Integrand size = 24, antiderivative size = 109 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac {1055}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
1055/567*(1-2*x)^(3/2)+211/441*(1-2*x)^(5/2)-1/126*(1-2*x)^(7/2)/(2+3*x)^2 +143/882*(1-2*x)^(7/2)/(2+3*x)-1055/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2) )*21^(1/2)+1055/81*(1-2*x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {1}{486} \left (\frac {3 \sqrt {1-2 x} \left (10007+25987 x+12828 x^2-3960 x^3+2160 x^4\right )}{(2+3 x)^2}-2110 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \]
((3*Sqrt[1 - 2*x]*(10007 + 25987*x + 12828*x^2 - 3960*x^3 + 2160*x^4))/(2 + 3*x)^2 - 2110*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/486
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {100, 87, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^2}{(3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {1}{126} \int \frac {(1-2 x)^{5/2} (1050 x+557)}{(3 x+2)^2}dx-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{126} \left (\frac {3165}{7} \int \frac {(1-2 x)^{5/2}}{3 x+2}dx+\frac {143 (1-2 x)^{7/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{126} \left (\frac {3165}{7} \left (\frac {7}{3} \int \frac {(1-2 x)^{3/2}}{3 x+2}dx+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {143 (1-2 x)^{7/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{126} \left (\frac {3165}{7} \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {143 (1-2 x)^{7/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{126} \left (\frac {3165}{7} \left (\frac {7}{3} \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {143 (1-2 x)^{7/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{126} \left (\frac {3165}{7} \left (\frac {7}{3} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {143 (1-2 x)^{7/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{126} \left (\frac {3165}{7} \left (\frac {7}{3} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{9} (1-2 x)^{3/2}\right )+\frac {2}{15} (1-2 x)^{5/2}\right )+\frac {143 (1-2 x)^{7/2}}{7 (3 x+2)}\right )-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}\) |
-1/126*(1 - 2*x)^(7/2)/(2 + 3*x)^2 + ((143*(1 - 2*x)^(7/2))/(7*(2 + 3*x)) + (3165*((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))/7 )/126
3.20.49.3.1 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*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*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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 = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {4320 x^{5}-10080 x^{4}+29616 x^{3}+39146 x^{2}-5973 x -10007}{162 \left (2+3 x \right )^{2} \sqrt {1-2 x}}-\frac {1055 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(61\) |
| pseudoelliptic | \(\frac {-2110 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}+3 \sqrt {1-2 x}\, \left (2160 x^{4}-3960 x^{3}+12828 x^{2}+25987 x +10007\right )}{486 \left (2+3 x \right )^{2}}\) | \(65\) |
| derivativedivides | \(\frac {10 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {130 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {1006 \sqrt {1-2 x}}{81}+\frac {-\frac {1043 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {2401 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{2}}-\frac {1055 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(75\) |
| default | \(\frac {10 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {130 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {1006 \sqrt {1-2 x}}{81}+\frac {-\frac {1043 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {2401 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{2}}-\frac {1055 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(75\) |
| trager | \(\frac {\left (2160 x^{4}-3960 x^{3}+12828 x^{2}+25987 x +10007\right ) \sqrt {1-2 x}}{162 \left (2+3 x \right )^{2}}-\frac {1055 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{486}\) | \(82\) |
-1/162*(4320*x^5-10080*x^4+29616*x^3+39146*x^2-5973*x-10007)/(2+3*x)^2/(1- 2*x)^(1/2)-1055/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {1055 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 3 \, {\left (2160 \, x^{4} - 3960 \, x^{3} + 12828 \, x^{2} + 25987 \, x + 10007\right )} \sqrt {-2 \, x + 1}}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/486*(1055*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*(2160*x^4 - 3960*x^3 + 12828*x^2 + 2598 7*x + 10007)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
Time = 104.94 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.36 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {10 \left (1 - 2 x\right )^{\frac {5}{2}}}{27} + \frac {130 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} + \frac {1006 \sqrt {1 - 2 x}}{81} + \frac {1657 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{729} + \frac {14896 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{243} + \frac {2744 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{243} \]
10*(1 - 2*x)**(5/2)/27 + 130*(1 - 2*x)**(3/2)/81 + 1006*sqrt(1 - 2*x)/81 + 1657*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3))/729 + 14896*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) > -sqr t(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/243 + 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)*s qrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2 *x) < sqrt(21)/3)))/243
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {130}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1055}{486} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1006}{81} \, \sqrt {-2 \, x + 1} - \frac {7 \, {\left (149 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}}{81 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
10/27*(-2*x + 1)^(5/2) + 130/81*(-2*x + 1)^(3/2) + 1055/486*sqrt(21)*log(- (sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1006/81*sqr t(-2*x + 1) - 7/81*(149*(-2*x + 1)^(3/2) - 343*sqrt(-2*x + 1))/(9*(2*x - 1 )^2 + 84*x + 7)
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {10}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {130}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1055}{486} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1006}{81} \, \sqrt {-2 \, x + 1} - \frac {7 \, {\left (149 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}}{324 \, {\left (3 \, x + 2\right )}^{2}} \]
10/27*(2*x - 1)^2*sqrt(-2*x + 1) + 130/81*(-2*x + 1)^(3/2) + 1055/486*sqrt (21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1006/81*sqrt(-2*x + 1) - 7/324*(149*(-2*x + 1)^(3/2) - 343*sqrt(-2 *x + 1))/(3*x + 2)^2
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {1006\,\sqrt {1-2\,x}}{81}+\frac {130\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {10\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {\frac {2401\,\sqrt {1-2\,x}}{729}-\frac {1043\,{\left (1-2\,x\right )}^{3/2}}{729}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,1055{}\mathrm {i}}{243} \]