Integrand size = 22, antiderivative size = 148 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\frac {(1-2 x)^{7/2}}{126 (2+3 x)^6}-\frac {41 (1-2 x)^{5/2}}{378 (2+3 x)^5}+\frac {205 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {205 \sqrt {1-2 x}}{13608 (2+3 x)^3}+\frac {205 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {205 \sqrt {1-2 x}}{444528 (2+3 x)}+\frac {205 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \]
1/126*(1-2*x)^(7/2)/(2+3*x)^6-41/378*(1-2*x)^(5/2)/(2+3*x)^5+205/4536*(1-2 *x)^(3/2)/(2+3*x)^4+205/4667544*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/ 2)-205/13608*(1-2*x)^(1/2)/(2+3*x)^3+205/190512*(1-2*x)^(1/2)/(2+3*x)^2+20 5/444528*(1-2*x)^(1/2)/(2+3*x)
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-51904+154312 x-176850 x^2-824526 x^3+204795 x^4+49815 x^5\right )}{2 (2+3 x)^6}+205 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4667544} \]
((21*Sqrt[1 - 2*x]*(-51904 + 154312*x - 176850*x^2 - 824526*x^3 + 204795*x ^4 + 49815*x^5))/(2*(2 + 3*x)^6) + 205*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/4667544
Time = 0.22 (sec) , antiderivative size = 173, 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, 51, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)}{(3 x+2)^7} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {205}{126} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^6}dx+\frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {205}{126} \left (-\frac {1}{3} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^5}dx-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {205}{126} \left (\frac {1}{3} \left (\frac {1}{4} \int \frac {\sqrt {1-2 x}}{(3 x+2)^4}dx+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {205}{126} \left (\frac {1}{3} \left (\frac {1}{4} \left (-\frac {1}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {205}{126} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {205}{126} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {205}{126} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {205}{126} \left (\frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{5/2}}{15 (3 x+2)^5}\right )+\frac {(1-2 x)^{7/2}}{126 (3 x+2)^6}\) |
(1 - 2*x)^(7/2)/(126*(2 + 3*x)^6) + (205*(-1/15*(1 - 2*x)^(5/2)/(2 + 3*x)^ 5 + ((1 - 2*x)^(3/2)/(12*(2 + 3*x)^4) + (-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*Ar cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14)/9)/4)/3))/126
3.20.41.3.1 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 = 3.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(-\frac {99630 x^{6}+359775 x^{5}-1853847 x^{4}+470826 x^{3}+485474 x^{2}-258120 x +51904}{444528 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {205 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\) | \(66\) |
| pseudoelliptic | \(\frac {410 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}+21 \sqrt {1-2 x}\, \left (49815 x^{5}+204795 x^{4}-824526 x^{3}-176850 x^{2}+154312 x -51904\right )}{9335088 \left (2+3 x \right )^{6}}\) | \(70\) |
| derivativedivides | \(-\frac {46656 \left (\frac {205 \left (1-2 x \right )^{\frac {11}{2}}}{42674688}-\frac {3485 \left (1-2 x \right )^{\frac {9}{2}}}{54867456}-\frac {439 \left (1-2 x \right )^{\frac {7}{2}}}{3919104}+\frac {451 \left (1-2 x \right )^{\frac {5}{2}}}{559872}-\frac {24395 \left (1-2 x \right )^{\frac {3}{2}}}{30233088}+\frac {10045 \sqrt {1-2 x}}{30233088}\right )}{\left (-4-6 x \right )^{6}}+\frac {205 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\) | \(84\) |
| default | \(-\frac {46656 \left (\frac {205 \left (1-2 x \right )^{\frac {11}{2}}}{42674688}-\frac {3485 \left (1-2 x \right )^{\frac {9}{2}}}{54867456}-\frac {439 \left (1-2 x \right )^{\frac {7}{2}}}{3919104}+\frac {451 \left (1-2 x \right )^{\frac {5}{2}}}{559872}-\frac {24395 \left (1-2 x \right )^{\frac {3}{2}}}{30233088}+\frac {10045 \sqrt {1-2 x}}{30233088}\right )}{\left (-4-6 x \right )^{6}}+\frac {205 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\) | \(84\) |
| trager | \(\frac {\left (49815 x^{5}+204795 x^{4}-824526 x^{3}-176850 x^{2}+154312 x -51904\right ) \sqrt {1-2 x}}{444528 \left (2+3 x \right )^{6}}-\frac {205 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{9335088}\) | \(87\) |
-1/444528*(99630*x^6+359775*x^5-1853847*x^4+470826*x^3+485474*x^2-258120*x +51904)/(2+3*x)^6/(1-2*x)^(1/2)+205/4667544*arctanh(1/7*21^(1/2)*(1-2*x)^( 1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\frac {205 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (49815 \, x^{5} + 204795 \, x^{4} - 824526 \, x^{3} - 176850 \, x^{2} + 154312 \, x - 51904\right )} \sqrt {-2 \, x + 1}}{9335088 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
1/9335088*(205*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x ^2 + 576*x + 64)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*( 49815*x^5 + 204795*x^4 - 824526*x^3 - 176850*x^2 + 154312*x - 51904)*sqrt( -2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=-\frac {205}{9335088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {49815 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 658665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 1161594 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 8353422 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 8367485 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 3445435 \, \sqrt {-2 \, x + 1}}{222264 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]
-205/9335088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr t(-2*x + 1))) - 1/222264*(49815*(-2*x + 1)^(11/2) - 658665*(-2*x + 1)^(9/2 ) - 1161594*(-2*x + 1)^(7/2) + 8353422*(-2*x + 1)^(5/2) - 8367485*(-2*x + 1)^(3/2) + 3445435*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 1 84877)
Time = 0.35 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=-\frac {205}{9335088} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {49815 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 658665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 1161594 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 8353422 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 8367485 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3445435 \, \sqrt {-2 \, x + 1}}{14224896 \, {\left (3 \, x + 2\right )}^{6}} \]
-205/9335088*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) + 1/14224896*(49815*(2*x - 1)^5*sqrt(-2*x + 1) + 65 8665*(2*x - 1)^4*sqrt(-2*x + 1) - 1161594*(2*x - 1)^3*sqrt(-2*x + 1) - 835 3422*(2*x - 1)^2*sqrt(-2*x + 1) + 8367485*(-2*x + 1)^(3/2) - 3445435*sqrt( -2*x + 1))/(3*x + 2)^6
Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^7} \, dx=\frac {205\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4667544}-\frac {\frac {10045\,\sqrt {1-2\,x}}{472392}-\frac {24395\,{\left (1-2\,x\right )}^{3/2}}{472392}+\frac {451\,{\left (1-2\,x\right )}^{5/2}}{8748}-\frac {439\,{\left (1-2\,x\right )}^{7/2}}{61236}-\frac {3485\,{\left (1-2\,x\right )}^{9/2}}{857304}+\frac {205\,{\left (1-2\,x\right )}^{11/2}}{666792}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]
(205*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/4667544 - ((10045*(1 - 2*x)^(1/2))/472392 - (24395*(1 - 2*x)^(3/2))/472392 + (451*(1 - 2*x)^(5/2) )/8748 - (439*(1 - 2*x)^(7/2))/61236 - (3485*(1 - 2*x)^(9/2))/857304 + (20 5*(1 - 2*x)^(11/2))/666792)/((67228*x)/81 + (12005*(2*x - 1)^2)/27 + (6860 *(2*x - 1)^3)/27 + (245*(2*x - 1)^4)/3 + 14*(2*x - 1)^5 + (2*x - 1)^6 - 18 4877/729)