Integrand size = 27, antiderivative size = 708 \[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {5}{648 c^3 x^4 \sqrt {c+d x^3}}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {253 \sqrt {c+d x^3}}{20736 c^4 x^4}+\frac {77 d \sqrt {c+d x^3}}{2592 c^5 x}-\frac {77 d^{4/3} \sqrt {c+d x^3}}{2592 c^5 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {11 d^{4/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{82944 \sqrt {3} c^{29/6}}+\frac {11 d^{4/3} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{248832 c^{29/6}}-\frac {11 d^{4/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{248832 c^{29/6}}+\frac {77 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{1728\ 3^{3/4} c^{14/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {77 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{1296 \sqrt {2} \sqrt [4]{3} c^{14/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
5/648/c^3/x^4/(d*x^3+c)^(1/2)+1/216/c^2/x^4/(-d*x^3+8*c)/(d*x^3+c)^(1/2)-2 53/20736*(d*x^3+c)^(1/2)/c^4/x^4+77/2592*d*(d*x^3+c)^(1/2)/c^5/x-77/2592*d ^(4/3)*(d*x^3+c)^(1/2)/c^5/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)-11/248832*d^(4/ 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^( 29/6)+11/248832*d^(4/3)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c )^(1/2))/c^(29/6)-11/248832*d^(4/3)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c ^(29/6)+77/5184*(1/2*6^(1/2)-1/2*2^(1/2))*d^(4/3)*(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)*3^(1/4)/c^(14/3)/(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)-77/7776*d^(4/3)*(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^(14/3)/(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 = 10.18 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.28 \[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {-24475 c d^2 x^6 \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-16 \left (10 c \left (648 c^3-2997 c^2 d x^3-4565 c d^2 x^6+616 d^3 x^9\right )+77 d^3 x^9 \left (-8 c+d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}{3317760 c^6 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \] Input:
Integrate[1/(x^5*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
(-24475*c*d^2*x^6*(8*c - d*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 16*(10*c*(648*c^3 - 2997*c^2*d*x^3 - 4 565*c*d^2*x^6 + 616*d^3*x^9) + 77*d^3*x^9*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/ c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(3317760*c^6* x^4*(8*c - d*x^3)*Sqrt[c + d*x^3])
Time = 2.07 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {972, 27, 1049, 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.
| \(\displaystyle \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {\int \frac {d \left (17 d x^3+62 c\right )}{2 x^5 \left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx}{216 c^2 d}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {17 d x^3+62 c}{x^5 \left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx}{432 c^2}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle \frac {\frac {10}{3 c x^4 \sqrt {c+d x^3}}-\frac {2 \int -\frac {99 c d \left (46 c-5 d x^3\right )}{2 x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{27 c^2 d}}{432 c^2}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {11 \int \frac {46 c-5 d x^3}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{3 c}+\frac {10}{3 c x^4 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {11 \left (-\frac {\int \frac {c d \left (896 c-115 d x^3\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{3 c}+\frac {10}{3 c x^4 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {11 \left (-\frac {d \int \frac {896 c-115 d x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{3 c}+\frac {10}{3 c x^4 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {11 \left (-\frac {d \left (-\frac {\int -\frac {8 c d x \left (445 c-56 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {112 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{3 c}+\frac {10}{3 c x^4 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {11 \left (-\frac {d \left (\frac {d \int \frac {x \left (445 c-56 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {112 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{3 c}+\frac {10}{3 c x^4 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {\frac {11 \left (-\frac {d \left (\frac {d \int \left (\frac {56 x}{\sqrt {d x^3+c}}-\frac {3 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}\right )dx}{c}-\frac {112 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{3 c}+\frac {10}{3 c x^4 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {11 \left (-\frac {d \left (\frac {d \left (\frac {112 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {56 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {\sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 \sqrt {3} d^{2/3}}-\frac {\sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{6 d^{2/3}}+\frac {\sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 d^{2/3}}+\frac {112 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{c}-\frac {112 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {23 \sqrt {c+d x^3}}{16 c x^4}\right )}{3 c}+\frac {10}{3 c x^4 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
Input:
Int[1/(x^5*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
1/(216*c^2*x^4*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (10/(3*c*x^4*Sqrt[c + d*x^ 3]) + (11*((-23*Sqrt[c + d*x^3])/(16*c*x^4) - (d*((-112*Sqrt[c + d*x^3])/( c*x) + (d*((112*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3) *x)) + (c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d* x^3]])/(2*Sqrt[3]*d^(2/3)) - (c^(1/6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2/(3*c ^(1/6)*Sqrt[c + d*x^3])])/(6*d^(2/3)) + (c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/( 3*Sqrt[c])])/(6*d^(2/3)) - (56*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]*Sq rt[c + d*x^3]) + (112*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]* 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]])/(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])))/c) )/(32*c)))/(3*c))/(432*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[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -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[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
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 = 4.80 (sec) , antiderivative size = 943, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(943\) |
| risch | \(\text {Expression too large to display}\) | \(2232\) |
| default | \(\text {Expression too large to display}\) | \(2775\) |
Input:
int(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/243*d^2*x^2/c^5/((x^3+c/d)*d)^(1/2)+1/124416*d^2*x^2/c^5*(d*x^3+c)^(1/2) /(-d*x^3+8*c)-1/256*(d*x^3+c)^(1/2)/c^4/x^4+11/512*d*(d*x^3+c)^(1/2)/c^5/x +77/7776*I*d/c^5*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))*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)))-11/37 3248*I/c^5/d*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)...
Leaf count of result is larger than twice the leaf count of optimal. 2692 vs. \(2 (507) = 1014\).
Time = 3.30 (sec) , antiderivative size = 2692, normalized size of antiderivative = 3.80 \[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="fricas")
Output:
1/2985984*(88704*(d^3*x^10 - 7*c*d^2*x^7 - 8*c^2*d*x^4)*sqrt(d)*weierstras sZeta(0, -4*c/d, weierstrassPInverse(0, -4*c/d, x)) + 11*(c^5*d^2*x^10 - 7 *c^6*d*x^7 - 8*c^7*x^4 + sqrt(-3)*(c^5*d^2*x^10 - 7*c^6*d*x^7 - 8*c^7*x^4) )*(d^8/c^29)^(1/6)*log(161051*(d^9*x^9 + 318*c*d^8*x^6 + 1200*c^2*d^7*x^3 + 640*c^3*d^6 - 9*(5*c^20*d^3*x^7 + 64*c^21*d^2*x^4 + 32*c^22*d*x + sqrt(- 3)*(5*c^20*d^3*x^7 + 64*c^21*d^2*x^4 + 32*c^22*d*x))*(d^8/c^29)^(2/3) + 3* sqrt(d*x^3 + c)*(6*(5*c^25*d*x^5 + 32*c^26*x^2 - sqrt(-3)*(5*c^25*d*x^5 + 32*c^26*x^2))*(d^8/c^29)^(5/6) - 2*(7*c^15*d^4*x^6 + 152*c^16*d^3*x^3 + 64 *c^17*d^2)*sqrt(d^8/c^29) + (c^5*d^7*x^7 + 80*c^6*d^6*x^4 + 160*c^7*d^5*x + sqrt(-3)*(c^5*d^7*x^7 + 80*c^6*d^6*x^4 + 160*c^7*d^5*x))*(d^8/c^29)^(1/6 )) - 9*(c^10*d^6*x^8 + 38*c^11*d^5*x^5 + 64*c^12*d^4*x^2 - sqrt(-3)*(c^10* d^6*x^8 + 38*c^11*d^5*x^5 + 64*c^12*d^4*x^2))*(d^8/c^29)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 11*(c^5*d^2*x^10 - 7*c^6*d*x^7 - 8*c^7*x^4 + sqrt(-3)*(c^5*d^2*x^10 - 7*c^6*d*x^7 - 8*c^7*x^4))*(d^8/c^2 9)^(1/6)*log(161051*(d^9*x^9 + 318*c*d^8*x^6 + 1200*c^2*d^7*x^3 + 640*c^3* d^6 - 9*(5*c^20*d^3*x^7 + 64*c^21*d^2*x^4 + 32*c^22*d*x + sqrt(-3)*(5*c^20 *d^3*x^7 + 64*c^21*d^2*x^4 + 32*c^22*d*x))*(d^8/c^29)^(2/3) - 3*sqrt(d*x^3 + c)*(6*(5*c^25*d*x^5 + 32*c^26*x^2 - sqrt(-3)*(5*c^25*d*x^5 + 32*c^26*x^ 2))*(d^8/c^29)^(5/6) - 2*(7*c^15*d^4*x^6 + 152*c^16*d^3*x^3 + 64*c^17*d^2) *sqrt(d^8/c^29) + (c^5*d^7*x^7 + 80*c^6*d^6*x^4 + 160*c^7*d^5*x + sqrt(...
\[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^{5} \left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**5/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)
Output:
Integral(1/(x**5*(-8*c + d*x**3)**2*(c + d*x**3)**(3/2)), x)
\[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{5}} \,d x } \] Input:
integrate(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^5), x)
\[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{5}} \,d x } \] Input:
integrate(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="giac")
Output:
integrate(1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)^2*x^5), x)
Timed out. \[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^5\,{\left (d\,x^3+c\right )}^{3/2}\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int(1/(x^5*(c + d*x^3)^(3/2)*(8*c - d*x^3)^2),x)
Output:
int(1/(x^5*(c + d*x^3)^(3/2)*(8*c - d*x^3)^2), x)
\[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {-16 \sqrt {d \,x^{3}+c}\, c +74 \sqrt {d \,x^{3}+c}\, d \,x^{3}-3256 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{3} x^{4}-2849 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{4} x^{7}+407 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) d^{5} x^{10}+10560 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{3} d^{2} x^{4}+9240 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{3} x^{7}-1320 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{4} x^{10}}{512 c^{3} x^{4} \left (-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}\right )} \] Input:
int(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x)
Output:
( - 16*sqrt(c + d*x**3)*c + 74*sqrt(c + d*x**3)*d*x**3 - 3256*int((sqrt(c + d*x**3)*x**4)/(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**3*x**4 - 2849*int((sqrt(c + d*x**3)*x**4)/( 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**1 2),x)*c*d**4*x**7 + 407*int((sqrt(c + d*x**3)*x**4)/(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)*d**5*x**10 + 10 560*int((sqrt(c + d*x**3)*x)/(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**2*x**4 + 9240*int((sqrt(c + d* x**3)*x)/(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**3*x**7 - 1320*int((sqrt(c + d*x**3)*x)/(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*d* *4*x**10)/(512*c**3*x**4*(8*c**2 + 7*c*d*x**3 - d**2*x**6))