Integrand size = 22, antiderivative size = 128 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {(1-2 x)^{7/2}}{105 (2+3 x)^5}-\frac {43 (1-2 x)^{5/2}}{315 (2+3 x)^4}+\frac {43 (1-2 x)^{3/2}}{567 (2+3 x)^3}-\frac {43 \sqrt {1-2 x}}{1134 (2+3 x)^2}+\frac {43 \sqrt {1-2 x}}{7938 (2+3 x)}+\frac {43 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}} \]
1/105*(1-2*x)^(7/2)/(2+3*x)^5-43/315*(1-2*x)^(5/2)/(2+3*x)^4+43/567*(1-2*x )^(3/2)/(2+3*x)^3+43/83349*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-43 /1134*(1-2*x)^(1/2)/(2+3*x)^2+43/7938*(1-2*x)^(1/2)/(2+3*x)
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-7018+3322 x-53772 x^2-116415 x^3+17415 x^4\right )}{(2+3 x)^5}+430 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{833490} \]
((21*Sqrt[1 - 2*x]*(-7018 + 3322*x - 53772*x^2 - 116415*x^3 + 17415*x^4))/ (2 + 3*x)^5 + 430*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/833490
Time = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {87, 51, 51, 51, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {172}{105} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^5}dx+\frac {(1-2 x)^{7/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {172}{105} \left (-\frac {5}{12} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^4}dx-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{7/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {172}{105} \left (-\frac {5}{12} \left (-\frac {1}{3} \int \frac {\sqrt {1-2 x}}{(3 x+2)^3}dx-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{7/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {172}{105} \left (-\frac {5}{12} \left (\frac {1}{3} \left (\frac {1}{6} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{7/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {172}{105} \left (-\frac {5}{12} \left (\frac {1}{3} \left (\frac {1}{6} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{7/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {172}{105} \left (-\frac {5}{12} \left (\frac {1}{3} \left (\frac {1}{6} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{7/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {172}{105} \left (-\frac {5}{12} \left (\frac {1}{3} \left (\frac {1}{6} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{7/2}}{105 (3 x+2)^5}\) |
(1 - 2*x)^(7/2)/(105*(2 + 3*x)^5) + (172*(-1/12*(1 - 2*x)^(5/2)/(2 + 3*x)^ 4 - (5*(-1/9*(1 - 2*x)^(3/2)/(2 + 3*x)^3 + (Sqrt[1 - 2*x]/(6*(2 + 3*x)^2) + (-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7* Sqrt[21]))/6)/3))/12))/105
3.20.40.3.1 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 + 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)^(-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 = 1.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(-\frac {34830 x^{5}-250245 x^{4}+8871 x^{3}+60416 x^{2}-17358 x +7018}{39690 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {43 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\) | \(61\) |
| pseudoelliptic | \(\frac {430 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}+21 \sqrt {1-2 x}\, \left (17415 x^{4}-116415 x^{3}-53772 x^{2}+3322 x -7018\right )}{833490 \left (2+3 x \right )^{5}}\) | \(65\) |
| derivativedivides | \(\frac {-\frac {43 \left (1-2 x \right )^{\frac {9}{2}}}{49}-\frac {74 \left (1-2 x \right )^{\frac {7}{2}}}{9}+\frac {5504 \left (1-2 x \right )^{\frac {5}{2}}}{135}-\frac {4214 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {2107 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{5}}+\frac {43 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\) | \(75\) |
| default | \(\frac {-\frac {43 \left (1-2 x \right )^{\frac {9}{2}}}{49}-\frac {74 \left (1-2 x \right )^{\frac {7}{2}}}{9}+\frac {5504 \left (1-2 x \right )^{\frac {5}{2}}}{135}-\frac {4214 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {2107 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{5}}+\frac {43 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{83349}\) | \(75\) |
| trager | \(\frac {\left (17415 x^{4}-116415 x^{3}-53772 x^{2}+3322 x -7018\right ) \sqrt {1-2 x}}{39690 \left (2+3 x \right )^{5}}+\frac {43 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{166698}\) | \(82\) |
-1/39690*(34830*x^5-250245*x^4+8871*x^3+60416*x^2-17358*x+7018)/(2+3*x)^5/ (1-2*x)^(1/2)+43/83349*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {215 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (17415 \, x^{4} - 116415 \, x^{3} - 53772 \, x^{2} + 3322 \, x - 7018\right )} \sqrt {-2 \, x + 1}}{833490 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/833490*(215*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 3 2)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(17415*x^4 - 11 6415*x^3 - 53772*x^2 + 3322*x - 7018)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^6} \, dx=-\frac {43}{166698} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {17415 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 163170 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 809088 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 1032430 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 516215 \, \sqrt {-2 \, x + 1}}{19845 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
-43/166698*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt( -2*x + 1))) + 1/19845*(17415*(-2*x + 1)^(9/2) + 163170*(-2*x + 1)^(7/2) - 809088*(-2*x + 1)^(5/2) + 1032430*(-2*x + 1)^(3/2) - 516215*sqrt(-2*x + 1) )/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1 )^2 + 72030*x - 19208)
Time = 0.40 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^6} \, dx=-\frac {43}{166698} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {17415 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 163170 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 809088 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 1032430 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 516215 \, \sqrt {-2 \, x + 1}}{635040 \, {\left (3 \, x + 2\right )}^{5}} \]
-43/166698*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/635040*(17415*(2*x - 1)^4*sqrt(-2*x + 1) - 163170 *(2*x - 1)^3*sqrt(-2*x + 1) - 809088*(2*x - 1)^2*sqrt(-2*x + 1) + 1032430* (-2*x + 1)^(3/2) - 516215*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {43\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{83349}+\frac {\frac {4214\,{\left (1-2\,x\right )}^{3/2}}{19683}-\frac {2107\,\sqrt {1-2\,x}}{19683}-\frac {5504\,{\left (1-2\,x\right )}^{5/2}}{32805}+\frac {74\,{\left (1-2\,x\right )}^{7/2}}{2187}+\frac {43\,{\left (1-2\,x\right )}^{9/2}}{11907}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
(43*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/83349 + ((4214*(1 - 2*x) ^(3/2))/19683 - (2107*(1 - 2*x)^(1/2))/19683 - (5504*(1 - 2*x)^(5/2))/3280 5 + (74*(1 - 2*x)^(7/2))/2187 + (43*(1 - 2*x)^(9/2))/11907)/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)