Integrand size = 22, antiderivative size = 96 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {365}{81} \sqrt {1-2 x}-\frac {365}{567} (1-2 x)^{3/2}+\frac {(1-2 x)^{7/2}}{42 (2+3 x)^2}-\frac {73 (1-2 x)^{5/2}}{126 (2+3 x)}+\frac {365}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
-365/567*(1-2*x)^(3/2)+1/42*(1-2*x)^(7/2)/(2+3*x)^2-73/126*(1-2*x)^(5/2)/( 2+3*x)+365/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-365/81*(1-2*x) ^(1/2)
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=\frac {1}{486} \left (\frac {3 \sqrt {1-2 x} \left (-3521-8731 x-4584 x^2+720 x^3\right )}{(2+3 x)^2}+730 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \]
((3*Sqrt[1 - 2*x]*(-3521 - 8731*x - 4584*x^2 + 720*x^3))/(2 + 3*x)^2 + 730 *Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/486
Time = 0.19 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {87, 51, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)}{(3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {73}{42} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^2}dx+\frac {(1-2 x)^{7/2}}{42 (3 x+2)^2}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {73}{42} \left (-\frac {5}{3} \int \frac {(1-2 x)^{3/2}}{3 x+2}dx-\frac {(1-2 x)^{5/2}}{3 (3 x+2)}\right )+\frac {(1-2 x)^{7/2}}{42 (3 x+2)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {73}{42} \left (-\frac {5}{3} \left (\frac {7}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{5/2}}{3 (3 x+2)}\right )+\frac {(1-2 x)^{7/2}}{42 (3 x+2)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {73}{42} \left (-\frac {5}{3} \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{5/2}}{3 (3 x+2)}\right )+\frac {(1-2 x)^{7/2}}{42 (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {73}{42} \left (-\frac {5}{3} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{5/2}}{3 (3 x+2)}\right )+\frac {(1-2 x)^{7/2}}{42 (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {73}{42} \left (-\frac {5}{3} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {(1-2 x)^{5/2}}{3 (3 x+2)}\right )+\frac {(1-2 x)^{7/2}}{42 (3 x+2)^2}\) |
(1 - 2*x)^(7/2)/(42*(2 + 3*x)^2) + (73*(-1/3*(1 - 2*x)^(5/2)/(2 + 3*x) - ( 5*((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))/42
3.20.37.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 1.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(-\frac {1440 x^{4}-9888 x^{3}-12878 x^{2}+1689 x +3521}{162 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {365 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(56\) |
| pseudoelliptic | \(\frac {730 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}+3 \sqrt {1-2 x}\, \left (720 x^{3}-4584 x^{2}-8731 x -3521\right )}{486 \left (2+3 x \right )^{2}}\) | \(60\) |
| derivativedivides | \(-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {32 \sqrt {1-2 x}}{9}-\frac {28 \left (-\frac {79 \left (1-2 x \right )^{\frac {3}{2}}}{36}+\frac {539 \sqrt {1-2 x}}{108}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {365 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(66\) |
| default | \(-\frac {20 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {32 \sqrt {1-2 x}}{9}-\frac {28 \left (-\frac {79 \left (1-2 x \right )^{\frac {3}{2}}}{36}+\frac {539 \sqrt {1-2 x}}{108}\right )}{3 \left (-4-6 x \right )^{2}}+\frac {365 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(66\) |
| trager | \(\frac {\left (720 x^{3}-4584 x^{2}-8731 x -3521\right ) \sqrt {1-2 x}}{162 \left (2+3 x \right )^{2}}-\frac {365 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{486}\) | \(77\) |
-1/162*(1440*x^4-9888*x^3-12878*x^2+1689*x+3521)/(2+3*x)^2/(1-2*x)^(1/2)+3 65/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=\frac {365 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 3 \, {\left (720 \, x^{3} - 4584 \, x^{2} - 8731 \, x - 3521\right )} \sqrt {-2 \, x + 1}}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/486*(365*sqrt(7)*sqrt(3)*(9*x^2 + 12*x + 4)*log(-(sqrt(7)*sqrt(3)*sqrt(- 2*x + 1) - 3*x + 5)/(3*x + 2)) + 3*(720*x^3 - 4584*x^2 - 8731*x - 3521)*sq rt(-2*x + 1))/(9*x^2 + 12*x + 4)
Time = 77.77 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.70 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=- \frac {20 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} - \frac {32 \sqrt {1 - 2 x}}{9} - \frac {74 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{81} - \frac {8036 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} - \frac {2744 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{81} \]
-20*(1 - 2*x)**(3/2)/81 - 32*sqrt(1 - 2*x)/9 - 74*sqrt(21)*(log(sqrt(1 - 2 *x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/81 - 8036*Piecewise(( sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x )/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt (21)/3)))/81 - 2744*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)* sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (s qrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/81
Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {365}{486} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {32}{9} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (237 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 539 \, \sqrt {-2 \, x + 1}\right )}}{81 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
-20/81*(-2*x + 1)^(3/2) - 365/486*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 32/9*sqrt(-2*x + 1) + 7/81*(237*(-2*x + 1)^(3/2) - 539*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {20}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {365}{486} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {32}{9} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (237 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 539 \, \sqrt {-2 \, x + 1}\right )}}{324 \, {\left (3 \, x + 2\right )}^{2}} \]
-20/81*(-2*x + 1)^(3/2) - 365/486*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqr t(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 32/9*sqrt(-2*x + 1) + 7/324* (237*(-2*x + 1)^(3/2) - 539*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 1.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^3} \, dx=-\frac {32\,\sqrt {1-2\,x}}{9}-\frac {20\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {\frac {3773\,\sqrt {1-2\,x}}{729}-\frac {553\,{\left (1-2\,x\right )}^{3/2}}{243}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,365{}\mathrm {i}}{243} \]