Integrand size = 25, antiderivative size = 141 \[ \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{6 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}} \] Output:
-1/18*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)*x)/(d*x^3+c)^(1/2))*3^(1/2)/ c^(5/6)/d^(2/3)+1/18*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^( 1/2))/c^(5/6)/d^(2/3)-1/18*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(5/6)/d^ (2/3)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.48 \[ \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {x^2 \sqrt {\frac {c+d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{16 c \sqrt {c+d x^3}} \] Input:
Integrate[x/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]
Output:
(x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/ (8*c)])/(16*c*Sqrt[c + d*x^3])
Time = 1.51 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, 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.
| \(\displaystyle \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx\) |
| \(\Big \downarrow \) 988 |
| \(\displaystyle -\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {d x^3+c}}dx}{12 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{d} \int \frac {x^2}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{4 \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 946 |
| \(\displaystyle -\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {d x^3+c}}dx}{12 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{d} \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{12 \sqrt [3]{c}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {d x^3+c}}dx}{12 c^{2/3} \sqrt [3]{d}}-\frac {\int \frac {1}{9 c-x^6}d\sqrt {d x^3+c}}{6 \sqrt [3]{c} d^{2/3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt {d x^3+c}}dx}{12 c^{2/3} \sqrt [3]{d}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\) |
| \(\Big \downarrow \) 2563 |
| \(\displaystyle \frac {\int \frac {1}{9-\frac {\left (\sqrt [3]{d} x+\sqrt [3]{c}\right )^4}{\sqrt [3]{c} \left (d x^3+c\right )}}d\frac {\left (\sqrt [3]{d} x+\sqrt [3]{c}\right )^2}{c^{2/3} \sqrt {d x^3+c}}}{6 \sqrt [3]{c} d^{2/3}}-\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\int \frac {-\frac {d^{4/3} x^2}{\sqrt [3]{c}}-2 d x+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^2}{c^{2/3}}+\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}+4\right ) \sqrt {d x^3+c}}dx}{12 c d}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\) |
| \(\Big \downarrow \) 2570 |
| \(\displaystyle \frac {d^{4/3} \int \frac {1}{-\frac {2 d^2}{c}-\frac {6 \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )^2 d^2}{c^{2/3} \left (d x^3+c\right )}}d\frac {\sqrt [3]{d} x+\sqrt [3]{c}}{\sqrt [3]{c} \sqrt {d x^3+c}}}{3 c^{4/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{6 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\) |
Input:
Int[x/((8*c - d*x^3)*Sqrt[c + d*x^3]),x]
Output:
-1/6*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]]/(Sqrt [3]*c^(5/6)*d^(2/3)) + ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])]/(18*c^(5/6)*d^(2/3)) - ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]/(18* c^(5/6)*d^(2/3))
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 = 1.07 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.95
| method | result | size |
| default | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {d \,x^{3}+c}}\right )}{27 d^{3} c}\) | \(416\) |
| elliptic | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {d \,x^{3}+c}}\right )}{27 d^{3} c}\) | \(416\) |
Input:
int(x/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/27*I/d^3/c*2^(1/2)*sum(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^ (1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c* d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d* (2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/ (d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/3 )+2*_alpha^2*d^2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^ (1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d /(-c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*(-c*d^2)^(1/3)*_alpha^2*3^(1/2)*d-I*(- c*d^2)^(2/3)*_alpha*3^(1/2)+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/c ,(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^ 2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d-8*c))
Leaf count of result is larger than twice the leaf count of optimal. 2285 vs. \(2 (95) = 190\).
Time = 0.49 (sec) , antiderivative size = 2285, normalized size of antiderivative = 16.21 \[ \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\text {Too large to display} \] Input:
integrate(x/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="fricas")
Output:
1/216*(sqrt(-3) + 1)*(1/(c^5*d^4))^(1/6)*log((d^3*x^9 + 318*c*d^2*x^6 + 12 00*c^2*d*x^3 + 640*c^3 - 9*(5*c^4*d^5*x^7 + 64*c^5*d^4*x^4 + 32*c^6*d^3*x + sqrt(-3)*(5*c^4*d^5*x^7 + 64*c^5*d^4*x^4 + 32*c^6*d^3*x))*(1/(c^5*d^4))^ (2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^5*d^5*x^5 + 32*c^6*d^4*x^2 - sqrt(-3)*(5 *c^5*d^5*x^5 + 32*c^6*d^4*x^2))*(1/(c^5*d^4))^(5/6) - 2*(7*c^3*d^4*x^6 + 1 52*c^4*d^3*x^3 + 64*c^5*d^2)*sqrt(1/(c^5*d^4)) + (c*d^3*x^7 + 80*c^2*d^2*x ^4 + 160*c^3*d*x + sqrt(-3)*(c*d^3*x^7 + 80*c^2*d^2*x^4 + 160*c^3*d*x))*(1 /(c^5*d^4))^(1/6)) - 9*(c^2*d^4*x^8 + 38*c^3*d^3*x^5 + 64*c^4*d^2*x^2 - sq rt(-3)*(c^2*d^4*x^8 + 38*c^3*d^3*x^5 + 64*c^4*d^2*x^2))*(1/(c^5*d^4))^(1/3 ))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 1/216*(sqrt(-3) + 1)*(1/(c^5*d^4))^(1/6)*log((d^3*x^9 + 318*c*d^2*x^6 + 1200*c^2*d*x^3 + 64 0*c^3 - 9*(5*c^4*d^5*x^7 + 64*c^5*d^4*x^4 + 32*c^6*d^3*x + sqrt(-3)*(5*c^4 *d^5*x^7 + 64*c^5*d^4*x^4 + 32*c^6*d^3*x))*(1/(c^5*d^4))^(2/3) - 3*sqrt(d* x^3 + c)*(6*(5*c^5*d^5*x^5 + 32*c^6*d^4*x^2 - sqrt(-3)*(5*c^5*d^5*x^5 + 32 *c^6*d^4*x^2))*(1/(c^5*d^4))^(5/6) - 2*(7*c^3*d^4*x^6 + 152*c^4*d^3*x^3 + 64*c^5*d^2)*sqrt(1/(c^5*d^4)) + (c*d^3*x^7 + 80*c^2*d^2*x^4 + 160*c^3*d*x + sqrt(-3)*(c*d^3*x^7 + 80*c^2*d^2*x^4 + 160*c^3*d*x))*(1/(c^5*d^4))^(1/6) ) - 9*(c^2*d^4*x^8 + 38*c^3*d^3*x^5 + 64*c^4*d^2*x^2 - sqrt(-3)*(c^2*d^4*x ^8 + 38*c^3*d^3*x^5 + 64*c^4*d^2*x^2))*(1/(c^5*d^4))^(1/3))/(d^3*x^9 - 24* c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 1/216*(sqrt(-3) - 1)*(1/(c^5*d^...
\[ \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=- \int \frac {x}{- 8 c \sqrt {c + d x^{3}} + d x^{3} \sqrt {c + d x^{3}}}\, dx \] Input:
integrate(x/(-d*x**3+8*c)/(d*x**3+c)**(1/2),x)
Output:
-Integral(x/(-8*c*sqrt(c + d*x**3) + d*x**3*sqrt(c + d*x**3)), x)
\[ \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\int { -\frac {x}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}} \,d x } \] Input:
integrate(x/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="maxima")
Output:
-integrate(x/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)), x)
\[ \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\int { -\frac {x}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}} \,d x } \] Input:
integrate(x/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="giac")
Output:
integrate(-x/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)), x)
Time = 40.04 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.93 \[ \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )\,{\left (\sqrt {d\,x^3+c}-\sqrt {c}+2\,c^{1/6}\,d^{1/3}\,x\right )}^3}{x^3\,{\left (d^{1/3}\,x-2\,c^{1/3}\right )}^3}\right )}{54\,c^{5/6}\,d^{2/3}}+\frac {\sqrt {2}\,\ln \left (\frac {\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )\,{\left (-\sqrt {3}\,c^{1/6}\,d^{1/3}\,x+\sqrt {d\,x^3+c}\,1{}\mathrm {i}+\sqrt {c}\,1{}\mathrm {i}+c^{1/6}\,d^{1/3}\,x\,1{}\mathrm {i}\right )}^3}{x^3\,{\left (d^{1/3}\,x+c^{1/3}-\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )}^3}\right )\,\sqrt {-1+\sqrt {3}\,1{}\mathrm {i}}}{108\,c^{5/6}\,d^{2/3}}+\frac {\sqrt {2}\,\ln \left (\frac {\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )\,{\left (\sqrt {3}\,c^{1/6}\,d^{1/3}\,x-\sqrt {d\,x^3+c}\,1{}\mathrm {i}+\sqrt {c}\,1{}\mathrm {i}+c^{1/6}\,d^{1/3}\,x\,1{}\mathrm {i}\right )}^3}{x^3\,{\left (d^{1/3}\,x+c^{1/3}+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )}^3}\right )\,\sqrt {1+\sqrt {3}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{108\,c^{5/6}\,d^{2/3}} \] Input:
int(x/((c + d*x^3)^(1/2)*(8*c - d*x^3)),x)
Output:
log((((c + d*x^3)^(1/2) + c^(1/2))*((c + d*x^3)^(1/2) - c^(1/2) + 2*c^(1/6 )*d^(1/3)*x)^3)/(x^3*(d^(1/3)*x - 2*c^(1/3))^3))/(54*c^(5/6)*d^(2/3)) + (2 ^(1/2)*log((((c + d*x^3)^(1/2) - c^(1/2))*((c + d*x^3)^(1/2)*1i + c^(1/2)* 1i + c^(1/6)*d^(1/3)*x*1i - 3^(1/2)*c^(1/6)*d^(1/3)*x)^3)/(x^3*(d^(1/3)*x - 3^(1/2)*c^(1/3)*1i + c^(1/3))^3))*(3^(1/2)*1i - 1)^(1/2))/(108*c^(5/6)*d ^(2/3)) + (2^(1/2)*log((((c + d*x^3)^(1/2) + c^(1/2))*(c^(1/2)*1i - (c + d *x^3)^(1/2)*1i + c^(1/6)*d^(1/3)*x*1i + 3^(1/2)*c^(1/6)*d^(1/3)*x)^3)/(x^3 *(3^(1/2)*c^(1/3)*1i + d^(1/3)*x + c^(1/3))^3))*(3^(1/2)*1i + 1)^(1/2)*1i) /(108*c^(5/6)*d^(2/3))
\[ \int \frac {x}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\int \frac {\sqrt {d \,x^{3}+c}\, x}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \] Input:
int(x/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x)
Output:
int((sqrt(c + d*x**3)*x)/(8*c**2 + 7*c*d*x**3 - d**2*x**6),x)