Integrand size = 27, antiderivative size = 164 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=-\frac {35 d^2 \sqrt {c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac {\sqrt {c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac {3 d \sqrt {c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac {31 d^2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{165888 c^{9/2}}-\frac {19 d^2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6144 c^{9/2}} \]
31/165888*d^2*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(9/2)-19/6144*d^2*arc tanh((d*x^3+c)^(1/2)/c^(1/2))/c^(9/2)-35/13824*d^2*(d*x^3+c)^(1/2)/c^4/(-d *x^3+8*c)-1/48*(d*x^3+c)^(1/2)/c^2/x^6/(-d*x^3+8*c)+3/128*d*(d*x^3+c)^(1/2 )/c^3/x^3/(-d*x^3+8*c)
Time = 0.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {\frac {12 \sqrt {c} \sqrt {c+d x^3} \left (288 c^2-324 c d x^3+35 d^2 x^6\right )}{-8 c x^6+d x^9}+31 d^2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-513 d^2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{165888 c^{9/2}} \]
((12*Sqrt[c]*Sqrt[c + d*x^3]*(288*c^2 - 324*c*d*x^3 + 35*d^2*x^6))/(-8*c*x ^6 + d*x^9) + 31*d^2*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] - 513*d^2*ArcTan h[Sqrt[c + d*x^3]/Sqrt[c]])/(165888*c^(9/2))
Time = 0.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {948, 114, 27, 168, 27, 168, 27, 174, 73, 219, 221}
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.
| \(\displaystyle \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{x^9 \left (8 c-d x^3\right )^2 \sqrt {d x^3+c}}dx^3\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {d \left (18 c-5 d x^3\right )}{2 x^6 \left (8 c-d x^3\right )^2 \sqrt {d x^3+c}}dx^3}{16 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {d \int \frac {18 c-5 d x^3}{x^6 \left (8 c-d x^3\right )^2 \sqrt {d x^3+c}}dx^3}{32 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {\int \frac {c d \left (76 c-27 d x^3\right )}{x^3 \left (8 c-d x^3\right )^2 \sqrt {d x^3+c}}dx^3}{8 c^2}-\frac {9 \sqrt {c+d x^3}}{4 c x^3 \left (8 c-d x^3\right )}\right )}{32 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \int \frac {76 c-27 d x^3}{x^3 \left (8 c-d x^3\right )^2 \sqrt {d x^3+c}}dx^3}{8 c}-\frac {9 \sqrt {c+d x^3}}{4 c x^3 \left (8 c-d x^3\right )}\right )}{32 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (-\frac {\int -\frac {2 c d \left (342 c-35 d x^3\right )}{x^3 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{72 c^2 d}-\frac {35 \sqrt {c+d x^3}}{18 c \left (8 c-d x^3\right )}\right )}{8 c}-\frac {9 \sqrt {c+d x^3}}{4 c x^3 \left (8 c-d x^3\right )}\right )}{32 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\int \frac {342 c-35 d x^3}{x^3 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{36 c}-\frac {35 \sqrt {c+d x^3}}{18 c \left (8 c-d x^3\right )}\right )}{8 c}-\frac {9 \sqrt {c+d x^3}}{4 c x^3 \left (8 c-d x^3\right )}\right )}{32 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {171}{4} \int \frac {1}{x^3 \sqrt {d x^3+c}}dx^3+\frac {31}{4} d \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{36 c}-\frac {35 \sqrt {c+d x^3}}{18 c \left (8 c-d x^3\right )}\right )}{8 c}-\frac {9 \sqrt {c+d x^3}}{4 c x^3 \left (8 c-d x^3\right )}\right )}{32 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {31}{2} \int \frac {1}{9 c-x^6}d\sqrt {d x^3+c}+\frac {171 \int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{2 d}}{36 c}-\frac {35 \sqrt {c+d x^3}}{18 c \left (8 c-d x^3\right )}\right )}{8 c}-\frac {9 \sqrt {c+d x^3}}{4 c x^3 \left (8 c-d x^3\right )}\right )}{32 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {171 \int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{2 d}+\frac {31 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 \sqrt {c}}}{36 c}-\frac {35 \sqrt {c+d x^3}}{18 c \left (8 c-d x^3\right )}\right )}{8 c}-\frac {9 \sqrt {c+d x^3}}{4 c x^3 \left (8 c-d x^3\right )}\right )}{32 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{3} \left (-\frac {d \left (-\frac {d \left (\frac {\frac {31 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 \sqrt {c}}-\frac {171 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{36 c}-\frac {35 \sqrt {c+d x^3}}{18 c \left (8 c-d x^3\right )}\right )}{8 c}-\frac {9 \sqrt {c+d x^3}}{4 c x^3 \left (8 c-d x^3\right )}\right )}{32 c^2}-\frac {\sqrt {c+d x^3}}{16 c^2 x^6 \left (8 c-d x^3\right )}\right )\) |
(-1/16*Sqrt[c + d*x^3]/(c^2*x^6*(8*c - d*x^3)) - (d*((-9*Sqrt[c + d*x^3])/ (4*c*x^3*(8*c - d*x^3)) - (d*((-35*Sqrt[c + d*x^3])/(18*c*(8*c - d*x^3)) + ((31*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(6*Sqrt[c]) - (171*ArcTanh[Sqr t[c + d*x^3]/Sqrt[c]])/(2*Sqrt[c]))/(36*c)))/(8*c)))/(32*c^2))/3
3.5.30.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 4.64 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.72
| method | result | size |
| pseudoelliptic | \(-\frac {4104 \left (-\frac {31 \left (c -\frac {d \,x^{3}}{8}\right ) d^{2} x^{6} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{513}+\left (c -\frac {d \,x^{3}}{8}\right ) d^{2} x^{6} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )+\frac {35 \left (d^{2} x^{6} \sqrt {c}-\frac {324 d \,x^{3} c^{\frac {3}{2}}}{35}+\frac {288 c^{\frac {5}{2}}}{35}\right ) \sqrt {d \,x^{3}+c}}{342}\right )}{c^{\frac {9}{2}} \left (-165888 d \,x^{9}+1327104 c \,x^{6}\right )}\) | \(118\) |
| risch | \(-\frac {\sqrt {d \,x^{3}+c}\, \left (-d \,x^{3}+c \right )}{384 c^{4} x^{6}}+\frac {d^{2} \left (-\frac {19 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{24 \sqrt {c}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{24 \sqrt {c}}+\frac {c \left (-\frac {\sqrt {d \,x^{3}+c}}{c \left (d \,x^{3}-8 c \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{54}\right )}{256 c^{4}}\) | \(124\) |
| default | \(\frac {-\frac {\sqrt {d \,x^{3}+c}}{6 c \,x^{6}}+\frac {d \sqrt {d \,x^{3}+c}}{4 c^{2} x^{3}}-\frac {d^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{4 c^{\frac {5}{2}}}}{64 c^{2}}+\frac {d \left (-\frac {\sqrt {d \,x^{3}+c}}{3 c \,x^{3}}+\frac {d \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{256 c^{3}}-\frac {d^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{2048 c^{\frac {9}{2}}}+\frac {d^{2} \left (\frac {\sqrt {d \,x^{3}+c}}{c \left (-d \,x^{3}+8 c \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{13824 c^{3}}+\frac {d^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{6144 c^{\frac {9}{2}}}\) | \(208\) |
| elliptic | \(\text {Expression too large to display}\) | \(1580\) |
-4104*(-31/513*(c-1/8*d*x^3)*d^2*x^6*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))+ (c-1/8*d*x^3)*d^2*x^6*arctanh((d*x^3+c)^(1/2)/c^(1/2))+35/342*(d^2*x^6*c^( 1/2)-324/35*d*x^3*c^(3/2)+288/35*c^(5/2))*(d*x^3+c)^(1/2))/c^(9/2)/(-16588 8*d*x^9+1327104*c*x^6)
Time = 0.31 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.89 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\left [\frac {31 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 513 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right ) + 24 \, {\left (35 \, c d^{2} x^{6} - 324 \, c^{2} d x^{3} + 288 \, c^{3}\right )} \sqrt {d x^{3} + c}}{331776 \, {\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )}}, \frac {513 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{c}\right ) - 31 \, {\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 12 \, {\left (35 \, c d^{2} x^{6} - 324 \, c^{2} d x^{3} + 288 \, c^{3}\right )} \sqrt {d x^{3} + c}}{165888 \, {\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )}}\right ] \]
[1/331776*(31*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(c)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) + 513*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(c)*lo g((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 24*(35*c*d^2*x^6 - 324* c^2*d*x^3 + 288*c^3)*sqrt(d*x^3 + c))/(c^5*d*x^9 - 8*c^6*x^6), 1/165888*(5 13*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(-c)*arctan(sqrt(d*x^3 + c)*sqrt(-c)/c) - 3 1*(d^3*x^9 - 8*c*d^2*x^6)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)*sqrt(-c)/c) + 12*(35*c*d^2*x^6 - 324*c^2*d*x^3 + 288*c^3)*sqrt(d*x^3 + c))/(c^5*d*x^9 - 8*c^6*x^6)]
\[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^{7} \left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \]
\[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}^{2} x^{7}} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {19 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{6144 \, \sqrt {-c} c^{4}} - \frac {31 \, d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{165888 \, \sqrt {-c} c^{4}} - \frac {\sqrt {d x^{3} + c} d^{2}}{13824 \, {\left (d x^{3} - 8 \, c\right )} c^{4}} + \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{2} - 2 \, \sqrt {d x^{3} + c} c d^{2}}{384 \, c^{4} d^{2} x^{6}} \]
19/6144*d^2*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^4) - 31/165888*d^ 2*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c^4) - 1/13824*sqrt(d*x^3 + c)*d^2/((d*x^3 - 8*c)*c^4) + 1/384*((d*x^3 + c)^(3/2)*d^2 - 2*sqrt(d*x^ 3 + c)*c*d^2)/(c^4*d^2*x^6)
Time = 8.43 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^7 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=-\frac {\frac {647\,d^2\,\sqrt {d\,x^3+c}}{4608\,c^2}-\frac {197\,d^2\,{\left (d\,x^3+c\right )}^{3/2}}{2304\,c^3}+\frac {35\,d^2\,{\left (d\,x^3+c\right )}^{5/2}}{4608\,c^4}}{33\,c\,{\left (d\,x^3+c\right )}^2-57\,c^2\,\left (d\,x^3+c\right )-3\,{\left (d\,x^3+c\right )}^3+27\,c^3}+\frac {d^2\,\left (\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{\sqrt {c^9}}\right )\,1{}\mathrm {i}-\frac {\mathrm {atanh}\left (\frac {c^4\,\sqrt {d\,x^3+c}}{3\,\sqrt {c^9}}\right )\,31{}\mathrm {i}}{513}\right )\,19{}\mathrm {i}}{6144\,\sqrt {c^9}} \]
(d^2*(atanh((c^4*(c + d*x^3)^(1/2))/(c^9)^(1/2))*1i - (atanh((c^4*(c + d*x ^3)^(1/2))/(3*(c^9)^(1/2)))*31i)/513)*19i)/(6144*(c^9)^(1/2)) - ((647*d^2* (c + d*x^3)^(1/2))/(4608*c^2) - (197*d^2*(c + d*x^3)^(3/2))/(2304*c^3) + ( 35*d^2*(c + d*x^3)^(5/2))/(4608*c^4))/(33*c*(c + d*x^3)^2 - 57*c^2*(c + d* x^3) - 3*(c + d*x^3)^3 + 27*c^3)