Integrand size = 24, antiderivative size = 95 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{2+3 x} \, dx=-\frac {14}{243} \sqrt {1-2 x}-\frac {2}{243} (1-2 x)^{3/2}-\frac {1027}{108} (1-2 x)^{5/2}+\frac {400}{63} (1-2 x)^{7/2}-\frac {125}{108} (1-2 x)^{9/2}+\frac {14}{243} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
-2/243*(1-2*x)^(3/2)-1027/108*(1-2*x)^(5/2)+400/63*(1-2*x)^(7/2)-125/108*( 1-2*x)^(9/2)+14/729*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-14/243*(1 -2*x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{2+3 x} \, dx=\frac {-3 \sqrt {1-2 x} \left (7456-15679 x-17649 x^2+23400 x^3+31500 x^4\right )+98 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5103} \]
(-3*Sqrt[1 - 2*x]*(7456 - 15679*x - 17649*x^2 + 23400*x^3 + 31500*x^4) + 9 8*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/5103
Time = 0.20 (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.
\(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^3}{3 x+2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {125}{12} (1-2 x)^{7/2}-\frac {400}{9} (1-2 x)^{5/2}-\frac {(1-2 x)^{3/2}}{27 (3 x+2)}+\frac {5135}{108} (1-2 x)^{3/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {14}{243} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {125}{108} (1-2 x)^{9/2}+\frac {400}{63} (1-2 x)^{7/2}-\frac {1027}{108} (1-2 x)^{5/2}-\frac {2}{243} (1-2 x)^{3/2}-\frac {14}{243} \sqrt {1-2 x}\) |
(-14*Sqrt[1 - 2*x])/243 - (2*(1 - 2*x)^(3/2))/243 - (1027*(1 - 2*x)^(5/2)) /108 + (400*(1 - 2*x)^(7/2))/63 - (125*(1 - 2*x)^(9/2))/108 + (14*Sqrt[7/3 ]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/243
3.19.87.3.1 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 = 0.96 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.52
method | result | size |
pseudoelliptic | \(\frac {14 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{729}-\frac {\sqrt {1-2 x}\, \left (31500 x^{4}+23400 x^{3}-17649 x^{2}-15679 x +7456\right )}{1701}\) | \(49\) |
risch | \(\frac {\left (31500 x^{4}+23400 x^{3}-17649 x^{2}-15679 x +7456\right ) \left (-1+2 x \right )}{1701 \sqrt {1-2 x}}+\frac {14 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{729}\) | \(54\) |
derivativedivides | \(-\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{243}-\frac {1027 \left (1-2 x \right )^{\frac {5}{2}}}{108}+\frac {400 \left (1-2 x \right )^{\frac {7}{2}}}{63}-\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{108}+\frac {14 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{729}-\frac {14 \sqrt {1-2 x}}{243}\) | \(65\) |
default | \(-\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{243}-\frac {1027 \left (1-2 x \right )^{\frac {5}{2}}}{108}+\frac {400 \left (1-2 x \right )^{\frac {7}{2}}}{63}-\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{108}+\frac {14 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{729}-\frac {14 \sqrt {1-2 x}}{243}\) | \(65\) |
trager | \(\left (-\frac {500}{27} x^{4}-\frac {2600}{189} x^{3}+\frac {1961}{189} x^{2}+\frac {15679}{1701} x -\frac {7456}{1701}\right ) \sqrt {1-2 x}-\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{729}\) | \(74\) |
14/729*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1/1701*(1-2*x)^(1/2)*( 31500*x^4+23400*x^3-17649*x^2-15679*x+7456)
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{2+3 x} \, dx=\frac {7}{729} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - \frac {1}{1701} \, {\left (31500 \, x^{4} + 23400 \, x^{3} - 17649 \, x^{2} - 15679 \, x + 7456\right )} \sqrt {-2 \, x + 1} \]
7/729*sqrt(7)*sqrt(3)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) - 1/1701*(31500*x^4 + 23400*x^3 - 17649*x^2 - 15679*x + 7456)*sqrt( -2*x + 1)
Time = 1.88 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{2+3 x} \, dx=- \frac {125 \left (1 - 2 x\right )^{\frac {9}{2}}}{108} + \frac {400 \left (1 - 2 x\right )^{\frac {7}{2}}}{63} - \frac {1027 \left (1 - 2 x\right )^{\frac {5}{2}}}{108} - \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{243} - \frac {14 \sqrt {1 - 2 x}}{243} - \frac {7 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{729} \]
-125*(1 - 2*x)**(9/2)/108 + 400*(1 - 2*x)**(7/2)/63 - 1027*(1 - 2*x)**(5/2 )/108 - 2*(1 - 2*x)**(3/2)/243 - 14*sqrt(1 - 2*x)/243 - 7*sqrt(21)*(log(sq rt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/729
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{2+3 x} \, dx=-\frac {125}{108} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {400}{63} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1027}{108} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {2}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7}{729} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {14}{243} \, \sqrt {-2 \, x + 1} \]
-125/108*(-2*x + 1)^(9/2) + 400/63*(-2*x + 1)^(7/2) - 1027/108*(-2*x + 1)^ (5/2) - 2/243*(-2*x + 1)^(3/2) - 7/729*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2 *x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 14/243*sqrt(-2*x + 1)
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.12 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{2+3 x} \, dx=-\frac {125}{108} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {400}{63} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1027}{108} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {2}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {7}{729} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {14}{243} \, \sqrt {-2 \, x + 1} \]
-125/108*(2*x - 1)^4*sqrt(-2*x + 1) - 400/63*(2*x - 1)^3*sqrt(-2*x + 1) - 1027/108*(2*x - 1)^2*sqrt(-2*x + 1) - 2/243*(-2*x + 1)^(3/2) - 7/729*sqrt( 21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 14/243*sqrt(-2*x + 1)
Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{2+3 x} \, dx=\frac {400\,{\left (1-2\,x\right )}^{7/2}}{63}-\frac {2\,{\left (1-2\,x\right )}^{3/2}}{243}-\frac {1027\,{\left (1-2\,x\right )}^{5/2}}{108}-\frac {14\,\sqrt {1-2\,x}}{243}-\frac {125\,{\left (1-2\,x\right )}^{9/2}}{108}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,14{}\mathrm {i}}{729} \]