Integrand size = 24, antiderivative size = 134 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx=\frac {22 \sqrt {1-2 x}}{390625}+\frac {2 (1-2 x)^{3/2}}{234375}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {243}{800} (1-2 x)^{15/2}-\frac {22 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{390625} \]
2/234375*(1-2*x)^(3/2)-167115051/2500000*(1-2*x)^(5/2)+70752609/700000*(1- 2*x)^(7/2)-665817/10000*(1-2*x)^(9/2)+507627/22000*(1-2*x)^(11/2)-43011/10 400*(1-2*x)^(13/2)+243/800*(1-2*x)^(15/2)-22/1953125*arctanh(1/11*55^(1/2) *(1-2*x)^(1/2))*55^(1/2)+22/390625*(1-2*x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx=\frac {-5 \sqrt {1-2 x} \left (15379193944-9645684935 x-56176961670 x^2-61883481375 x^3+49094797500 x^4+174123928125 x^5+150857437500 x^6+45608062500 x^7\right )-66066 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5865234375} \]
(-5*Sqrt[1 - 2*x]*(15379193944 - 9645684935*x - 56176961670*x^2 - 61883481 375*x^3 + 49094797500*x^4 + 174123928125*x^5 + 150857437500*x^6 + 45608062 500*x^7) - 66066*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5865234375
Time = 0.23 (sec) , antiderivative size = 134, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (3 x+2)^6}{5 x+3} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {729}{160} (1-2 x)^{13/2}+\frac {43011}{800} (1-2 x)^{11/2}-\frac {507627 (1-2 x)^{9/2}}{2000}+\frac {5992353 (1-2 x)^{7/2}}{10000}-\frac {70752609 (1-2 x)^{5/2}}{100000}+\frac {(1-2 x)^{3/2}}{15625 (5 x+3)}+\frac {167115051 (1-2 x)^{3/2}}{500000}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {22 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{390625}+\frac {243}{800} (1-2 x)^{15/2}-\frac {43011 (1-2 x)^{13/2}}{10400}+\frac {507627 (1-2 x)^{11/2}}{22000}-\frac {665817 (1-2 x)^{9/2}}{10000}+\frac {70752609 (1-2 x)^{7/2}}{700000}-\frac {167115051 (1-2 x)^{5/2}}{2500000}+\frac {2 (1-2 x)^{3/2}}{234375}+\frac {22 \sqrt {1-2 x}}{390625}\) |
(22*Sqrt[1 - 2*x])/390625 + (2*(1 - 2*x)^(3/2))/234375 - (167115051*(1 - 2 *x)^(5/2))/2500000 + (70752609*(1 - 2*x)^(7/2))/700000 - (665817*(1 - 2*x) ^(9/2))/10000 + (507627*(1 - 2*x)^(11/2))/22000 - (43011*(1 - 2*x)^(13/2)) /10400 + (243*(1 - 2*x)^(15/2))/800 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sq rt[1 - 2*x]])/390625
3.19.95.3.1 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 = 1.00 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.48
| method | result | size |
| pseudoelliptic | \(-\frac {22 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1953125}-\frac {\sqrt {1-2 x}\, \left (45608062500 x^{7}+150857437500 x^{6}+174123928125 x^{5}+49094797500 x^{4}-61883481375 x^{3}-56176961670 x^{2}-9645684935 x +15379193944\right )}{1173046875}\) | \(64\) |
| risch | \(\frac {\left (45608062500 x^{7}+150857437500 x^{6}+174123928125 x^{5}+49094797500 x^{4}-61883481375 x^{3}-56176961670 x^{2}-9645684935 x +15379193944\right ) \left (-1+2 x \right )}{1173046875 \sqrt {1-2 x}}-\frac {22 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1953125}\) | \(69\) |
| trager | \(\left (-\frac {972}{25} x^{7}-\frac {41796}{325} x^{6}-\frac {2653317}{17875} x^{5}-\frac {3740556}{89375} x^{4}+\frac {165022617}{3128125} x^{3}+\frac {3745130778}{78203125} x^{2}+\frac {1929136987}{234609375} x -\frac {15379193944}{1173046875}\right ) \sqrt {1-2 x}-\frac {11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1953125}\) | \(89\) |
| derivativedivides | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{234375}-\frac {167115051 \left (1-2 x \right )^{\frac {5}{2}}}{2500000}+\frac {70752609 \left (1-2 x \right )^{\frac {7}{2}}}{700000}-\frac {665817 \left (1-2 x \right )^{\frac {9}{2}}}{10000}+\frac {507627 \left (1-2 x \right )^{\frac {11}{2}}}{22000}-\frac {43011 \left (1-2 x \right )^{\frac {13}{2}}}{10400}+\frac {243 \left (1-2 x \right )^{\frac {15}{2}}}{800}-\frac {22 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1953125}+\frac {22 \sqrt {1-2 x}}{390625}\) | \(92\) |
| default | \(\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{234375}-\frac {167115051 \left (1-2 x \right )^{\frac {5}{2}}}{2500000}+\frac {70752609 \left (1-2 x \right )^{\frac {7}{2}}}{700000}-\frac {665817 \left (1-2 x \right )^{\frac {9}{2}}}{10000}+\frac {507627 \left (1-2 x \right )^{\frac {11}{2}}}{22000}-\frac {43011 \left (1-2 x \right )^{\frac {13}{2}}}{10400}+\frac {243 \left (1-2 x \right )^{\frac {15}{2}}}{800}-\frac {22 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1953125}+\frac {22 \sqrt {1-2 x}}{390625}\) | \(92\) |
-22/1953125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-1/1173046875*(1- 2*x)^(1/2)*(45608062500*x^7+150857437500*x^6+174123928125*x^5+49094797500* x^4-61883481375*x^3-56176961670*x^2-9645684935*x+15379193944)
Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx=\frac {11}{1953125} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{1173046875} \, {\left (45608062500 \, x^{7} + 150857437500 \, x^{6} + 174123928125 \, x^{5} + 49094797500 \, x^{4} - 61883481375 \, x^{3} - 56176961670 \, x^{2} - 9645684935 \, x + 15379193944\right )} \sqrt {-2 \, x + 1} \]
11/1953125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8 )/(5*x + 3)) - 1/1173046875*(45608062500*x^7 + 150857437500*x^6 + 17412392 8125*x^5 + 49094797500*x^4 - 61883481375*x^3 - 56176961670*x^2 - 964568493 5*x + 15379193944)*sqrt(-2*x + 1)
Time = 2.72 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.01 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx=\frac {243 \left (1 - 2 x\right )^{\frac {15}{2}}}{800} - \frac {43011 \left (1 - 2 x\right )^{\frac {13}{2}}}{10400} + \frac {507627 \left (1 - 2 x\right )^{\frac {11}{2}}}{22000} - \frac {665817 \left (1 - 2 x\right )^{\frac {9}{2}}}{10000} + \frac {70752609 \left (1 - 2 x\right )^{\frac {7}{2}}}{700000} - \frac {167115051 \left (1 - 2 x\right )^{\frac {5}{2}}}{2500000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{234375} + \frac {22 \sqrt {1 - 2 x}}{390625} + \frac {11 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{1953125} \]
243*(1 - 2*x)**(15/2)/800 - 43011*(1 - 2*x)**(13/2)/10400 + 507627*(1 - 2* x)**(11/2)/22000 - 665817*(1 - 2*x)**(9/2)/10000 + 70752609*(1 - 2*x)**(7/ 2)/700000 - 167115051*(1 - 2*x)**(5/2)/2500000 + 2*(1 - 2*x)**(3/2)/234375 + 22*sqrt(1 - 2*x)/390625 + 11*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/1953125
Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx=\frac {243}{800} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {43011}{10400} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {507627}{22000} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {665817}{10000} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {70752609}{700000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {167115051}{2500000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{234375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{1953125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{390625} \, \sqrt {-2 \, x + 1} \]
243/800*(-2*x + 1)^(15/2) - 43011/10400*(-2*x + 1)^(13/2) + 507627/22000*( -2*x + 1)^(11/2) - 665817/10000*(-2*x + 1)^(9/2) + 70752609/700000*(-2*x + 1)^(7/2) - 167115051/2500000*(-2*x + 1)^(5/2) + 2/234375*(-2*x + 1)^(3/2) + 11/1953125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sq rt(-2*x + 1))) + 22/390625*sqrt(-2*x + 1)
Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.15 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx=-\frac {243}{800} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {43011}{10400} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {507627}{22000} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {665817}{10000} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {70752609}{700000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {167115051}{2500000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{234375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{1953125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {22}{390625} \, \sqrt {-2 \, x + 1} \]
-243/800*(2*x - 1)^7*sqrt(-2*x + 1) - 43011/10400*(2*x - 1)^6*sqrt(-2*x + 1) - 507627/22000*(2*x - 1)^5*sqrt(-2*x + 1) - 665817/10000*(2*x - 1)^4*sq rt(-2*x + 1) - 70752609/700000*(2*x - 1)^3*sqrt(-2*x + 1) - 167115051/2500 000*(2*x - 1)^2*sqrt(-2*x + 1) + 2/234375*(-2*x + 1)^(3/2) + 11/1953125*sq rt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* x + 1))) + 22/390625*sqrt(-2*x + 1)
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx=\frac {22\,\sqrt {1-2\,x}}{390625}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{234375}-\frac {167115051\,{\left (1-2\,x\right )}^{5/2}}{2500000}+\frac {70752609\,{\left (1-2\,x\right )}^{7/2}}{700000}-\frac {665817\,{\left (1-2\,x\right )}^{9/2}}{10000}+\frac {507627\,{\left (1-2\,x\right )}^{11/2}}{22000}-\frac {43011\,{\left (1-2\,x\right )}^{13/2}}{10400}+\frac {243\,{\left (1-2\,x\right )}^{15/2}}{800}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{1953125} \]
(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*22i)/1953125 + (22*(1 - 2 *x)^(1/2))/390625 + (2*(1 - 2*x)^(3/2))/234375 - (167115051*(1 - 2*x)^(5/2 ))/2500000 + (70752609*(1 - 2*x)^(7/2))/700000 - (665817*(1 - 2*x)^(9/2))/ 10000 + (507627*(1 - 2*x)^(11/2))/22000 - (43011*(1 - 2*x)^(13/2))/10400 + (243*(1 - 2*x)^(15/2))/800