Integrand size = 24, antiderivative size = 135 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {935}{81} \sqrt {1-2 x}-\frac {220}{21} (1-2 x)^{3/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)}+\frac {55 (1-2 x)^{3/2} (209+603 x)}{1134}+\frac {935}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
-220/21*(1-2*x)^(3/2)*(3+5*x)^2-1/6*(1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^2+55/9 *(1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)+55/1134*(1-2*x)^(3/2)*(209+603*x)+935/243 *arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-935/81*(1-2*x)^(1/2)
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.56 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^3} \, dx=\frac {\sqrt {1-2 x} \left (-64943-152833 x-67962 x^2-17460 x^3-24120 x^4+54000 x^5\right )}{1134 (2+3 x)^2}+\frac {935}{81} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
(Sqrt[1 - 2*x]*(-64943 - 152833*x - 67962*x^2 - 17460*x^3 - 24120*x^4 + 54 000*x^5))/(1134*(2 + 3*x)^2) + (935*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2 *x]])/81
Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {108, 27, 166, 27, 170, 27, 164, 60, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^3}{(3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{6} \int -\frac {55 (1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^2}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{6} \int \frac {(1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^2}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {55}{6} \left (-\frac {1}{3} \int -\frac {3 (1-24 x) \sqrt {1-2 x} (5 x+3)^2}{3 x+2}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{6} \left (\int \frac {(1-24 x) \sqrt {1-2 x} (5 x+3)^2}{3 x+2}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle -\frac {55}{6} \left (-\frac {1}{21} \int -\frac {3 \sqrt {1-2 x} (5 x+3) (67 x+5)}{3 x+2}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {8}{7} (1-2 x)^{3/2} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{6} \left (\frac {1}{7} \int \frac {\sqrt {1-2 x} (5 x+3) (67 x+5)}{3 x+2}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {8}{7} (1-2 x)^{3/2} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {55}{6} \left (\frac {1}{7} \left (\frac {119}{9} \int \frac {\sqrt {1-2 x}}{3 x+2}dx-\frac {1}{27} (1-2 x)^{3/2} (603 x+209)\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {8}{7} (1-2 x)^{3/2} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {55}{6} \left (\frac {1}{7} \left (\frac {119}{9} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )-\frac {1}{27} (1-2 x)^{3/2} (603 x+209)\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {8}{7} (1-2 x)^{3/2} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {55}{6} \left (\frac {1}{7} \left (\frac {119}{9} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{27} (1-2 x)^{3/2} (603 x+209)\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {8}{7} (1-2 x)^{3/2} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {55}{6} \left (\frac {1}{7} \left (\frac {119}{9} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {1}{27} (1-2 x)^{3/2} (603 x+209)\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {8}{7} (1-2 x)^{3/2} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{6 (3 x+2)^2}\) |
-1/6*((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^2 - (55*((8*(1 - 2*x)^(3/2)*( 3 + 5*x)^2)/7 - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^3)/(3*(2 + 3*x)) + (-1/27*((1 - 2*x)^(3/2)*(209 + 603*x)) + (119*((2*Sqrt[1 - 2*x])/3 - (2*Sqrt[7/3]*Ar cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3))/9)/7))/6
3.20.62.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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 = 3.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(-\frac {108000 x^{6}-102240 x^{5}-10800 x^{4}-118464 x^{3}-237704 x^{2}+22947 x +64943}{1134 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {935 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(66\) |
| pseudoelliptic | \(\frac {13090 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}+3 \sqrt {1-2 x}\, \left (54000 x^{5}-24120 x^{4}-17460 x^{3}-67962 x^{2}-152833 x -64943\right )}{3402 \left (2+3 x \right )^{2}}\) | \(70\) |
| derivativedivides | \(-\frac {125 \left (1-2 x \right )^{\frac {7}{2}}}{189}-\frac {10 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {370 \left (1-2 x \right )^{\frac {3}{2}}}{243}-\frac {8198 \sqrt {1-2 x}}{729}-\frac {14 \left (-\frac {73 \left (1-2 x \right )^{\frac {3}{2}}}{2}+\frac {1519 \sqrt {1-2 x}}{18}\right )}{81 \left (-4-6 x \right )^{2}}+\frac {935 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(84\) |
| default | \(-\frac {125 \left (1-2 x \right )^{\frac {7}{2}}}{189}-\frac {10 \left (1-2 x \right )^{\frac {5}{2}}}{27}-\frac {370 \left (1-2 x \right )^{\frac {3}{2}}}{243}-\frac {8198 \sqrt {1-2 x}}{729}-\frac {14 \left (-\frac {73 \left (1-2 x \right )^{\frac {3}{2}}}{2}+\frac {1519 \sqrt {1-2 x}}{18}\right )}{81 \left (-4-6 x \right )^{2}}+\frac {935 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{243}\) | \(84\) |
| trager | \(\frac {\left (54000 x^{5}-24120 x^{4}-17460 x^{3}-67962 x^{2}-152833 x -64943\right ) \sqrt {1-2 x}}{1134 \left (2+3 x \right )^{2}}+\frac {935 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{486}\) | \(87\) |
-1/1134*(108000*x^6-102240*x^5-10800*x^4-118464*x^3-237704*x^2+22947*x+649 43)/(2+3*x)^2/(1-2*x)^(1/2)+935/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21 ^(1/2)
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^3} \, dx=\frac {6545 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 3 \, {\left (54000 \, x^{5} - 24120 \, x^{4} - 17460 \, x^{3} - 67962 \, x^{2} - 152833 \, x - 64943\right )} \sqrt {-2 \, x + 1}}{3402 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/3402*(6545*sqrt(7)*sqrt(3)*(9*x^2 + 12*x + 4)*log(-(sqrt(7)*sqrt(3)*sqrt (-2*x + 1) - 3*x + 5)/(3*x + 2)) + 3*(54000*x^5 - 24120*x^4 - 17460*x^3 - 67962*x^2 - 152833*x - 64943)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
Time = 133.99 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.81 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^3} \, dx=- \frac {125 \left (1 - 2 x\right )^{\frac {7}{2}}}{189} - \frac {10 \left (1 - 2 x\right )^{\frac {5}{2}}}{27} - \frac {370 \left (1 - 2 x\right )^{\frac {3}{2}}}{243} - \frac {8198 \sqrt {1 - 2 x}}{729} - \frac {1439 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{729} - \frac {7252 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{243} - \frac {2744 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{729} \]
-125*(1 - 2*x)**(7/2)/189 - 10*(1 - 2*x)**(5/2)/27 - 370*(1 - 2*x)**(3/2)/ 243 - 8198*sqrt(1 - 2*x)/729 - 1439*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21) /3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/729 - 7252*Piecewise((sqrt(21)*(-lo g(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 - 2744*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*lo g(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)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/729
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {125}{189} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {370}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {935}{486} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8198}{729} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (657 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1519 \, \sqrt {-2 \, x + 1}\right )}}{729 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
-125/189*(-2*x + 1)^(7/2) - 10/27*(-2*x + 1)^(5/2) - 370/243*(-2*x + 1)^(3 /2) - 935/486*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq rt(-2*x + 1))) - 8198/729*sqrt(-2*x + 1) + 7/729*(657*(-2*x + 1)^(3/2) - 1 519*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)
Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^3} \, dx=\frac {125}{189} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {10}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {370}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {935}{486} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {8198}{729} \, \sqrt {-2 \, x + 1} + \frac {7 \, {\left (657 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1519 \, \sqrt {-2 \, x + 1}\right )}}{2916 \, {\left (3 \, x + 2\right )}^{2}} \]
125/189*(2*x - 1)^3*sqrt(-2*x + 1) - 10/27*(2*x - 1)^2*sqrt(-2*x + 1) - 37 0/243*(-2*x + 1)^(3/2) - 935/486*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt (-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 8198/729*sqrt(-2*x + 1) + 7/2 916*(657*(-2*x + 1)^(3/2) - 1519*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^3} \, dx=-\frac {8198\,\sqrt {1-2\,x}}{729}-\frac {370\,{\left (1-2\,x\right )}^{3/2}}{243}-\frac {10\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {125\,{\left (1-2\,x\right )}^{7/2}}{189}-\frac {\frac {10633\,\sqrt {1-2\,x}}{6561}-\frac {511\,{\left (1-2\,x\right )}^{3/2}}{729}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,935{}\mathrm {i}}{243} \]