Integrand size = 24, antiderivative size = 148 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}+\frac {1613 \sqrt {1-2 x}}{1037232 (2+3 x)}+\frac {1613 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{518616 \sqrt {21}} \]
-1/378*(1-2*x)^(3/2)/(2+3*x)^6+137/4410*(1-2*x)^(3/2)/(2+3*x)^5+1613/10890 936*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1613/7560*(1-2*x)^(1/2)/( 2+3*x)^4+1613/158760*(1-2*x)^(1/2)/(2+3*x)^3+1613/444528*(1-2*x)^(1/2)/(2+ 3*x)^2+1613/1037232*(1-2*x)^(1/2)/(2+3*x)
Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-3136864-7772840 x+1791558 x^2+14197626 x^3+8056935 x^4+1959795 x^5\right )}{2 (2+3 x)^6}+8065 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{54454680} \]
((21*Sqrt[1 - 2*x]*(-3136864 - 7772840*x + 1791558*x^2 + 14197626*x^3 + 80 56935*x^4 + 1959795*x^5))/(2*(2 + 3*x)^6) + 8065*Sqrt[21]*ArcTanh[Sqrt[3/7 ]*Sqrt[1 - 2*x]])/54454680
Time = 0.23 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {100, 27, 87, 51, 52, 52, 52, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^7} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {1}{378} \int \frac {3 \sqrt {1-2 x} (1050 x+563)}{(3 x+2)^6}dx-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{126} \int \frac {\sqrt {1-2 x} (1050 x+563)}{(3 x+2)^6}dx-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{126} \left (\frac {1613}{5} \int \frac {\sqrt {1-2 x}}{(3 x+2)^5}dx+\frac {137 (1-2 x)^{3/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{126} \left (\frac {1613}{5} \left (-\frac {1}{12} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {137 (1-2 x)^{3/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{126} \left (\frac {1613}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {137 (1-2 x)^{3/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{126} \left (\frac {1613}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {137 (1-2 x)^{3/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{126} \left (\frac {1613}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {137 (1-2 x)^{3/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{126} \left (\frac {1613}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {137 (1-2 x)^{3/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{126} \left (\frac {1613}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {137 (1-2 x)^{3/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}\) |
-1/378*(1 - 2*x)^(3/2)/(2 + 3*x)^6 + ((137*(1 - 2*x)^(3/2))/(35*(2 + 3*x)^ 5) + (1613*(-1/12*Sqrt[1 - 2*x]/(2 + 3*x)^4 + (Sqrt[1 - 2*x]/(21*(2 + 3*x) ^3) - (5*(-1/14*Sqrt[1 - 2*x]/(2 + 3*x)^2 + (3*(-1/7*Sqrt[1 - 2*x]/(2 + 3* x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14))/21)/12))/5)/ 126
3.19.14.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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 = 2.98 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(-\frac {3919590 x^{6}+14154075 x^{5}+20338317 x^{4}-10614510 x^{3}-17337238 x^{2}+1499112 x +3136864}{5186160 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {1613 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{10890936}\) | \(66\) |
| pseudoelliptic | \(\frac {16130 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}+21 \sqrt {1-2 x}\, \left (1959795 x^{5}+8056935 x^{4}+14197626 x^{3}+1791558 x^{2}-7772840 x -3136864\right )}{108909360 \left (2+3 x \right )^{6}}\) | \(70\) |
| derivativedivides | \(\frac {-\frac {14517 \left (1-2 x \right )^{\frac {11}{2}}}{19208}+\frac {27421 \left (1-2 x \right )^{\frac {9}{2}}}{2744}-\frac {53229 \left (1-2 x \right )^{\frac {7}{2}}}{980}+\frac {113751 \left (1-2 x \right )^{\frac {5}{2}}}{980}-\frac {86837 \left (1-2 x \right )^{\frac {3}{2}}}{1512}-\frac {11291 \sqrt {1-2 x}}{216}}{\left (-4-6 x \right )^{6}}+\frac {1613 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{10890936}\) | \(84\) |
| default | \(\frac {-\frac {14517 \left (1-2 x \right )^{\frac {11}{2}}}{19208}+\frac {27421 \left (1-2 x \right )^{\frac {9}{2}}}{2744}-\frac {53229 \left (1-2 x \right )^{\frac {7}{2}}}{980}+\frac {113751 \left (1-2 x \right )^{\frac {5}{2}}}{980}-\frac {86837 \left (1-2 x \right )^{\frac {3}{2}}}{1512}-\frac {11291 \sqrt {1-2 x}}{216}}{\left (-4-6 x \right )^{6}}+\frac {1613 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{10890936}\) | \(84\) |
| trager | \(\frac {\left (1959795 x^{5}+8056935 x^{4}+14197626 x^{3}+1791558 x^{2}-7772840 x -3136864\right ) \sqrt {1-2 x}}{5186160 \left (2+3 x \right )^{6}}-\frac {1613 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{21781872}\) | \(87\) |
-1/5186160*(3919590*x^6+14154075*x^5+20338317*x^4-10614510*x^3-17337238*x^ 2+1499112*x+3136864)/(2+3*x)^6/(1-2*x)^(1/2)+1613/10890936*arctanh(1/7*21^ (1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {8065 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (1959795 \, x^{5} + 8056935 \, x^{4} + 14197626 \, x^{3} + 1791558 \, x^{2} - 7772840 \, x - 3136864\right )} \sqrt {-2 \, x + 1}}{108909360 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
1/108909360*(8065*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 216 0*x^2 + 576*x + 64)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 2 1*(1959795*x^5 + 8056935*x^4 + 14197626*x^3 + 1791558*x^2 - 7772840*x - 31 36864)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^ 2 + 576*x + 64)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {1613}{21781872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1959795 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 25912845 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 140843934 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 300985146 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 148925455 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 135548455 \, \sqrt {-2 \, x + 1}}{2593080 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]
-1613/21781872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*s qrt(-2*x + 1))) - 1/2593080*(1959795*(-2*x + 1)^(11/2) - 25912845*(-2*x + 1)^(9/2) + 140843934*(-2*x + 1)^(7/2) - 300985146*(-2*x + 1)^(5/2) + 14892 5455*(-2*x + 1)^(3/2) + 135548455*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206 *(2*x - 1)^5 + 59535*(2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {1613}{21781872} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1959795 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 25912845 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 140843934 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 300985146 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 148925455 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 135548455 \, \sqrt {-2 \, x + 1}}{165957120 \, {\left (3 \, x + 2\right )}^{6}} \]
-1613/21781872*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt( 21) + 3*sqrt(-2*x + 1))) + 1/165957120*(1959795*(2*x - 1)^5*sqrt(-2*x + 1) + 25912845*(2*x - 1)^4*sqrt(-2*x + 1) + 140843934*(2*x - 1)^3*sqrt(-2*x + 1) + 300985146*(2*x - 1)^2*sqrt(-2*x + 1) - 148925455*(-2*x + 1)^(3/2) - 135548455*sqrt(-2*x + 1))/(3*x + 2)^6
Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {1613\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{10890936}-\frac {\frac {11291\,\sqrt {1-2\,x}}{157464}+\frac {86837\,{\left (1-2\,x\right )}^{3/2}}{1102248}-\frac {4213\,{\left (1-2\,x\right )}^{5/2}}{26460}+\frac {17743\,{\left (1-2\,x\right )}^{7/2}}{238140}-\frac {27421\,{\left (1-2\,x\right )}^{9/2}}{2000376}+\frac {1613\,{\left (1-2\,x\right )}^{11/2}}{1555848}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]
(1613*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/10890936 - ((11291*(1 - 2*x)^(1/2))/157464 + (86837*(1 - 2*x)^(3/2))/1102248 - (4213*(1 - 2*x)^( 5/2))/26460 + (17743*(1 - 2*x)^(7/2))/238140 - (27421*(1 - 2*x)^(9/2))/200 0376 + (1613*(1 - 2*x)^(11/2))/1555848)/((67228*x)/81 + (12005*(2*x - 1)^2 )/27 + (6860*(2*x - 1)^3)/27 + (245*(2*x - 1)^4)/3 + 14*(2*x - 1)^5 + (2*x - 1)^6 - 184877/729)