Integrand size = 27, antiderivative size = 102 \[ \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {2 c}{243 d^4 \sqrt {c+d x^3}}+\frac {2 \sqrt {c+d x^3}}{3 d^4}+\frac {512 c \sqrt {c+d x^3}}{243 d^4 \left (8 c-d x^3\right )}-\frac {640 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{243 d^4} \] Output:
2/243*c/d^4/(d*x^3+c)^(1/2)+2/3*(d*x^3+c)^(1/2)/d^4+512/243*c*(d*x^3+c)^(1 /2)/d^4/(-d*x^3+8*c)-640/243*c^(1/2)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/ d^4
Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97 \[ \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=-\frac {2 \left (912 c^2+822 c d x^3-81 d^2 x^6-320 \sqrt {c} \left (8 c-d x^3\right ) \sqrt {c+d x^3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )\right )}{243 d^4 \left (-8 c+d x^3\right ) \sqrt {c+d x^3}} \] Input:
Integrate[x^11/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
(-2*(912*c^2 + 822*c*d*x^3 - 81*d^2*x^6 - 320*Sqrt[c]*(8*c - d*x^3)*Sqrt[c + d*x^3]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])]))/(243*d^4*(-8*c + d*x^3)*S qrt[c + d*x^3])
Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {948, 109, 27, 163, 73, 219}
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.
| \(\displaystyle \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {x^9}{\left (8 c-d x^3\right )^2 \left (d x^3+c\right )^{3/2}}dx^3\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 x^6}{9 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\int \frac {c x^3 \left (13 d x^3+16 c\right )}{\left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx^3}{9 c d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 x^6}{9 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\int \frac {x^3 \left (13 d x^3+16 c\right )}{\left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx^3}{9 d^2}\right )\) |
| \(\Big \downarrow \) 163 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 x^6}{9 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\frac {320 c \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{3 d}-\frac {2 \left (38 c+39 d x^3\right )}{3 d^2 \sqrt {c+d x^3}}}{9 d^2}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 x^6}{9 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\frac {640 c \int \frac {1}{9 c-x^6}d\sqrt {d x^3+c}}{3 d^2}-\frac {2 \left (38 c+39 d x^3\right )}{3 d^2 \sqrt {c+d x^3}}}{9 d^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 x^6}{9 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\frac {640 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^2}-\frac {2 \left (38 c+39 d x^3\right )}{3 d^2 \sqrt {c+d x^3}}}{9 d^2}\right )\) |
Input:
Int[x^11/((8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
((8*x^6)/(9*d^2*(8*c - d*x^3)*Sqrt[c + d*x^3]) - ((-2*(38*c + 39*d*x^3))/( 3*d^2*Sqrt[c + d*x^3]) + (640*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])] )/(9*d^2))/(9*d^2))/3
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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* (m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f *h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* d*(b*c - a*d)*(m + 1)*(m + n + 3)) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
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[(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 = 1.43 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74
| method | result | size |
| pseudoelliptic | \(\frac {\frac {2 \sqrt {d \,x^{3}+c}}{3}+\frac {128 c \left (\frac {4 \sqrt {d \,x^{3}+c}}{-d \,x^{3}+8 c}-\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{\sqrt {c}}\right )}{243}+\frac {2 c}{243 \sqrt {d \,x^{3}+c}}}{d^{4}}\) | \(75\) |
| risch | \(\frac {2 \sqrt {d \,x^{3}+c}}{3 d^{4}}+\frac {c \left (\frac {2}{243 d \sqrt {d \,x^{3}+c}}-\frac {2432 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{729 \sqrt {c}\, d}+\frac {512 c \left (-\frac {\sqrt {d \,x^{3}+c}}{c \left (d \,x^{3}-8 c \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{243 d}\right )}{d^{3}}\) | \(111\) |
| default | \(\frac {d \left (\frac {2 c}{3 d^{2} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}+\frac {2 \sqrt {d \,x^{3}+c}}{3 d^{2}}\right )-\frac {32 c}{3 d \sqrt {d \,x^{3}+c}}}{d^{3}}-\frac {128 c^{2} \left (-\frac {1}{c \sqrt {d \,x^{3}+c}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{9 d^{4}}+\frac {512 c \left (-\frac {2}{\sqrt {d \,x^{3}+c}}+\frac {\sqrt {d \,x^{3}+c}}{-d \,x^{3}+8 c}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{\sqrt {c}}\right )}{243 d^{4}}\) | \(160\) |
| elliptic | \(\frac {2 c}{243 d^{4} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}+\frac {512 c \sqrt {d \,x^{3}+c}}{243 d^{4} \left (-d \,x^{3}+8 c \right )}+\frac {2 \sqrt {d \,x^{3}+c}}{3 d^{4}}+\frac {320 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \sqrt {d \,x^{3}+c}}\right )}{729 d^{6}}\) | \(471\) |
Input:
int(x^11/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/3*((d*x^3+c)^(1/2)+64/81*c*(4*(d*x^3+c)^(1/2)/(-d*x^3+8*c)-5*arctanh(1/3 *(d*x^3+c)^(1/2)/c^(1/2))/c^(1/2))+1/81*c/(d*x^3+c)^(1/2))/d^4
Time = 0.08 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.25 \[ \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\left [\frac {2 \, {\left (160 \, {\left (d^{2} x^{6} - 7 \, c d x^{3} - 8 \, c^{2}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 3 \, {\left (27 \, d^{2} x^{6} - 274 \, c d x^{3} - 304 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{243 \, {\left (d^{6} x^{6} - 7 \, c d^{5} x^{3} - 8 \, c^{2} d^{4}\right )}}, \frac {2 \, {\left (320 \, {\left (d^{2} x^{6} - 7 \, c d x^{3} - 8 \, c^{2}\right )} \sqrt {-c} \arctan \left (\frac {3 \, \sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) + 3 \, {\left (27 \, d^{2} x^{6} - 274 \, c d x^{3} - 304 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{243 \, {\left (d^{6} x^{6} - 7 \, c d^{5} x^{3} - 8 \, c^{2} d^{4}\right )}}\right ] \] Input:
integrate(x^11/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="fricas")
Output:
[2/243*(160*(d^2*x^6 - 7*c*d*x^3 - 8*c^2)*sqrt(c)*log((d*x^3 - 6*sqrt(d*x^ 3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) + 3*(27*d^2*x^6 - 274*c*d*x^3 - 304* c^2)*sqrt(d*x^3 + c))/(d^6*x^6 - 7*c*d^5*x^3 - 8*c^2*d^4), 2/243*(320*(d^2 *x^6 - 7*c*d*x^3 - 8*c^2)*sqrt(-c)*arctan(3*sqrt(-c)/sqrt(d*x^3 + c)) + 3* (27*d^2*x^6 - 274*c*d*x^3 - 304*c^2)*sqrt(d*x^3 + c))/(d^6*x^6 - 7*c*d^5*x ^3 - 8*c^2*d^4)]
\[ \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {x^{11}}{\left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**11/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)
Output:
Integral(x**11/((-8*c + d*x**3)**2*(c + d*x**3)**(3/2)), x)
Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.96 \[ \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {2 \, {\left (160 \, \sqrt {c} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 81 \, \sqrt {d x^{3} + c} - \frac {3 \, {\left (85 \, {\left (d x^{3} + c\right )} c + 3 \, c^{2}\right )}}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {d x^{3} + c} c}\right )}}{243 \, d^{4}} \] Input:
integrate(x^11/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="maxima")
Output:
2/243*(160*sqrt(c)*log((sqrt(d*x^3 + c) - 3*sqrt(c))/(sqrt(d*x^3 + c) + 3* sqrt(c))) + 81*sqrt(d*x^3 + c) - 3*(85*(d*x^3 + c)*c + 3*c^2)/((d*x^3 + c) ^(3/2) - 9*sqrt(d*x^3 + c)*c))/d^4
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.86 \[ \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {640 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{243 \, \sqrt {-c} d^{4}} + \frac {2 \, \sqrt {d x^{3} + c}}{3 \, d^{4}} - \frac {2 \, {\left (85 \, {\left (d x^{3} + c\right )} c + 3 \, c^{2}\right )}}{81 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {d x^{3} + c} c\right )} d^{4}} \] Input:
integrate(x^11/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="giac")
Output:
640/243*c*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^4) + 2/3*sqrt(d *x^3 + c)/d^4 - 2/81*(85*(d*x^3 + c)*c + 3*c^2)/(((d*x^3 + c)^(3/2) - 9*sq rt(d*x^3 + c)*c)*d^4)
Time = 2.90 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09 \[ \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {2\,\sqrt {d\,x^3+c}}{3\,d^4}+\frac {320\,\sqrt {c}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{243\,d^4}+\frac {\sqrt {d\,x^3+c}\,\left (\frac {176\,c^2}{81\,d^4}+\frac {170\,c\,x^3}{81\,d^3}\right )}{8\,c^2+7\,c\,d\,x^3-d^2\,x^6} \] Input:
int(x^11/((c + d*x^3)^(3/2)*(8*c - d*x^3)^2),x)
Output:
(2*(c + d*x^3)^(1/2))/(3*d^4) + (320*c^(1/2)*log((10*c + d*x^3 - 6*c^(1/2) *(c + d*x^3)^(1/2))/(8*c - d*x^3)))/(243*d^4) + ((c + d*x^3)^(1/2)*((176*c ^2)/(81*d^4) + (170*c*x^3)/(81*d^3)))/(8*c^2 - d^2*x^6 + 7*c*d*x^3)
\[ \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {\frac {352 \sqrt {d \,x^{3}+c}\, c^{2}}{9}+\frac {44 \sqrt {d \,x^{3}+c}\, c d \,x^{3}}{3}-\frac {2 \sqrt {d \,x^{3}+c}\, d^{2} x^{6}}{3}-2560 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{4} d^{2}-2240 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{3} d^{3} x^{3}+320 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{4} x^{6}}{d^{4} \left (-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}\right )} \] Input:
int(x^11/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x)
Output:
(2*(176*sqrt(c + d*x**3)*c**2 + 66*sqrt(c + d*x**3)*c*d*x**3 - 3*sqrt(c + d*x**3)*d**2*x**6 - 11520*int((sqrt(c + d*x**3)*x**5)/(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**4*d**2 - 1 0080*int((sqrt(c + d*x**3)*x**5)/(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**3*x**3 + 1440*int((sqrt(c + d*x**3)*x**5)/(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**4*x**6))/(9*d**4*(8*c**2 + 7*c*d*x**3 - d** 2*x**6))