Integrand size = 24, antiderivative size = 148 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {324 \sqrt {1-2 x}}{78125}-\frac {4016 (1-2 x)^{3/2} (2+3 x)^2}{48125}+\frac {38 (1-2 x)^{3/2} (2+3 x)^3}{4125}+\frac {39}{275} (1-2 x)^{3/2} (2+3 x)^4-\frac {(1-2 x)^{3/2} (2+3 x)^5}{5 (3+5 x)}-\frac {2 (1-2 x)^{3/2} (298462+204777 x)}{515625}-\frac {324 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{78125} \]
-4016/48125*(1-2*x)^(3/2)*(2+3*x)^2+38/4125*(1-2*x)^(3/2)*(2+3*x)^3+39/275 *(1-2*x)^(3/2)*(2+3*x)^4-1/5*(1-2*x)^(3/2)*(2+3*x)^5/(3+5*x)-2/515625*(1-2 *x)^(3/2)*(298462+204777*x)-324/390625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2) )*55^(1/2)+324/78125*(1-2*x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {-\frac {5 \sqrt {1-2 x} \left (23061496-2532130 x-135193430 x^2-76760550 x^3+181738125 x^4+270112500 x^5+106312500 x^6\right )}{3+5 x}-24948 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{30078125} \]
((-5*Sqrt[1 - 2*x]*(23061496 - 2532130*x - 135193430*x^2 - 76760550*x^3 + 181738125*x^4 + 270112500*x^5 + 106312500*x^6))/(3 + 5*x) - 24948*Sqrt[55] *ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/30078125
Time = 0.26 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.17, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {108, 27, 170, 27, 170, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (3 x+2)^5}{(5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{5} \int \frac {3 (3-13 x) \sqrt {1-2 x} (3 x+2)^4}{5 x+3}dx-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \int \frac {(3-13 x) \sqrt {1-2 x} (3 x+2)^4}{5 x+3}dx-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {3}{5} \left (\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4-\frac {1}{55} \int -\frac {2 (48-19 x) \sqrt {1-2 x} (3 x+2)^3}{5 x+3}dx\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{55} \int \frac {(48-19 x) \sqrt {1-2 x} (3 x+2)^3}{5 x+3}dx+\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{55} \left (\frac {19}{45} (1-2 x)^{3/2} (3 x+2)^3-\frac {1}{45} \int -\frac {3 \sqrt {1-2 x} (3 x+2)^2 (2008 x+1383)}{5 x+3}dx\right )+\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{55} \left (\frac {1}{15} \int \frac {\sqrt {1-2 x} (3 x+2)^2 (2008 x+1383)}{5 x+3}dx+\frac {19}{45} (1-2 x)^{3/2} (3 x+2)^3\right )+\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{55} \left (\frac {1}{15} \left (-\frac {1}{35} \int -\frac {7 \sqrt {1-2 x} (3 x+2) (22753 x+13830)}{5 x+3}dx-\frac {2008}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {19}{45} (1-2 x)^{3/2} (3 x+2)^3\right )+\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{55} \left (\frac {1}{15} \left (\frac {1}{5} \int \frac {\sqrt {1-2 x} (3 x+2) (22753 x+13830)}{5 x+3}dx-\frac {2008}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {19}{45} (1-2 x)^{3/2} (3 x+2)^3\right )+\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{55} \left (\frac {1}{15} \left (\frac {1}{5} \left (\frac {891}{25} \int \frac {\sqrt {1-2 x}}{5 x+3}dx-\frac {1}{75} (1-2 x)^{3/2} (204777 x+298462)\right )-\frac {2008}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {19}{45} (1-2 x)^{3/2} (3 x+2)^3\right )+\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{55} \left (\frac {1}{15} \left (\frac {1}{5} \left (\frac {891}{25} \left (\frac {11}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {2}{5} \sqrt {1-2 x}\right )-\frac {1}{75} (1-2 x)^{3/2} (204777 x+298462)\right )-\frac {2008}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {19}{45} (1-2 x)^{3/2} (3 x+2)^3\right )+\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{55} \left (\frac {1}{15} \left (\frac {1}{5} \left (\frac {891}{25} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {11}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{75} (1-2 x)^{3/2} (204777 x+298462)\right )-\frac {2008}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {19}{45} (1-2 x)^{3/2} (3 x+2)^3\right )+\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{55} \left (\frac {1}{15} \left (\frac {1}{5} \left (\frac {891}{25} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {2}{5} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {1}{75} (1-2 x)^{3/2} (204777 x+298462)\right )-\frac {2008}{35} (1-2 x)^{3/2} (3 x+2)^2\right )+\frac {19}{45} (1-2 x)^{3/2} (3 x+2)^3\right )+\frac {13}{55} (1-2 x)^{3/2} (3 x+2)^4\right )-\frac {(1-2 x)^{3/2} (3 x+2)^5}{5 (5 x+3)}\) |
-1/5*((1 - 2*x)^(3/2)*(2 + 3*x)^5)/(3 + 5*x) + (3*((13*(1 - 2*x)^(3/2)*(2 + 3*x)^4)/55 + (2*((19*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/45 + ((-2008*(1 - 2*x) ^(3/2)*(2 + 3*x)^2)/35 + (-1/75*((1 - 2*x)^(3/2)*(298462 + 204777*x)) + (8 91*((2*Sqrt[1 - 2*x])/5 - (2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]) /5))/25)/5)/15))/55))/5
3.20.8.3.1 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[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 = 1.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {212625000 x^{7}+433912500 x^{6}+93363750 x^{5}-335259225 x^{4}-193626310 x^{3}+130129170 x^{2}+48655122 x -23061496}{6015625 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {324 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(71\) |
| pseudoelliptic | \(\frac {-24948 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-5 \sqrt {1-2 x}\, \left (106312500 x^{6}+270112500 x^{5}+181738125 x^{4}-76760550 x^{3}-135193430 x^{2}-2532130 x +23061496\right )}{90234375+150390625 x}\) | \(72\) |
| derivativedivides | \(\frac {243 \left (1-2 x \right )^{\frac {11}{2}}}{2200}-\frac {981 \left (1-2 x \right )^{\frac {9}{2}}}{1000}+\frac {107109 \left (1-2 x \right )^{\frac {7}{2}}}{35000}-\frac {434043 \left (1-2 x \right )^{\frac {5}{2}}}{125000}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {326 \sqrt {1-2 x}}{78125}+\frac {22 \sqrt {1-2 x}}{390625 \left (-\frac {6}{5}-2 x \right )}-\frac {324 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(90\) |
| default | \(\frac {243 \left (1-2 x \right )^{\frac {11}{2}}}{2200}-\frac {981 \left (1-2 x \right )^{\frac {9}{2}}}{1000}+\frac {107109 \left (1-2 x \right )^{\frac {7}{2}}}{35000}-\frac {434043 \left (1-2 x \right )^{\frac {5}{2}}}{125000}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {326 \sqrt {1-2 x}}{78125}+\frac {22 \sqrt {1-2 x}}{390625 \left (-\frac {6}{5}-2 x \right )}-\frac {324 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{390625}\) | \(90\) |
| trager | \(-\frac {\left (106312500 x^{6}+270112500 x^{5}+181738125 x^{4}-76760550 x^{3}-135193430 x^{2}-2532130 x +23061496\right ) \sqrt {1-2 x}}{6015625 \left (3+5 x \right )}+\frac {162 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{390625}\) | \(92\) |
1/6015625*(212625000*x^7+433912500*x^6+93363750*x^5-335259225*x^4-19362631 0*x^3+130129170*x^2+48655122*x-23061496)/(3+5*x)/(1-2*x)^(1/2)-324/390625* arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {12474 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (106312500 \, x^{6} + 270112500 \, x^{5} + 181738125 \, x^{4} - 76760550 \, x^{3} - 135193430 \, x^{2} - 2532130 \, x + 23061496\right )} \sqrt {-2 \, x + 1}}{30078125 \, {\left (5 \, x + 3\right )}} \]
1/30078125*(12474*sqrt(11)*sqrt(5)*(5*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2 *x + 1) + 5*x - 8)/(5*x + 3)) - 5*(106312500*x^6 + 270112500*x^5 + 1817381 25*x^4 - 76760550*x^3 - 135193430*x^2 - 2532130*x + 23061496)*sqrt(-2*x + 1))/(5*x + 3)
Time = 49.59 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.57 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {243 \left (1 - 2 x\right )^{\frac {11}{2}}}{2200} - \frac {981 \left (1 - 2 x\right )^{\frac {9}{2}}}{1000} + \frac {107109 \left (1 - 2 x\right )^{\frac {7}{2}}}{35000} - \frac {434043 \left (1 - 2 x\right )^{\frac {5}{2}}}{125000} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{3125} + \frac {326 \sqrt {1 - 2 x}}{78125} + \frac {161 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{390625} - \frac {484 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{78125} \]
243*(1 - 2*x)**(11/2)/2200 - 981*(1 - 2*x)**(9/2)/1000 + 107109*(1 - 2*x)* *(7/2)/35000 - 434043*(1 - 2*x)**(5/2)/125000 + 2*(1 - 2*x)**(3/2)/3125 + 326*sqrt(1 - 2*x)/78125 + 161*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/390625 - 484*Piecewise((sqrt(55)*(-log(sq rt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/ (4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1) ))/605, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/781 25
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {243}{2200} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {981}{1000} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {107109}{35000} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {434043}{125000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {162}{390625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {326}{78125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \]
243/2200*(-2*x + 1)^(11/2) - 981/1000*(-2*x + 1)^(9/2) + 107109/35000*(-2* x + 1)^(7/2) - 434043/125000*(-2*x + 1)^(5/2) + 2/3125*(-2*x + 1)^(3/2) + 162/390625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( -2*x + 1))) + 326/78125*sqrt(-2*x + 1) - 11/78125*sqrt(-2*x + 1)/(5*x + 3)
Time = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {243}{2200} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {981}{1000} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {107109}{35000} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {434043}{125000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {162}{390625} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {326}{78125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{78125 \, {\left (5 \, x + 3\right )}} \]
-243/2200*(2*x - 1)^5*sqrt(-2*x + 1) - 981/1000*(2*x - 1)^4*sqrt(-2*x + 1) - 107109/35000*(2*x - 1)^3*sqrt(-2*x + 1) - 434043/125000*(2*x - 1)^2*sqr t(-2*x + 1) + 2/3125*(-2*x + 1)^(3/2) + 162/390625*sqrt(55)*log(1/2*abs(-2 *sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 326/78125* sqrt(-2*x + 1) - 11/78125*sqrt(-2*x + 1)/(5*x + 3)
Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {326\,\sqrt {1-2\,x}}{78125}-\frac {22\,\sqrt {1-2\,x}}{390625\,\left (2\,x+\frac {6}{5}\right )}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {434043\,{\left (1-2\,x\right )}^{5/2}}{125000}+\frac {107109\,{\left (1-2\,x\right )}^{7/2}}{35000}-\frac {981\,{\left (1-2\,x\right )}^{9/2}}{1000}+\frac {243\,{\left (1-2\,x\right )}^{11/2}}{2200}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,324{}\mathrm {i}}{390625} \]