Integrand size = 27, antiderivative size = 522 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )^2} \, dx=-\frac {\sqrt {c+d x^3}}{16 c x}+\frac {\sqrt [3]{d} \sqrt {c+d x^3}}{16 c \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {3 \sqrt {c+d x^3}}{8 x \left (8 c-d x^3\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{32 c^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {\sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{8 \sqrt {2} \sqrt [4]{3} c^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-1/16*(d*x^3+c)^(1/2)/c/x+1/16*d^(1/3)*(d*x^3+c)^(1/2)/c/((1+3^(1/2))*c^(1 /3)+d^(1/3)*x)+3/8*(d*x^3+c)^(1/2)/x/(-d*x^3+8*c)-1/32*3^(1/4)*(1/2*6^(1/2 )-1/2*2^(1/2))*d^(1/3)*(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^(2/3)/ (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)+1/48*d^(1/3)*(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^(2/3)/(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 complex when optimal does not.
Time = 11.50 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.46 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )^2} \, dx=\frac {\left (2 c-d x^3\right ) \sqrt {c+d x^3}}{16 c x \left (-8 c+d x^3\right )}-\frac {\sqrt [6]{-1} \sqrt [3]{-d} \sqrt {(-1)^{5/6} \left (-1+\frac {\sqrt [3]{-d} x}{\sqrt [3]{c}}\right )} \sqrt {1+\frac {\sqrt [3]{-d} x}{\sqrt [3]{c}}+\frac {(-d)^{2/3} x^2}{c^{2/3}}} \left (-i \sqrt {3} E\left (\arcsin \left (\frac {\sqrt {-(-1)^{5/6}-\frac {i \sqrt [3]{-d} x}{\sqrt [3]{c}}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt [3]{-1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-(-1)^{5/6}-\frac {i \sqrt [3]{-d} x}{\sqrt [3]{c}}}}{\sqrt [4]{3}}\right ),\sqrt [3]{-1}\right )\right )}{16 \sqrt [4]{3} \sqrt [3]{c} \sqrt {c+d x^3}} \] Input:
Integrate[(c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)^2),x]
Output:
((2*c - d*x^3)*Sqrt[c + d*x^3])/(16*c*x*(-8*c + d*x^3)) - ((-1)^(1/6)*(-d) ^(1/3)*Sqrt[(-1)^(5/6)*(-1 + ((-d)^(1/3)*x)/c^(1/3))]*Sqrt[1 + ((-d)^(1/3) *x)/c^(1/3) + ((-d)^(2/3)*x^2)/c^(2/3)]*((-I)*Sqrt[3]*EllipticE[ArcSin[Sqr t[-(-1)^(5/6) - (I*(-d)^(1/3)*x)/c^(1/3)]/3^(1/4)], (-1)^(1/3)] + (-1)^(1/ 3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-d)^(1/3)*x)/c^(1/3)]/3^(1/4)], (-1)^(1/3)]))/(16*3^(1/4)*c^(1/3)*Sqrt[c + d*x^3])
Time = 1.02 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {968, 27, 847, 832, 759, 2416}
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.
| \(\displaystyle \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 968 |
| \(\displaystyle \frac {\int \frac {3 c d}{2 x^2 \sqrt {d x^3+c}}dx}{24 c d}+\frac {3 \sqrt {c+d x^3}}{8 x \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {1}{x^2 \sqrt {d x^3+c}}dx+\frac {3 \sqrt {c+d x^3}}{8 x \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \int \frac {x}{\sqrt {d x^3+c}}dx}{2 c}-\frac {\sqrt {c+d x^3}}{c x}\right )+\frac {3 \sqrt {c+d x^3}}{8 x \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (\frac {\int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c} \int \frac {1}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}\right )}{2 c}-\frac {\sqrt {c+d x^3}}{c x}\right )+\frac {3 \sqrt {c+d x^3}}{8 x \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (\frac {\int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\right )}{2 c}-\frac {\sqrt {c+d x^3}}{c x}\right )+\frac {3 \sqrt {c+d x^3}}{8 x \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (\frac {\frac {2 \sqrt {c+d x^3}}{\sqrt [3]{d} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}}{\sqrt [3]{d}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\right )}{2 c}-\frac {\sqrt {c+d x^3}}{c x}\right )+\frac {3 \sqrt {c+d x^3}}{8 x \left (8 c-d x^3\right )}\) |
Input:
Int[(c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)^2),x]
Output:
(3*Sqrt[c + d*x^3])/(8*x*(8*c - d*x^3)) + (-(Sqrt[c + d*x^3]/(c*x)) + (d*( ((2*Sqrt[c + d*x^3])/(d^(1/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (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]*Elliptic E[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^(1/3)*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]))/d^(1/3) - (2*(1 - S qrt[3])*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]*Elli pticF[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])))/(2*c)) /16
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-(c*b - a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1) *((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Simp[1/(a*b*n*(p + 1)) Int [(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c *b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 3.18 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.93
| method | result | size |
| elliptic | \(\frac {3 d \,x^{2} \sqrt {d \,x^{3}+c}}{64 c \left (-d \,x^{3}+8 c \right )}-\frac {\sqrt {d \,x^{3}+c}}{64 c x}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{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}}}}\, \sqrt {\frac {x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}}{-\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}}}\, \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}}}}\, \left (\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 ) \operatorname {EllipticE}\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}, \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 )+\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \operatorname {EllipticF}\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}, \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 )}{d}\right )}{48 c \sqrt {d \,x^{3}+c}}\) | \(483\) |
| risch | \(\text {Expression too large to display}\) | \(1758\) |
| default | \(\text {Expression too large to display}\) | \(2218\) |
Input:
int((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)
Output:
3/64*d*x^2/c*(d*x^3+c)^(1/2)/(-d*x^3+8*c)-1/64*(d*x^3+c)^(1/2)/c/x-1/48*I/ c*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))*Elli pticE(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)*Ell ipticF(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)))
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.13 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )^2} \, dx=-\frac {{\left (d x^{4} - 8 \, c x\right )} \sqrt {d} {\rm weierstrassZeta}\left (0, -\frac {4 \, c}{d}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, c}{d}, x\right )\right ) + \sqrt {d x^{3} + c} {\left (d x^{3} - 2 \, c\right )}}{16 \, {\left (c d x^{4} - 8 \, c^{2} x\right )}} \] Input:
integrate((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c)^2,x, algorithm="fricas")
Output:
-1/16*((d*x^4 - 8*c*x)*sqrt(d)*weierstrassZeta(0, -4*c/d, weierstrassPInve rse(0, -4*c/d, x)) + sqrt(d*x^3 + c)*(d*x^3 - 2*c))/(c*d*x^4 - 8*c^2*x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )^2} \, dx=\int \frac {\left (c + d x^{3}\right )^{\frac {3}{2}}}{x^{2} \left (- 8 c + d x^{3}\right )^{2}}\, dx \] Input:
integrate((d*x**3+c)**(3/2)/x**2/(-d*x**3+8*c)**2,x)
Output:
Integral((c + d*x**3)**(3/2)/(x**2*(-8*c + d*x**3)**2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{2}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c)^2,x, algorithm="maxima")
Output:
integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)^2*x^2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{2}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c)^2,x, algorithm="giac")
Output:
integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)^2*x^2), x)
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )^2} \, dx=\int \frac {{\left (d\,x^3+c\right )}^{3/2}}{x^2\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int((c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)^2),x)
Output:
int((c + d*x^3)^(3/2)/(x^2*(8*c - d*x^3)^2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^2 \left (8 c-d x^3\right )^2} \, dx=\frac {-2 \sqrt {d \,x^{3}+c}+168 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c \,d^{2} x -21 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) d^{3} x^{4}+384 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{2} d x -48 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c \,d^{2} x^{4}}{16 x \left (-d \,x^{3}+8 c \right )} \] Input:
int((d*x^3+c)^(3/2)/x^2/(-d*x^3+8*c)^2,x)
Output:
( - 2*sqrt(c + d*x**3) + 168*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**2*x - 21*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**3* x**4 + 384*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*x - 48*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**2*x**4)/(16*x*(8*c - d*x **3))