Integrand size = 27, antiderivative size = 699 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}-\frac {139 \sqrt {c+d x^3}}{1512 c^3 x^7}+\frac {6095 d \sqrt {c+d x^3}}{48384 c^4 x^4}-\frac {953 d^2 \sqrt {c+d x^3}}{3024 c^5 x}+\frac {953 d^{7/3} \sqrt {c+d x^3}}{3024 c^5 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {d^{7/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{27648 \sqrt {3} c^{29/6}}+\frac {d^{7/3} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{82944 c^{29/6}}-\frac {d^{7/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{82944 c^{29/6}}-\frac {953 \sqrt {2-\sqrt {3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{2016\ 3^{3/4} c^{14/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {953 d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{1512 \sqrt {2} \sqrt [4]{3} c^{14/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
2/27/c^2/x^7/(d*x^3+c)^(1/2)-139/1512*(d*x^3+c)^(1/2)/c^3/x^7+6095/48384*d *(d*x^3+c)^(1/2)/c^4/x^4-953/3024*d^2*(d*x^3+c)^(1/2)/c^5/x+953/3024*d^(7/ 3)*(d*x^3+c)^(1/2)/c^5/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)-1/82944*d^(7/3)*arc tan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)*x)/(d*x^3+c)^(1/2))*3^(1/2)/c^(29/6)+ 1/82944*d^(7/3)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2)) /c^(29/6)-1/82944*d^(7/3)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(29/6)-95 3/6048*(1/2*6^(1/2)-1/2*2^(1/2))*d^(7/3)*(c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^( 1/3)*d^(1/3)*x+d^(2/3)*x^2)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)*Ellip ticE(((1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x),I*3^( 1/2)+2*I)*3^(1/4)/c^(14/3)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/ 3)+d^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)+953/9072*d^(7/3)*(c^(1/3)+d^(1/3)*x )*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x) ^2)^(1/2)*EllipticF(((1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d ^(1/3)*x),I*3^(1/2)+2*I)*2^(1/2)*3^(3/4)/c^(14/3)/(c^(1/3)*(c^(1/3)+d^(1/3 )*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.24 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\frac {610025 c d^3 x^9 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-32 \left (5 c \left (864 c^3-1647 c^2 d x^3+9153 c d^2 x^6+15248 d^3 x^9\right )+953 d^4 x^{12} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}{7741440 c^6 x^7 \sqrt {c+d x^3}} \] Input:
Integrate[1/(x^8*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]
Output:
(610025*c*d^3*x^9*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3) /c), (d*x^3)/(8*c)] - 32*(5*c*(864*c^3 - 1647*c^2*d*x^3 + 9153*c*d^2*x^6 + 15248*d^3*x^9) + 953*d^4*x^12*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8 /3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(7741440*c^6*x^7*Sqrt[c + d*x^3])
Time = 2.00 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {972, 27, 1053, 27, 1053, 27, 1053, 27, 1054, 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.
| \(\displaystyle \int \frac {1}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}-\frac {2 \int -\frac {d \left (139 c-17 d x^3\right )}{2 x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{27 c^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {139 c-17 d x^3}{x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{27 c^2}+\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {\int \frac {c d \left (12190 c-1529 d x^3\right )}{2 x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{56 c^2}-\frac {139 \sqrt {c+d x^3}}{56 c x^7}}{27 c^2}+\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {d \int \frac {12190 c-1529 d x^3}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{112 c}-\frac {139 \sqrt {c+d x^3}}{56 c x^7}}{27 c^2}+\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {d \left (-\frac {\int \frac {c d \left (243968 c-30475 d x^3\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {6095 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c}-\frac {139 \sqrt {c+d x^3}}{56 c x^7}}{27 c^2}+\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {d \left (-\frac {d \int \frac {243968 c-30475 d x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {6095 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c}-\frac {139 \sqrt {c+d x^3}}{56 c x^7}}{27 c^2}+\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {d \left (-\frac {d \left (-\frac {\int -\frac {8 c d x \left (122005 c-15248 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {30496 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {6095 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c}-\frac {139 \sqrt {c+d x^3}}{56 c x^7}}{27 c^2}+\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {d \left (-\frac {d \left (\frac {d \int \frac {x \left (122005 c-15248 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {30496 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {6095 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c}-\frac {139 \sqrt {c+d x^3}}{56 c x^7}}{27 c^2}+\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {-\frac {d \left (-\frac {d \left (\frac {d \int \left (\frac {21 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {15248 x}{\sqrt {d x^3+c}}\right )dx}{c}-\frac {30496 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {6095 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c}-\frac {139 \sqrt {c+d x^3}}{56 c x^7}}{27 c^2}+\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {d \left (-\frac {d \left (\frac {d \left (\frac {30496 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {15248 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {7 \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 \sqrt {3} d^{2/3}}+\frac {7 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{6 d^{2/3}}-\frac {7 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 d^{2/3}}+\frac {30496 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{c}-\frac {30496 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {6095 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c}-\frac {139 \sqrt {c+d x^3}}{56 c x^7}}{27 c^2}+\frac {2}{27 c^2 x^7 \sqrt {c+d x^3}}\) |
Input:
Int[1/(x^8*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]
Output:
2/(27*c^2*x^7*Sqrt[c + d*x^3]) + ((-139*Sqrt[c + d*x^3])/(56*c*x^7) - (d*( (-6095*Sqrt[c + d*x^3])/(16*c*x^4) - (d*((-30496*Sqrt[c + d*x^3])/(c*x) + (d*((30496*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (7*c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3 ]])/(2*Sqrt[3]*d^(2/3)) + (7*c^(1/6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c^ (1/6)*Sqrt[c + d*x^3])])/(6*d^(2/3)) - (7*c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/ (3*Sqrt[c])])/(6*d^(2/3)) - (15248*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(1/3)*(c^(1 /3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sq rt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*c^(1/3) + d ^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(d^(2/3)* Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2 ]*Sqrt[c + d*x^3]) + (30496*Sqrt[2]*c^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^ (2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)* x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])* c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*d^(2/3)*Sqrt[(c^(1/3)*(c^ (1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3] )))/c))/(32*c)))/(112*c))/(27*c^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.87 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(930\) |
| risch | \(\text {Expression too large to display}\) | \(1357\) |
| default | \(\text {Expression too large to display}\) | \(2389\) |
Input:
int(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/56*(d*x^3+c)^(1/2)/c^3/x^7+93/1792*d*(d*x^3+c)^(1/2)/c^4/x^4-27/112*d^2 *(d*x^3+c)^(1/2)/c^5/x-2/27*d^3/c^5*x^2/((x^3+c/d)*d)^(1/2)-953/9072*I*d^2 /c^5*3^(1/2)*(-c*d^2)^(1/3)*(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)*((x-1/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)*(-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) /(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*E llipticE(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),(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))+1/d*(-c*d^2)^(1/3)* EllipticF(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),(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)))-1/124416*I/c^5*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/...
Leaf count of result is larger than twice the leaf count of optimal. 2564 vs. \(2 (499) = 998\).
Time = 4.66 (sec) , antiderivative size = 2564, normalized size of antiderivative = 3.67 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x, algorithm="fricas")
Output:
-1/6967296*(2195712*(d^3*x^10 + c*d^2*x^7)*sqrt(d)*weierstrassZeta(0, -4*c /d, weierstrassPInverse(0, -4*c/d, x)) - 7*(c^5*d*x^10 + c^6*x^7 + sqrt(-3 )*(c^5*d*x^10 + c^6*x^7))*(d^14/c^29)^(1/6)*log((d^14*x^9 + 318*c*d^13*x^6 + 1200*c^2*d^12*x^3 + 640*c^3*d^11 - 9*(5*c^20*d^4*x^7 + 64*c^21*d^3*x^4 + 32*c^22*d^2*x + sqrt(-3)*(5*c^20*d^4*x^7 + 64*c^21*d^3*x^4 + 32*c^22*d^2 *x))*(d^14/c^29)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^25*d*x^5 + 32*c^26*x^2 - sqrt(-3)*(5*c^25*d*x^5 + 32*c^26*x^2))*(d^14/c^29)^(5/6) - 2*(7*c^15*d^6 *x^6 + 152*c^16*d^5*x^3 + 64*c^17*d^4)*sqrt(d^14/c^29) + (c^5*d^11*x^7 + 8 0*c^6*d^10*x^4 + 160*c^7*d^9*x + sqrt(-3)*(c^5*d^11*x^7 + 80*c^6*d^10*x^4 + 160*c^7*d^9*x))*(d^14/c^29)^(1/6)) - 9*(c^10*d^9*x^8 + 38*c^11*d^8*x^5 + 64*c^12*d^7*x^2 - sqrt(-3)*(c^10*d^9*x^8 + 38*c^11*d^8*x^5 + 64*c^12*d^7* x^2))*(d^14/c^29)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3 )) + 7*(c^5*d*x^10 + c^6*x^7 + sqrt(-3)*(c^5*d*x^10 + c^6*x^7))*(d^14/c^29 )^(1/6)*log((d^14*x^9 + 318*c*d^13*x^6 + 1200*c^2*d^12*x^3 + 640*c^3*d^11 - 9*(5*c^20*d^4*x^7 + 64*c^21*d^3*x^4 + 32*c^22*d^2*x + sqrt(-3)*(5*c^20*d ^4*x^7 + 64*c^21*d^3*x^4 + 32*c^22*d^2*x))*(d^14/c^29)^(2/3) - 3*sqrt(d*x^ 3 + c)*(6*(5*c^25*d*x^5 + 32*c^26*x^2 - sqrt(-3)*(5*c^25*d*x^5 + 32*c^26*x ^2))*(d^14/c^29)^(5/6) - 2*(7*c^15*d^6*x^6 + 152*c^16*d^5*x^3 + 64*c^17*d^ 4)*sqrt(d^14/c^29) + (c^5*d^11*x^7 + 80*c^6*d^10*x^4 + 160*c^7*d^9*x + sqr t(-3)*(c^5*d^11*x^7 + 80*c^6*d^10*x^4 + 160*c^7*d^9*x))*(d^14/c^29)^(1/...
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=- \int \frac {1}{- 8 c^{2} x^{8} \sqrt {c + d x^{3}} - 7 c d x^{11} \sqrt {c + d x^{3}} + d^{2} x^{14} \sqrt {c + d x^{3}}}\, dx \] Input:
integrate(1/x**8/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)
Output:
-Integral(1/(-8*c**2*x**8*sqrt(c + d*x**3) - 7*c*d*x**11*sqrt(c + d*x**3) + d**2*x**14*sqrt(c + d*x**3)), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\int { -\frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )} x^{8}} \,d x } \] Input:
integrate(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x, algorithm="maxima")
Output:
-integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x^8), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\int { -\frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )} x^{8}} \,d x } \] Input:
integrate(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x, algorithm="giac")
Output:
integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x^8), x)
Timed out. \[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^8\,{\left (d\,x^3+c\right )}^{3/2}\,\left (8\,c-d\,x^3\right )} \,d x \] Input:
int(1/(x^8*(c + d*x^3)^(3/2)*(8*c - d*x^3)),x)
Output:
int(1/(x^8*(c + d*x^3)^(3/2)*(8*c - d*x^3)), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\frac {-64 \sqrt {d \,x^{3}+c}\, c^{2}+122 \sqrt {d \,x^{3}+c}\, c d \,x^{3}-68 \sqrt {d \,x^{3}+c}\, d^{2} x^{6}+4880 \left (\int \frac {\sqrt {d \,x^{3}+c}}{-d^{3} x^{11}+6 c \,d^{2} x^{8}+15 c^{2} d \,x^{5}+8 c^{3} x^{2}}d x \right ) c^{3} d^{2} x^{7}+4880 \left (\int \frac {\sqrt {d \,x^{3}+c}}{-d^{3} x^{11}+6 c \,d^{2} x^{8}+15 c^{2} d \,x^{5}+8 c^{3} x^{2}}d x \right ) c^{2} d^{3} x^{10}+170 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{-d^{3} x^{9}+6 c \,d^{2} x^{6}+15 c^{2} d \,x^{3}+8 c^{3}}d x \right ) c \,d^{4} x^{7}+170 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{-d^{3} x^{9}+6 c \,d^{2} x^{6}+15 c^{2} d \,x^{3}+8 c^{3}}d x \right ) d^{5} x^{10}-1963 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{-d^{3} x^{9}+6 c \,d^{2} x^{6}+15 c^{2} d \,x^{3}+8 c^{3}}d x \right ) c^{2} d^{3} x^{7}-1963 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{-d^{3} x^{9}+6 c \,d^{2} x^{6}+15 c^{2} d \,x^{3}+8 c^{3}}d x \right ) c \,d^{4} x^{10}}{3584 c^{4} x^{7} \left (d \,x^{3}+c \right )} \] Input:
int(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x)
Output:
( - 64*sqrt(c + d*x**3)*c**2 + 122*sqrt(c + d*x**3)*c*d*x**3 - 68*sqrt(c + d*x**3)*d**2*x**6 + 4880*int(sqrt(c + d*x**3)/(8*c**3*x**2 + 15*c**2*d*x* *5 + 6*c*d**2*x**8 - d**3*x**11),x)*c**3*d**2*x**7 + 4880*int(sqrt(c + d*x **3)/(8*c**3*x**2 + 15*c**2*d*x**5 + 6*c*d**2*x**8 - d**3*x**11),x)*c**2*d **3*x**10 + 170*int((sqrt(c + d*x**3)*x**4)/(8*c**3 + 15*c**2*d*x**3 + 6*c *d**2*x**6 - d**3*x**9),x)*c*d**4*x**7 + 170*int((sqrt(c + d*x**3)*x**4)/( 8*c**3 + 15*c**2*d*x**3 + 6*c*d**2*x**6 - d**3*x**9),x)*d**5*x**10 - 1963* int((sqrt(c + d*x**3)*x)/(8*c**3 + 15*c**2*d*x**3 + 6*c*d**2*x**6 - d**3*x **9),x)*c**2*d**3*x**7 - 1963*int((sqrt(c + d*x**3)*x)/(8*c**3 + 15*c**2*d *x**3 + 6*c*d**2*x**6 - d**3*x**9),x)*c*d**4*x**10)/(3584*c**4*x**7*(c + d *x**3))