Integrand size = 24, antiderivative size = 108 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx=-\frac {98}{729} \sqrt {1-2 x}-\frac {14}{729} (1-2 x)^{3/2}-\frac {2}{405} (1-2 x)^{5/2}-\frac {5135}{756} (1-2 x)^{7/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {125}{132} (1-2 x)^{11/2}+\frac {98}{729} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
-14/729*(1-2*x)^(3/2)-2/405*(1-2*x)^(5/2)-5135/756*(1-2*x)^(7/2)+400/81*(1 -2*x)^(9/2)-125/132*(1-2*x)^(11/2)+98/2187*arctanh(1/7*21^(1/2)*(1-2*x)^(1 /2))*21^(1/2)-98/729*(1-2*x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx=\frac {\sqrt {1-2 x} \left (-830656+3024349 x-249219 x^2-7838550 x^3+913500 x^4+8505000 x^5\right )}{280665}+\frac {98}{729} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(-830656 + 3024349*x - 249219*x^2 - 7838550*x^3 + 913500*x^ 4 + 8505000*x^5))/280665 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]) /729
Time = 0.21 (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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^3}{3 x+2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {125}{12} (1-2 x)^{9/2}-\frac {400}{9} (1-2 x)^{7/2}-\frac {(1-2 x)^{5/2}}{27 (3 x+2)}+\frac {5135}{108} (1-2 x)^{5/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {98}{729} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {125}{132} (1-2 x)^{11/2}+\frac {400}{81} (1-2 x)^{9/2}-\frac {5135}{756} (1-2 x)^{7/2}-\frac {2}{405} (1-2 x)^{5/2}-\frac {14}{729} (1-2 x)^{3/2}-\frac {98}{729} \sqrt {1-2 x}\) |
(-98*Sqrt[1 - 2*x])/729 - (14*(1 - 2*x)^(3/2))/729 - (2*(1 - 2*x)^(5/2))/4 05 - (5135*(1 - 2*x)^(7/2))/756 + (400*(1 - 2*x)^(9/2))/81 - (125*(1 - 2*x )^(11/2))/132 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/729
3.20.60.3.1 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 = 1.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50
method | result | size |
pseudoelliptic | \(\frac {98 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}+\frac {\sqrt {1-2 x}\, \left (8505000 x^{5}+913500 x^{4}-7838550 x^{3}-249219 x^{2}+3024349 x -830656\right )}{280665}\) | \(54\) |
risch | \(-\frac {\left (8505000 x^{5}+913500 x^{4}-7838550 x^{3}-249219 x^{2}+3024349 x -830656\right ) \left (-1+2 x \right )}{280665 \sqrt {1-2 x}}+\frac {98 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}\) | \(59\) |
derivativedivides | \(-\frac {14 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{405}-\frac {5135 \left (1-2 x \right )^{\frac {7}{2}}}{756}+\frac {400 \left (1-2 x \right )^{\frac {9}{2}}}{81}-\frac {125 \left (1-2 x \right )^{\frac {11}{2}}}{132}+\frac {98 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}-\frac {98 \sqrt {1-2 x}}{729}\) | \(74\) |
default | \(-\frac {14 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{405}-\frac {5135 \left (1-2 x \right )^{\frac {7}{2}}}{756}+\frac {400 \left (1-2 x \right )^{\frac {9}{2}}}{81}-\frac {125 \left (1-2 x \right )^{\frac {11}{2}}}{132}+\frac {98 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}-\frac {98 \sqrt {1-2 x}}{729}\) | \(74\) |
trager | \(\left (\frac {1000}{33} x^{5}+\frac {2900}{891} x^{4}-\frac {174190}{6237} x^{3}-\frac {27691}{31185} x^{2}+\frac {3024349}{280665} x -\frac {830656}{280665}\right ) \sqrt {1-2 x}+\frac {49 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2187}\) | \(79\) |
98/2187*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/280665*(1-2*x)^(1/2 )*(8505000*x^5+913500*x^4-7838550*x^3-249219*x^2+3024349*x-830656)
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx=\frac {49}{2187} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac {1}{280665} \, {\left (8505000 \, x^{5} + 913500 \, x^{4} - 7838550 \, x^{3} - 249219 \, x^{2} + 3024349 \, x - 830656\right )} \sqrt {-2 \, x + 1} \]
49/2187*sqrt(7)*sqrt(3)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3 *x + 2)) + 1/280665*(8505000*x^5 + 913500*x^4 - 7838550*x^3 - 249219*x^2 + 3024349*x - 830656)*sqrt(-2*x + 1)
Time = 2.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx=- \frac {125 \left (1 - 2 x\right )^{\frac {11}{2}}}{132} + \frac {400 \left (1 - 2 x\right )^{\frac {9}{2}}}{81} - \frac {5135 \left (1 - 2 x\right )^{\frac {7}{2}}}{756} - \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{405} - \frac {14 \left (1 - 2 x\right )^{\frac {3}{2}}}{729} - \frac {98 \sqrt {1 - 2 x}}{729} - \frac {49 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{2187} \]
-125*(1 - 2*x)**(11/2)/132 + 400*(1 - 2*x)**(9/2)/81 - 5135*(1 - 2*x)**(7/ 2)/756 - 2*(1 - 2*x)**(5/2)/405 - 14*(1 - 2*x)**(3/2)/729 - 98*sqrt(1 - 2* x)/729 - 49*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/2187
Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx=-\frac {125}{132} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {400}{81} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {5135}{756} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {2}{405} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {14}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {49}{2187} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {98}{729} \, \sqrt {-2 \, x + 1} \]
-125/132*(-2*x + 1)^(11/2) + 400/81*(-2*x + 1)^(9/2) - 5135/756*(-2*x + 1) ^(7/2) - 2/405*(-2*x + 1)^(5/2) - 14/729*(-2*x + 1)^(3/2) - 49/2187*sqrt(2 1)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 98/ 729*sqrt(-2*x + 1)
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx=\frac {125}{132} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {400}{81} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {5135}{756} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {2}{405} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {14}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {49}{2187} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {98}{729} \, \sqrt {-2 \, x + 1} \]
125/132*(2*x - 1)^5*sqrt(-2*x + 1) + 400/81*(2*x - 1)^4*sqrt(-2*x + 1) + 5 135/756*(2*x - 1)^3*sqrt(-2*x + 1) - 2/405*(2*x - 1)^2*sqrt(-2*x + 1) - 14 /729*(-2*x + 1)^(3/2) - 49/2187*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt( -2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 98/729*sqrt(-2*x + 1)
Time = 1.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx=\frac {400\,{\left (1-2\,x\right )}^{9/2}}{81}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{729}-\frac {2\,{\left (1-2\,x\right )}^{5/2}}{405}-\frac {5135\,{\left (1-2\,x\right )}^{7/2}}{756}-\frac {98\,\sqrt {1-2\,x}}{729}-\frac {125\,{\left (1-2\,x\right )}^{11/2}}{132}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,98{}\mathrm {i}}{2187} \]