28431 x^3}{100 \sqrt {1-2 x}}-\frac {479439 x^2}{1000 \sqrt {1-2 x}}-\frac {4693491 x}{10000 \sqrt {1-2 x}}-\frac {31289679}{100000 \sqrt {1-2 x}}+\frac {1}{34375 \sqrt {1-2 x} (5 x+3)}+\frac {117649}{352 (1-2 x)^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{34375 \sqrt {55}}+\frac {81}{160} (1-2 x)^{9/2}-\frac {43011 (1-2 x)^{7/2}}{5600}+\frac {507627 (1-2 x)^{5/2}}{10000}-\frac {1997451 (1-2 x)^{3/2}}{10000}+\frac {70752609 \sqrt {1-2 x}}{100000}+\frac {117649}{352 \sqrt {1-2 x}}\) |
117649/(352*Sqrt[1 - 2*x]) + (70752609*Sqrt[1 - 2*x])/100000 - (1997451*(1 - 2*x)^(3/2))/10000 + (507627*(1 - 2*x)^(5/2))/10000 - (43011*(1 - 2*x)^( 7/2))/5600 + (81*(1 - 2*x)^(9/2))/160 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x ]])/(34375*Sqrt[55])
3.22.4.3.1 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 = 1.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.51
method | result | size |
risch | \(-\frac {3898125 x^{5}+19824750 x^{4}+48323385 x^{3}+85159800 x^{2}+207964053 x -213097384}{240625 \sqrt {1-2 x}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1890625}\) | \(54\) |
pseudoelliptic | \(\frac {-214396875 x^{5}-1090361250 x^{4}-14 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}\, \sqrt {1-2 x}-2657786175 x^{3}-4683789000 x^{2}-11438022915 x +11720356120}{13234375 \sqrt {1-2 x}}\) | \(60\) |
derivativedivides | \(-\frac {1997451 \left (1-2 x \right )^{\frac {3}{2}}}{10000}+\frac {507627 \left (1-2 x \right )^{\frac {5}{2}}}{10000}-\frac {43011 \left (1-2 x \right )^{\frac {7}{2}}}{5600}+\frac {81 \left (1-2 x \right )^{\frac {9}{2}}}{160}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1890625}+\frac {117649}{352 \sqrt {1-2 x}}+\frac {70752609 \sqrt {1-2 x}}{100000}\) | \(74\) |
default | \(-\frac {1997451 \left (1-2 x \right )^{\frac {3}{2}}}{10000}+\frac {507627 \left (1-2 x \right )^{\frac {5}{2}}}{10000}-\frac {43011 \left (1-2 x \right )^{\frac {7}{2}}}{5600}+\frac {81 \left (1-2 x \right )^{\frac {9}{2}}}{160}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1890625}+\frac {117649}{352 \sqrt {1-2 x}}+\frac {70752609 \sqrt {1-2 x}}{100000}\) | \(74\) |
trager | \(\frac {\left (3898125 x^{5}+19824750 x^{4}+48323385 x^{3}+85159800 x^{2}+207964053 x -213097384\right ) \sqrt {1-2 x}}{-240625+481250 x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1890625}\) | \(87\) |
-1/240625*(3898125*x^5+19824750*x^4+48323385*x^3+85159800*x^2+207964053*x- 213097384)/(1-2*x)^(1/2)-2/1890625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55 ^(1/2)
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.75 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {7 \, \sqrt {55} {\left (2 \, x - 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (3898125 \, x^{5} + 19824750 \, x^{4} + 48323385 \, x^{3} + 85159800 \, x^{2} + 207964053 \, x - 213097384\right )} \sqrt {-2 \, x + 1}}{13234375 \, {\left (2 \, x - 1\right )}} \]
1/13234375*(7*sqrt(55)*(2*x - 1)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/( 5*x + 3)) + 55*(3898125*x^5 + 19824750*x^4 + 48323385*x^3 + 85159800*x^2 + 207964053*x - 213097384)*sqrt(-2*x + 1))/(2*x - 1)
Time = 3.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {81 \left (1 - 2 x\right )^{\frac {9}{2}}}{160} - \frac {43011 \left (1 - 2 x\right )^{\frac {7}{2}}}{5600} + \frac {507627 \left (1 - 2 x\right )^{\frac {5}{2}}}{10000} - \frac {1997451 \left (1 - 2 x\right )^{\frac {3}{2}}}{10000} + \frac {70752609 \sqrt {1 - 2 x}}{100000} + \frac {\sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{1890625} + \frac {117649}{352 \sqrt {1 - 2 x}} \]
81*(1 - 2*x)**(9/2)/160 - 43011*(1 - 2*x)**(7/2)/5600 + 507627*(1 - 2*x)** (5/2)/10000 - 1997451*(1 - 2*x)**(3/2)/10000 + 70752609*sqrt(1 - 2*x)/1000 00 + sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt( 55)/5))/1890625 + 117649/(352*sqrt(1 - 2*x))
Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {81}{160} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {43011}{5600} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {507627}{10000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1997451}{10000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1890625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {70752609}{100000} \, \sqrt {-2 \, x + 1} + \frac {117649}{352 \, \sqrt {-2 \, x + 1}} \]
81/160*(-2*x + 1)^(9/2) - 43011/5600*(-2*x + 1)^(7/2) + 507627/10000*(-2*x + 1)^(5/2) - 1997451/10000*(-2*x + 1)^(3/2) + 1/1890625*sqrt(55)*log(-(sq rt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 70752609/10000 0*sqrt(-2*x + 1) + 117649/352/sqrt(-2*x + 1)
Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.08 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {81}{160} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {43011}{5600} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {507627}{10000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {1997451}{10000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{1890625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {70752609}{100000} \, \sqrt {-2 \, x + 1} + \frac {117649}{352 \, \sqrt {-2 \, x + 1}} \]
81/160*(2*x - 1)^4*sqrt(-2*x + 1) + 43011/5600*(2*x - 1)^3*sqrt(-2*x + 1) + 507627/10000*(2*x - 1)^2*sqrt(-2*x + 1) - 1997451/10000*(-2*x + 1)^(3/2) + 1/1890625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(5 5) + 5*sqrt(-2*x + 1))) + 70752609/100000*sqrt(-2*x + 1) + 117649/352/sqrt (-2*x + 1)
Time = 1.51 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx=\frac {117649}{352\,\sqrt {1-2\,x}}+\frac {70752609\,\sqrt {1-2\,x}}{100000}-\frac {1997451\,{\left (1-2\,x\right )}^{3/2}}{10000}+\frac {507627\,{\left (1-2\,x\right )}^{5/2}}{10000}-\frac {43011\,{\left (1-2\,x\right )}^{7/2}}{5600}+\frac {81\,{\left (1-2\,x\right )}^{9/2}}{160}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{1890625} \]