Integrand size = 27, antiderivative size = 687 \[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=-\frac {31 \sqrt {c+d x^3}}{6912 c^3 x^4}+\frac {5 d \sqrt {c+d x^3}}{864 c^4 x}-\frac {5 d^{4/3} \sqrt {c+d x^3}}{864 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}-\frac {25 d^{4/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{27648 \sqrt {3} c^{23/6}}+\frac {25 d^{4/3} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{82944 c^{23/6}}-\frac {25 d^{4/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{82944 c^{23/6}}+\frac {5 \sqrt {2-\sqrt {3}} d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{576\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {5 d^{4/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{432 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-31/6912*(d*x^3+c)^(1/2)/c^3/x^4+5/864*d*(d*x^3+c)^(1/2)/c^4/x-5/864*d^(4/ 3)*(d*x^3+c)^(1/2)/c^4/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)+1/216*(d*x^3+c)^(1/ 2)/c^2/x^4/(-d*x^3+8*c)-25/82944*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^(23/6)+25/82944*d^(4/3)*arctanh(1/3*( c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/c^(23/6)-25/82944*d^(4/3)*ar ctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(23/6)+5/1728*(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^(11/3)/(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)-5/2592*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^(11/3)/(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 = 196, normalized size of antiderivative = 0.29 \[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {245 c d^2 x^6 \left (-8 c+d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-16 \left (2 c \left (216 c^3-135 c^2 d x^3-311 c d^2 x^6+40 d^3 x^9\right )+d^3 x^9 \left (-8 c+d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}{221184 c^5 x^4 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \] Input:
Integrate[1/(x^5*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]
Output:
(245*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*(2*c*(216*c^3 - 135*c^2*d*x^3 - 311*c *d^2*x^6 + 40*d^3*x^9) + d^3*x^9*(-8*c + d*x^3)*Sqrt[1 + (d*x^3)/c]*Appell F1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))/(221184*c^5*x^4*(8*c - d*x^3)*Sqrt[c + d*x^3])
Time = 1.95 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {972, 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.
| \(\displaystyle \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {\int \frac {d \left (11 d x^3+62 c\right )}{2 x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{216 c^2 d}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {11 d x^3+62 c}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {\int \frac {5 c d \left (128 c-31 d x^3\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {31 \sqrt {c+d x^3}}{16 c x^4}}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {5 d \int \frac {128 c-31 d x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {31 \sqrt {c+d x^3}}{16 c x^4}}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {5 d \left (-\frac {\int -\frac {8 c d x \left (49 c-8 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {16 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {31 \sqrt {c+d x^3}}{16 c x^4}}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {5 d \left (\frac {d \int \frac {x \left (49 c-8 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {16 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {31 \sqrt {c+d x^3}}{16 c x^4}}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {-\frac {5 d \left (\frac {d \int \left (\frac {8 x}{\sqrt {d x^3+c}}-\frac {15 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}\right )dx}{c}-\frac {16 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {31 \sqrt {c+d x^3}}{16 c x^4}}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {5 d \left (\frac {d \left (\frac {16 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {8 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {5 \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 \sqrt {3} d^{2/3}}-\frac {5 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{6 d^{2/3}}+\frac {5 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 d^{2/3}}+\frac {16 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{c}-\frac {16 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {31 \sqrt {c+d x^3}}{16 c x^4}}{432 c^2}+\frac {\sqrt {c+d x^3}}{216 c^2 x^4 \left (8 c-d x^3\right )}\) |
Input:
Int[1/(x^5*(8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]
Output:
Sqrt[c + d*x^3]/(216*c^2*x^4*(8*c - d*x^3)) + ((-31*Sqrt[c + d*x^3])/(16*c *x^4) - (5*d*((-16*Sqrt[c + d*x^3])/(c*x) + (d*((16*Sqrt[c + d*x^3])/(d^(2 /3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) + (5*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)) - (5*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)) + (5*c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(6*d^(2/3)) - (8*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]*Ellipt icE[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]) + (16*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))/(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[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 = 3.78 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.34
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(919\) |
| risch | \(\text {Expression too large to display}\) | \(1770\) |
| default | \(\text {Expression too large to display}\) | \(2241\) |
Input:
int(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/13824*x^2*d^2/c^4*(d*x^3+c)^(1/2)/(-d*x^3+8*c)-1/256*(d*x^3+c)^(1/2)/c^3 /x^4+3/512*d*(d*x^3+c)^(1/2)/c^4/x+5/2592*I*d/c^4*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)))-25/124416*I/d/c^4*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...
Leaf count of result is larger than twice the leaf count of optimal. 2549 vs. \(2 (490) = 980\).
Time = 1.94 (sec) , antiderivative size = 2549, normalized size of antiderivative = 3.71 \[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="fricas")
Output:
1/995328*(5760*(d^2*x^7 - 8*c*d*x^4)*sqrt(d)*weierstrassZeta(0, -4*c/d, we ierstrassPInverse(0, -4*c/d, x)) + 25*(c^4*d*x^7 - 8*c^5*x^4 + sqrt(-3)*(c ^4*d*x^7 - 8*c^5*x^4))*(d^8/c^23)^(1/6)*log(9765625*(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^16*d^3*x^7 + 64*c^17*d^2*x^4 + 32*c^18*d*x + sqrt(-3)*(5*c^16*d^3*x^7 + 64*c^17*d^2*x^4 + 32*c^18*d*x)) *(d^8/c^23)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^20*d*x^5 + 32*c^21*x^2 - sqr t(-3)*(5*c^20*d*x^5 + 32*c^21*x^2))*(d^8/c^23)^(5/6) - 2*(7*c^12*d^4*x^6 + 152*c^13*d^3*x^3 + 64*c^14*d^2)*sqrt(d^8/c^23) + (c^4*d^7*x^7 + 80*c^5*d^ 6*x^4 + 160*c^6*d^5*x + sqrt(-3)*(c^4*d^7*x^7 + 80*c^5*d^6*x^4 + 160*c^6*d ^5*x))*(d^8/c^23)^(1/6)) - 9*(c^8*d^6*x^8 + 38*c^9*d^5*x^5 + 64*c^10*d^4*x ^2 - sqrt(-3)*(c^8*d^6*x^8 + 38*c^9*d^5*x^5 + 64*c^10*d^4*x^2))*(d^8/c^23) ^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) - 25*(c^4*d*x^ 7 - 8*c^5*x^4 + sqrt(-3)*(c^4*d*x^7 - 8*c^5*x^4))*(d^8/c^23)^(1/6)*log(976 5625*(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^16 *d^3*x^7 + 64*c^17*d^2*x^4 + 32*c^18*d*x + sqrt(-3)*(5*c^16*d^3*x^7 + 64*c ^17*d^2*x^4 + 32*c^18*d*x))*(d^8/c^23)^(2/3) - 3*sqrt(d*x^3 + c)*(6*(5*c^2 0*d*x^5 + 32*c^21*x^2 - sqrt(-3)*(5*c^20*d*x^5 + 32*c^21*x^2))*(d^8/c^23)^ (5/6) - 2*(7*c^12*d^4*x^6 + 152*c^13*d^3*x^3 + 64*c^14*d^2)*sqrt(d^8/c^23) + (c^4*d^7*x^7 + 80*c^5*d^6*x^4 + 160*c^6*d^5*x + sqrt(-3)*(c^4*d^7*x^7 + 80*c^5*d^6*x^4 + 160*c^6*d^5*x))*(d^8/c^23)^(1/6)) - 9*(c^8*d^6*x^8 + ...
\[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^{5} \left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \] Input:
integrate(1/x**5/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)
Output:
Integral(1/(x**5*(-8*c + d*x**3)**2*sqrt(c + d*x**3)), x)
\[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}^{2} x^{5}} \,d x } \] Input:
integrate(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2*x^5), x)
\[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {d x^{3} + c} {\left (d x^{3} - 8 \, c\right )}^{2} x^{5}} \,d x } \] Input:
integrate(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2*x^5), x)
Timed out. \[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^5\,\sqrt {d\,x^3+c}\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int(1/(x^5*(c + d*x^3)^(1/2)*(8*c - d*x^3)^2),x)
Output:
int(1/(x^5*(c + d*x^3)^(1/2)*(8*c - d*x^3)^2), x)
\[ \int \frac {1}{x^5 \left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {-16 \sqrt {d \,x^{3}+c}\, c +26 \sqrt {d \,x^{3}+c}\, d \,x^{3}-520 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c \,d^{3} x^{4}+65 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) d^{4} x^{7}-960 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{2} d^{2} x^{4}+120 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c \,d^{3} x^{7}}{512 c^{3} x^{4} \left (-d \,x^{3}+8 c \right )} \] Input:
int(1/x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x)
Output:
( - 16*sqrt(c + d*x**3)*c + 26*sqrt(c + d*x**3)*d*x**3 - 520*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**3*x**4 + 65*int((sqrt(c + d*x**3)*x**4)/(64*c**3 + 48*c**2*d*x**3 - 1 5*c*d**2*x**6 + d**3*x**9),x)*d**4*x**7 - 960*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**2*x**4 + 1 20*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**3*x**7)/(512*c**3*x**4*(8*c - d*x**3))