Integrand size = 24, antiderivative size = 140 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {463344 \sqrt {1-2 x} (2+3 x)^2}{166375}-\frac {10283 (2+3 x)^3}{6655 \sqrt {1-2 x}}-\frac {38 (2+3 x)^4}{1815 \sqrt {1-2 x} (3+5 x)}+\frac {7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)}-\frac {21 \sqrt {1-2 x} (4633904+1544625 x)}{831875}-\frac {406 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{831875 \sqrt {55}} \]
7/33*(2+3*x)^5/(1-2*x)^(3/2)/(3+5*x)-406/45753125*arctanh(1/11*55^(1/2)*(1 -2*x)^(1/2))*55^(1/2)-10283/6655*(2+3*x)^3/(1-2*x)^(1/2)-38/1815*(2+3*x)^4 /(3+5*x)/(1-2*x)^(1/2)-463344/166375*(2+3*x)^2*(1-2*x)^(1/2)-21/831875*(46 33904+1544625*x)*(1-2*x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {55 \left (1035652776-1434109759 x-3837745731 x^2+2644064775 x^3+480298005 x^4+72772425 x^5\right )-1218 \sqrt {55} \sqrt {1-2 x} \left (-3+x+10 x^2\right ) \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{137259375 (1-2 x)^{3/2} (3+5 x)} \]
-1/137259375*(55*(1035652776 - 1434109759*x - 3837745731*x^2 + 2644064775* x^3 + 480298005*x^4 + 72772425*x^5) - 1218*Sqrt[55]*Sqrt[1 - 2*x]*(-3 + x + 10*x^2)*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/((1 - 2*x)^(3/2)*(3 + 5*x))
Time = 0.23 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {109, 166, 27, 167, 25, 170, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^6}{(1-2 x)^{5/2} (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}-\frac {1}{33} \int \frac {(3 x+2)^4 (411 x+239)}{(1-2 x)^{3/2} (5 x+3)^2}dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{33} \left (-\frac {1}{55} \int \frac {21 (3 x+2)^3 (673 x+398)}{(1-2 x)^{3/2} (5 x+3)}dx-\frac {38 (3 x+2)^4}{55 \sqrt {1-2 x} (5 x+3)}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{33} \left (-\frac {21}{55} \int \frac {(3 x+2)^3 (673 x+398)}{(1-2 x)^{3/2} (5 x+3)}dx-\frac {38 (3 x+2)^4}{55 \sqrt {1-2 x} (5 x+3)}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{33} \left (-\frac {21}{55} \left (\frac {1}{11} \int -\frac {(3 x+2)^2 (66192 x+39721)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {1469 (3 x+2)^3}{11 \sqrt {1-2 x}}\right )-\frac {38 (3 x+2)^4}{55 \sqrt {1-2 x} (5 x+3)}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{33} \left (-\frac {21}{55} \left (\frac {1469 (3 x+2)^3}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(3 x+2)^2 (66192 x+39721)}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {38 (3 x+2)^4}{55 \sqrt {1-2 x} (5 x+3)}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {1}{33} \left (-\frac {21}{55} \left (\frac {1}{11} \left (\frac {1}{25} \int -\frac {(3 x+2) (4633875 x+2780354)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {66192}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {1469 (3 x+2)^3}{11 \sqrt {1-2 x}}\right )-\frac {38 (3 x+2)^4}{55 \sqrt {1-2 x} (5 x+3)}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{33} \left (-\frac {21}{55} \left (\frac {1}{11} \left (\frac {66192}{25} \sqrt {1-2 x} (3 x+2)^2-\frac {1}{25} \int \frac {(3 x+2) (4633875 x+2780354)}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {1469 (3 x+2)^3}{11 \sqrt {1-2 x}}\right )-\frac {38 (3 x+2)^4}{55 \sqrt {1-2 x} (5 x+3)}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {1}{33} \left (-\frac {21}{55} \left (\frac {1}{11} \left (\frac {1}{25} \left (\frac {3}{5} \sqrt {1-2 x} (1544625 x+4633904)-\frac {29}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {66192}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {1469 (3 x+2)^3}{11 \sqrt {1-2 x}}\right )-\frac {38 (3 x+2)^4}{55 \sqrt {1-2 x} (5 x+3)}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{33} \left (-\frac {21}{55} \left (\frac {1}{11} \left (\frac {1}{25} \left (\frac {29}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {3}{5} \sqrt {1-2 x} (1544625 x+4633904)\right )+\frac {66192}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {1469 (3 x+2)^3}{11 \sqrt {1-2 x}}\right )-\frac {38 (3 x+2)^4}{55 \sqrt {1-2 x} (5 x+3)}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{33} \left (-\frac {21}{55} \left (\frac {1}{11} \left (\frac {1}{25} \left (\frac {58 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}+\frac {3}{5} \sqrt {1-2 x} (1544625 x+4633904)\right )+\frac {66192}{25} \sqrt {1-2 x} (3 x+2)^2\right )+\frac {1469 (3 x+2)^3}{11 \sqrt {1-2 x}}\right )-\frac {38 (3 x+2)^4}{55 \sqrt {1-2 x} (5 x+3)}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}\) |
(7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)) + ((-38*(2 + 3*x)^4)/(55*Sq rt[1 - 2*x]*(3 + 5*x)) - (21*((1469*(2 + 3*x)^3)/(11*Sqrt[1 - 2*x]) + ((66 192*Sqrt[1 - 2*x]*(2 + 3*x)^2)/25 + ((3*Sqrt[1 - 2*x]*(4633904 + 1544625*x ))/5 + (58*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55]))/25)/11))/55)/3 3
3.22.80.3.1 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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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[(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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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 = 1.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(\frac {72772425 x^{5}+480298005 x^{4}+2644064775 x^{3}-3837745731 x^{2}-1434109759 x +1035652776}{2495625 \sqrt {1-2 x}\, \left (3+5 x \right ) \left (-1+2 x \right )}-\frac {406 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{45753125}\) | \(68\) |
| pseudoelliptic | \(\frac {1218 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (10 x^{2}+x -3\right ) \sqrt {55}-4002483375 x^{5}-26416390275 x^{4}-145423562625 x^{3}+211076015205 x^{2}+78876036745 x -56960902680}{\left (1-2 x \right )^{\frac {3}{2}} \left (411778125+686296875 x \right )}\) | \(75\) |
| derivativedivides | \(-\frac {729 \left (1-2 x \right )^{\frac {5}{2}}}{2000}+\frac {729 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {315171 \sqrt {1-2 x}}{5000}+\frac {2 \sqrt {1-2 x}}{4159375 \left (-\frac {6}{5}-2 x \right )}-\frac {406 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{45753125}+\frac {117649}{5808 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {134456}{1331 \sqrt {1-2 x}}\) | \(81\) |
| default | \(-\frac {729 \left (1-2 x \right )^{\frac {5}{2}}}{2000}+\frac {729 \left (1-2 x \right )^{\frac {3}{2}}}{125}-\frac {315171 \sqrt {1-2 x}}{5000}+\frac {2 \sqrt {1-2 x}}{4159375 \left (-\frac {6}{5}-2 x \right )}-\frac {406 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{45753125}+\frac {117649}{5808 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {134456}{1331 \sqrt {1-2 x}}\) | \(81\) |
| trager | \(-\frac {\left (72772425 x^{5}+480298005 x^{4}+2644064775 x^{3}-3837745731 x^{2}-1434109759 x +1035652776\right ) \sqrt {1-2 x}}{2495625 \left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {203 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{45753125}\) | \(94\) |
1/2495625*(72772425*x^5+480298005*x^4+2644064775*x^3-3837745731*x^2-143410 9759*x+1035652776)/(1-2*x)^(1/2)/(3+5*x)/(-1+2*x)-406/45753125*arctanh(1/1 1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.71 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {609 \, \sqrt {55} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (72772425 \, x^{5} + 480298005 \, x^{4} + 2644064775 \, x^{3} - 3837745731 \, x^{2} - 1434109759 \, x + 1035652776\right )} \sqrt {-2 \, x + 1}}{137259375 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
1/137259375*(609*sqrt(55)*(20*x^3 - 8*x^2 - 7*x + 3)*log((5*x + sqrt(55)*s qrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(72772425*x^5 + 480298005*x^4 + 2644064 775*x^3 - 3837745731*x^2 - 1434109759*x + 1035652776)*sqrt(-2*x + 1))/(20* x^3 - 8*x^2 - 7*x + 3)
Time = 96.03 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.58 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=- \frac {729 \left (1 - 2 x\right )^{\frac {5}{2}}}{2000} + \frac {729 \left (1 - 2 x\right )^{\frac {3}{2}}}{125} - \frac {315171 \sqrt {1 - 2 x}}{5000} + \frac {202 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{45753125} - \frac {4 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{75625} - \frac {134456}{1331 \sqrt {1 - 2 x}} + \frac {117649}{5808 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
-729*(1 - 2*x)**(5/2)/2000 + 729*(1 - 2*x)**(3/2)/125 - 315171*sqrt(1 - 2* x)/5000 + 202*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x ) + sqrt(55)/5))/45753125 - 4*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqr t(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)))/75625 - 134456/(133 1*sqrt(1 - 2*x)) + 117649/(5808*(1 - 2*x)**(3/2))
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {729}{2000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {729}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {203}{45753125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {315171}{5000} \, \sqrt {-2 \, x + 1} - \frac {10084199952 \, {\left (2 \, x - 1\right )}^{2} + 48414664375 \, x - 19758729375}{19965000 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
-729/2000*(-2*x + 1)^(5/2) + 729/125*(-2*x + 1)^(3/2) + 203/45753125*sqrt( 55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 31 5171/5000*sqrt(-2*x + 1) - 1/19965000*(10084199952*(2*x - 1)^2 + 484146643 75*x - 19758729375)/(5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.79 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {729}{2000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {729}{125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {203}{45753125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {315171}{5000} \, \sqrt {-2 \, x + 1} - \frac {16807 \, {\left (768 \, x - 307\right )}}{63888 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{831875 \, {\left (5 \, x + 3\right )}} \]
-729/2000*(2*x - 1)^2*sqrt(-2*x + 1) + 729/125*(-2*x + 1)^(3/2) + 203/4575 3125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s qrt(-2*x + 1))) - 315171/5000*sqrt(-2*x + 1) - 16807/63888*(768*x - 307)/( (2*x - 1)*sqrt(-2*x + 1)) - 1/831875*sqrt(-2*x + 1)/(5*x + 3)
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.60 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {\frac {7042133\,x}{14520}+\frac {420174998\,{\left (2\,x-1\right )}^2}{4159375}-\frac {957999}{4840}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {315171\,\sqrt {1-2\,x}}{5000}+\frac {729\,{\left (1-2\,x\right )}^{3/2}}{125}-\frac {729\,{\left (1-2\,x\right )}^{5/2}}{2000}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,406{}\mathrm {i}}{45753125} \]