Integrand size = 24, antiderivative size = 82 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx=\frac {14}{81} \sqrt {1-2 x}+\frac {2}{81} (1-2 x)^{3/2}-\frac {31}{18} (1-2 x)^{5/2}+\frac {25}{42} (1-2 x)^{7/2}-\frac {14}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \] Output:
14/81*(1-2*x)^(1/2)+2/81*(1-2*x)^(3/2)-31/18*(1-2*x)^(5/2)+25/42*(1-2*x)^( 7/2)-14/243*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx=\frac {3 \sqrt {1-2 x} \left (-527+1853 x+144 x^2-2700 x^3\right )-98 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1701} \] Input:
Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^2)/(2 + 3*x),x]
Output:
(3*Sqrt[1 - 2*x]*(-527 + 1853*x + 144*x^2 - 2700*x^3) - 98*Sqrt[21]*ArcTan h[Sqrt[3/7]*Sqrt[1 - 2*x]])/1701
Time = 0.20 (sec) , antiderivative size = 82, 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.
\(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^2}{3 x+2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {25}{6} (1-2 x)^{5/2}+\frac {(1-2 x)^{3/2}}{9 (3 x+2)}+\frac {155}{18} (1-2 x)^{3/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {14}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {25}{42} (1-2 x)^{7/2}-\frac {31}{18} (1-2 x)^{5/2}+\frac {2}{81} (1-2 x)^{3/2}+\frac {14}{81} \sqrt {1-2 x}\) |
Input:
Int[((1 - 2*x)^(3/2)*(3 + 5*x)^2)/(2 + 3*x),x]
Output:
(14*Sqrt[1 - 2*x])/81 + (2*(1 - 2*x)^(3/2))/81 - (31*(1 - 2*x)^(5/2))/18 + (25*(1 - 2*x)^(7/2))/42 - (14*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]) /81
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 = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54
method | result | size |
pseudoelliptic | \(-\frac {14 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{243}-\frac {\sqrt {1-2 x}\, \left (2700 x^{3}-144 x^{2}-1853 x +527\right )}{567}\) | \(44\) |
risch | \(\frac {\left (2700 x^{3}-144 x^{2}-1853 x +527\right ) \left (-1+2 x \right )}{567 \sqrt {1-2 x}}-\frac {14 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{243}\) | \(49\) |
derivativedivides | \(\frac {14 \sqrt {1-2 x}}{81}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {31 \left (1-2 x \right )^{\frac {5}{2}}}{18}+\frac {25 \left (1-2 x \right )^{\frac {7}{2}}}{42}-\frac {14 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{243}\) | \(56\) |
default | \(\frac {14 \sqrt {1-2 x}}{81}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {31 \left (1-2 x \right )^{\frac {5}{2}}}{18}+\frac {25 \left (1-2 x \right )^{\frac {7}{2}}}{42}-\frac {14 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{243}\) | \(56\) |
trager | \(\left (-\frac {100}{21} x^{3}+\frac {16}{63} x^{2}+\frac {1853}{567} x -\frac {527}{567}\right ) \sqrt {1-2 x}-\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{243}\) | \(69\) |
Input:
int((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x),x,method=_RETURNVERBOSE)
Output:
-14/243*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-1/567*(1-2*x)^(1/2)*( 2700*x^3-144*x^2-1853*x+527)
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx=-\frac {1}{567} \, {\left (2700 \, x^{3} - 144 \, x^{2} - 1853 \, x + 527\right )} \sqrt {-2 \, x + 1} + \frac {7}{81} \, \sqrt {\frac {7}{3}} \log \left (\frac {3 \, x + 3 \, \sqrt {\frac {7}{3}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x),x, algorithm="fricas")
Output:
-1/567*(2700*x^3 - 144*x^2 - 1853*x + 527)*sqrt(-2*x + 1) + 7/81*sqrt(7/3) *log((3*x + 3*sqrt(7/3)*sqrt(-2*x + 1) - 5)/(3*x + 2))
Time = 1.84 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx=\frac {25 \left (1 - 2 x\right )^{\frac {7}{2}}}{42} - \frac {31 \left (1 - 2 x\right )^{\frac {5}{2}}}{18} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} + \frac {14 \sqrt {1 - 2 x}}{81} + \frac {7 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{243} \] Input:
integrate((1-2*x)**(3/2)*(3+5*x)**2/(2+3*x),x)
Output:
25*(1 - 2*x)**(7/2)/42 - 31*(1 - 2*x)**(5/2)/18 + 2*(1 - 2*x)**(3/2)/81 + 14*sqrt(1 - 2*x)/81 + 7*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sq rt(1 - 2*x) + sqrt(21)/3))/243
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx=\frac {25}{42} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {31}{18} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {7}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {14}{81} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x),x, algorithm="maxima")
Output:
25/42*(-2*x + 1)^(7/2) - 31/18*(-2*x + 1)^(5/2) + 2/81*(-2*x + 1)^(3/2) + 7/243*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 14/81*sqrt(-2*x + 1)
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx=-\frac {25}{42} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {31}{18} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {7}{243} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {14}{81} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x),x, algorithm="giac")
Output:
-25/42*(2*x - 1)^3*sqrt(-2*x + 1) - 31/18*(2*x - 1)^2*sqrt(-2*x + 1) + 2/8 1*(-2*x + 1)^(3/2) + 7/243*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 14/81*sqrt(-2*x + 1)
Time = 1.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx=\frac {14\,\sqrt {1-2\,x}}{81}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {31\,{\left (1-2\,x\right )}^{5/2}}{18}+\frac {25\,{\left (1-2\,x\right )}^{7/2}}{42}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,14{}\mathrm {i}}{243} \] Input:
int(((1 - 2*x)^(3/2)*(5*x + 3)^2)/(3*x + 2),x)
Output:
(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*14i)/243 + (14*(1 - 2*x)^( 1/2))/81 + (2*(1 - 2*x)^(3/2))/81 - (31*(1 - 2*x)^(5/2))/18 + (25*(1 - 2*x )^(7/2))/42
Time = 0.15 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{2+3 x} \, dx=-\frac {100 \sqrt {-2 x +1}\, x^{3}}{21}+\frac {16 \sqrt {-2 x +1}\, x^{2}}{63}+\frac {1853 \sqrt {-2 x +1}\, x}{567}-\frac {527 \sqrt {-2 x +1}}{567}+\frac {7 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )}{243}-\frac {7 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{243} \] Input:
int((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x),x)
Output:
( - 8100*sqrt( - 2*x + 1)*x**3 + 432*sqrt( - 2*x + 1)*x**2 + 5559*sqrt( - 2*x + 1)*x - 1581*sqrt( - 2*x + 1) + 49*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 49*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/1701