Integrand size = 23, antiderivative size = 103 \[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \left (1+\sqrt [3]{d} x\right )}{\sqrt {1+d x^3}}\right )}{6 \sqrt {3} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (1+\sqrt [3]{d} x\right )^2}{3 \sqrt {1+d x^3}}\right )}{18 d^{2/3}}-\frac {\text {arctanh}\left (\frac {1}{3} \sqrt {1+d x^3}\right )}{18 d^{2/3}} \]
1/18*arctanh(1/3*(1+d^(1/3)*x)^2/(d*x^3+1)^(1/2))/d^(2/3)-1/18*arctanh(1/3 *(d*x^3+1)^(1/2))/d^(2/3)-1/18*arctan((1+d^(1/3)*x)*3^(1/2)/(d*x^3+1)^(1/2 ))/d^(2/3)*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.31 \[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=\frac {1}{16} x^2 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-d x^3,\frac {d x^3}{8}\right ) \]
Time = 0.65 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {988, 946, 73, 219, 2563, 219, 2570, 218}
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.
| \(\displaystyle \int \frac {x}{\left (8-d x^3\right ) \sqrt {d x^3+1}} \, dx\) |
| \(\Big \downarrow \) 988 |
| \(\displaystyle -\frac {\int \frac {-d^{4/3} x^2-2 d x+2 d^{2/3}}{\left (d^{2/3} x^2+2 \sqrt [3]{d} x+4\right ) \sqrt {d x^3+1}}dx}{12 d}+\frac {\int \frac {\sqrt [3]{d} x+1}{\left (2-\sqrt [3]{d} x\right ) \sqrt {d x^3+1}}dx}{12 \sqrt [3]{d}}-\frac {1}{4} \sqrt [3]{d} \int \frac {x^2}{\left (8-d x^3\right ) \sqrt {d x^3+1}}dx\) |
| \(\Big \downarrow \) 946 |
| \(\displaystyle -\frac {\int \frac {-d^{4/3} x^2-2 d x+2 d^{2/3}}{\left (d^{2/3} x^2+2 \sqrt [3]{d} x+4\right ) \sqrt {d x^3+1}}dx}{12 d}+\frac {\int \frac {\sqrt [3]{d} x+1}{\left (2-\sqrt [3]{d} x\right ) \sqrt {d x^3+1}}dx}{12 \sqrt [3]{d}}-\frac {1}{12} \sqrt [3]{d} \int \frac {1}{\left (8-d x^3\right ) \sqrt {d x^3+1}}dx^3\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\int \frac {1}{9-x^6}d\sqrt {d x^3+1}}{6 d^{2/3}}-\frac {\int \frac {-d^{4/3} x^2-2 d x+2 d^{2/3}}{\left (d^{2/3} x^2+2 \sqrt [3]{d} x+4\right ) \sqrt {d x^3+1}}dx}{12 d}+\frac {\int \frac {\sqrt [3]{d} x+1}{\left (2-\sqrt [3]{d} x\right ) \sqrt {d x^3+1}}dx}{12 \sqrt [3]{d}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\int \frac {-d^{4/3} x^2-2 d x+2 d^{2/3}}{\left (d^{2/3} x^2+2 \sqrt [3]{d} x+4\right ) \sqrt {d x^3+1}}dx}{12 d}+\frac {\int \frac {\sqrt [3]{d} x+1}{\left (2-\sqrt [3]{d} x\right ) \sqrt {d x^3+1}}dx}{12 \sqrt [3]{d}}-\frac {\text {arctanh}\left (\frac {1}{3} \sqrt {d x^3+1}\right )}{18 d^{2/3}}\) |
| \(\Big \downarrow \) 2563 |
| \(\displaystyle \frac {\int \frac {1}{9-\frac {\left (\sqrt [3]{d} x+1\right )^4}{d x^3+1}}d\frac {\left (\sqrt [3]{d} x+1\right )^2}{\sqrt {d x^3+1}}}{6 d^{2/3}}-\frac {\int \frac {-d^{4/3} x^2-2 d x+2 d^{2/3}}{\left (d^{2/3} x^2+2 \sqrt [3]{d} x+4\right ) \sqrt {d x^3+1}}dx}{12 d}-\frac {\text {arctanh}\left (\frac {1}{3} \sqrt {d x^3+1}\right )}{18 d^{2/3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\int \frac {-d^{4/3} x^2-2 d x+2 d^{2/3}}{\left (d^{2/3} x^2+2 \sqrt [3]{d} x+4\right ) \sqrt {d x^3+1}}dx}{12 d}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{d} x+1\right )^2}{3 \sqrt {d x^3+1}}\right )}{18 d^{2/3}}-\frac {\text {arctanh}\left (\frac {1}{3} \sqrt {d x^3+1}\right )}{18 d^{2/3}}\) |
| \(\Big \downarrow \) 2570 |
| \(\displaystyle \frac {1}{3} d^{4/3} \int \frac {1}{-\frac {6 \left (\sqrt [3]{d} x+1\right )^2 d^2}{d x^3+1}-2 d^2}d\frac {\sqrt [3]{d} x+1}{\sqrt {d x^3+1}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{d} x+1\right )^2}{3 \sqrt {d x^3+1}}\right )}{18 d^{2/3}}-\frac {\text {arctanh}\left (\frac {1}{3} \sqrt {d x^3+1}\right )}{18 d^{2/3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt {3} \left (\sqrt [3]{d} x+1\right )}{\sqrt {d x^3+1}}\right )}{6 \sqrt {3} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{d} x+1\right )^2}{3 \sqrt {d x^3+1}}\right )}{18 d^{2/3}}-\frac {\text {arctanh}\left (\frac {1}{3} \sqrt {d x^3+1}\right )}{18 d^{2/3}}\) |
-1/6*ArcTan[(Sqrt[3]*(1 + d^(1/3)*x))/Sqrt[1 + d*x^3]]/(Sqrt[3]*d^(2/3)) + ArcTanh[(1 + d^(1/3)*x)^2/(3*Sqrt[1 + d*x^3])]/(18*d^(2/3)) - ArcTanh[Sqr t[1 + d*x^3]/3]/(18*d^(2/3))
3.1.76.3.1 Defintions of rubi rules used
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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n + 1, 0]
Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> Wi th[{q = Rt[d/c, 3]}, Simp[d*(q/(4*b)) Int[x^2/((8*c - d*x^3)*Sqrt[c + d*x ^3]), x], x] + (-Simp[q^2/(12*b) Int[(1 + q*x)/((2 - q*x)*Sqrt[c + d*x^3] ), x], x] + Simp[1/(12*b*c) Int[(2*c*q^2 - 2*d*x - d*q*x^2)/((4 + 2*q*x + q^2*x^2)*Sqrt[c + d*x^3]), x], x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[8*b*c + a*d, 0]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[-2*(e/d) Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & & EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)* Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Simp[-2*g*h Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /; Fre eQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8 *a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.09 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.72
| method | result | size |
| default | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-8\right )}{\sum }\frac {\left (-d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}+\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}+\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-d^{2}\right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -3 \left (-d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, d -3 d}{18 d}, \sqrt {\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {d \,x^{3}+1}}\right )}{27 d^{3}}\) | \(383\) |
| elliptic | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-8\right )}{\sum }\frac {\left (-d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}+\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}+\left (-d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-d^{2}\right )^{\frac {2}{3}}\right ) \Pi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha -3 \left (-d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, d -3 d}{18 d}, \sqrt {\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {d \,x^{3}+1}}\right )}{27 d^{3}}\) | \(383\) |
-1/27*I/d^3*2^(1/2)*sum(1/_alpha*(-d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2 )*(-d^2)^(1/3)+(-d^2)^(1/3)))/(-d^2)^(1/3))^(1/2)*(d*(x-1/d*(-d^2)^(1/3))/ (-3*(-d^2)^(1/3)+I*3^(1/2)*(-d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1 /2)*(-d^2)^(1/3)+(-d^2)^(1/3)))/(-d^2)^(1/3))^(1/2)/(d*x^3+1)^(1/2)*(I*(-d ^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-d^2)^(2/3)+2*_alpha^2*d^2-(-d^2)^(1 /3)*_alpha*d-(-d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-d^2)^(1/3) -1/2*I*3^(1/2)/d*(-d^2)^(1/3))*3^(1/2)*d/(-d^2)^(1/3))^(1/2),-1/18/d*(2*I* (-d^2)^(1/3)*3^(1/2)*_alpha^2*d-I*(-d^2)^(2/3)*3^(1/2)*_alpha-3*(-d^2)^(2/ 3)*_alpha+I*3^(1/2)*d-3*d),(I*3^(1/2)/d*(-d^2)^(1/3)/(-3/2/d*(-d^2)^(1/3)+ 1/2*I*3^(1/2)/d*(-d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d-8))
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (73) = 146\).
Time = 0.42 (sec) , antiderivative size = 497, normalized size of antiderivative = 4.83 \[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (-\frac {{\left (9 \, \sqrt {3} d^{3} x^{5} - \sqrt {3} {\left (d^{2} x^{6} - 40 \, d x^{3} - 32\right )} {\left (d^{2}\right )}^{\frac {2}{3}} + 3 \, \sqrt {3} {\left (5 \, d^{2} x^{4} + 8 \, d x\right )} {\left (d^{2}\right )}^{\frac {1}{3}}\right )} \sqrt {d x^{3} + 1} {\left (d^{2}\right )}^{\frac {1}{6}}}{9 \, {\left (d^{4} x^{7} - 7 \, d^{3} x^{4} - 8 \, d^{2} x\right )}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {d^{4} x^{9} + 318 \, d^{3} x^{6} + 1200 \, d^{2} x^{3} + 18 \, {\left (5 \, d^{2} x^{7} + 64 \, d x^{4} + 32 \, x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} + 6 \, {\left (7 \, d^{3} x^{6} + 152 \, d^{2} x^{3} + {\left (d^{2} x^{7} + 80 \, d x^{4} + 160 \, x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} + 6 \, {\left (5 \, d^{2} x^{5} + 32 \, d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 64 \, d\right )} \sqrt {d x^{3} + 1} + 18 \, {\left (d^{3} x^{8} + 38 \, d^{2} x^{5} + 64 \, d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 640 \, d}{d^{3} x^{9} - 24 \, d^{2} x^{6} + 192 \, d x^{3} - 512}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {d^{4} x^{9} - 276 \, d^{3} x^{6} - 1608 \, d^{2} x^{3} - 18 \, {\left (d^{2} x^{7} - 52 \, d x^{4} - 80 \, x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} - 6 \, {\left (4 \, d^{3} x^{6} + 164 \, d^{2} x^{3} + {\left (d^{2} x^{7} - 28 \, d x^{4} - 272 \, x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} - 24 \, {\left (d^{2} x^{5} + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 160 \, d\right )} \sqrt {d x^{3} + 1} + 18 \, {\left (d^{3} x^{8} + 20 \, d^{2} x^{5} - 8 \, d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - 1088 \, d}{d^{3} x^{9} - 24 \, d^{2} x^{6} + 192 \, d x^{3} - 512}\right )}{108 \, d^{2}} \]
1/108*(2*sqrt(3)*(d^2)^(1/6)*d*arctan(-1/9*(9*sqrt(3)*d^3*x^5 - sqrt(3)*(d ^2*x^6 - 40*d*x^3 - 32)*(d^2)^(2/3) + 3*sqrt(3)*(5*d^2*x^4 + 8*d*x)*(d^2)^ (1/3))*sqrt(d*x^3 + 1)*(d^2)^(1/6)/(d^4*x^7 - 7*d^3*x^4 - 8*d^2*x)) + 2*(d ^2)^(2/3)*log((d^4*x^9 + 318*d^3*x^6 + 1200*d^2*x^3 + 18*(5*d^2*x^7 + 64*d *x^4 + 32*x)*(d^2)^(2/3) + 6*(7*d^3*x^6 + 152*d^2*x^3 + (d^2*x^7 + 80*d*x^ 4 + 160*x)*(d^2)^(2/3) + 6*(5*d^2*x^5 + 32*d*x^2)*(d^2)^(1/3) + 64*d)*sqrt (d*x^3 + 1) + 18*(d^3*x^8 + 38*d^2*x^5 + 64*d*x^2)*(d^2)^(1/3) + 640*d)/(d ^3*x^9 - 24*d^2*x^6 + 192*d*x^3 - 512)) - (d^2)^(2/3)*log((d^4*x^9 - 276*d ^3*x^6 - 1608*d^2*x^3 - 18*(d^2*x^7 - 52*d*x^4 - 80*x)*(d^2)^(2/3) - 6*(4* d^3*x^6 + 164*d^2*x^3 + (d^2*x^7 - 28*d*x^4 - 272*x)*(d^2)^(2/3) - 24*(d^2 *x^5 + d*x^2)*(d^2)^(1/3) + 160*d)*sqrt(d*x^3 + 1) + 18*(d^3*x^8 + 20*d^2* x^5 - 8*d*x^2)*(d^2)^(1/3) - 1088*d)/(d^3*x^9 - 24*d^2*x^6 + 192*d*x^3 - 5 12)))/d^2
\[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=- \int \frac {x}{d x^{3} \sqrt {d x^{3} + 1} - 8 \sqrt {d x^{3} + 1}}\, dx \]
\[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=\int { -\frac {x}{\sqrt {d x^{3} + 1} {\left (d x^{3} - 8\right )}} \,d x } \]
\[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=\int { -\frac {x}{\sqrt {d x^{3} + 1} {\left (d x^{3} - 8\right )}} \,d x } \]
Timed out. \[ \int \frac {x}{\left (8-d x^3\right ) \sqrt {1+d x^3}} \, dx=-\int \frac {x}{\sqrt {d\,x^3+1}\,\left (d\,x^3-8\right )} \,d x \]