Integrand size = 27, antiderivative size = 686 \[ \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {5}{648 c^3 x \sqrt {c+d x^3}}+\frac {1}{216 c^2 x \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {31 \sqrt {c+d x^3}}{1296 c^4 x}+\frac {31 \sqrt [3]{d} \sqrt {c+d x^3}}{1296 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [3]{d} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{1296 \sqrt {3} c^{23/6}}+\frac {\sqrt [3]{d} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{3888 c^{23/6}}-\frac {\sqrt [3]{d} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{3888 c^{23/6}}-\frac {31 \sqrt {2-\sqrt {3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{864\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {31 \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{648 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
5/648/c^3/x/(d*x^3+c)^(1/2)+1/216/c^2/x/(-d*x^3+8*c)/(d*x^3+c)^(1/2)-31/12 96*(d*x^3+c)^(1/2)/c^4/x+31/1296*d^(1/3)*(d*x^3+c)^(1/2)/c^4/((1+3^(1/2))* c^(1/3)+d^(1/3)*x)-1/3888*d^(1/3)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)* x)/(d*x^3+c)^(1/2))*3^(1/2)/c^(23/6)+1/3888*d^(1/3)*arctanh(1/3*(c^(1/3)+d ^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/c^(23/6)-1/3888*d^(1/3)*arctanh(1/3*( d*x^3+c)^(1/2)/c^(1/2))/c^(23/6)-31/2592*(1/2*6^(1/2)-1/2*2^(1/2))*d^(1/3) *(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)*EllipticE(((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^(11/3)/(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)+3 1/3888*d^(1/3)*(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^(11/3)/(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 = 11.17 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {-80 c \left (162 c^2+227 c d x^3-31 d^2 x^6\right )+650 c d x^3 \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+31 d^2 x^6 \left (-8 c+d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{103680 c^5 \sqrt {c+d x^3} \left (8 c x-d x^4\right )} \] Input:
Integrate[1/(x^2*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
(-80*c*(162*c^2 + 227*c*d*x^3 - 31*d^2*x^6) + 650*c*d*x^3*(8*c - d*x^3)*Sq rt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 31*d^2*x^6*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(103680*c^5*Sqrt[c + d*x^3]*(8*c*x - d*x^4) )
Time = 1.91 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {972, 27, 1049, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {\int \frac {d \left (11 d x^3+56 c\right )}{2 x^2 \left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx}{216 c^2 d}+\frac {1}{216 c^2 x \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {11 d x^3+56 c}{x^2 \left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx}{432 c^2}+\frac {1}{216 c^2 x \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle \frac {\frac {10}{3 c x \sqrt {c+d x^3}}-\frac {2 \int -\frac {9 c d \left (248 c-25 d x^3\right )}{2 x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{27 c^2 d}}{432 c^2}+\frac {1}{216 c^2 x \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {248 c-25 d x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{3 c}+\frac {10}{3 c x \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {4 c d x \left (260 c-31 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {31 \sqrt {c+d x^3}}{c x}}{3 c}+\frac {10}{3 c x \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {d \int \frac {x \left (260 c-31 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{2 c}-\frac {31 \sqrt {c+d x^3}}{c x}}{3 c}+\frac {10}{3 c x \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {\frac {\frac {d \int \left (\frac {12 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {31 x}{\sqrt {d x^3+c}}\right )dx}{2 c}-\frac {31 \sqrt {c+d x^3}}{c x}}{3 c}+\frac {10}{3 c x \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {d \left (\frac {62 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {31 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {2 \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{\sqrt {3} d^{2/3}}+\frac {2 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{3 d^{2/3}}-\frac {2 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{3 d^{2/3}}+\frac {62 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{2 c}-\frac {31 \sqrt {c+d x^3}}{c x}}{3 c}+\frac {10}{3 c x \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
Input:
Int[1/(x^2*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
1/(216*c^2*x*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (10/(3*c*x*Sqrt[c + d*x^3]) + ((-31*Sqrt[c + d*x^3])/(c*x) + (d*((62*Sqrt[c + d*x^3])/(d^(2/3)*((1 + S qrt[3])*c^(1/3) + d^(1/3)*x)) - (2*c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3 ) + d^(1/3)*x))/Sqrt[c + d*x^3]])/(Sqrt[3]*d^(2/3)) + (2*c^(1/6)*ArcTanh[( c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/(3*d^(2/3)) - (2*c^(1 /6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(3*d^(2/3)) - (31*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 + Sqrt[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]) + (62*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])))/(2*c))/(3*c))/(432*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[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
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 = 3.93 (sec) , antiderivative size = 920, normalized size of antiderivative = 1.34
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(920\) |
| risch | \(\text {Expression too large to display}\) | \(2220\) |
| default | \(\text {Expression too large to display}\) | \(2270\) |
Input:
int(1/x^2/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/243*d*x^2/c^4/((x^3+c/d)*d)^(1/2)+1/15552*d*x^2/c^4*(d*x^3+c)^(1/2)/(-d *x^3+8*c)-1/64*(d*x^3+c)^(1/2)/c^4/x-31/3888*I/c^4*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))*EllipticE(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)))-1/5832*I/c^4/d^2*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...
Leaf count of result is larger than twice the leaf count of optimal. 2534 vs. \(2 (489) = 978\).
Time = 0.98 (sec) , antiderivative size = 2534, normalized size of antiderivative = 3.69 \[ \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="fricas")
Output:
-1/46656*(1116*(d^2*x^7 - 7*c*d*x^4 - 8*c^2*x)*sqrt(d)*weierstrassZeta(0, -4*c/d, weierstrassPInverse(0, -4*c/d, x)) - (c^4*d^2*x^7 - 7*c^5*d*x^4 - 8*c^6*x + sqrt(-3)*(c^4*d^2*x^7 - 7*c^5*d*x^4 - 8*c^6*x))*(d^2/c^23)^(1/6) *log((d^4*x^9 + 318*c*d^3*x^6 + 1200*c^2*d^2*x^3 + 640*c^3*d - 9*(5*c^16*d ^2*x^7 + 64*c^17*d*x^4 + 32*c^18*x + sqrt(-3)*(5*c^16*d^2*x^7 + 64*c^17*d* x^4 + 32*c^18*x))*(d^2/c^23)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^20*d*x^5 + 32*c^21*x^2 - sqrt(-3)*(5*c^20*d*x^5 + 32*c^21*x^2))*(d^2/c^23)^(5/6) - 2* (7*c^12*d^2*x^6 + 152*c^13*d*x^3 + 64*c^14)*sqrt(d^2/c^23) + (c^4*d^3*x^7 + 80*c^5*d^2*x^4 + 160*c^6*d*x + sqrt(-3)*(c^4*d^3*x^7 + 80*c^5*d^2*x^4 + 160*c^6*d*x))*(d^2/c^23)^(1/6)) - 9*(c^8*d^3*x^8 + 38*c^9*d^2*x^5 + 64*c^1 0*d*x^2 - sqrt(-3)*(c^8*d^3*x^8 + 38*c^9*d^2*x^5 + 64*c^10*d*x^2))*(d^2/c^ 23)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + (c^4*d^2* x^7 - 7*c^5*d*x^4 - 8*c^6*x + sqrt(-3)*(c^4*d^2*x^7 - 7*c^5*d*x^4 - 8*c^6* x))*(d^2/c^23)^(1/6)*log((d^4*x^9 + 318*c*d^3*x^6 + 1200*c^2*d^2*x^3 + 640 *c^3*d - 9*(5*c^16*d^2*x^7 + 64*c^17*d*x^4 + 32*c^18*x + sqrt(-3)*(5*c^16* d^2*x^7 + 64*c^17*d*x^4 + 32*c^18*x))*(d^2/c^23)^(2/3) - 3*sqrt(d*x^3 + c) *(6*(5*c^20*d*x^5 + 32*c^21*x^2 - sqrt(-3)*(5*c^20*d*x^5 + 32*c^21*x^2))*( d^2/c^23)^(5/6) - 2*(7*c^12*d^2*x^6 + 152*c^13*d*x^3 + 64*c^14)*sqrt(d^2/c ^23) + (c^4*d^3*x^7 + 80*c^5*d^2*x^4 + 160*c^6*d*x + sqrt(-3)*(c^4*d^3*x^7 + 80*c^5*d^2*x^4 + 160*c^6*d*x))*(d^2/c^23)^(1/6)) - 9*(c^8*d^3*x^8 + ...
\[ \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**2/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)
Output:
Integral(1/(x**2*(-8*c + d*x**3)**2*(c + d*x**3)**(3/2)), x)
\[ \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^2), x)
\[ \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{2}} \,d x } \] Input:
integrate(1/x^2/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="giac")
Output:
integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (d\,x^3+c\right )}^{3/2}\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int(1/(x^2*(c + d*x^3)^(3/2)*(8*c - d*x^3)^2),x)
Output:
int(1/(x^2*(c + d*x^3)^(3/2)*(8*c - d*x^3)^2), x)
\[ \int \frac {1}{x^2 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {-2 \sqrt {d \,x^{3}+c}+88 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{2} x +77 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{3} x^{4}-11 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) d^{4} x^{7}-256 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{3} d x -224 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{2} x^{4}+32 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{3} x^{7}}{16 c^{2} x \left (-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}\right )} \] Input:
int(1/x^2/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x)
Output:
( - 2*sqrt(c + d*x**3) + 88*int((sqrt(c + d*x**3)*x**4)/(64*c**4 + 112*c** 3*d*x**3 + 33*c**2*d**2*x**6 - 14*c*d**3*x**9 + d**4*x**12),x)*c**2*d**2*x + 77*int((sqrt(c + d*x**3)*x**4)/(64*c**4 + 112*c**3*d*x**3 + 33*c**2*d** 2*x**6 - 14*c*d**3*x**9 + d**4*x**12),x)*c*d**3*x**4 - 11*int((sqrt(c + d* x**3)*x**4)/(64*c**4 + 112*c**3*d*x**3 + 33*c**2*d**2*x**6 - 14*c*d**3*x** 9 + d**4*x**12),x)*d**4*x**7 - 256*int((sqrt(c + d*x**3)*x)/(64*c**4 + 112 *c**3*d*x**3 + 33*c**2*d**2*x**6 - 14*c*d**3*x**9 + d**4*x**12),x)*c**3*d* x - 224*int((sqrt(c + d*x**3)*x)/(64*c**4 + 112*c**3*d*x**3 + 33*c**2*d**2 *x**6 - 14*c*d**3*x**9 + d**4*x**12),x)*c**2*d**2*x**4 + 32*int((sqrt(c + d*x**3)*x)/(64*c**4 + 112*c**3*d*x**3 + 33*c**2*d**2*x**6 - 14*c*d**3*x**9 + d**4*x**12),x)*c*d**3*x**7)/(16*c**2*x*(8*c**2 + 7*c*d*x**3 - d**2*x**6 ))