Integrand size = 24, antiderivative size = 121 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx=\frac {242 \sqrt {1-2 x}}{78125}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {2 (1-2 x)^{5/2}}{15625}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {81}{520} (1-2 x)^{13/2}-\frac {242 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \] Output:
242/78125*(1-2*x)^(1/2)+22/46875*(1-2*x)^(3/2)+2/15625*(1-2*x)^(5/2)-13641 9/35000*(1-2*x)^(7/2)+3819/1000*(1-2*x)^(9/2)-2889/2200*(1-2*x)^(11/2)+81/ 520*(1-2*x)^(13/2)-242/390625*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2) )
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.59 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx=\frac {5 \sqrt {1-2 x} \left (-289133384+960784285 x+399578370 x^2-2556079875 x^3-1540428750 x^4+2842087500 x^5+2338875000 x^6\right )-726726 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1173046875} \] Input:
Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(3 + 5*x),x]
Output:
(5*Sqrt[1 - 2*x]*(-289133384 + 960784285*x + 399578370*x^2 - 2556079875*x^ 3 - 1540428750*x^4 + 2842087500*x^5 + 2338875000*x^6) - 726726*Sqrt[55]*Ar cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1173046875
Time = 0.23 (sec) , antiderivative size = 121, 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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^4}{5 x+3} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {81}{40} (1-2 x)^{11/2}+\frac {2889}{200} (1-2 x)^{9/2}-\frac {34371 (1-2 x)^{7/2}}{1000}+\frac {(1-2 x)^{5/2}}{625 (5 x+3)}+\frac {136419 (1-2 x)^{5/2}}{5000}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {242 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125}+\frac {81}{520} (1-2 x)^{13/2}-\frac {2889 (1-2 x)^{11/2}}{2200}+\frac {3819 (1-2 x)^{9/2}}{1000}-\frac {136419 (1-2 x)^{7/2}}{35000}+\frac {2 (1-2 x)^{5/2}}{15625}+\frac {22 (1-2 x)^{3/2}}{46875}+\frac {242 \sqrt {1-2 x}}{78125}\) |
Input:
Int[((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(3 + 5*x),x]
Output:
(242*Sqrt[1 - 2*x])/78125 + (22*(1 - 2*x)^(3/2))/46875 + (2*(1 - 2*x)^(5/2 ))/15625 - (136419*(1 - 2*x)^(7/2))/35000 + (3819*(1 - 2*x)^(9/2))/1000 - (2889*(1 - 2*x)^(11/2))/2200 + (81*(1 - 2*x)^(13/2))/520 - (242*Sqrt[11/5] *ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125
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 = 59, normalized size of antiderivative = 0.49
method | result | size |
pseudoelliptic | \(-\frac {242 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{390625}+\frac {\sqrt {1-2 x}\, \left (2338875000 x^{6}+2842087500 x^{5}-1540428750 x^{4}-2556079875 x^{3}+399578370 x^{2}+960784285 x -289133384\right )}{234609375}\) | \(59\) |
risch | \(-\frac {\left (2338875000 x^{6}+2842087500 x^{5}-1540428750 x^{4}-2556079875 x^{3}+399578370 x^{2}+960784285 x -289133384\right ) \left (-1+2 x \right )}{234609375 \sqrt {1-2 x}}-\frac {242 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{390625}\) | \(64\) |
derivativedivides | \(\frac {242 \sqrt {1-2 x}}{78125}+\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{46875}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{15625}-\frac {136419 \left (1-2 x \right )^{\frac {7}{2}}}{35000}+\frac {3819 \left (1-2 x \right )^{\frac {9}{2}}}{1000}-\frac {2889 \left (1-2 x \right )^{\frac {11}{2}}}{2200}+\frac {81 \left (1-2 x \right )^{\frac {13}{2}}}{520}-\frac {242 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{390625}\) | \(83\) |
default | \(\frac {242 \sqrt {1-2 x}}{78125}+\frac {22 \left (1-2 x \right )^{\frac {3}{2}}}{46875}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{15625}-\frac {136419 \left (1-2 x \right )^{\frac {7}{2}}}{35000}+\frac {3819 \left (1-2 x \right )^{\frac {9}{2}}}{1000}-\frac {2889 \left (1-2 x \right )^{\frac {11}{2}}}{2200}+\frac {81 \left (1-2 x \right )^{\frac {13}{2}}}{520}-\frac {242 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{390625}\) | \(83\) |
trager | \(\left (\frac {648}{65} x^{6}+\frac {43308}{3575} x^{5}-\frac {117366}{17875} x^{4}-\frac {6816213}{625625} x^{3}+\frac {26638558}{15640625} x^{2}+\frac {192156857}{46921875} x -\frac {289133384}{234609375}\right ) \sqrt {1-2 x}-\frac {121 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{390625}\) | \(84\) |
Input:
int((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x),x,method=_RETURNVERBOSE)
Output:
-242/390625*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))+1/234609375*(1-2 *x)^(1/2)*(2338875000*x^6+2842087500*x^5-1540428750*x^4-2556079875*x^3+399 578370*x^2+960784285*x-289133384)
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.59 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx=\frac {1}{234609375} \, {\left (2338875000 \, x^{6} + 2842087500 \, x^{5} - 1540428750 \, x^{4} - 2556079875 \, x^{3} + 399578370 \, x^{2} + 960784285 \, x - 289133384\right )} \sqrt {-2 \, x + 1} + \frac {121}{78125} \, \sqrt {\frac {11}{5}} \log \left (\frac {5 \, x + 5 \, \sqrt {\frac {11}{5}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x),x, algorithm="fricas")
Output:
1/234609375*(2338875000*x^6 + 2842087500*x^5 - 1540428750*x^4 - 2556079875 *x^3 + 399578370*x^2 + 960784285*x - 289133384)*sqrt(-2*x + 1) + 121/78125 *sqrt(11/5)*log((5*x + 5*sqrt(11/5)*sqrt(-2*x + 1) - 8)/(5*x + 3))
Time = 2.44 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx=\frac {81 \left (1 - 2 x\right )^{\frac {13}{2}}}{520} - \frac {2889 \left (1 - 2 x\right )^{\frac {11}{2}}}{2200} + \frac {3819 \left (1 - 2 x\right )^{\frac {9}{2}}}{1000} - \frac {136419 \left (1 - 2 x\right )^{\frac {7}{2}}}{35000} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{15625} + \frac {22 \left (1 - 2 x\right )^{\frac {3}{2}}}{46875} + \frac {242 \sqrt {1 - 2 x}}{78125} + \frac {121 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{390625} \] Input:
integrate((1-2*x)**(5/2)*(2+3*x)**4/(3+5*x),x)
Output:
81*(1 - 2*x)**(13/2)/520 - 2889*(1 - 2*x)**(11/2)/2200 + 3819*(1 - 2*x)**( 9/2)/1000 - 136419*(1 - 2*x)**(7/2)/35000 + 2*(1 - 2*x)**(5/2)/15625 + 22* (1 - 2*x)**(3/2)/46875 + 242*sqrt(1 - 2*x)/78125 + 121*sqrt(55)*(log(sqrt( 1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/390625
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx=\frac {81}{520} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {2889}{2200} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {3819}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {136419}{35000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{15625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {22}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {242}{78125} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x),x, algorithm="maxima")
Output:
81/520*(-2*x + 1)^(13/2) - 2889/2200*(-2*x + 1)^(11/2) + 3819/1000*(-2*x + 1)^(9/2) - 136419/35000*(-2*x + 1)^(7/2) + 2/15625*(-2*x + 1)^(5/2) + 22/ 46875*(-2*x + 1)^(3/2) + 121/390625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/78125*sqrt(-2*x + 1)
Time = 0.13 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.14 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx=\frac {81}{520} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {2889}{2200} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {3819}{1000} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {136419}{35000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{15625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {22}{46875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {121}{390625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {242}{78125} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x),x, algorithm="giac")
Output:
81/520*(2*x - 1)^6*sqrt(-2*x + 1) + 2889/2200*(2*x - 1)^5*sqrt(-2*x + 1) + 3819/1000*(2*x - 1)^4*sqrt(-2*x + 1) + 136419/35000*(2*x - 1)^3*sqrt(-2*x + 1) + 2/15625*(2*x - 1)^2*sqrt(-2*x + 1) + 22/46875*(-2*x + 1)^(3/2) + 1 21/390625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/78125*sqrt(-2*x + 1)
Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx=\frac {242\,\sqrt {1-2\,x}}{78125}+\frac {22\,{\left (1-2\,x\right )}^{3/2}}{46875}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{15625}-\frac {136419\,{\left (1-2\,x\right )}^{7/2}}{35000}+\frac {3819\,{\left (1-2\,x\right )}^{9/2}}{1000}-\frac {2889\,{\left (1-2\,x\right )}^{11/2}}{2200}+\frac {81\,{\left (1-2\,x\right )}^{13/2}}{520}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,242{}\mathrm {i}}{390625} \] Input:
int(((1 - 2*x)^(5/2)*(3*x + 2)^4)/(5*x + 3),x)
Output:
(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*242i)/390625 + (242*(1 - 2*x)^(1/2))/78125 + (22*(1 - 2*x)^(3/2))/46875 + (2*(1 - 2*x)^(5/2))/15625 - (136419*(1 - 2*x)^(7/2))/35000 + (3819*(1 - 2*x)^(9/2))/1000 - (2889*(1 - 2*x)^(11/2))/2200 + (81*(1 - 2*x)^(13/2))/520
Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{3+5 x} \, dx=\frac {648 \sqrt {-2 x +1}\, x^{6}}{65}+\frac {43308 \sqrt {-2 x +1}\, x^{5}}{3575}-\frac {117366 \sqrt {-2 x +1}\, x^{4}}{17875}-\frac {6816213 \sqrt {-2 x +1}\, x^{3}}{625625}+\frac {26638558 \sqrt {-2 x +1}\, x^{2}}{15640625}+\frac {192156857 \sqrt {-2 x +1}\, x}{46921875}-\frac {289133384 \sqrt {-2 x +1}}{234609375}+\frac {121 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )}{390625}-\frac {121 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )}{390625} \] Input:
int((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x),x)
Output:
(11694375000*sqrt( - 2*x + 1)*x**6 + 14210437500*sqrt( - 2*x + 1)*x**5 - 7 702143750*sqrt( - 2*x + 1)*x**4 - 12780399375*sqrt( - 2*x + 1)*x**3 + 1997 891850*sqrt( - 2*x + 1)*x**2 + 4803921425*sqrt( - 2*x + 1)*x - 1445666920* sqrt( - 2*x + 1) + 363363*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 36 3363*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)))/1173046875