Integrand size = 27, antiderivative size = 711 \[ \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx=-\frac {5 \sqrt {c+d x^3}}{672 c^2 x^7}-\frac {53 d \sqrt {c+d x^3}}{21504 c^3 x^4}-\frac {d^2 \sqrt {c+d x^3}}{5376 c^4 x}+\frac {d^{7/3} \sqrt {c+d x^3}}{5376 c^4 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac {13 d^{7/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{12288 \sqrt {3} c^{23/6}}+\frac {13 d^{7/3} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{36864 c^{23/6}}-\frac {13 d^{7/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{36864 c^{23/6}}-\frac {\sqrt {2-\sqrt {3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{3584\ 3^{3/4} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{2688 \sqrt {2} \sqrt [4]{3} c^{11/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-5/672*(d*x^3+c)^(1/2)/c^2/x^7-53/21504*d*(d*x^3+c)^(1/2)/c^3/x^4-1/5376*d ^2*(d*x^3+c)^(1/2)/c^4/x+1/5376*d^(7/3)*(d*x^3+c)^(1/2)/c^4/((1+3^(1/2))*c ^(1/3)+d^(1/3)*x)+1/24*(d*x^3+c)^(1/2)/c/x^7/(-d*x^3+8*c)-13/36864*d^(7/3) *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)+13/36864*d^(7/3)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^( 1/2))/c^(23/6)-13/36864*d^(7/3)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(23 /6)-1/10752*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*d^(7/3)*(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^(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)+1/16128*3^(3/4)*d^(7/3)*(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)/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 = 10.15 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\frac {1525 c d^3 x^9 \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-8 \left (20 c \left (384 c^4+648 c^3 d x^3+243 c^2 d^2 x^6-25 c d^3 x^9-4 d^4 x^{12}\right )+d^4 x^{12} \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}{3440640 c^5 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \] Input:
Integrate[Sqrt[c + d*x^3]/(x^8*(8*c - d*x^3)^2),x]
Output:
(1525*c*d^3*x^9*(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)] - 8*(20*c*(384*c^4 + 648*c^3*d*x^3 + 243*c ^2*d^2*x^6 - 25*c*d^3*x^9 - 4*d^4*x^12) + d^4*x^12*(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)]))/(3440 640*c^5*x^7*(8*c - d*x^3)*Sqrt[c + d*x^3])
Time = 2.03 (sec) , antiderivative size = 728, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {969, 27, 1053, 27, 1053, 25, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 969 |
| \(\displaystyle \frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}-\frac {\int -\frac {17 d x^3+20 c}{2 x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{24 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {17 d x^3+20 c}{x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 c d \left (55 d x^3+106 c\right )}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{56 c^2}-\frac {5 \sqrt {c+d x^3}}{14 c x^7}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \int \frac {55 d x^3+106 c}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{28 c}-\frac {5 \sqrt {c+d x^3}}{14 c x^7}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {d \left (-\frac {\int -\frac {c d \left (265 d x^3+64 c\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {53 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {5 \sqrt {c+d x^3}}{14 c x^7}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {d \left (\frac {\int \frac {c d \left (265 d x^3+64 c\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {53 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {5 \sqrt {c+d x^3}}{14 c x^7}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \int \frac {265 d x^3+64 c}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {53 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {5 \sqrt {c+d x^3}}{14 c x^7}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \left (-\frac {\int -\frac {8 c d x \left (305 c-4 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {8 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {53 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {5 \sqrt {c+d x^3}}{14 c x^7}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \left (\frac {d \int \frac {x \left (305 c-4 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {8 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {53 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {5 \sqrt {c+d x^3}}{14 c x^7}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \left (\frac {d \int \left (\frac {273 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {4 x}{\sqrt {d x^3+c}}\right )dx}{c}-\frac {8 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {53 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {5 \sqrt {c+d x^3}}{14 c x^7}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \left (\frac {d \left (\frac {8 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {91 \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 \sqrt {3} d^{2/3}}+\frac {91 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{6 d^{2/3}}-\frac {91 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 d^{2/3}}+\frac {8 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{c}-\frac {8 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {53 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {5 \sqrt {c+d x^3}}{14 c x^7}}{48 c}+\frac {\sqrt {c+d x^3}}{24 c x^7 \left (8 c-d x^3\right )}\) |
Input:
Int[Sqrt[c + d*x^3]/(x^8*(8*c - d*x^3)^2),x]
Output:
Sqrt[c + d*x^3]/(24*c*x^7*(8*c - d*x^3)) + ((-5*Sqrt[c + d*x^3])/(14*c*x^7 ) + (d*((-53*Sqrt[c + d*x^3])/(16*c*x^4) + (d*((-8*Sqrt[c + d*x^3])/(c*x) + (d*((8*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (91*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)) + (91*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)) - (91*c^(1/6)*ArcTanh[Sqrt[c + d*x^3] /(3*Sqrt[c])])/(6*d^(2/3)) - (4*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)*Sqr t[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*S qrt[c + d*x^3]) + (8*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]*E llipticF[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)))/(28*c))/(48*c)
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[(-(e*x)^(m + 1))*(a + b*x^n)^(p + 1)*((c + d*x^n )^q/(a*e*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[(e*x)^m*(a + b*x^n)^( p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + 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] && LtQ[0, q, 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 = 5.11 (sec) , antiderivative size = 938, normalized size of antiderivative = 1.32
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(938\) |
| risch | \(\text {Expression too large to display}\) | \(1781\) |
| default | \(\text {Expression too large to display}\) | \(3170\) |
Input:
int((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)
Output:
-1/448*(d*x^3+c)^(1/2)/c^2/x^7-13/7168*d*(d*x^3+c)^(1/2)/c^3/x^4-3/28672*d ^2*(d*x^3+c)^(1/2)/c^4/x+1/12288*x^2*d^3*(d*x^3+c)^(1/2)/(-d*x^3+8*c)/c^4- 1/16128*I*d^2/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)))-13/5 5296*I/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...
Leaf count of result is larger than twice the leaf count of optimal. 2582 vs. \(2 (510) = 1020\).
Time = 3.18 (sec) , antiderivative size = 2582, normalized size of antiderivative = 3.63 \[ \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c)^2,x, algorithm="fricas")
Output:
-1/3096576*(576*(d^3*x^10 - 8*c*d^2*x^7)*sqrt(d)*weierstrassZeta(0, -4*c/d , weierstrassPInverse(0, -4*c/d, x)) - 91*(c^4*d*x^10 - 8*c^5*x^7 + sqrt(- 3)*(c^4*d*x^10 - 8*c^5*x^7))*(d^14/c^23)^(1/6)*log(371293*(d^14*x^9 + 318* c*d^13*x^6 + 1200*c^2*d^12*x^3 + 640*c^3*d^11 - 9*(5*c^16*d^4*x^7 + 64*c^1 7*d^3*x^4 + 32*c^18*d^2*x + sqrt(-3)*(5*c^16*d^4*x^7 + 64*c^17*d^3*x^4 + 3 2*c^18*d^2*x))*(d^14/c^23)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^20*d*x^5 + 32 *c^21*x^2 - sqrt(-3)*(5*c^20*d*x^5 + 32*c^21*x^2))*(d^14/c^23)^(5/6) - 2*( 7*c^12*d^6*x^6 + 152*c^13*d^5*x^3 + 64*c^14*d^4)*sqrt(d^14/c^23) + (c^4*d^ 11*x^7 + 80*c^5*d^10*x^4 + 160*c^6*d^9*x + sqrt(-3)*(c^4*d^11*x^7 + 80*c^5 *d^10*x^4 + 160*c^6*d^9*x))*(d^14/c^23)^(1/6)) - 9*(c^8*d^9*x^8 + 38*c^9*d ^8*x^5 + 64*c^10*d^7*x^2 - sqrt(-3)*(c^8*d^9*x^8 + 38*c^9*d^8*x^5 + 64*c^1 0*d^7*x^2))*(d^14/c^23)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 5 12*c^3)) + 91*(c^4*d*x^10 - 8*c^5*x^7 + sqrt(-3)*(c^4*d*x^10 - 8*c^5*x^7)) *(d^14/c^23)^(1/6)*log(371293*(d^14*x^9 + 318*c*d^13*x^6 + 1200*c^2*d^12*x ^3 + 640*c^3*d^11 - 9*(5*c^16*d^4*x^7 + 64*c^17*d^3*x^4 + 32*c^18*d^2*x + sqrt(-3)*(5*c^16*d^4*x^7 + 64*c^17*d^3*x^4 + 32*c^18*d^2*x))*(d^14/c^23)^( 2/3) - 3*sqrt(d*x^3 + c)*(6*(5*c^20*d*x^5 + 32*c^21*x^2 - sqrt(-3)*(5*c^20 *d*x^5 + 32*c^21*x^2))*(d^14/c^23)^(5/6) - 2*(7*c^12*d^6*x^6 + 152*c^13*d^ 5*x^3 + 64*c^14*d^4)*sqrt(d^14/c^23) + (c^4*d^11*x^7 + 80*c^5*d^10*x^4 + 1 60*c^6*d^9*x + sqrt(-3)*(c^4*d^11*x^7 + 80*c^5*d^10*x^4 + 160*c^6*d^9*x...
Timed out. \[ \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate((d*x**3+c)**(1/2)/x**8/(-d*x**3+8*c)**2,x)
Output:
Timed out
\[ \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{8}} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c)^2,x, algorithm="maxima")
Output:
integrate(sqrt(d*x^3 + c)/((d*x^3 - 8*c)^2*x^8), x)
\[ \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{8}} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c)^2,x, algorithm="giac")
Output:
integrate(sqrt(d*x^3 + c)/((d*x^3 - 8*c)^2*x^8), x)
Timed out. \[ \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\int \frac {\sqrt {d\,x^3+c}}{x^8\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int((c + d*x^3)^(1/2)/(x^8*(8*c - d*x^3)^2),x)
Output:
int((c + d*x^3)^(1/2)/(x^8*(8*c - d*x^3)^2), x)
\[ \int \frac {\sqrt {c+d x^3}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\int \frac {\sqrt {d \,x^{3}+c}}{d^{2} x^{14}-16 c d \,x^{11}+64 c^{2} x^{8}}d x \] Input:
int((d*x^3+c)^(1/2)/x^8/(-d*x^3+8*c)^2,x)
Output:
int(sqrt(c + d*x**3)/(64*c**2*x**8 - 16*c*d*x**11 + d**2*x**14),x)