Integrand size = 24, antiderivative size = 118 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {31}{3} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {6 \sqrt {1-2 x} (3+5 x)^3}{2+3 x}+\frac {1}{54} \sqrt {1-2 x} (367+1715 x)+\frac {887 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \]
-1/6*(1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^2+887/567*arctanh(1/7*21^(1/2)*(1-2*x )^(1/2))*21^(1/2)-31/3*(3+5*x)^2*(1-2*x)^(1/2)+6*(3+5*x)^3*(1-2*x)^(1/2)/( 2+3*x)+1/54*(367+1715*x)*(1-2*x)^(1/2)
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {\sqrt {1-2 x} \left (1367+2965 x+570 x^2+1800 x^4\right )}{54 (2+3 x)^2}+\frac {887 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{27 \sqrt {21}} \]
-1/54*(Sqrt[1 - 2*x]*(1367 + 2965*x + 570*x^2 + 1800*x^4))/(2 + 3*x)^2 + ( 887*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])
Time = 0.21 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {108, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^3}{(3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{6} \int \frac {3 (2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {(2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{2} \left (\frac {12 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {1}{3} \int \frac {3 (89-310 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {12 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\int \frac {(89-310 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{15} \int \frac {5 (67-343 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)}dx+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {62}{3} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {(67-343 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)}dx+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {62}{3} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{9} \sqrt {1-2 x} (1715 x+367)-\frac {887}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {62}{3} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {887}{9} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {1}{9} \sqrt {1-2 x} (1715 x+367)\right )+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {62}{3} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1774 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}+\frac {1}{9} \sqrt {1-2 x} (1715 x+367)\right )+\frac {12 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {62}{3} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
-1/6*((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^2 + ((-62*Sqrt[1 - 2*x]*(3 + 5*x)^2)/3 + (12*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x) + ((Sqrt[1 - 2*x]*(36 7 + 1715*x))/9 + (1774*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21]))/3)/ 2
3.19.89.3.1 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 = 1.00 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.51
| method | result | size |
| pseudoelliptic | \(\frac {1774 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}-21 \sqrt {1-2 x}\, \left (1800 x^{4}+570 x^{2}+2965 x +1367\right )}{1134 \left (2+3 x \right )^{2}}\) | \(60\) |
| risch | \(\frac {3600 x^{5}-1800 x^{4}+1140 x^{3}+5360 x^{2}-231 x -1367}{54 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {887 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{567}\) | \(61\) |
| derivativedivides | \(-\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {50 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {370 \sqrt {1-2 x}}{81}-\frac {2 \left (-\frac {215 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {497 \sqrt {1-2 x}}{18}\right )}{9 \left (-4-6 x \right )^{2}}+\frac {887 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{567}\) | \(75\) |
| default | \(-\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {50 \left (1-2 x \right )^{\frac {3}{2}}}{81}-\frac {370 \sqrt {1-2 x}}{81}-\frac {2 \left (-\frac {215 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {497 \sqrt {1-2 x}}{18}\right )}{9 \left (-4-6 x \right )^{2}}+\frac {887 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{567}\) | \(75\) |
| trager | \(-\frac {\left (1800 x^{4}+570 x^{2}+2965 x +1367\right ) \sqrt {1-2 x}}{54 \left (2+3 x \right )^{2}}+\frac {887 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1134}\) | \(77\) |
1/1134*(1774*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*(2+3*x)^2*21^(1/2)-21*(1- 2*x)^(1/2)*(1800*x^4+570*x^2+2965*x+1367))/(2+3*x)^2
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx=\frac {887 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1800 \, x^{4} + 570 \, x^{2} + 2965 \, x + 1367\right )} \sqrt {-2 \, x + 1}}{1134 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/1134*(887*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(1800*x^4 + 570*x^2 + 2965*x + 1367)*sqrt(-2*x + 1)) /(9*x^2 + 12*x + 4)
Time = 135.07 (sec) , antiderivative size = 367, normalized size of antiderivative = 3.11 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx=- \frac {25 \left (1 - 2 x\right )^{\frac {5}{2}}}{27} - \frac {50 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} - \frac {370 \sqrt {1 - 2 x}}{81} - \frac {4099 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{5103} - \frac {3052 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{243} - \frac {392 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{243} \]
-25*(1 - 2*x)**(5/2)/27 - 50*(1 - 2*x)**(3/2)/81 - 370*sqrt(1 - 2*x)/81 - 4099*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 21)/3))/5103 - 3052*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1 )/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt (21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/243 - 392*Piecewise((sqrt(21)*(3* log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1) /16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x )/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqr t(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x ) < sqrt(21)/3)))/243
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {25}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {887}{1134} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {370}{81} \, \sqrt {-2 \, x + 1} + \frac {215 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 497 \, \sqrt {-2 \, x + 1}}{81 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
-25/27*(-2*x + 1)^(5/2) - 50/81*(-2*x + 1)^(3/2) - 887/1134*sqrt(21)*log(- (sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 370/81*sqrt (-2*x + 1) + 1/81*(215*(-2*x + 1)^(3/2) - 497*sqrt(-2*x + 1))/(9*(2*x - 1) ^2 + 84*x + 7)
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {25}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {887}{1134} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {370}{81} \, \sqrt {-2 \, x + 1} + \frac {215 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 497 \, \sqrt {-2 \, x + 1}}{324 \, {\left (3 \, x + 2\right )}^{2}} \]
-25/27*(2*x - 1)^2*sqrt(-2*x + 1) - 50/81*(-2*x + 1)^(3/2) - 887/1134*sqrt (21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 370/81*sqrt(-2*x + 1) + 1/324*(215*(-2*x + 1)^(3/2) - 497*sqrt(-2* x + 1))/(3*x + 2)^2
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {370\,\sqrt {1-2\,x}}{81}-\frac {50\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {25\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {\frac {497\,\sqrt {1-2\,x}}{729}-\frac {215\,{\left (1-2\,x\right )}^{3/2}}{729}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,887{}\mathrm {i}}{567} \]