Integrand size = 22, antiderivative size = 128 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {(1-2 x)^{5/2}}{105 (2+3 x)^5}-\frac {17 (1-2 x)^{3/2}}{126 (2+3 x)^4}+\frac {17 \sqrt {1-2 x}}{378 (2+3 x)^3}-\frac {17 \sqrt {1-2 x}}{5292 (2+3 x)^2}-\frac {17 \sqrt {1-2 x}}{12348 (2+3 x)}-\frac {17 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{6174 \sqrt {21}} \]
1/105*(1-2*x)^(5/2)/(2+3*x)^5-17/126*(1-2*x)^(3/2)/(2+3*x)^4-17/129654*arc tanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+17/378*(1-2*x)^(1/2)/(2+3*x)^3-1 7/5292*(1-2*x)^(1/2)/(2+3*x)^2-17/12348*(1-2*x)^(1/2)/(2+3*x)
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (7912-23998 x-48252 x^2+23715 x^3+6885 x^4\right )}{2 (2+3 x)^5}-85 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{648270} \]
((-21*Sqrt[1 - 2*x]*(7912 - 23998*x - 48252*x^2 + 23715*x^3 + 6885*x^4))/( 2*(2 + 3*x)^5) - 85*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/648270
Time = 0.20 (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, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {34}{21} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^5}dx+\frac {(1-2 x)^{5/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {34}{21} \left (-\frac {1}{4} \int \frac {\sqrt {1-2 x}}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {34}{21} \left (\frac {1}{4} \left (\frac {1}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {34}{21} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {34}{21} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {34}{21} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {34}{21} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {(1-2 x)^{5/2}}{105 (3 x+2)^5}\) |
(1 - 2*x)^(5/2)/(105*(2 + 3*x)^5) + (34*(-1/12*(1 - 2*x)^(3/2)/(2 + 3*x)^4 + (Sqrt[1 - 2*x]/(9*(2 + 3*x)^3) + (-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*Sq rt[21])))/14)/9)/4))/21
3.19.69.3.1 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 = 1.00 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {13770 x^{5}+40545 x^{4}-120219 x^{3}+256 x^{2}+39822 x -7912}{61740 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {17 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{129654}\) | \(61\) |
| pseudoelliptic | \(\frac {-170 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}-21 \sqrt {1-2 x}\, \left (6885 x^{4}+23715 x^{3}-48252 x^{2}-23998 x +7912\right )}{1296540 \left (2+3 x \right )^{5}}\) | \(65\) |
| derivativedivides | \(\frac {\frac {153 \left (1-2 x \right )^{\frac {9}{2}}}{686}-\frac {17 \left (1-2 x \right )^{\frac {7}{2}}}{7}-\frac {32 \left (1-2 x \right )^{\frac {5}{2}}}{105}+\frac {119 \left (1-2 x \right )^{\frac {3}{2}}}{9}-\frac {119 \sqrt {1-2 x}}{18}}{\left (-4-6 x \right )^{5}}-\frac {17 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{129654}\) | \(75\) |
| default | \(\frac {\frac {153 \left (1-2 x \right )^{\frac {9}{2}}}{686}-\frac {17 \left (1-2 x \right )^{\frac {7}{2}}}{7}-\frac {32 \left (1-2 x \right )^{\frac {5}{2}}}{105}+\frac {119 \left (1-2 x \right )^{\frac {3}{2}}}{9}-\frac {119 \sqrt {1-2 x}}{18}}{\left (-4-6 x \right )^{5}}-\frac {17 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{129654}\) | \(75\) |
| trager | \(-\frac {\left (6885 x^{4}+23715 x^{3}-48252 x^{2}-23998 x +7912\right ) \sqrt {1-2 x}}{61740 \left (2+3 x \right )^{5}}-\frac {17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{259308}\) | \(82\) |
1/61740*(13770*x^5+40545*x^4-120219*x^3+256*x^2+39822*x-7912)/(2+3*x)^5/(1 -2*x)^(1/2)-17/129654*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {85 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (6885 \, x^{4} + 23715 \, x^{3} - 48252 \, x^{2} - 23998 \, x + 7912\right )} \sqrt {-2 \, x + 1}}{1296540 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/1296540*(85*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*(6885*x^4 + 237 15*x^3 - 48252*x^2 - 23998*x + 7912)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {17}{259308} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {6885 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 74970 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 9408 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 408170 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 204085 \, \sqrt {-2 \, x + 1}}{30870 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
17/259308*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(- 2*x + 1))) - 1/30870*(6885*(-2*x + 1)^(9/2) - 74970*(-2*x + 1)^(7/2) - 940 8*(-2*x + 1)^(5/2) + 408170*(-2*x + 1)^(3/2) - 204085*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 = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {17}{259308} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {6885 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 74970 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 9408 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 408170 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 204085 \, \sqrt {-2 \, x + 1}}{987840 \, {\left (3 \, x + 2\right )}^{5}} \]
17/259308*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/987840*(6885*(2*x - 1)^4*sqrt(-2*x + 1) + 74970*(2 *x - 1)^3*sqrt(-2*x + 1) - 9408*(2*x - 1)^2*sqrt(-2*x + 1) + 408170*(-2*x + 1)^(3/2) - 204085*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 1.50 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)}{(2+3 x)^6} \, dx=\frac {\frac {119\,\sqrt {1-2\,x}}{4374}-\frac {119\,{\left (1-2\,x\right )}^{3/2}}{2187}+\frac {32\,{\left (1-2\,x\right )}^{5/2}}{25515}+\frac {17\,{\left (1-2\,x\right )}^{7/2}}{1701}-\frac {17\,{\left (1-2\,x\right )}^{9/2}}{18522}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}}-\frac {17\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{129654} \]
((119*(1 - 2*x)^(1/2))/4374 - (119*(1 - 2*x)^(3/2))/2187 + (32*(1 - 2*x)^( 5/2))/25515 + (17*(1 - 2*x)^(7/2))/1701 - (17*(1 - 2*x)^(9/2))/18522)/((24 010*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) - (17*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/ 2))/7))/129654