Integrand size = 22, antiderivative size = 143 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {123}{16807 \sqrt {1-2 x}}+\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}-\frac {41}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {41}{1715 \sqrt {1-2 x} (2+3 x)^3}-\frac {41}{3430 \sqrt {1-2 x} (2+3 x)^2}-\frac {41}{4802 \sqrt {1-2 x} (2+3 x)}-\frac {123 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{16807} \]
-123/117649*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+123/16807/(1-2*x) ^(1/2)+1/105/(2+3*x)^5/(1-2*x)^(1/2)-41/735/(2+3*x)^4/(1-2*x)^(1/2)-41/171 5/(2+3*x)^3/(1-2*x)^(1/2)-41/3430/(2+3*x)^2/(1-2*x)^(1/2)-41/4802/(2+3*x)/ (1-2*x)^(1/2)
Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.52 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {\frac {7 \left (-32894+8774 x+430992 x^2+964197 x^3+880065 x^4+298890 x^5\right )}{2 \sqrt {1-2 x} (2+3 x)^5}-615 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588245} \]
((7*(-32894 + 8774*x + 430992*x^2 + 964197*x^3 + 880065*x^4 + 298890*x^5)) /(2*Sqrt[1 - 2*x]*(2 + 3*x)^5) - 615*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2 *x]])/588245
Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {87, 52, 52, 52, 52, 61, 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.
| \(\displaystyle \int \frac {5 x+3}{(1-2 x)^{3/2} (3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {164}{105} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^5}dx+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {164}{105} \left (\frac {9}{28} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^4}dx-\frac {1}{28 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {164}{105} \left (\frac {9}{28} \left (\frac {1}{3} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^3}dx-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {1}{28 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {164}{105} \left (\frac {9}{28} \left (\frac {1}{3} \left (\frac {5}{14} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2}dx-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {1}{28 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {164}{105} \left (\frac {9}{28} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {1}{28 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {164}{105} \left (\frac {9}{28} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {1}{28 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {164}{105} \left (\frac {9}{28} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {1}{28 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {164}{105} \left (\frac {9}{28} \left (\frac {1}{3} \left (\frac {5}{14} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {1}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {1}{14 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {1}{21 \sqrt {1-2 x} (3 x+2)^3}\right )-\frac {1}{28 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}\) |
1/(105*Sqrt[1 - 2*x]*(2 + 3*x)^5) + (164*(-1/28*1/(Sqrt[1 - 2*x]*(2 + 3*x) ^4) + (9*(-1/21*1/(Sqrt[1 - 2*x]*(2 + 3*x)^3) + (-1/14*1/(Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*(-1/7*1/(Sqrt[1 - 2*x]*(2 + 3*x)) + (3*(2/(7*Sqrt[1 - 2*x] ) - (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7))/7))/14)/3))/28))/10 5
3.21.82.3.1 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 + 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] :> 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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 = 3.42 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.43
| method | result | size |
| risch | \(\frac {298890 x^{5}+880065 x^{4}+964197 x^{3}+430992 x^{2}+8774 x -32894}{168070 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {123 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{117649}\) | \(61\) |
| pseudoelliptic | \(-\frac {29889 \left (\sqrt {21}\, \left (\frac {2}{3}+x \right )^{5} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )-7 x^{5}-\frac {371 x^{4}}{18}-\frac {6097 x^{3}}{270}-\frac {4088 x^{2}}{405}-\frac {749 x}{3645}+\frac {115129}{149445}\right )}{117649 \sqrt {1-2 x}\, \left (2+3 x \right )^{5}}\) | \(71\) |
| derivativedivides | \(\frac {\frac {123687 \left (1-2 x \right )^{\frac {9}{2}}}{117649}-\frac {182898 \left (1-2 x \right )^{\frac {7}{2}}}{16807}+\frac {516672 \left (1-2 x \right )^{\frac {5}{2}}}{12005}-\frac {26622 \left (1-2 x \right )^{\frac {3}{2}}}{343}+\frac {2633 \sqrt {1-2 x}}{49}}{\left (-4-6 x \right )^{5}}-\frac {123 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{117649}+\frac {352}{117649 \sqrt {1-2 x}}\) | \(84\) |
| default | \(\frac {\frac {123687 \left (1-2 x \right )^{\frac {9}{2}}}{117649}-\frac {182898 \left (1-2 x \right )^{\frac {7}{2}}}{16807}+\frac {516672 \left (1-2 x \right )^{\frac {5}{2}}}{12005}-\frac {26622 \left (1-2 x \right )^{\frac {3}{2}}}{343}+\frac {2633 \sqrt {1-2 x}}{49}}{\left (-4-6 x \right )^{5}}-\frac {123 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{117649}+\frac {352}{117649 \sqrt {1-2 x}}\) | \(84\) |
| trager | \(-\frac {\left (298890 x^{5}+880065 x^{4}+964197 x^{3}+430992 x^{2}+8774 x -32894\right ) \sqrt {1-2 x}}{168070 \left (2+3 x \right )^{5} \left (-1+2 x \right )}-\frac {123 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{235298}\) | \(94\) |
1/168070*(298890*x^5+880065*x^4+964197*x^3+430992*x^2+8774*x-32894)/(2+3*x )^5/(1-2*x)^(1/2)-123/117649*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {615 \, \sqrt {7} \sqrt {3} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (298890 \, x^{5} + 880065 \, x^{4} + 964197 \, x^{3} + 430992 \, x^{2} + 8774 \, x - 32894\right )} \sqrt {-2 \, x + 1}}{1176490 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \]
1/1176490*(615*sqrt(7)*sqrt(3)*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 7*(298890*x^5 + 880065*x^4 + 964197*x^3 + 430992*x^2 + 8774*x - 32 894)*sqrt(-2*x + 1))/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)
Timed out. \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.96 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {123}{235298} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {149445 \, {\left (2 \, x - 1\right )}^{5} + 1627290 \, {\left (2 \, x - 1\right )}^{4} + 6943104 \, {\left (2 \, x - 1\right )}^{3} + 14283990 \, {\left (2 \, x - 1\right )}^{2} + 27141590 \, x - 9345035}{84035 \, {\left (243 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2835 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 13230 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 30870 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 36015 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 16807 \, \sqrt {-2 \, x + 1}\right )}} \]
123/235298*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt( -2*x + 1))) - 1/84035*(149445*(2*x - 1)^5 + 1627290*(2*x - 1)^4 + 6943104* (2*x - 1)^3 + 14283990*(2*x - 1)^2 + 27141590*x - 9345035)/(243*(-2*x + 1) ^(11/2) - 2835*(-2*x + 1)^(9/2) + 13230*(-2*x + 1)^(7/2) - 30870*(-2*x + 1 )^(5/2) + 36015*(-2*x + 1)^(3/2) - 16807*sqrt(-2*x + 1))
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.87 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {123}{235298} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {352}{117649 \, \sqrt {-2 \, x + 1}} - \frac {618435 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 6401430 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 25316928 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 45656730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 31609165 \, \sqrt {-2 \, x + 1}}{18823840 \, {\left (3 \, x + 2\right )}^{5}} \]
123/235298*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 352/117649/sqrt(-2*x + 1) - 1/18823840*(618435*(2*x - 1)^4*sqrt(-2*x + 1) + 6401430*(2*x - 1)^3*sqrt(-2*x + 1) + 25316928*(2* x - 1)^2*sqrt(-2*x + 1) - 45656730*(-2*x + 1)^(3/2) + 31609165*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 1.35 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.83 \[ \int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx=\frac {\frac {15826\,x}{11907}+\frac {6478\,{\left (2\,x-1\right )}^2}{9261}+\frac {5248\,{\left (2\,x-1\right )}^3}{15435}+\frac {82\,{\left (2\,x-1\right )}^4}{1029}+\frac {123\,{\left (2\,x-1\right )}^5}{16807}-\frac {5449}{11907}}{\frac {16807\,\sqrt {1-2\,x}}{243}-\frac {12005\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {3430\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {490\,{\left (1-2\,x\right )}^{7/2}}{9}+\frac {35\,{\left (1-2\,x\right )}^{9/2}}{3}-{\left (1-2\,x\right )}^{11/2}}-\frac {123\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{117649} \]
((15826*x)/11907 + (6478*(2*x - 1)^2)/9261 + (5248*(2*x - 1)^3)/15435 + (8 2*(2*x - 1)^4)/1029 + (123*(2*x - 1)^5)/16807 - 5449/11907)/((16807*(1 - 2 *x)^(1/2))/243 - (12005*(1 - 2*x)^(3/2))/81 + (3430*(1 - 2*x)^(5/2))/27 - (490*(1 - 2*x)^(7/2))/9 + (35*(1 - 2*x)^(9/2))/3 - (1 - 2*x)^(11/2)) - (12 3*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/117649