Integrand size = 27, antiderivative size = 678 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=-\frac {\sqrt {c+d x^3}}{56 c^2 x^7}+\frac {37 d \sqrt {c+d x^3}}{1792 c^3 x^4}-\frac {3 d^2 \sqrt {c+d x^3}}{56 c^4 x}+\frac {3 d^{7/3} \sqrt {c+d x^3}}{56 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {d^{7/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{3072 \sqrt {3} c^{23/6}}+\frac {d^{7/3} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{9216 c^{23/6}}-\frac {d^{7/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9216 c^{23/6}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{112 c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {3^{3/4} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{28 \sqrt {2} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-1/56*(d*x^3+c)^(1/2)/c^2/x^7+37/1792*d*(d*x^3+c)^(1/2)/c^3/x^4-3/56*d^2*( d*x^3+c)^(1/2)/c^4/x+3/56*d^(7/3)*(d*x^3+c)^(1/2)/c^4/((1+3^(1/2))*c^(1/3) +d^(1/3)*x)-1/9216*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^(23/6)+1/9216*d^(7/3)*arctanh(1/3*(c^(1/3)+d^(1/3)* x)^2/c^(1/6)/(d*x^3+c)^(1/2))/c^(23/6)-1/9216*d^(7/3)*arctanh(1/3*(d*x^3+c )^(1/2)/c^(1/2))/c^(23/6)-3/112*3^(1/4)*(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)/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)+1/56*3^(3/ 4)*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)/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 = 10.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {3875 c d^3 x^9 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-32 \left (5 c \left (32 c^3-5 c^2 d x^3+59 c d^2 x^6+96 d^3 x^9\right )+6 d^4 x^{12} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}{286720 c^5 x^7 \sqrt {c+d x^3}} \] Input:
Integrate[1/(x^8*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]
Output:
(3875*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*(32*c^3 - 5*c^2*d*x^3 + 59*c*d^2*x^6 + 96*d^3* x^9) + 6*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)]))/(286720*c^5*x^7*Sqrt[c + d*x^3])
Time = 1.85 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {980, 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.
| \(\displaystyle \int \frac {1}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx\) |
| \(\Big \downarrow \) 980 |
| \(\displaystyle \frac {\int -\frac {d \left (74 c-11 d x^3\right )}{2 x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{56 c^2}-\frac {\sqrt {c+d x^3}}{56 c^2 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {d \int \frac {74 c-11 d x^3}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{112 c^2}-\frac {\sqrt {c+d x^3}}{56 c^2 x^7}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle -\frac {d \left (-\frac {\int \frac {c d \left (1536 c-185 d x^3\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {37 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c^2}-\frac {\sqrt {c+d x^3}}{56 c^2 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {d \left (-\frac {d \int \frac {1536 c-185 d x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {37 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c^2}-\frac {\sqrt {c+d x^3}}{56 c^2 x^7}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle -\frac {d \left (-\frac {d \left (-\frac {\int -\frac {8 c d x \left (775 c-96 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {192 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {37 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c^2}-\frac {\sqrt {c+d x^3}}{56 c^2 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {d \left (-\frac {d \left (\frac {d \int \frac {x \left (775 c-96 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {192 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {37 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c^2}-\frac {\sqrt {c+d x^3}}{56 c^2 x^7}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle -\frac {d \left (-\frac {d \left (\frac {d \int \left (\frac {7 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {96 x}{\sqrt {d x^3+c}}\right )dx}{c}-\frac {192 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {37 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c^2}-\frac {\sqrt {c+d x^3}}{56 c^2 x^7}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d \left (-\frac {d \left (\frac {d \left (\frac {64 \sqrt {2} 3^{3/4} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {96 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {7 \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{6 \sqrt {3} d^{2/3}}+\frac {7 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{18 d^{2/3}}-\frac {7 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{18 d^{2/3}}+\frac {192 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{c}-\frac {192 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {37 \sqrt {c+d x^3}}{16 c x^4}\right )}{112 c^2}-\frac {\sqrt {c+d x^3}}{56 c^2 x^7}\) |
Input:
Int[1/(x^8*(8*c - d*x^3)*Sqrt[c + d*x^3]),x]
Output:
-1/56*Sqrt[c + d*x^3]/(c^2*x^7) - (d*((-37*Sqrt[c + d*x^3])/(16*c*x^4) - ( d*((-192*Sqrt[c + d*x^3])/(c*x) + (d*((192*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]])/(6*Sqrt[3]*d^(2/3)) + (7*c^(1/6)*ArcTa nh[(c^(1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/(18*d^(2/3)) - (7 *c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(18*d^(2/3)) - (96*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[Ar cSin[((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]) + (64*Sqrt[2]*3^(3/4)*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]] )/(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^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[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^n*(m + 1)) Int[(e*x)^(m + n)*( a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -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 = 3.36 (sec) , antiderivative size = 895, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(895\) |
| elliptic | \(\text {Expression too large to display}\) | \(906\) |
| default | \(\text {Expression too large to display}\) | \(1849\) |
Input:
int(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/1792*(d*x^3+c)^(1/2)*(96*d^2*x^6-37*c*d*x^3+32*c^2)/c^4/x^7+1/3584*d^3/ c^4*(-64*I*3^(1/2)/d*(-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)))-7/27*I/d^ 3*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...
Leaf count of result is larger than twice the leaf count of optimal. 2436 vs. \(2 (482) = 964\).
Time = 3.05 (sec) , antiderivative size = 2436, normalized size of antiderivative = 3.59 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="fricas")
Output:
1/774144*(14*c^4*x^7*(d^14/c^23)^(1/6)*log((d^14*x^9 + 318*c*d^13*x^6 + 12 00*c^2*d^12*x^3 + 640*c^3*d^11 + 18*(5*c^16*d^4*x^7 + 64*c^17*d^3*x^4 + 32 *c^18*d^2*x)*(d^14/c^23)^(2/3) + 6*sqrt(d*x^3 + c)*(6*(5*c^20*d*x^5 + 32*c ^21*x^2)*(d^14/c^23)^(5/6) + (7*c^12*d^6*x^6 + 152*c^13*d^5*x^3 + 64*c^14* d^4)*sqrt(d^14/c^23) + (c^4*d^11*x^7 + 80*c^5*d^10*x^4 + 160*c^6*d^9*x)*(d ^14/c^23)^(1/6)) + 18*(c^8*d^9*x^8 + 38*c^9*d^8*x^5 + 64*c^10*d^7*x^2)*(d^ 14/c^23)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 14*c ^4*x^7*(d^14/c^23)^(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 + 18*(5*c^16*d^4*x^7 + 64*c^17*d^3*x^4 + 32*c^18*d^2*x)*( d^14/c^23)^(2/3) - 6*sqrt(d*x^3 + c)*(6*(5*c^20*d*x^5 + 32*c^21*x^2)*(d^14 /c^23)^(5/6) + (7*c^12*d^6*x^6 + 152*c^13*d^5*x^3 + 64*c^14*d^4)*sqrt(d^14 /c^23) + (c^4*d^11*x^7 + 80*c^5*d^10*x^4 + 160*c^6*d^9*x)*(d^14/c^23)^(1/6 )) + 18*(c^8*d^9*x^8 + 38*c^9*d^8*x^5 + 64*c^10*d^7*x^2)*(d^14/c^23)^(1/3) )/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 41472*d^(5/2)*x^7* weierstrassZeta(0, -4*c/d, weierstrassPInverse(0, -4*c/d, x)) + 7*(sqrt(-3 )*c^4*x^7 + c^4*x^7)*(d^14/c^23)^(1/6)*log((d^14*x^9 + 318*c*d^13*x^6 + 12 00*c^2*d^12*x^3 + 640*c^3*d^11 - 9*(5*c^16*d^4*x^7 + 64*c^17*d^3*x^4 + 32* c^18*d^2*x + sqrt(-3)*(5*c^16*d^4*x^7 + 64*c^17*d^3*x^4 + 32*c^18*d^2*x))* (d^14/c^23)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^20*d*x^5 + 32*c^21*x^2 - sqr t(-3)*(5*c^20*d*x^5 + 32*c^21*x^2))*(d^14/c^23)^(5/6) - 2*(7*c^12*d^6*x...
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=- \int \frac {1}{- 8 c x^{8} \sqrt {c + d x^{3}} + d x^{11} \sqrt {c + d x^{3}}}\, dx \] Input:
integrate(1/x**8/(-d*x**3+8*c)/(d*x**3+c)**(1/2),x)
Output:
-Integral(1/(-8*c*x**8*sqrt(c + d*x**3) + d*x**11*sqrt(c + d*x**3)), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\int { -\frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )} x^{8}} \,d x } \] Input:
integrate(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="maxima")
Output:
-integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^8), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\int { -\frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )} x^{8}} \,d x } \] Input:
integrate(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x, algorithm="giac")
Output:
integrate(-1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*x^8), x)
Timed out. \[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^8\,\sqrt {d\,x^3+c}\,\left (8\,c-d\,x^3\right )} \,d x \] Input:
int(1/(x^8*(c + d*x^3)^(1/2)*(8*c - d*x^3)),x)
Output:
int(1/(x^8*(c + d*x^3)^(1/2)*(8*c - d*x^3)), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {-64 \sqrt {d \,x^{3}+c}\, c +74 \sqrt {d \,x^{3}+c}\, d \,x^{3}+1536 \left (\int \frac {\sqrt {d \,x^{3}+c}}{-d^{2} x^{8}+7 c d \,x^{5}+8 c^{2} x^{2}}d x \right ) c \,d^{2} x^{7}-185 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) d^{3} x^{7}}{3584 c^{3} x^{7}} \] Input:
int(1/x^8/(-d*x^3+8*c)/(d*x^3+c)^(1/2),x)
Output:
( - 64*sqrt(c + d*x**3)*c + 74*sqrt(c + d*x**3)*d*x**3 + 1536*int(sqrt(c + d*x**3)/(8*c**2*x**2 + 7*c*d*x**5 - d**2*x**8),x)*c*d**2*x**7 - 185*int(( sqrt(c + d*x**3)*x)/(8*c**2 + 7*c*d*x**3 - d**2*x**6),x)*d**3*x**7)/(3584* c**3*x**7)