Integrand size = 22, antiderivative size = 82 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{2+3 x} \, dx=-\frac {98}{81} \sqrt {1-2 x}-\frac {14}{81} (1-2 x)^{3/2}-\frac {2}{45} (1-2 x)^{5/2}-\frac {5}{21} (1-2 x)^{7/2}+\frac {98}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
-14/81*(1-2*x)^(3/2)-2/45*(1-2*x)^(5/2)-5/21*(1-2*x)^(7/2)+98/243*arctanh( 1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-98/81*(1-2*x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{2+3 x} \, dx=\frac {\sqrt {1-2 x} \left (-4721+5534 x-8604 x^2+5400 x^3\right )}{2835}+\frac {98}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(-4721 + 5534*x - 8604*x^2 + 5400*x^3))/2835 + (98*Sqrt[7/3 ]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81
Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {90, 60, 60, 60, 73, 219}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)}{3 x+2} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle -\frac {1}{3} \int \frac {(1-2 x)^{5/2}}{3 x+2}dx-\frac {5}{21} (1-2 x)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{3} \left (-\frac {7}{3} \int \frac {(1-2 x)^{3/2}}{3 x+2}dx-\frac {2}{15} (1-2 x)^{5/2}\right )-\frac {5}{21} (1-2 x)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{3} \left (-\frac {7}{3} \left (\frac {7}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {2}{15} (1-2 x)^{5/2}\right )-\frac {5}{21} (1-2 x)^{7/2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{3} \left (-\frac {7}{3} \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {2}{15} (1-2 x)^{5/2}\right )-\frac {5}{21} (1-2 x)^{7/2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (-\frac {7}{3} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {2}{15} (1-2 x)^{5/2}\right )-\frac {5}{21} (1-2 x)^{7/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (-\frac {7}{3} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {2}{15} (1-2 x)^{5/2}\right )-\frac {5}{21} (1-2 x)^{7/2}\) |
(-5*(1 - 2*x)^(7/2))/21 + ((-2*(1 - 2*x)^(5/2))/15 - (7*((2*(1 - 2*x)^(3/2 ))/9 + (7*((2*Sqrt[1 - 2*x])/3 - (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2 *x]])/3))/3))/3)/3
3.20.35.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 0.97 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54
| method | result | size |
| pseudoelliptic | \(\frac {98 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}+\frac {\sqrt {1-2 x}\, \left (5400 x^{3}-8604 x^{2}+5534 x -4721\right )}{2835}\) | \(44\) |
| risch | \(-\frac {\left (5400 x^{3}-8604 x^{2}+5534 x -4721\right ) \left (-1+2 x \right )}{2835 \sqrt {1-2 x}}+\frac {98 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(49\) |
| derivativedivides | \(-\frac {14 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{45}-\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{21}+\frac {98 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}-\frac {98 \sqrt {1-2 x}}{81}\) | \(56\) |
| default | \(-\frac {14 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{45}-\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{21}+\frac {98 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}-\frac {98 \sqrt {1-2 x}}{81}\) | \(56\) |
| trager | \(\left (\frac {40}{21} x^{3}-\frac {956}{315} x^{2}+\frac {5534}{2835} x -\frac {4721}{2835}\right ) \sqrt {1-2 x}-\frac {49 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{243}\) | \(69\) |
98/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/2835*(1-2*x)^(1/2)*( 5400*x^3-8604*x^2+5534*x-4721)
Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{2+3 x} \, dx=\frac {49}{243} \, \sqrt {7} \sqrt {3} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac {1}{2835} \, {\left (5400 \, x^{3} - 8604 \, x^{2} + 5534 \, x - 4721\right )} \sqrt {-2 \, x + 1} \]
49/243*sqrt(7)*sqrt(3)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3* x + 2)) + 1/2835*(5400*x^3 - 8604*x^2 + 5534*x - 4721)*sqrt(-2*x + 1)
Time = 1.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{2+3 x} \, dx=- \frac {5 \left (1 - 2 x\right )^{\frac {7}{2}}}{21} - \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{45} - \frac {14 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} - \frac {98 \sqrt {1 - 2 x}}{81} - \frac {49 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{243} \]
-5*(1 - 2*x)**(7/2)/21 - 2*(1 - 2*x)**(5/2)/45 - 14*(1 - 2*x)**(3/2)/81 - 98*sqrt(1 - 2*x)/81 - 49*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(s qrt(1 - 2*x) + sqrt(21)/3))/243
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{2+3 x} \, dx=-\frac {5}{21} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {2}{45} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {14}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {49}{243} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {98}{81} \, \sqrt {-2 \, x + 1} \]
-5/21*(-2*x + 1)^(7/2) - 2/45*(-2*x + 1)^(5/2) - 14/81*(-2*x + 1)^(3/2) - 49/243*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 98/81*sqrt(-2*x + 1)
Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{2+3 x} \, dx=\frac {5}{21} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {2}{45} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {14}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {49}{243} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {98}{81} \, \sqrt {-2 \, x + 1} \]
5/21*(2*x - 1)^3*sqrt(-2*x + 1) - 2/45*(2*x - 1)^2*sqrt(-2*x + 1) - 14/81* (-2*x + 1)^(3/2) - 49/243*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 98/81*sqrt(-2*x + 1)
Time = 1.41 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{2+3 x} \, dx=-\frac {98\,\sqrt {1-2\,x}}{81}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {2\,{\left (1-2\,x\right )}^{5/2}}{45}-\frac {5\,{\left (1-2\,x\right )}^{7/2}}{21}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,98{}\mathrm {i}}{243} \]