Integrand size = 24, antiderivative size = 140 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {217152 \sqrt {1-2 x} (2+3 x)^2}{75625}+\frac {14517 \sqrt {1-2 x} (2+3 x)^3}{21175}-\frac {36 \sqrt {1-2 x} (2+3 x)^4}{605 (3+5 x)}+\frac {7 (2+3 x)^5}{11 \sqrt {1-2 x} (3+5 x)}+\frac {9 \sqrt {1-2 x} (5065808+1688625 x)}{378125}-\frac {402 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{378125 \sqrt {55}} \]
-402/20796875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+7/11*(2+3*x)^5 /(3+5*x)/(1-2*x)^(1/2)+217152/75625*(2+3*x)^2*(1-2*x)^(1/2)+14517/21175*(2 +3*x)^3*(1-2*x)^(1/2)-36/605*(2+3*x)^4*(1-2*x)^(1/2)/(3+5*x)+9/378125*(506 5808+1688625*x)*(1-2*x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {-\frac {55 \left (-1143572552-818846961 x+2195407665 x^2+795400155 x^3+293294925 x^4+55130625 x^5\right )}{\sqrt {1-2 x} (3+5 x)}-2814 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{145578125} \]
((-55*(-1143572552 - 818846961*x + 2195407665*x^2 + 795400155*x^3 + 293294 925*x^4 + 55130625*x^5))/(Sqrt[1 - 2*x]*(3 + 5*x)) - 2814*Sqrt[55]*ArcTanh [Sqrt[5/11]*Sqrt[1 - 2*x]])/145578125
Time = 0.24 (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, 27, 166, 170, 27, 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.
| \(\displaystyle \int \frac {(3 x+2)^6}{(1-2 x)^{3/2} (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {1}{11} \int \frac {3 (3 x+2)^4 (139 x+81)}{\sqrt {1-2 x} (5 x+3)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {3}{11} \int \frac {(3 x+2)^4 (139 x+81)}{\sqrt {1-2 x} (5 x+3)^2}dx\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {3}{11} \left (\frac {1}{55} \int \frac {(3 x+2)^3 (4839 x+2890)}{\sqrt {1-2 x} (5 x+3)}dx+\frac {12 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {3}{11} \left (\frac {1}{55} \left (-\frac {1}{35} \int -\frac {7 (3 x+2)^2 (72384 x+43417)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {4839}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {12 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {3}{11} \left (\frac {1}{55} \left (\frac {1}{5} \int \frac {(3 x+2)^2 (72384 x+43417)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {4839}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {12 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {3}{11} \left (\frac {1}{55} \left (\frac {1}{5} \left (-\frac {1}{25} \int -\frac {(3 x+2) (5065875 x+3039458)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {72384}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {4839}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {12 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {3}{11} \left (\frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{25} \int \frac {(3 x+2) (5065875 x+3039458)}{\sqrt {1-2 x} (5 x+3)}dx-\frac {72384}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {4839}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {12 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {3}{11} \left (\frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{25} \left (-\frac {67}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-\frac {3}{5} \sqrt {1-2 x} (1688625 x+5065808)\right )-\frac {72384}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {4839}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {12 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {3}{11} \left (\frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{25} \left (\frac {67}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {3}{5} \sqrt {1-2 x} (1688625 x+5065808)\right )-\frac {72384}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {4839}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {12 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7 (3 x+2)^5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {3}{11} \left (\frac {1}{55} \left (\frac {1}{5} \left (\frac {1}{25} \left (\frac {134 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{5 \sqrt {55}}-\frac {3}{5} \sqrt {1-2 x} (1688625 x+5065808)\right )-\frac {72384}{25} \sqrt {1-2 x} (3 x+2)^2\right )-\frac {4839}{35} \sqrt {1-2 x} (3 x+2)^3\right )+\frac {12 \sqrt {1-2 x} (3 x+2)^4}{55 (5 x+3)}\right )\) |
(7*(2 + 3*x)^5)/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (3*((12*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(55*(3 + 5*x)) + ((-4839*Sqrt[1 - 2*x]*(2 + 3*x)^3)/35 + ((-72384* Sqrt[1 - 2*x]*(2 + 3*x)^2)/25 + ((-3*Sqrt[1 - 2*x]*(5065808 + 1688625*x))/ 5 + (134*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55]))/25)/5)/55))/11
3.22.15.3.1 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[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 = 1.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(-\frac {55130625 x^{5}+293294925 x^{4}+795400155 x^{3}+2195407665 x^{2}-818846961 x -1143572552}{2646875 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {402 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{20796875}\) | \(61\) |
| pseudoelliptic | \(-\frac {3032184375 \left (\frac {938 \sqrt {55}\, \left (x +\frac {3}{5}\right ) \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{202145625}+x^{5}+\frac {133 x^{4}}{25}+\frac {16231 x^{3}}{1125}+\frac {134399 x^{2}}{3375}-\frac {272948987 x}{18376875}-\frac {1143572552}{55130625}\right )}{\sqrt {1-2 x}\, \left (436734375+727890625 x \right )}\) | \(68\) |
| derivativedivides | \(-\frac {729 \left (1-2 x \right )^{\frac {7}{2}}}{2800}+\frac {2187 \left (1-2 x \right )^{\frac {5}{2}}}{625}-\frac {105057 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {315684 \sqrt {1-2 x}}{3125}+\frac {2 \sqrt {1-2 x}}{1890625 \left (-\frac {6}{5}-2 x \right )}-\frac {402 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{20796875}+\frac {117649}{1936 \sqrt {1-2 x}}\) | \(81\) |
| default | \(-\frac {729 \left (1-2 x \right )^{\frac {7}{2}}}{2800}+\frac {2187 \left (1-2 x \right )^{\frac {5}{2}}}{625}-\frac {105057 \left (1-2 x \right )^{\frac {3}{2}}}{5000}+\frac {315684 \sqrt {1-2 x}}{3125}+\frac {2 \sqrt {1-2 x}}{1890625 \left (-\frac {6}{5}-2 x \right )}-\frac {402 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{20796875}+\frac {117649}{1936 \sqrt {1-2 x}}\) | \(81\) |
| trager | \(\frac {\left (55130625 x^{5}+293294925 x^{4}+795400155 x^{3}+2195407665 x^{2}-818846961 x -1143572552\right ) \sqrt {1-2 x}}{26468750 x^{2}+2646875 x -7940625}+\frac {201 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{20796875}\) | \(90\) |
-1/2646875*(55130625*x^5+293294925*x^4+795400155*x^3+2195407665*x^2-818846 961*x-1143572552)/(3+5*x)/(1-2*x)^(1/2)-402/20796875*arctanh(1/11*55^(1/2) *(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.61 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {1407 \, \sqrt {55} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (55130625 \, x^{5} + 293294925 \, x^{4} + 795400155 \, x^{3} + 2195407665 \, x^{2} - 818846961 \, x - 1143572552\right )} \sqrt {-2 \, x + 1}}{145578125 \, {\left (10 \, x^{2} + x - 3\right )}} \]
1/145578125*(1407*sqrt(55)*(10*x^2 + x - 3)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 55*(55130625*x^5 + 293294925*x^4 + 795400155*x^3 + 2195407665*x^2 - 818846961*x - 1143572552)*sqrt(-2*x + 1))/(10*x^2 + x - 3 )
Time = 95.54 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.58 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=- \frac {729 \left (1 - 2 x\right )^{\frac {7}{2}}}{2800} + \frac {2187 \left (1 - 2 x\right )^{\frac {5}{2}}}{625} - \frac {105057 \left (1 - 2 x\right )^{\frac {3}{2}}}{5000} + \frac {315684 \sqrt {1 - 2 x}}{3125} + \frac {8 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{831875} - \frac {4 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{34375} + \frac {117649}{1936 \sqrt {1 - 2 x}} \]
-729*(1 - 2*x)**(7/2)/2800 + 2187*(1 - 2*x)**(5/2)/625 - 105057*(1 - 2*x)* *(3/2)/5000 + 315684*sqrt(1 - 2*x)/3125 + 8*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/831875 - 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)*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)))/34375 + 117649/(1936*sqrt(1 - 2*x))
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=-\frac {729}{2800} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2187}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {105057}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {201}{20796875} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {315684}{3125} \, \sqrt {-2 \, x + 1} - \frac {1838265657 \, x + 1102959359}{3025000 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
-729/2800*(-2*x + 1)^(7/2) + 2187/625*(-2*x + 1)^(5/2) - 105057/5000*(-2*x + 1)^(3/2) + 201/20796875*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sq rt(55) + 5*sqrt(-2*x + 1))) + 315684/3125*sqrt(-2*x + 1) - 1/3025000*(1838 265657*x + 1102959359)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {729}{2800} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2187}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {105057}{5000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {201}{20796875} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {315684}{3125} \, \sqrt {-2 \, x + 1} - \frac {1838265657 \, x + 1102959359}{3025000 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} \]
729/2800*(2*x - 1)^3*sqrt(-2*x + 1) + 2187/625*(2*x - 1)^2*sqrt(-2*x + 1) - 105057/5000*(-2*x + 1)^(3/2) + 201/20796875*sqrt(55)*log(1/2*abs(-2*sqrt (55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 315684/3125*sqr t(-2*x + 1) - 1/3025000*(1838265657*x + 1102959359)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.60 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx=\frac {315684\,\sqrt {1-2\,x}}{3125}-\frac {105057\,{\left (1-2\,x\right )}^{3/2}}{5000}+\frac {2187\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {729\,{\left (1-2\,x\right )}^{7/2}}{2800}+\frac {\frac {1838265657\,x}{15125000}+\frac {1102959359}{15125000}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,402{}\mathrm {i}}{20796875} \]