Integrand size = 22, antiderivative size = 128 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^6} \, dx=\frac {(1-2 x)^{3/2}}{105 (2+3 x)^5}-\frac {2 \sqrt {1-2 x}}{15 (2+3 x)^4}+\frac {2 \sqrt {1-2 x}}{315 (2+3 x)^3}+\frac {\sqrt {1-2 x}}{441 (2+3 x)^2}+\frac {\sqrt {1-2 x}}{1029 (2+3 x)}+\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
1/105*(1-2*x)^(3/2)/(2+3*x)^5+2/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))* 21^(1/2)-2/15*(1-2*x)^(1/2)/(2+3*x)^4+2/315*(1-2*x)^(1/2)/(2+3*x)^3+1/441* (1-2*x)^(1/2)/(2+3*x)^2+1/1029*(1-2*x)^(1/2)/(2+3*x)
Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^6} \, dx=\frac {\sqrt {1-2 x} \left (-1019-864 x+2004 x^2+1395 x^3+405 x^4\right )}{5145 (2+3 x)^5}+\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
(Sqrt[1 - 2*x]*(-1019 - 864*x + 2004*x^2 + 1395*x^3 + 405*x^4))/(5145*(2 + 3*x)^5) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])
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, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {8}{5} \int \frac {\sqrt {1-2 x}}{(3 x+2)^5}dx+\frac {(1-2 x)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {8}{5} \left (-\frac {1}{12} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {8}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {8}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {8}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {8}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {8}{5} \left (\frac {1}{12} \left (\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}-\frac {5}{21} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )\right )-\frac {\sqrt {1-2 x}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{3/2}}{105 (3 x+2)^5}\) |
(1 - 2*x)^(3/2)/(105*(2 + 3*x)^5) + (8*(-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*S qrt[21])))/14))/21)/12))/5
3.19.2.3.1 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 = 0.98 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(-\frac {810 x^{5}+2385 x^{4}+2613 x^{3}-3732 x^{2}-1174 x +1019}{5145 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(61\) |
| pseudoelliptic | \(\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}+21 \sqrt {1-2 x}\, \left (405 x^{4}+1395 x^{3}+2004 x^{2}-864 x -1019\right )}{108045 \left (2+3 x \right )^{5}}\) | \(65\) |
| derivativedivides | \(\frac {-\frac {54 \left (1-2 x \right )^{\frac {9}{2}}}{343}+\frac {12 \left (1-2 x \right )^{\frac {7}{2}}}{7}-\frac {256 \left (1-2 x \right )^{\frac {5}{2}}}{35}+\frac {52 \left (1-2 x \right )^{\frac {3}{2}}}{7}+\frac {14 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{5}}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(75\) |
| default | \(\frac {-\frac {54 \left (1-2 x \right )^{\frac {9}{2}}}{343}+\frac {12 \left (1-2 x \right )^{\frac {7}{2}}}{7}-\frac {256 \left (1-2 x \right )^{\frac {5}{2}}}{35}+\frac {52 \left (1-2 x \right )^{\frac {3}{2}}}{7}+\frac {14 \sqrt {1-2 x}}{3}}{\left (-4-6 x \right )^{5}}+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(75\) |
| trager | \(\frac {\left (405 x^{4}+1395 x^{3}+2004 x^{2}-864 x -1019\right ) \sqrt {1-2 x}}{5145 \left (2+3 x \right )^{5}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{21609}\) | \(82\) |
-1/5145*(810*x^5+2385*x^4+2613*x^3-3732*x^2-1174*x+1019)/(2+3*x)^5/(1-2*x) ^(1/2)+2/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^6} \, dx=\frac {5 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (405 \, x^{4} + 1395 \, x^{3} + 2004 \, x^{2} - 864 \, x - 1019\right )} \sqrt {-2 \, x + 1}}{108045 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/108045*(5*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) *log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(405*x^4 + 1395*x ^3 + 2004*x^2 - 864*x - 1019)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^ 3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^6} \, dx=-\frac {1}{21609} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (405 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 4410 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 18816 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 19110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 12005 \, \sqrt {-2 \, x + 1}\right )}}{5145 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
-1/21609*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 *x + 1))) + 2/5145*(405*(-2*x + 1)^(9/2) - 4410*(-2*x + 1)^(7/2) + 18816*( -2*x + 1)^(5/2) - 19110*(-2*x + 1)^(3/2) - 12005*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 = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^6} \, dx=-\frac {1}{21609} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {405 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 4410 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 18816 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 19110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 12005 \, \sqrt {-2 \, x + 1}}{82320 \, {\left (3 \, x + 2\right )}^{5}} \]
-1/21609*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/82320*(405*(2*x - 1)^4*sqrt(-2*x + 1) + 4410*(2*x - 1)^3*sqrt(-2*x + 1) + 18816*(2*x - 1)^2*sqrt(-2*x + 1) - 19110*(-2*x + 1) ^(3/2) - 12005*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 1.36 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^6} \, dx=\frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}-\frac {\frac {14\,\sqrt {1-2\,x}}{729}+\frac {52\,{\left (1-2\,x\right )}^{3/2}}{1701}-\frac {256\,{\left (1-2\,x\right )}^{5/2}}{8505}+\frac {4\,{\left (1-2\,x\right )}^{7/2}}{567}-\frac {2\,{\left (1-2\,x\right )}^{9/2}}{3087}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
(2*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/21609 - ((14*(1 - 2*x)^(1 /2))/729 + (52*(1 - 2*x)^(3/2))/1701 - (256*(1 - 2*x)^(5/2))/8505 + (4*(1 - 2*x)^(7/2))/567 - (2*(1 - 2*x)^(9/2))/3087)/((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)