Integrand size = 24, antiderivative size = 128 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {(1-2 x)^{3/2}}{315 (2+3 x)^5}+\frac {7 (1-2 x)^{3/2}}{180 (2+3 x)^4}-\frac {31 \sqrt {1-2 x}}{108 (2+3 x)^3}+\frac {31 \sqrt {1-2 x}}{1512 (2+3 x)^2}+\frac {31 \sqrt {1-2 x}}{3528 (2+3 x)}+\frac {31 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1764 \sqrt {21}} \]
-1/315*(1-2*x)^(3/2)/(2+3*x)^5+7/180*(1-2*x)^(3/2)/(2+3*x)^4+31/37044*arct anh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-31/108*(1-2*x)^(1/2)/(2+3*x)^3+31 /1512*(1-2*x)^(1/2)/(2+3*x)^2+31/3528*(1-2*x)^(1/2)/(2+3*x)
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-13564-33434 x+3324 x^2+43245 x^3+12555 x^4\right )}{(2+3 x)^5}+310 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{370440} \]
((21*Sqrt[1 - 2*x]*(-13564 - 33434*x + 3324*x^2 + 43245*x^3 + 12555*x^4))/ (2 + 3*x)^5 + 310*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/370440
Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {100, 27, 87, 51, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {1}{315} \int \frac {21 \sqrt {1-2 x} (125 x+67)}{(3 x+2)^5}dx-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \int \frac {\sqrt {1-2 x} (125 x+67)}{(3 x+2)^5}dx-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{15} \left (\frac {155}{4} \int \frac {\sqrt {1-2 x}}{(3 x+2)^4}dx+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{15} \left (\frac {155}{4} \left (-\frac {1}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{15} \left (\frac {155}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{15} \left (\frac {155}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{15} \left (\frac {155}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{15} \left (\frac {155}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2}}{315 (3 x+2)^5}\) |
-1/315*(1 - 2*x)^(3/2)/(2 + 3*x)^5 + ((7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4) + (155*(-1/9*Sqrt[1 - 2*x]/(2 + 3*x)^3 + (Sqrt[1 - 2*x]/(14*(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)/9))/4)/15
3.19.13.3.1 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 = 0.99 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(-\frac {25110 x^{5}+73935 x^{4}-36597 x^{3}-70192 x^{2}+6306 x +13564}{17640 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {31 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{37044}\) | \(61\) |
| pseudoelliptic | \(\frac {310 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}+21 \sqrt {1-2 x}\, \left (12555 x^{4}+43245 x^{3}+3324 x^{2}-33434 x -13564\right )}{370440 \left (2+3 x \right )^{5}}\) | \(65\) |
| derivativedivides | \(-\frac {3888 \left (\frac {31 \left (1-2 x \right )^{\frac {9}{2}}}{84672}-\frac {31 \left (1-2 x \right )^{\frac {7}{2}}}{7776}+\frac {37 \left (1-2 x \right )^{\frac {5}{2}}}{3645}-\frac {983 \left (1-2 x \right )^{\frac {3}{2}}}{489888}-\frac {1519 \sqrt {1-2 x}}{139968}\right )}{\left (-4-6 x \right )^{5}}+\frac {31 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{37044}\) | \(75\) |
| default | \(-\frac {3888 \left (\frac {31 \left (1-2 x \right )^{\frac {9}{2}}}{84672}-\frac {31 \left (1-2 x \right )^{\frac {7}{2}}}{7776}+\frac {37 \left (1-2 x \right )^{\frac {5}{2}}}{3645}-\frac {983 \left (1-2 x \right )^{\frac {3}{2}}}{489888}-\frac {1519 \sqrt {1-2 x}}{139968}\right )}{\left (-4-6 x \right )^{5}}+\frac {31 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{37044}\) | \(75\) |
| trager | \(\frac {\left (12555 x^{4}+43245 x^{3}+3324 x^{2}-33434 x -13564\right ) \sqrt {1-2 x}}{17640 \left (2+3 x \right )^{5}}+\frac {31 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{74088}\) | \(82\) |
-1/17640*(25110*x^5+73935*x^4-36597*x^3-70192*x^2+6306*x+13564)/(2+3*x)^5/ (1-2*x)^(1/2)+31/37044*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 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {155 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (12555 \, x^{4} + 43245 \, x^{3} + 3324 \, x^{2} - 33434 \, x - 13564\right )} \sqrt {-2 \, x + 1}}{370440 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/370440*(155*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*(12555*x^4 + 43 245*x^3 + 3324*x^2 - 33434*x - 13564)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {31}{74088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {12555 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 136710 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 348096 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 68810 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 372155 \, \sqrt {-2 \, x + 1}}{8820 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
-31/74088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(- 2*x + 1))) + 1/8820*(12555*(-2*x + 1)^(9/2) - 136710*(-2*x + 1)^(7/2) + 34 8096*(-2*x + 1)^(5/2) - 68810*(-2*x + 1)^(3/2) - 372155*sqrt(-2*x + 1))/(2 43*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)
Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx=-\frac {31}{74088} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {12555 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 136710 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 348096 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 68810 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 372155 \, \sqrt {-2 \, x + 1}}{282240 \, {\left (3 \, x + 2\right )}^{5}} \]
-31/74088*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/282240*(12555*(2*x - 1)^4*sqrt(-2*x + 1) + 136710* (2*x - 1)^3*sqrt(-2*x + 1) + 348096*(2*x - 1)^2*sqrt(-2*x + 1) - 68810*(-2 *x + 1)^(3/2) - 372155*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^6} \, dx=\frac {31\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{37044}-\frac {\frac {1519\,\sqrt {1-2\,x}}{8748}+\frac {983\,{\left (1-2\,x\right )}^{3/2}}{30618}-\frac {592\,{\left (1-2\,x\right )}^{5/2}}{3645}+\frac {31\,{\left (1-2\,x\right )}^{7/2}}{486}-\frac {31\,{\left (1-2\,x\right )}^{9/2}}{5292}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
(31*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/37044 - ((1519*(1 - 2*x) ^(1/2))/8748 + (983*(1 - 2*x)^(3/2))/30618 - (592*(1 - 2*x)^(5/2))/3645 + (31*(1 - 2*x)^(7/2))/486 - (31*(1 - 2*x)^(9/2))/5292)/((24010*x)/81 + (343 0*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)