Integrand size = 47, antiderivative size = 258 \[ \int \frac {1-x^3 \sqrt [3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt {5+4 x}} \, dx=\frac {3}{64} (-15+4 x) \sqrt [3]{5+4 x}+\frac {3}{160} (5+4 x)^{2/3} (-15+8 x)+\frac {3}{748} (5+4 x)^{5/6} \left (45-30 x+22 x^2\right )+\frac {3 \sqrt [6]{5+4 x} \left (4583-150 x+70 x^2+728 x^3\right )}{6916}-3 \log \left (1+\sqrt [6]{5+4 x}\right )+\frac {1}{2} \text {RootSum}\left [-4-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-4 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right )+3 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}-8 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^2+2 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^3-\log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^5}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \]
Time = 0.14 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.15 \[ \int \frac {1-x^3 \sqrt [3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt {5+4 x}} \, dx=\frac {3 \left (51731680 \sqrt [6]{5+4 x}-32332300 \sqrt [3]{5+4 x}-16166150 (5+4 x)^{2/3}+16166150 (5+4 x)^{5/6}+11547250 (5+4 x)^{7/6}+1616615 (5+4 x)^{4/3}+1293292 (5+4 x)^{5/3}-2939300 (5+4 x)^{11/6}-2487100 (5+4 x)^{13/6}+190190 (5+4 x)^{17/6}+170170 (5+4 x)^{19/6}\right )}{103463360}-3 \log \left (1+\sqrt [6]{5+4 x}\right )+\frac {1}{2} \text {RootSum}\left [-4-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-4 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right )+3 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}-8 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^2+2 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^3-\log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (\sqrt [6]{5+4 x}-\text {$\#$1}\right ) \text {$\#$1}^5}{-\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \]
(3*(51731680*(5 + 4*x)^(1/6) - 32332300*(5 + 4*x)^(1/3) - 16166150*(5 + 4* x)^(2/3) + 16166150*(5 + 4*x)^(5/6) + 11547250*(5 + 4*x)^(7/6) + 1616615*( 5 + 4*x)^(4/3) + 1293292*(5 + 4*x)^(5/3) - 2939300*(5 + 4*x)^(11/6) - 2487 100*(5 + 4*x)^(13/6) + 190190*(5 + 4*x)^(17/6) + 170170*(5 + 4*x)^(19/6))) /103463360 - 3*Log[1 + (5 + 4*x)^(1/6)] + RootSum[-4 - #1^3 + #1^6 & , (-4 *Log[(5 + 4*x)^(1/6) - #1] + 3*Log[(5 + 4*x)^(1/6) - #1]*#1 - 8*Log[(5 + 4 *x)^(1/6) - #1]*#1^2 + 2*Log[(5 + 4*x)^(1/6) - #1]*#1^3 - Log[(5 + 4*x)^(1 /6) - #1]*#1^4 + 2*Log[(5 + 4*x)^(1/6) - #1]*#1^5)/(-#1^2 + 2*#1^5) & ]/2
Leaf count is larger than twice the leaf count of optimal. \(948\) vs. \(2(258)=516\).
Time = 1.58 (sec) , antiderivative size = 948, normalized size of antiderivative = 3.67, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {7267, 27, 2457, 2462, 2009}
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.
| \(\displaystyle \int \frac {-(4 x+5)^{2/3} x^3-\sqrt [3]{4 x+5} x^3+1}{1-x \sqrt {4 x+5}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 6 \int \frac {(4 x+5)^{5/6} \left (-64 (4 x+5)^{2/3} x^3-64 \sqrt [3]{4 x+5} x^3+64\right )}{64 \left (-(4 x+5)^{3/2}+5 \sqrt {4 x+5}+4\right )}d\sqrt [6]{4 x+5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{32} \int \frac {(4 x+5)^{5/6} \left (-64 (4 x+5)^{2/3} x^3-64 \sqrt [3]{4 x+5} x^3+64\right )}{-(4 x+5)^{3/2}+5 \sqrt {4 x+5}+4}d\sqrt [6]{4 x+5}\) |
| \(\Big \downarrow \) 2457 |
| \(\displaystyle \frac {3}{32} \int \frac {(4 x+5)^{5/6} \left (-(4 x+5)^{10/3}-(4 x+5)^{19/6}-(4 x+5)^3+(4 x+5)^{8/3}+(4 x+5)^{5/2}+15 (4 x+5)^{7/3}+14 (4 x+5)^{13/6}+14 (4 x+5)^2-14 (4 x+5)^{5/3}-14 (4 x+5)^{3/2}-75 (4 x+5)^{4/3}-61 (4 x+5)^{7/6}-61 (4 x+5)+61 (4 x+5)^{2/3}+61 \sqrt {4 x+5}+125 \sqrt [3]{4 x+5}+64 \sqrt [6]{4 x+5}+64\right )}{-(4 x+5)^{7/6}+(4 x+5)^{2/3}+\sqrt {4 x+5}+4 \sqrt [6]{4 x+5}-4 x-1}d\sqrt [6]{4 x+5}\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \frac {3}{32} \int \left ((4 x+5)^3+(4 x+5)^{8/3}-10 (4 x+5)^2-10 (4 x+5)^{5/3}+4 (4 x+5)^{3/2}+4 (4 x+5)^{7/6}+25 (4 x+5)+25 (4 x+5)^{2/3}-20 \sqrt {4 x+5}-20 \sqrt [6]{4 x+5}+\frac {16 \left (2 (4 x+5)^{5/6}-(4 x+5)^{2/3}+2 \sqrt {4 x+5}-8 \sqrt [3]{4 x+5}+3 \sqrt [6]{4 x+5}-4\right )}{4 x-\sqrt {4 x+5}+1}-\frac {32}{\sqrt [6]{4 x+5}+1}+16\right )d\sqrt [6]{4 x+5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{32} \left (\frac {1}{19} (4 x+5)^{19/6}+\frac {1}{17} (4 x+5)^{17/6}-\frac {10}{13} (4 x+5)^{13/6}-\frac {10}{11} (4 x+5)^{11/6}+\frac {2}{5} (4 x+5)^{5/3}+\frac {1}{2} (4 x+5)^{4/3}+\frac {25}{7} (4 x+5)^{7/6}+5 (4 x+5)^{5/6}-5 (4 x+5)^{2/3}-10 \sqrt [3]{4 x+5}+16 \sqrt [6]{4 x+5}+\frac {8 \sqrt [3]{243+59 \sqrt {17}} \arctan \left (\frac {1-2 \sqrt [3]{\frac {2}{-1+\sqrt {17}}} \sqrt [6]{4 x+5}}{\sqrt {3}}\right )}{\sqrt {51}}-\frac {16 \sqrt [3]{2 \left (37+9 \sqrt {17}\right )} \arctan \left (\frac {1-2 \sqrt [3]{\frac {2}{-1+\sqrt {17}}} \sqrt [6]{4 x+5}}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {8 \sqrt [3]{-243+59 \sqrt {17}} \arctan \left (\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {17}}} \sqrt [6]{4 x+5}+1}{\sqrt {3}}\right )}{\sqrt {51}}-\frac {16 \sqrt [3]{2 \left (-37+9 \sqrt {17}\right )} \arctan \left (\frac {2 \sqrt [3]{\frac {2}{1+\sqrt {17}}} \sqrt [6]{4 x+5}+1}{\sqrt {3}}\right )}{\sqrt {51}}-32 \log \left (\sqrt [6]{4 x+5}+1\right )+\frac {8 \sqrt [3]{-243+59 \sqrt {17}} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{4 x+5}\right )}{3 \sqrt {17}}+\frac {16 \sqrt [3]{2 \left (-37+9 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{2} \sqrt [6]{4 x+5}\right )}{3 \sqrt {17}}+\frac {8 \sqrt [3]{243+59 \sqrt {17}} \log \left (\sqrt [3]{2} \sqrt [6]{4 x+5}+\sqrt [3]{-1+\sqrt {17}}\right )}{3 \sqrt {17}}+\frac {16 \sqrt [3]{2 \left (37+9 \sqrt {17}\right )} \log \left (\sqrt [3]{2} \sqrt [6]{4 x+5}+\sqrt [3]{-1+\sqrt {17}}\right )}{3 \sqrt {17}}-\frac {4 \sqrt [3]{243+59 \sqrt {17}} \log \left (2^{2/3} \sqrt [3]{4 x+5}-\sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [6]{4 x+5}+\left (-1+\sqrt {17}\right )^{2/3}\right )}{3 \sqrt {17}}-\frac {8 \sqrt [3]{2 \left (37+9 \sqrt {17}\right )} \log \left (2^{2/3} \sqrt [3]{4 x+5}-\sqrt [3]{2 \left (-1+\sqrt {17}\right )} \sqrt [6]{4 x+5}+\left (-1+\sqrt {17}\right )^{2/3}\right )}{3 \sqrt {17}}-\frac {4 \sqrt [3]{-243+59 \sqrt {17}} \log \left (2^{2/3} \sqrt [3]{4 x+5}+\sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [6]{4 x+5}+\left (1+\sqrt {17}\right )^{2/3}\right )}{3 \sqrt {17}}-\frac {8 \sqrt [3]{2 \left (-37+9 \sqrt {17}\right )} \log \left (2^{2/3} \sqrt [3]{4 x+5}+\sqrt [3]{2 \left (1+\sqrt {17}\right )} \sqrt [6]{4 x+5}+\left (1+\sqrt {17}\right )^{2/3}\right )}{3 \sqrt {17}}+\frac {16}{51} \left (17+7 \sqrt {17}\right ) \log \left (-2 \sqrt {4 x+5}-\sqrt {17}+1\right )+\frac {16}{51} \left (17-7 \sqrt {17}\right ) \log \left (-2 \sqrt {4 x+5}+\sqrt {17}+1\right )\right )\) |
(3*(16*(5 + 4*x)^(1/6) - 10*(5 + 4*x)^(1/3) - 5*(5 + 4*x)^(2/3) + 5*(5 + 4 *x)^(5/6) + (25*(5 + 4*x)^(7/6))/7 + (5 + 4*x)^(4/3)/2 + (2*(5 + 4*x)^(5/3 ))/5 - (10*(5 + 4*x)^(11/6))/11 - (10*(5 + 4*x)^(13/6))/13 + (5 + 4*x)^(17 /6)/17 + (5 + 4*x)^(19/6)/19 - (16*(2*(37 + 9*Sqrt[17]))^(1/3)*ArcTan[(1 - 2*(2/(-1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6))/Sqrt[3]])/Sqrt[51] + (8*(243 + 59*Sqrt[17])^(1/3)*ArcTan[(1 - 2*(2/(-1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1 /6))/Sqrt[3]])/Sqrt[51] - (16*(2*(-37 + 9*Sqrt[17]))^(1/3)*ArcTan[(1 + 2*( 2/(1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6))/Sqrt[3]])/Sqrt[51] + (8*(-243 + 5 9*Sqrt[17])^(1/3)*ArcTan[(1 + 2*(2/(1 + Sqrt[17]))^(1/3)*(5 + 4*x)^(1/6))/ Sqrt[3]])/Sqrt[51] - 32*Log[1 + (5 + 4*x)^(1/6)] + (16*(2*(-37 + 9*Sqrt[17 ]))^(1/3)*Log[(1 + Sqrt[17])^(1/3) - 2^(1/3)*(5 + 4*x)^(1/6)])/(3*Sqrt[17] ) + (8*(-243 + 59*Sqrt[17])^(1/3)*Log[(1 + Sqrt[17])^(1/3) - 2^(1/3)*(5 + 4*x)^(1/6)])/(3*Sqrt[17]) + (16*(2*(37 + 9*Sqrt[17]))^(1/3)*Log[(-1 + Sqrt [17])^(1/3) + 2^(1/3)*(5 + 4*x)^(1/6)])/(3*Sqrt[17]) + (8*(243 + 59*Sqrt[1 7])^(1/3)*Log[(-1 + Sqrt[17])^(1/3) + 2^(1/3)*(5 + 4*x)^(1/6)])/(3*Sqrt[17 ]) - (8*(2*(37 + 9*Sqrt[17]))^(1/3)*Log[(-1 + Sqrt[17])^(2/3) - (2*(-1 + S qrt[17]))^(1/3)*(5 + 4*x)^(1/6) + 2^(2/3)*(5 + 4*x)^(1/3)])/(3*Sqrt[17]) - (4*(243 + 59*Sqrt[17])^(1/3)*Log[(-1 + Sqrt[17])^(2/3) - (2*(-1 + Sqrt[17 ]))^(1/3)*(5 + 4*x)^(1/6) + 2^(2/3)*(5 + 4*x)^(1/3)])/(3*Sqrt[17]) - (8*(2 *(-37 + 9*Sqrt[17]))^(1/3)*Log[(1 + Sqrt[17])^(2/3) + (2*(1 + Sqrt[17])...
3.28.58.3.1 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[(u_.)*(Px_)*(Qx_)^(q_), x_Symbol] :> Module[{Rx = PolyGCD[Px, Qx, x]}, Int[u*Rx^(q + 1)*PolynomialQuotient[Px, Rx, x]*PolynomialQuotient[Qx, Rx, x ]^q, x] /; NeQ[Rx, 1]] /; ILtQ[q, 0] && PolyQ[Px, x] && PolyQ[Qx, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {3 \left (5+4 x \right )^{\frac {19}{6}}}{608}+\frac {3 \left (5+4 x \right )^{\frac {17}{6}}}{544}-\frac {15 \left (5+4 x \right )^{\frac {13}{6}}}{208}-\frac {15 \left (5+4 x \right )^{\frac {11}{6}}}{176}+\frac {3 \left (5+4 x \right )^{\frac {5}{3}}}{80}+\frac {3 \left (5+4 x \right )^{\frac {4}{3}}}{64}+\frac {75 \left (5+4 x \right )^{\frac {7}{6}}}{224}+\frac {15 \left (5+4 x \right )^{\frac {5}{6}}}{32}-\frac {15 \left (5+4 x \right )^{\frac {2}{3}}}{32}-\frac {15 \left (5+4 x \right )^{\frac {1}{3}}}{16}+\frac {3 \left (5+4 x \right )^{\frac {1}{6}}}{2}-3 \ln \left (\left (5+4 x \right )^{\frac {1}{6}}+1\right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}-4\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+2 \textit {\_R}^{3}-8 \textit {\_R}^{2}+3 \textit {\_R} -4\right ) \ln \left (\left (5+4 x \right )^{\frac {1}{6}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\) | \(180\) |
| default | \(\frac {3 \left (5+4 x \right )^{\frac {19}{6}}}{608}+\frac {3 \left (5+4 x \right )^{\frac {17}{6}}}{544}-\frac {15 \left (5+4 x \right )^{\frac {13}{6}}}{208}-\frac {15 \left (5+4 x \right )^{\frac {11}{6}}}{176}+\frac {3 \left (5+4 x \right )^{\frac {5}{3}}}{80}+\frac {3 \left (5+4 x \right )^{\frac {4}{3}}}{64}+\frac {75 \left (5+4 x \right )^{\frac {7}{6}}}{224}+\frac {15 \left (5+4 x \right )^{\frac {5}{6}}}{32}-\frac {15 \left (5+4 x \right )^{\frac {2}{3}}}{32}-\frac {15 \left (5+4 x \right )^{\frac {1}{3}}}{16}+\frac {3 \left (5+4 x \right )^{\frac {1}{6}}}{2}-3 \ln \left (\left (5+4 x \right )^{\frac {1}{6}}+1\right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}-4\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-\textit {\_R}^{4}+2 \textit {\_R}^{3}-8 \textit {\_R}^{2}+3 \textit {\_R} -4\right ) \ln \left (\left (5+4 x \right )^{\frac {1}{6}}-\textit {\_R} \right )}{2 \textit {\_R}^{5}-\textit {\_R}^{2}}\right )}{2}\) | \(180\) |
3/608*(5+4*x)^(19/6)+3/544*(5+4*x)^(17/6)-15/208*(5+4*x)^(13/6)-15/176*(5+ 4*x)^(11/6)+3/80*(5+4*x)^(5/3)+3/64*(5+4*x)^(4/3)+75/224*(5+4*x)^(7/6)+15/ 32*(5+4*x)^(5/6)-15/32*(5+4*x)^(2/3)-15/16*(5+4*x)^(1/3)+3/2*(5+4*x)^(1/6) -3*ln((5+4*x)^(1/6)+1)+1/2*sum((2*_R^5-_R^4+2*_R^3-8*_R^2+3*_R-4)/(2*_R^5- _R^2)*ln((5+4*x)^(1/6)-_R),_R=RootOf(_Z^6-_Z^3-4))
Timed out. \[ \int \frac {1-x^3 \sqrt [3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt {5+4 x}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1-x^3 \sqrt [3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt {5+4 x}} \, dx=\text {Timed out} \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.97 \[ \int \frac {1-x^3 \sqrt [3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt {5+4 x}} \, dx=\int { \frac {{\left (4 \, x + 5\right )}^{\frac {2}{3}} x^{3} + {\left (4 \, x + 5\right )}^{\frac {1}{3}} x^{3} - 1}{\sqrt {4 \, x + 5} x - 1} \,d x } \]
-3/315392*(11264*x^4 + 2560*x^3 - 3600*x^2 + 5400*x - 10125)*(4*x + 5)^(2/ 3) - 7/34*sqrt(17)*log(-(sqrt(17) - 2*sqrt(4*x + 5) + 1)/(sqrt(17) + 2*sqr t(4*x + 5) - 1)) - 3/186368*(7168*x^4 + 896*x^3 - 1440*x^2 + 2700*x - 1012 5)*(4*x + 5)^(1/3) - 3/15865304*(857584*x^6 + 1314040*x^5 + 12103*x^4 - 43 6618*x^3 - 7070*x^2 + 15150*x - 113625)*(4*x + 5)^(1/6) - integrate(-1/4*( 2*(4*x^6 + 5*x^5 - x^3)*(4*x + 5)^(1/3) + (16*x^8 + 40*x^7 + 25*x^6 - x^2) *(4*x + 5)^(1/6))/(4*x^3 + 5*x^2 - 2*sqrt(4*x + 5)*x + 1), x) + 1/2*log(4* x - sqrt(4*x + 5) + 1) - log(sqrt(4*x + 5) + 1)
Not integrable
Time = 0.37 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.16 \[ \int \frac {1-x^3 \sqrt [3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt {5+4 x}} \, dx=\int { \frac {{\left (4 \, x + 5\right )}^{\frac {2}{3}} x^{3} + {\left (4 \, x + 5\right )}^{\frac {1}{3}} x^{3} - 1}{\sqrt {4 \, x + 5} x - 1} \,d x } \]
Time = 6.67 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.64 \[ \int \frac {1-x^3 \sqrt [3]{5+4 x}-x^3 (5+4 x)^{2/3}}{1-x \sqrt {5+4 x}} \, dx=\left (\sum _{k=1}^6\ln \left (\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (-\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (-\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (-\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\,\left (89890560\,{\left (4\,x+5\right )}^{1/6}+95508720\right )+92137824\,{\left (4\,x+5\right )}^{1/6}+79777872\right )+37240965\,{\left (4\,x+5\right )}^{1/6}+52777656\right )+42123807\,{\left (4\,x+5\right )}^{1/6}+37377288\right )+8945559\,{\left (4\,x+5\right )}^{1/6}+13837149\right )+5031558\,{\left (4\,x+5\right )}^{1/6}+2990358\right )+1119744\,{\left (4\,x+5\right )}^{1/6}+874800\right )\,\mathrm {root}\left (z^6-3\,z^5+\frac {45\,z^4}{68}+\frac {957\,z^3}{544}-\frac {1863\,z^2}{9248}-\frac {225\,z}{578}-\frac {2439}{39304},z,k\right )\right )-3\,\ln \left (-83860333479\,{\left (4\,x+5\right )}^{1/6}-83860333479\right )-\frac {15\,{\left (4\,x+5\right )}^{1/3}}{16}-\frac {15\,{\left (4\,x+5\right )}^{2/3}}{32}+\frac {3\,{\left (4\,x+5\right )}^{1/6}}{2}+\frac {3\,{\left (4\,x+5\right )}^{4/3}}{64}+\frac {3\,{\left (4\,x+5\right )}^{5/3}}{80}+\frac {15\,{\left (4\,x+5\right )}^{5/6}}{32}+\frac {75\,{\left (4\,x+5\right )}^{7/6}}{224}-\frac {15\,{\left (4\,x+5\right )}^{11/6}}{176}-\frac {15\,{\left (4\,x+5\right )}^{13/6}}{208}+\frac {3\,{\left (4\,x+5\right )}^{17/6}}{544}+\frac {3\,{\left (4\,x+5\right )}^{19/6}}{608} \]
symsum(log(root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/924 8 - (225*z)/578 - 2439/39304, z, k)*(5031558*(4*x + 5)^(1/6) - root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/578 - 2439 /39304, z, k)*(root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2) /9248 - (225*z)/578 - 2439/39304, z, k)*(42123807*(4*x + 5)^(1/6) - root(z ^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/578 - 2439/39304, z, k)*(root(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863 *z^2)/9248 - (225*z)/578 - 2439/39304, z, k)*(92137824*(4*x + 5)^(1/6) - r oot(z^6 - 3*z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/ 578 - 2439/39304, z, k)*(89890560*(4*x + 5)^(1/6) + 95508720) + 79777872) + 37240965*(4*x + 5)^(1/6) + 52777656) + 37377288) + 8945559*(4*x + 5)^(1/ 6) + 13837149) + 2990358) + 1119744*(4*x + 5)^(1/6) + 874800)*root(z^6 - 3 *z^5 + (45*z^4)/68 + (957*z^3)/544 - (1863*z^2)/9248 - (225*z)/578 - 2439/ 39304, z, k), k, 1, 6) - 3*log(- 83860333479*(4*x + 5)^(1/6) - 83860333479 ) - (15*(4*x + 5)^(1/3))/16 - (15*(4*x + 5)^(2/3))/32 + (3*(4*x + 5)^(1/6) )/2 + (3*(4*x + 5)^(4/3))/64 + (3*(4*x + 5)^(5/3))/80 + (15*(4*x + 5)^(5/6 ))/32 + (75*(4*x + 5)^(7/6))/224 - (15*(4*x + 5)^(11/6))/176 - (15*(4*x + 5)^(13/6))/208 + (3*(4*x + 5)^(17/6))/544 + (3*(4*x + 5)^(19/6))/608