Integrand size = 27, antiderivative size = 648 \[ \int \frac {x^7 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=-\frac {214 c x^2 \sqrt {c+d x^3}}{91 d^2}-\frac {2 x^5 \sqrt {c+d x^3}}{13 d}-\frac {12248 c^2 \sqrt {c+d x^3}}{91 d^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {32 \sqrt {3} c^{13/6} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{8/3}}+\frac {32 c^{13/6} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{8/3}}-\frac {32 c^{13/6} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{8/3}}+\frac {6124 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{91 d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {12248 \sqrt {2} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{91 \sqrt [4]{3} d^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \]
32*c^(13/6)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/d^( 8/3)-32*c^(13/6)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/d^(8/3)-32*c^(13/6)* arctan(c^(1/6)*(c^(1/3)+d^(1/3)*x)*3^(1/2)/(d*x^3+c)^(1/2))*3^(1/2)/d^(8/3 )-214/91*c*x^2*(d*x^3+c)^(1/2)/d^2-2/13*x^5*(d*x^3+c)^(1/2)/d-12248/91*c^2 *(d*x^3+c)^(1/2)/d^(8/3)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))-12248/273*c^(7/3) *(c^(1/3)+d^(1/3)*x)*EllipticF((d^(1/3)*x+c^(1/3)*(1-3^(1/2)))/(d^(1/3)*x+ c^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*2^(1/2)*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^ (2/3)*x^2)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(3/4)/d^(8/3)/(d*x^3 +c)^(1/2)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))^2)^ (1/2)+6124/91*3^(1/4)*c^(7/3)*(c^(1/3)+d^(1/3)*x)*EllipticE((d^(1/3)*x+c^( 1/3)*(1-3^(1/2)))/(d^(1/3)*x+c^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1 /2)-1/2*2^(1/2))*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/(d^(1/3)*x+c^(1/ 3)*(1+3^(1/2)))^2)^(1/2)/d^(8/3)/(d*x^3+c)^(1/2)/(c^(1/3)*(c^(1/3)+d^(1/3) *x)/(d^(1/3)*x+c^(1/3)*(1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.23 \[ \int \frac {x^7 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\frac {-20 \left (107 c^2 x^2+114 c d x^5+7 d^2 x^8\right )+2140 c^2 x^2 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )+1531 c d x^5 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{910 d^2 \sqrt {c+d x^3}} \]
(-20*(107*c^2*x^2 + 114*c*d*x^5 + 7*d^2*x^8) + 2140*c^2*x^2*Sqrt[1 + (d*x^ 3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 1531*c*d*x ^5*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8 *c)])/(910*d^2*Sqrt[c + d*x^3])
Time = 1.10 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {978, 27, 1052, 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.
| \(\displaystyle \int \frac {x^7 \sqrt {c+d x^3}}{8 c-d x^3} \, dx\) |
| \(\Big \downarrow \) 978 |
| \(\displaystyle \frac {2 \int \frac {c x^4 \left (107 d x^3+80 c\right )}{2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{13 d}-\frac {2 x^5 \sqrt {c+d x^3}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \int \frac {x^4 \left (107 d x^3+80 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{13 d}-\frac {2 x^5 \sqrt {c+d x^3}}{13 d}\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle \frac {c \left (\frac {2 \int \frac {2 c d x \left (1531 d x^3+856 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{7 d^2}-\frac {214 x^2 \sqrt {c+d x^3}}{7 d}\right )}{13 d}-\frac {2 x^5 \sqrt {c+d x^3}}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {4 c \int \frac {x \left (1531 d x^3+856 c\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{7 d}-\frac {214 x^2 \sqrt {c+d x^3}}{7 d}\right )}{13 d}-\frac {2 x^5 \sqrt {c+d x^3}}{13 d}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {c \left (\frac {4 c \int \left (\frac {13104 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}-\frac {1531 x}{\sqrt {d x^3+c}}\right )dx}{7 d}-\frac {214 x^2 \sqrt {c+d x^3}}{7 d}\right )}{13 d}-\frac {2 x^5 \sqrt {c+d x^3}}{13 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c \left (\frac {4 c \left (-\frac {3062 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {1531 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {728 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{d^{2/3}}+\frac {728 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{d^{2/3}}-\frac {728 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^{2/3}}-\frac {3062 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{7 d}-\frac {214 x^2 \sqrt {c+d x^3}}{7 d}\right )}{13 d}-\frac {2 x^5 \sqrt {c+d x^3}}{13 d}\) |
(-2*x^5*Sqrt[c + d*x^3])/(13*d) + (c*((-214*x^2*Sqrt[c + d*x^3])/(7*d) + ( 4*c*((-3062*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (728*Sqrt[3]*c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqr t[c + d*x^3]])/d^(2/3) + (728*c^(1/6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c ^(1/6)*Sqrt[c + d*x^3])])/d^(2/3) - (728*c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/( 3*Sqrt[c])])/d^(2/3) + (1531*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]) - (3062*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]*E llipticF[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])))/(7* d)))/(13*d)
3.3.89.3.1 Defintions of rubi rules used
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^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Simp[e^n/(b*(m + n*(p + q) + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c , d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[m - n + 1, 0] && 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[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q + 1) + 1))), x] - Simp[g^n/(b*d*(m + n*(p + q + 1) + 1)) Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*( f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 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 = 4.91 (sec) , antiderivative size = 884, normalized size of antiderivative = 1.36
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(884\) |
| elliptic | \(\text {Expression too large to display}\) | \(889\) |
| default | \(\text {Expression too large to display}\) | \(1788\) |
-2/91*x^2*(7*d*x^3+107*c)*(d*x^3+c)^(1/2)/d^2-4/91/d^2*c^2*(-3062/3*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))*Elliptic E(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)*Ellipti cF(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)))+1456/3*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)*3^(1/2)*_alpha^2*d-I*(-c*d^2)^(2/3)*3^(1/2)*_...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 13.58 (sec) , antiderivative size = 2442, normalized size of antiderivative = 3.77 \[ \int \frac {x^7 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\text {Too large to display} \]
2/273*(728*d^3*(c^13/d^16)^(1/6)*log(33554432*((d^16*x^9 + 318*c*d^15*x^6 + 1200*c^2*d^14*x^3 + 640*c^3*d^13)*(c^13/d^16)^(5/6) + 6*(c^11*d^2*x^7 + 80*c^12*d*x^4 + 160*c^13*x + 6*(5*c^3*d^12*x^5 + 32*c^4*d^11*x^2)*(c^13/d^ 16)^(2/3) + (7*c^7*d^7*x^6 + 152*c^8*d^6*x^3 + 64*c^9*d^5)*(c^13/d^16)^(1/ 3))*sqrt(d*x^3 + c) + 18*(5*c^5*d^10*x^7 + 64*c^6*d^9*x^4 + 32*c^7*d^8*x)* sqrt(c^13/d^16) + 18*(c^9*d^5*x^8 + 38*c^10*d^4*x^5 + 64*c^11*d^3*x^2)*(c^ 13/d^16)^(1/6))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 728* d^3*(c^13/d^16)^(1/6)*log(-33554432*((d^16*x^9 + 318*c*d^15*x^6 + 1200*c^2 *d^14*x^3 + 640*c^3*d^13)*(c^13/d^16)^(5/6) - 6*(c^11*d^2*x^7 + 80*c^12*d* x^4 + 160*c^13*x + 6*(5*c^3*d^12*x^5 + 32*c^4*d^11*x^2)*(c^13/d^16)^(2/3) + (7*c^7*d^7*x^6 + 152*c^8*d^6*x^3 + 64*c^9*d^5)*(c^13/d^16)^(1/3))*sqrt(d *x^3 + c) + 18*(5*c^5*d^10*x^7 + 64*c^6*d^9*x^4 + 32*c^7*d^8*x)*sqrt(c^13/ d^16) + 18*(c^9*d^5*x^8 + 38*c^10*d^4*x^5 + 64*c^11*d^3*x^2)*(c^13/d^16)^( 1/6))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 18372*c^2*sqrt (d)*weierstrassZeta(0, -4*c/d, weierstrassPInverse(0, -4*c/d, x)) - 364*(s qrt(-3)*d^3 - d^3)*(c^13/d^16)^(1/6)*log(33554432*((d^16*x^9 + 318*c*d^15* x^6 + 1200*c^2*d^14*x^3 + 640*c^3*d^13 + sqrt(-3)*(d^16*x^9 + 318*c*d^15*x ^6 + 1200*c^2*d^14*x^3 + 640*c^3*d^13))*(c^13/d^16)^(5/6) + 6*(2*c^11*d^2* x^7 + 160*c^12*d*x^4 + 320*c^13*x - 6*(5*c^3*d^12*x^5 + 32*c^4*d^11*x^2 - sqrt(-3)*(5*c^3*d^12*x^5 + 32*c^4*d^11*x^2))*(c^13/d^16)^(2/3) - (7*c^7...
\[ \int \frac {x^7 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=- \int \frac {x^{7} \sqrt {c + d x^{3}}}{- 8 c + d x^{3}}\, dx \]
\[ \int \frac {x^7 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\int { -\frac {\sqrt {d x^{3} + c} x^{7}}{d x^{3} - 8 \, c} \,d x } \]
\[ \int \frac {x^7 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\int { -\frac {\sqrt {d x^{3} + c} x^{7}}{d x^{3} - 8 \, c} \,d x } \]
Timed out. \[ \int \frac {x^7 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\int \frac {x^7\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3} \,d x \]