Integrand size = 24, antiderivative size = 95 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx=\frac {98}{243} \sqrt {1-2 x}+\frac {14}{243} (1-2 x)^{3/2}+\frac {2}{135} (1-2 x)^{5/2}-\frac {155}{126} (1-2 x)^{7/2}+\frac {25}{54} (1-2 x)^{9/2}-\frac {98}{243} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \] Output:
98/243*(1-2*x)^(1/2)+14/243*(1-2*x)^(3/2)+2/135*(1-2*x)^(5/2)-155/126*(1-2 *x)^(7/2)+25/54*(1-2*x)^(9/2)-98/729*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x) ^(1/2))
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx=\frac {3 \sqrt {1-2 x} \left (-2479+29791 x-30546 x^2-42300 x^3+63000 x^4\right )-3430 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{25515} \] Input:
Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x),x]
Output:
(3*Sqrt[1 - 2*x]*(-2479 + 29791*x - 30546*x^2 - 42300*x^3 + 63000*x^4) - 3 430*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/25515
Time = 0.21 (sec) , antiderivative size = 95, 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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^2}{3 x+2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {25}{6} (1-2 x)^{7/2}+\frac {(1-2 x)^{5/2}}{9 (3 x+2)}+\frac {155}{18} (1-2 x)^{5/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {98}{243} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {25}{54} (1-2 x)^{9/2}-\frac {155}{126} (1-2 x)^{7/2}+\frac {2}{135} (1-2 x)^{5/2}+\frac {14}{243} (1-2 x)^{3/2}+\frac {98}{243} \sqrt {1-2 x}\) |
Input:
Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x),x]
Output:
(98*Sqrt[1 - 2*x])/243 + (14*(1 - 2*x)^(3/2))/243 + (2*(1 - 2*x)^(5/2))/13 5 - (155*(1 - 2*x)^(7/2))/126 + (25*(1 - 2*x)^(9/2))/54 - (98*Sqrt[7/3]*Ar cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/243
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 = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.52
method | result | size |
pseudoelliptic | \(-\frac {98 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{729}+\frac {\sqrt {1-2 x}\, \left (63000 x^{4}-42300 x^{3}-30546 x^{2}+29791 x -2479\right )}{8505}\) | \(49\) |
risch | \(-\frac {\left (63000 x^{4}-42300 x^{3}-30546 x^{2}+29791 x -2479\right ) \left (-1+2 x \right )}{8505 \sqrt {1-2 x}}-\frac {98 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{729}\) | \(54\) |
derivativedivides | \(\frac {98 \sqrt {1-2 x}}{243}+\frac {14 \left (1-2 x \right )^{\frac {3}{2}}}{243}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{135}-\frac {155 \left (1-2 x \right )^{\frac {7}{2}}}{126}+\frac {25 \left (1-2 x \right )^{\frac {9}{2}}}{54}-\frac {98 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{729}\) | \(65\) |
default | \(\frac {98 \sqrt {1-2 x}}{243}+\frac {14 \left (1-2 x \right )^{\frac {3}{2}}}{243}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{135}-\frac {155 \left (1-2 x \right )^{\frac {7}{2}}}{126}+\frac {25 \left (1-2 x \right )^{\frac {9}{2}}}{54}-\frac {98 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{729}\) | \(65\) |
trager | \(\left (\frac {200}{27} x^{4}-\frac {940}{189} x^{3}-\frac {3394}{945} x^{2}+\frac {29791}{8505} x -\frac {2479}{8505}\right ) \sqrt {1-2 x}+\frac {49 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{729}\) | \(74\) |
Input:
int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x),x,method=_RETURNVERBOSE)
Output:
-98/729*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))+1/8505*(1-2*x)^(1/2)* (63000*x^4-42300*x^3-30546*x^2+29791*x-2479)
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx=\frac {1}{8505} \, {\left (63000 \, x^{4} - 42300 \, x^{3} - 30546 \, x^{2} + 29791 \, x - 2479\right )} \sqrt {-2 \, x + 1} + \frac {49}{243} \, \sqrt {\frac {7}{3}} \log \left (\frac {3 \, x + 3 \, \sqrt {\frac {7}{3}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x),x, algorithm="fricas")
Output:
1/8505*(63000*x^4 - 42300*x^3 - 30546*x^2 + 29791*x - 2479)*sqrt(-2*x + 1) + 49/243*sqrt(7/3)*log((3*x + 3*sqrt(7/3)*sqrt(-2*x + 1) - 5)/(3*x + 2))
Time = 1.87 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx=\frac {25 \left (1 - 2 x\right )^{\frac {9}{2}}}{54} - \frac {155 \left (1 - 2 x\right )^{\frac {7}{2}}}{126} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{135} + \frac {14 \left (1 - 2 x\right )^{\frac {3}{2}}}{243} + \frac {98 \sqrt {1 - 2 x}}{243} + \frac {49 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{729} \] Input:
integrate((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x),x)
Output:
25*(1 - 2*x)**(9/2)/54 - 155*(1 - 2*x)**(7/2)/126 + 2*(1 - 2*x)**(5/2)/135 + 14*(1 - 2*x)**(3/2)/243 + 98*sqrt(1 - 2*x)/243 + 49*sqrt(21)*(log(sqrt( 1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/729
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx=\frac {25}{54} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {155}{126} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{135} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {14}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {49}{729} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {98}{243} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x),x, algorithm="maxima")
Output:
25/54*(-2*x + 1)^(9/2) - 155/126*(-2*x + 1)^(7/2) + 2/135*(-2*x + 1)^(5/2) + 14/243*(-2*x + 1)^(3/2) + 49/729*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 98/243*sqrt(-2*x + 1)
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.12 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx=\frac {25}{54} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {155}{126} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{135} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {14}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {49}{729} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {98}{243} \, \sqrt {-2 \, x + 1} \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x),x, algorithm="giac")
Output:
25/54*(2*x - 1)^4*sqrt(-2*x + 1) + 155/126*(2*x - 1)^3*sqrt(-2*x + 1) + 2/ 135*(2*x - 1)^2*sqrt(-2*x + 1) + 14/243*(-2*x + 1)^(3/2) + 49/729*sqrt(21) *log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)) ) + 98/243*sqrt(-2*x + 1)
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx=\frac {98\,\sqrt {1-2\,x}}{243}+\frac {14\,{\left (1-2\,x\right )}^{3/2}}{243}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{135}-\frac {155\,{\left (1-2\,x\right )}^{7/2}}{126}+\frac {25\,{\left (1-2\,x\right )}^{9/2}}{54}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,98{}\mathrm {i}}{729} \] Input:
int(((1 - 2*x)^(5/2)*(5*x + 3)^2)/(3*x + 2),x)
Output:
(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*98i)/729 + (98*(1 - 2*x)^( 1/2))/243 + (14*(1 - 2*x)^(3/2))/243 + (2*(1 - 2*x)^(5/2))/135 - (155*(1 - 2*x)^(7/2))/126 + (25*(1 - 2*x)^(9/2))/54
Time = 0.15 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx=\frac {200 \sqrt {-2 x +1}\, x^{4}}{27}-\frac {940 \sqrt {-2 x +1}\, x^{3}}{189}-\frac {3394 \sqrt {-2 x +1}\, x^{2}}{945}+\frac {29791 \sqrt {-2 x +1}\, x}{8505}-\frac {2479 \sqrt {-2 x +1}}{8505}+\frac {49 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )}{729}-\frac {49 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{729} \] Input:
int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x),x)
Output:
(189000*sqrt( - 2*x + 1)*x**4 - 126900*sqrt( - 2*x + 1)*x**3 - 91638*sqrt( - 2*x + 1)*x**2 + 89373*sqrt( - 2*x + 1)*x - 7437*sqrt( - 2*x + 1) + 1715 *sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 1715*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/25515