Integrand size = 24, antiderivative size = 108 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 (1-2 x)^{3/2}}{9375}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {321}{200} (1-2 x)^{9/2}+\frac {81}{440} (1-2 x)^{11/2}-\frac {22 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \] Output:
22/15625*(1-2*x)^(1/2)+2/9375*(1-2*x)^(3/2)-136419/25000*(1-2*x)^(5/2)+343 71/7000*(1-2*x)^(7/2)-321/200*(1-2*x)^(9/2)+81/440*(1-2*x)^(11/2)-22/78125 *55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {-5 \sqrt {1-2 x} \left (7095688-12144995 x-21433590 x^2+9559125 x^3+39532500 x^4+21262500 x^5\right )-5082 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{18046875} \] Input:
Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]
Output:
(-5*Sqrt[1 - 2*x]*(7095688 - 12144995*x - 21433590*x^2 + 9559125*x^3 + 395 32500*x^4 + 21262500*x^5) - 5082*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x] ])/18046875
Time = 0.23 (sec) , antiderivative size = 108, 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.
\(\displaystyle \int \frac {(1-2 x)^{3/2} (3 x+2)^4}{5 x+3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {81}{40} (1-2 x)^{9/2}+\frac {2889}{200} (1-2 x)^{7/2}-\frac {34371 (1-2 x)^{5/2}}{1000}+\frac {(1-2 x)^{3/2}}{625 (5 x+3)}+\frac {136419 (1-2 x)^{3/2}}{5000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {22 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625}+\frac {81}{440} (1-2 x)^{11/2}-\frac {321}{200} (1-2 x)^{9/2}+\frac {34371 (1-2 x)^{7/2}}{7000}-\frac {136419 (1-2 x)^{5/2}}{25000}+\frac {2 (1-2 x)^{3/2}}{9375}+\frac {22 \sqrt {1-2 x}}{15625}\) |
Input:
Int[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]
Output:
(22*Sqrt[1 - 2*x])/15625 + (2*(1 - 2*x)^(3/2))/9375 - (136419*(1 - 2*x)^(5 /2))/25000 + (34371*(1 - 2*x)^(7/2))/7000 - (321*(1 - 2*x)^(9/2))/200 + (8 1*(1 - 2*x)^(11/2))/440 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]] )/15625
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 = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50
method | result | size |
pseudoelliptic | \(-\frac {22 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{78125}-\frac {\sqrt {1-2 x}\, \left (21262500 x^{5}+39532500 x^{4}+9559125 x^{3}-21433590 x^{2}-12144995 x +7095688\right )}{3609375}\) | \(54\) |
risch | \(\frac {\left (21262500 x^{5}+39532500 x^{4}+9559125 x^{3}-21433590 x^{2}-12144995 x +7095688\right ) \left (-1+2 x \right )}{3609375 \sqrt {1-2 x}}-\frac {22 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{78125}\) | \(59\) |
derivativedivides | \(\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{9375}-\frac {136419 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {34371 \left (1-2 x \right )^{\frac {7}{2}}}{7000}-\frac {321 \left (1-2 x \right )^{\frac {9}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {11}{2}}}{440}-\frac {22 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{78125}\) | \(74\) |
default | \(\frac {22 \sqrt {1-2 x}}{15625}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{9375}-\frac {136419 \left (1-2 x \right )^{\frac {5}{2}}}{25000}+\frac {34371 \left (1-2 x \right )^{\frac {7}{2}}}{7000}-\frac {321 \left (1-2 x \right )^{\frac {9}{2}}}{200}+\frac {81 \left (1-2 x \right )^{\frac {11}{2}}}{440}-\frac {22 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{78125}\) | \(74\) |
trager | \(\left (-\frac {324}{55} x^{5}-\frac {3012}{275} x^{4}-\frac {25491}{9625} x^{3}+\frac {1428906}{240625} x^{2}+\frac {2428999}{721875} x -\frac {7095688}{3609375}\right ) \sqrt {1-2 x}+\frac {11 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{78125}\) | \(79\) |
Input:
int((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x,method=_RETURNVERBOSE)
Output:
-22/78125*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))-1/3609375*(1-2*x)^ (1/2)*(21262500*x^5+39532500*x^4+9559125*x^3-21433590*x^2-12144995*x+70956 88)
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=-\frac {1}{3609375} \, {\left (21262500 \, x^{5} + 39532500 \, x^{4} + 9559125 \, x^{3} - 21433590 \, x^{2} - 12144995 \, x + 7095688\right )} \sqrt {-2 \, x + 1} + \frac {11}{15625} \, \sqrt {\frac {11}{5}} \log \left (\frac {5 \, x + 5 \, \sqrt {\frac {11}{5}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x, algorithm="fricas")
Output:
-1/3609375*(21262500*x^5 + 39532500*x^4 + 9559125*x^3 - 21433590*x^2 - 121 44995*x + 7095688)*sqrt(-2*x + 1) + 11/15625*sqrt(11/5)*log((5*x + 5*sqrt( 11/5)*sqrt(-2*x + 1) - 8)/(5*x + 3))
Time = 2.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {81 \left (1 - 2 x\right )^{\frac {11}{2}}}{440} - \frac {321 \left (1 - 2 x\right )^{\frac {9}{2}}}{200} + \frac {34371 \left (1 - 2 x\right )^{\frac {7}{2}}}{7000} - \frac {136419 \left (1 - 2 x\right )^{\frac {5}{2}}}{25000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{9375} + \frac {22 \sqrt {1 - 2 x}}{15625} + \frac {11 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{78125} \] Input:
integrate((1-2*x)**(3/2)*(2+3*x)**4/(3+5*x),x)
Output:
81*(1 - 2*x)**(11/2)/440 - 321*(1 - 2*x)**(9/2)/200 + 34371*(1 - 2*x)**(7/ 2)/7000 - 136419*(1 - 2*x)**(5/2)/25000 + 2*(1 - 2*x)**(3/2)/9375 + 22*sqr t(1 - 2*x)/15625 + 11*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt (1 - 2*x) + sqrt(55)/5))/78125
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {81}{440} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {321}{200} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {34371}{7000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {136419}{25000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{15625} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x, algorithm="maxima")
Output:
81/440*(-2*x + 1)^(11/2) - 321/200*(-2*x + 1)^(9/2) + 34371/7000*(-2*x + 1 )^(7/2) - 136419/25000*(-2*x + 1)^(5/2) + 2/9375*(-2*x + 1)^(3/2) + 11/781 25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1 ))) + 22/15625*sqrt(-2*x + 1)
Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=-\frac {81}{440} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {321}{200} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {34371}{7000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {136419}{25000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{9375} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{78125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {22}{15625} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x, algorithm="giac")
Output:
-81/440*(2*x - 1)^5*sqrt(-2*x + 1) - 321/200*(2*x - 1)^4*sqrt(-2*x + 1) - 34371/7000*(2*x - 1)^3*sqrt(-2*x + 1) - 136419/25000*(2*x - 1)^2*sqrt(-2*x + 1) + 2/9375*(-2*x + 1)^(3/2) + 11/78125*sqrt(55)*log(1/2*abs(-2*sqrt(55 ) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/15625*sqrt(-2*x + 1)
Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=\frac {22\,\sqrt {1-2\,x}}{15625}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{9375}-\frac {136419\,{\left (1-2\,x\right )}^{5/2}}{25000}+\frac {34371\,{\left (1-2\,x\right )}^{7/2}}{7000}-\frac {321\,{\left (1-2\,x\right )}^{9/2}}{200}+\frac {81\,{\left (1-2\,x\right )}^{11/2}}{440}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{78125} \] Input:
int(((1 - 2*x)^(3/2)*(3*x + 2)^4)/(5*x + 3),x)
Output:
(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*22i)/78125 + (22*(1 - 2*x )^(1/2))/15625 + (2*(1 - 2*x)^(3/2))/9375 - (136419*(1 - 2*x)^(5/2))/25000 + (34371*(1 - 2*x)^(7/2))/7000 - (321*(1 - 2*x)^(9/2))/200 + (81*(1 - 2*x )^(11/2))/440
Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx=-\frac {324 \sqrt {-2 x +1}\, x^{5}}{55}-\frac {3012 \sqrt {-2 x +1}\, x^{4}}{275}-\frac {25491 \sqrt {-2 x +1}\, x^{3}}{9625}+\frac {1428906 \sqrt {-2 x +1}\, x^{2}}{240625}+\frac {2428999 \sqrt {-2 x +1}\, x}{721875}-\frac {7095688 \sqrt {-2 x +1}}{3609375}+\frac {11 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )}{78125}-\frac {11 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )}{78125} \] Input:
int((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x)
Output:
( - 106312500*sqrt( - 2*x + 1)*x**5 - 197662500*sqrt( - 2*x + 1)*x**4 - 47 795625*sqrt( - 2*x + 1)*x**3 + 107167950*sqrt( - 2*x + 1)*x**2 + 60724975* sqrt( - 2*x + 1)*x - 35478440*sqrt( - 2*x + 1) + 2541*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 2541*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) )/18046875