Integrand size = 24, antiderivative size = 147 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {1441}{27} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac {275 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac {22}{243} \sqrt {1-2 x} (578+1885 x)-\frac {41360 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{243 \sqrt {21}} \]
-1/9*(1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^3+55/27*(1-2*x)^(3/2)*(3+5*x)^3/(2+3* x)^2-41360/5103*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1441/27*(3+5* x)^2*(1-2*x)^(1/2)-275/9*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)-22/243*(578+1885* x)*(1-2*x)^(1/2)
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (56141+229336 x+289719 x^2+87030 x^3-20700 x^4+16200 x^5\right )}{(2+3 x)^3}-41360 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5103} \]
((21*Sqrt[1 - 2*x]*(56141 + 229336*x + 289719*x^2 + 87030*x^3 - 20700*x^4 + 16200*x^5))/(2 + 3*x)^3 - 41360*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x] ])/5103
Time = 0.24 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {108, 27, 166, 27, 166, 27, 170, 27, 164, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^3}{(3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{9} \int -\frac {55 (1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^3}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{9} \int \frac {(1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^3}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {55}{9} \left (-\frac {1}{6} \int \frac {6 \sqrt {1-2 x} (5 x+3)^2 (9 x+1)}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{9} \left (-\int \frac {\sqrt {1-2 x} (5 x+3)^2 (9 x+1)}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {55}{9} \left (\frac {1}{3} \int -\frac {3 (38-131 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)}dx+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{9} \left (-\int \frac {(38-131 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)}dx+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle -\frac {55}{9} \left (\frac {1}{15} \int \frac {2 (62-377 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)}dx+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{9} \left (\frac {2}{15} \int \frac {(62-377 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)}dx+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {55}{9} \left (\frac {2}{15} \left (\frac {1}{9} \sqrt {1-2 x} (1885 x+578)-\frac {940}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {55}{9} \left (\frac {2}{15} \left (\frac {940}{9} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {1}{9} \sqrt {1-2 x} (1885 x+578)\right )+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {55}{9} \left (\frac {2}{15} \left (\frac {1880 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}+\frac {1}{9} \sqrt {1-2 x} (1885 x+578)\right )+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\) |
-1/9*((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^3 - (55*((-131*Sqrt[1 - 2*x]* (3 + 5*x)^2)/15 - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(3*(2 + 3*x)^2) + (5*Sqrt[ 1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x) + (2*((Sqrt[1 - 2*x]*(578 + 1885*x))/9 + ( 1880*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21])))/15))/9
3.20.63.3.1 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] :> 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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 = 1.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(-\frac {32400 x^{6}-57600 x^{5}+194760 x^{4}+492408 x^{3}+168953 x^{2}-117054 x -56141}{243 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {41360 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{5103}\) | \(66\) |
| pseudoelliptic | \(\frac {-41360 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}+21 \sqrt {1-2 x}\, \left (16200 x^{5}-20700 x^{4}+87030 x^{3}+289719 x^{2}+229336 x +56141\right )}{5103 \left (2+3 x \right )^{3}}\) | \(70\) |
| derivativedivides | \(\frac {50 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {2050 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {16570 \sqrt {1-2 x}}{729}+\frac {-\frac {8306 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {344260 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {396410 \sqrt {1-2 x}}{729}}{\left (-4-6 x \right )^{3}}-\frac {41360 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{5103}\) | \(84\) |
| default | \(\frac {50 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {2050 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {16570 \sqrt {1-2 x}}{729}+\frac {-\frac {8306 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {344260 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {396410 \sqrt {1-2 x}}{729}}{\left (-4-6 x \right )^{3}}-\frac {41360 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{5103}\) | \(84\) |
| trager | \(\frac {\left (16200 x^{5}-20700 x^{4}+87030 x^{3}+289719 x^{2}+229336 x +56141\right ) \sqrt {1-2 x}}{243 \left (2+3 x \right )^{3}}+\frac {20680 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{5103}\) | \(87\) |
-1/243*(32400*x^6-57600*x^5+194760*x^4+492408*x^3+168953*x^2-117054*x-5614 1)/(2+3*x)^3/(1-2*x)^(1/2)-41360/5103*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))* 21^(1/2)
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {20680 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (16200 \, x^{5} - 20700 \, x^{4} + 87030 \, x^{3} + 289719 \, x^{2} + 229336 \, x + 56141\right )} \sqrt {-2 \, x + 1}}{5103 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
1/5103*(20680*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sq rt(-2*x + 1) - 5)/(3*x + 2)) + 21*(16200*x^5 - 20700*x^4 + 87030*x^3 + 289 719*x^2 + 229336*x + 56141)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2050}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {20680}{5103} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {16570}{729} \, \sqrt {-2 \, x + 1} + \frac {2 \, {\left (37377 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 172130 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 198205 \, \sqrt {-2 \, x + 1}\right )}}{729 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
50/81*(-2*x + 1)^(5/2) + 2050/729*(-2*x + 1)^(3/2) + 20680/5103*sqrt(21)*l og(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16570/7 29*sqrt(-2*x + 1) + 2/729*(37377*(-2*x + 1)^(5/2) - 172130*(-2*x + 1)^(3/2 ) + 198205*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {50}{81} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2050}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {20680}{5103} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16570}{729} \, \sqrt {-2 \, x + 1} + \frac {37377 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 172130 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 198205 \, \sqrt {-2 \, x + 1}}{2916 \, {\left (3 \, x + 2\right )}^{3}} \]
50/81*(2*x - 1)^2*sqrt(-2*x + 1) + 2050/729*(-2*x + 1)^(3/2) + 20680/5103* sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 *x + 1))) + 16570/729*sqrt(-2*x + 1) + 1/2916*(37377*(2*x - 1)^2*sqrt(-2*x + 1) - 172130*(-2*x + 1)^(3/2) + 198205*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {16570\,\sqrt {1-2\,x}}{729}+\frac {2050\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {50\,{\left (1-2\,x\right )}^{5/2}}{81}+\frac {\frac {396410\,\sqrt {1-2\,x}}{19683}-\frac {344260\,{\left (1-2\,x\right )}^{3/2}}{19683}+\frac {8306\,{\left (1-2\,x\right )}^{5/2}}{2187}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,41360{}\mathrm {i}}{5103} \]
(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*41360i)/5103 + (16570*(1 - 2*x)^(1/2))/729 + (2050*(1 - 2*x)^(3/2))/729 + (50*(1 - 2*x)^(5/2))/81 + ((396410*(1 - 2*x)^(1/2))/19683 - (344260*(1 - 2*x)^(3/2))/19683 + (8306*( 1 - 2*x)^(5/2))/2187)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 98/27)