Integrand size = 27, antiderivative size = 711 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=-\frac {17 \sqrt {c+d x^3}}{6048 c^3 x^7}+\frac {391 d \sqrt {c+d x^3}}{193536 c^4 x^4}-\frac {289 d^2 \sqrt {c+d x^3}}{48384 c^5 x}+\frac {289 d^{7/3} \sqrt {c+d x^3}}{48384 c^5 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}-\frac {17 d^{7/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{110592 \sqrt {3} c^{29/6}}+\frac {17 d^{7/3} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{331776 c^{29/6}}-\frac {17 d^{7/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{331776 c^{29/6}}-\frac {289 \sqrt {2-\sqrt {3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{32256\ 3^{3/4} c^{14/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {289 d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{24192 \sqrt {2} \sqrt [4]{3} c^{14/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-17/6048*(d*x^3+c)^(1/2)/c^3/x^7+391/193536*d*(d*x^3+c)^(1/2)/c^4/x^4-289/ 48384*d^2*(d*x^3+c)^(1/2)/c^5/x+289/48384*d^(7/3)*(d*x^3+c)^(1/2)/c^5/((1+ 3^(1/2))*c^(1/3)+d^(1/3)*x)+1/216*(d*x^3+c)^(1/2)/c^2/x^7/(-d*x^3+8*c)-17/ 331776*d^(7/3)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)*x)/(d*x^3+c)^(1/2)) *3^(1/2)/c^(29/6)+17/331776*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)-17/331776*d^(7/3)*arctanh(1/3*(d*x^3+c)^(1/2 )/c^(1/2))/c^(29/6)-289/96768*(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)*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^(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)+289/145152*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 = 10.22 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.30 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\sqrt {c+d x^3} \left (-\frac {1}{448 c^3 x^7}+\frac {15 d}{7168 c^4 x^4}-\frac {171 d^2}{28672 c^5 x}-\frac {d^3 x^2}{110592 c^5 \left (-8 c+d x^3\right )}\right )+\frac {9605 d^3 x^2 \sqrt {\frac {c+d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{6193152 c^5 \sqrt {c+d x^3}}-\frac {289 d^4 x^5 \sqrt {\frac {c+d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{3870720 c^6 \sqrt {c+d x^3}} \] Input:
Integrate[1/(x^8*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]
Output:
Sqrt[c + d*x^3]*(-1/448*1/(c^3*x^7) + (15*d)/(7168*c^4*x^4) - (171*d^2)/(2 8672*c^5*x) - (d^3*x^2)/(110592*c^5*(-8*c + d*x^3))) + (9605*d^3*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)])/(6 193152*c^5*Sqrt[c + d*x^3]) - (289*d^4*x^5*Sqrt[(c + d*x^3)/c]*AppellF1[5/ 3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(3870720*c^6*Sqrt[c + d*x^3] )
Time = 2.02 (sec) , antiderivative size = 728, 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.
| \(\displaystyle \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {\int \frac {17 d \left (d x^3+4 c\right )}{2 x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{216 c^2 d}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {17 \int \frac {d x^3+4 c}{x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {17 \left (-\frac {\int \frac {2 c d \left (46 c-11 d x^3\right )}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{56 c^2}-\frac {\sqrt {c+d x^3}}{14 c x^7}\right )}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {17 \left (-\frac {d \int \frac {46 c-11 d x^3}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{28 c}-\frac {\sqrt {c+d x^3}}{14 c x^7}\right )}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {17 \left (-\frac {d \left (-\frac {\int \frac {c d \left (1088 c-115 d x^3\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {\sqrt {c+d x^3}}{14 c x^7}\right )}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {17 \left (-\frac {d \left (-\frac {d \int \frac {1088 c-115 d x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {\sqrt {c+d x^3}}{14 c x^7}\right )}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {17 \left (-\frac {d \left (-\frac {d \left (-\frac {\int -\frac {8 c d x \left (565 c-68 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {136 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {\sqrt {c+d x^3}}{14 c x^7}\right )}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {17 \left (-\frac {d \left (-\frac {d \left (\frac {d \int \frac {x \left (565 c-68 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {136 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {\sqrt {c+d x^3}}{14 c x^7}\right )}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {17 \left (-\frac {d \left (-\frac {d \left (\frac {d \int \left (\frac {21 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {68 x}{\sqrt {d x^3+c}}\right )dx}{c}-\frac {136 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {\sqrt {c+d x^3}}{14 c x^7}\right )}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {17 \left (-\frac {d \left (-\frac {d \left (\frac {d \left (\frac {136 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {68 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {7 \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 \sqrt {3} d^{2/3}}+\frac {7 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{6 d^{2/3}}-\frac {7 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 d^{2/3}}+\frac {136 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{c}-\frac {136 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {\sqrt {c+d x^3}}{14 c x^7}\right )}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^7 \left (8 c-d x^3\right )}\) |
Input:
Int[1/(x^8*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]
Output:
Sqrt[c + d*x^3]/(216*c^2*x^7*(8*c - d*x^3)) + (17*(-1/14*Sqrt[c + d*x^3]/( c*x^7) - (d*((-23*Sqrt[c + d*x^3])/(16*c*x^4) - (d*((-136*Sqrt[c + d*x^3]) /(c*x) + (d*((136*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)) - (68*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]) + (136*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)))/(28*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[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 = 5.02 (sec) , antiderivative size = 938, normalized size of antiderivative = 1.32
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(938\) |
| risch | \(\text {Expression too large to display}\) | \(1781\) |
| default | \(\text {Expression too large to display}\) | \(2739\) |
Input:
int(1/x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/448*(d*x^3+c)^(1/2)/c^3/x^7+15/7168*d*(d*x^3+c)^(1/2)/c^4/x^4-171/28672 *d^2*(d*x^3+c)^(1/2)/c^5/x+1/110592*d^3*x^2/c^5*(d*x^3+c)^(1/2)/(-d*x^3+8* c)-289/145152*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))*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)) )-17/497664*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^...
Leaf count of result is larger than twice the leaf count of optimal. 2582 vs. \(2 (510) = 1020\).
Time = 4.44 (sec) , antiderivative size = 2582, normalized size of antiderivative = 3.63 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="fricas")
Output:
-1/27869184*(166464*(d^3*x^10 - 8*c*d^2*x^7)*sqrt(d)*weierstrassZeta(0, -4 *c/d, weierstrassPInverse(0, -4*c/d, x)) - 119*(c^5*d*x^10 - 8*c^6*x^7 + s qrt(-3)*(c^5*d*x^10 - 8*c^6*x^7))*(d^14/c^29)^(1/6)*log(1419857*(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 + 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 + 3 8*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)) + 119*(c^5*d*x^10 - 8*c^6*x^7 + sqrt(-3)*(c^5*d*x^10 - 8*c^6*x^7))*(d^14/c^29)^(1/6)*log(1419857*(d^14*x^9 + 318*c*d^13*x^6 + 120 0*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 + sqrt(-3)*(c^5*d^11*x^7 + 80*c^6*d^10*x^4 + 1...
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^{8} \left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \] Input:
integrate(1/x**8/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)
Output:
Integral(1/(x**8*(-8*c + d*x**3)**2*sqrt(c + d*x**3)), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}^{2} x^{8}} \,d x } \] Input:
integrate(1/x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2*x^8), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}^{2} x^{8}} \,d x } \] Input:
integrate(1/x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2*x^8), x)
Timed out. \[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^8\,\sqrt {d\,x^3+c}\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int(1/(x^8*(c + d*x^3)^(1/2)*(8*c - d*x^3)^2),x)
Output:
int(1/(x^8*(c + d*x^3)^(1/2)*(8*c - d*x^3)^2), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {-32 \sqrt {d \,x^{3}+c}\, c^{2}+34 \sqrt {d \,x^{3}+c}\, c d \,x^{3}-34 \sqrt {d \,x^{3}+c}\, d^{2} x^{6}+3536 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{3} x^{11}-15 c \,d^{2} x^{8}+48 c^{2} d \,x^{5}+64 c^{3} x^{2}}d x \right ) c^{3} d^{2} x^{7}-442 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{3} x^{11}-15 c \,d^{2} x^{8}+48 c^{2} d \,x^{5}+64 c^{3} x^{2}}d x \right ) c^{2} d^{3} x^{10}+680 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c \,d^{4} x^{7}-85 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) d^{5} x^{10}+680 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{2} d^{3} x^{7}-85 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c \,d^{4} x^{10}}{1792 c^{4} x^{7} \left (-d \,x^{3}+8 c \right )} \] Input:
int(1/x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x)
Output:
( - 32*sqrt(c + d*x**3)*c**2 + 34*sqrt(c + d*x**3)*c*d*x**3 - 34*sqrt(c + d*x**3)*d**2*x**6 + 3536*int(sqrt(c + d*x**3)/(64*c**3*x**2 + 48*c**2*d*x* *5 - 15*c*d**2*x**8 + d**3*x**11),x)*c**3*d**2*x**7 - 442*int(sqrt(c + d*x **3)/(64*c**3*x**2 + 48*c**2*d*x**5 - 15*c*d**2*x**8 + d**3*x**11),x)*c**2 *d**3*x**10 + 680*int((sqrt(c + d*x**3)*x**4)/(64*c**3 + 48*c**2*d*x**3 - 15*c*d**2*x**6 + d**3*x**9),x)*c*d**4*x**7 - 85*int((sqrt(c + d*x**3)*x**4 )/(64*c**3 + 48*c**2*d*x**3 - 15*c*d**2*x**6 + d**3*x**9),x)*d**5*x**10 + 680*int((sqrt(c + d*x**3)*x)/(64*c**3 + 48*c**2*d*x**3 - 15*c*d**2*x**6 + d**3*x**9),x)*c**2*d**3*x**7 - 85*int((sqrt(c + d*x**3)*x)/(64*c**3 + 48*c **2*d*x**3 - 15*c*d**2*x**6 + d**3*x**9),x)*c*d**4*x**10)/(1792*c**4*x**7* (8*c - d*x**3))