Integrand size = 27, antiderivative size = 708 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx=-\frac {11 \sqrt {c+d x^3}}{224 c x^7}-\frac {83 d \sqrt {c+d x^3}}{7168 c^2 x^4}-\frac {19 d^2 \sqrt {c+d x^3}}{1792 c^3 x}+\frac {19 d^{7/3} \sqrt {c+d x^3}}{1792 c^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}-\frac {9 \sqrt {3} d^{7/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{4096 c^{17/6}}+\frac {9 d^{7/3} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{4096 c^{17/6}}-\frac {9 d^{7/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{4096 c^{17/6}}-\frac {19 \sqrt [4]{3} \sqrt {2-\sqrt {3}} d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt {3}\right )}{3584 c^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}+\frac {19 d^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{896 \sqrt {2} \sqrt [4]{3} c^{8/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
-11/224*(d*x^3+c)^(1/2)/c/x^7-83/7168*d*(d*x^3+c)^(1/2)/c^2/x^4-19/1792*d^ 2*(d*x^3+c)^(1/2)/c^3/x+19/1792*d^(7/3)*(d*x^3+c)^(1/2)/c^3/((1+3^(1/2))*c ^(1/3)+d^(1/3)*x)+3/8*(d*x^3+c)^(1/2)/x^7/(-d*x^3+8*c)-9/4096*3^(1/2)*d^(7 /3)*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+d^(1/3)*x)/(d*x^3+c)^(1/2))/c^(17/6)+9 /4096*d^(7/3)*arctanh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/c ^(17/6)-9/4096*d^(7/3)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(17/6)-19/35 84*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)*E llipticE(((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^(8/3)/(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)+19/5376*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)*3^(3/4)/c^(8/3)/(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 = 10.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.30 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\sqrt {c+d x^3} \left (-\frac {1}{448 c x^7}-\frac {41 d}{7168 c^2 x^4}-\frac {283 d^2}{28672 c^3 x}-\frac {3 d^3 x^2}{4096 c^3 \left (-8 c+d x^3\right )}\right )+\frac {1175 d^3 x^2 \sqrt {\frac {c+d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{229376 c^3 \sqrt {c+d x^3}}-\frac {19 d^4 x^5 \sqrt {\frac {c+d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )}{143360 c^4 \sqrt {c+d x^3}} \] Input:
Integrate[(c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)^2),x]
Output:
Sqrt[c + d*x^3]*(-1/448*1/(c*x^7) - (41*d)/(7168*c^2*x^4) - (283*d^2)/(286 72*c^3*x) - (3*d^3*x^2)/(4096*c^3*(-8*c + d*x^3))) + (1175*d^3*x^2*Sqrt[(c + d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(229 376*c^3*Sqrt[c + d*x^3]) - (19*d^4*x^5*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1 /2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(143360*c^4*Sqrt[c + d*x^3])
Time = 2.03 (sec) , antiderivative size = 722, 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 = {968, 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.
| \(\displaystyle \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 968 |
| \(\displaystyle \frac {\int \frac {3 c d \left (35 d x^3+44 c\right )}{2 x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{24 c d}+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {35 d x^3+44 c}{x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{16} \left (-\frac {\int -\frac {2 c d \left (121 d x^3+166 c\right )}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{56 c^2}-\frac {11 \sqrt {c+d x^3}}{14 c x^7}\right )+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \int \frac {121 d x^3+166 c}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{28 c}-\frac {11 \sqrt {c+d x^3}}{14 c x^7}\right )+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (-\frac {\int -\frac {c d \left (415 d x^3+1216 c\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {83 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {11 \sqrt {c+d x^3}}{14 c x^7}\right )+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (\frac {\int \frac {c d \left (415 d x^3+1216 c\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {83 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {11 \sqrt {c+d x^3}}{14 c x^7}\right )+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (\frac {d \int \frac {415 d x^3+1216 c}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {83 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {11 \sqrt {c+d x^3}}{14 c x^7}\right )+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (\frac {d \left (-\frac {\int -\frac {8 c d x \left (1175 c-76 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {152 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {83 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {11 \sqrt {c+d x^3}}{14 c x^7}\right )+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (\frac {d \left (\frac {d \int \frac {x \left (1175 c-76 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {152 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {83 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {11 \sqrt {c+d x^3}}{14 c x^7}\right )+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (\frac {d \left (\frac {d \int \left (\frac {567 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {76 x}{\sqrt {d x^3+c}}\right )dx}{c}-\frac {152 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {83 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {11 \sqrt {c+d x^3}}{14 c x^7}\right )+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{16} \left (\frac {d \left (\frac {d \left (\frac {d \left (\frac {152 \sqrt {2} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {76 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{d^{2/3} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {63 \sqrt {3} \sqrt [6]{c} \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt {c+d x^3}}\right )}{2 d^{2/3}}+\frac {63 \sqrt [6]{c} \text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x^3}}\right )}{2 d^{2/3}}-\frac {63 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{2 d^{2/3}}+\frac {152 \sqrt {c+d x^3}}{d^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}\right )}{c}-\frac {152 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {83 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {11 \sqrt {c+d x^3}}{14 c x^7}\right )+\frac {3 \sqrt {c+d x^3}}{8 x^7 \left (8 c-d x^3\right )}\) |
Input:
Int[(c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)^2),x]
Output:
(3*Sqrt[c + d*x^3])/(8*x^7*(8*c - d*x^3)) + ((-11*Sqrt[c + d*x^3])/(14*c*x ^7) + (d*((-83*Sqrt[c + d*x^3])/(16*c*x^4) + (d*((-152*Sqrt[c + d*x^3])/(c *x) + (d*((152*Sqrt[c + d*x^3])/(d^(2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)* x)) - (63*Sqrt[3]*c^(1/6)*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x))/S qrt[c + d*x^3]])/(2*d^(2/3)) + (63*c^(1/6)*ArcTanh[(c^(1/3) + d^(1/3)*x)^2 /(3*c^(1/6)*Sqrt[c + d*x^3])])/(2*d^(2/3)) - (63*c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(2*d^(2/3)) - (76*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]*Sqrt[c + d*x^3]) + (152*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)))/(28*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[((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[((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 = 5.08 (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 | \(\text {Expression too large to display}\) | \(3187\) |
Input:
int((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)
Output:
-1/448*(d*x^3+c)^(1/2)/c/x^7-41/7168*d*(d*x^3+c)^(1/2)/c^2/x^4-283/28672*d ^2*(d*x^3+c)^(1/2)/c^3/x+3/4096/c^3*x^2*d^3*(d*x^3+c)^(1/2)/(-d*x^3+8*c)-1 9/5376*I/c^3*d^2*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)))-3/204 8*I/c^3*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*_a lpha^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. \(2 (507) = 1014\).
Time = 1.87 (sec) , antiderivative size = 2582, normalized size of antiderivative = 3.65 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c)^2,x, algorithm="fricas")
Output:
-1/114688*(1216*(d^3*x^10 - 8*c*d^2*x^7)*sqrt(d)*weierstrassZeta(0, -4*c/d , weierstrassPInverse(0, -4*c/d, x)) - 21*(c^3*d*x^10 - 8*c^4*x^7 + sqrt(- 3)*(c^3*d*x^10 - 8*c^4*x^7))*(d^14/c^17)^(1/6)*log(6561*(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^12*d^4*x^7 + 64*c^13* d^3*x^4 + 32*c^14*d^2*x + sqrt(-3)*(5*c^12*d^4*x^7 + 64*c^13*d^3*x^4 + 32* c^14*d^2*x))*(d^14/c^17)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^15*d*x^5 + 32*c ^16*x^2 - sqrt(-3)*(5*c^15*d*x^5 + 32*c^16*x^2))*(d^14/c^17)^(5/6) - 2*(7* c^9*d^6*x^6 + 152*c^10*d^5*x^3 + 64*c^11*d^4)*sqrt(d^14/c^17) + (c^3*d^11* x^7 + 80*c^4*d^10*x^4 + 160*c^5*d^9*x + sqrt(-3)*(c^3*d^11*x^7 + 80*c^4*d^ 10*x^4 + 160*c^5*d^9*x))*(d^14/c^17)^(1/6)) - 9*(c^6*d^9*x^8 + 38*c^7*d^8* x^5 + 64*c^8*d^7*x^2 - sqrt(-3)*(c^6*d^9*x^8 + 38*c^7*d^8*x^5 + 64*c^8*d^7 *x^2))*(d^14/c^17)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^ 3)) + 21*(c^3*d*x^10 - 8*c^4*x^7 + sqrt(-3)*(c^3*d*x^10 - 8*c^4*x^7))*(d^1 4/c^17)^(1/6)*log(6561*(d^14*x^9 + 318*c*d^13*x^6 + 1200*c^2*d^12*x^3 + 64 0*c^3*d^11 - 9*(5*c^12*d^4*x^7 + 64*c^13*d^3*x^4 + 32*c^14*d^2*x + sqrt(-3 )*(5*c^12*d^4*x^7 + 64*c^13*d^3*x^4 + 32*c^14*d^2*x))*(d^14/c^17)^(2/3) - 3*sqrt(d*x^3 + c)*(6*(5*c^15*d*x^5 + 32*c^16*x^2 - sqrt(-3)*(5*c^15*d*x^5 + 32*c^16*x^2))*(d^14/c^17)^(5/6) - 2*(7*c^9*d^6*x^6 + 152*c^10*d^5*x^3 + 64*c^11*d^4)*sqrt(d^14/c^17) + (c^3*d^11*x^7 + 80*c^4*d^10*x^4 + 160*c^5*d ^9*x + sqrt(-3)*(c^3*d^11*x^7 + 80*c^4*d^10*x^4 + 160*c^5*d^9*x))*(d^14...
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate((d*x**3+c)**(3/2)/x**8/(-d*x**3+8*c)**2,x)
Output:
Timed out
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{8}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c)^2,x, algorithm="maxima")
Output:
integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)^2*x^8), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (d x^{3} - 8 \, c\right )}^{2} x^{8}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c)^2,x, algorithm="giac")
Output:
integrate((d*x^3 + c)^(3/2)/((d*x^3 - 8*c)^2*x^8), x)
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\int \frac {{\left (d\,x^3+c\right )}^{3/2}}{x^8\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int((c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)^2),x)
Output:
int((c + d*x^3)^(3/2)/(x^8*(8*c - d*x^3)^2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{x^8 \left (8 c-d x^3\right )^2} \, dx=\frac {-4 \sqrt {d \,x^{3}+c}\, c^{2}-42 \sqrt {d \,x^{3}+c}\, d^{2} x^{6}+2496 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{3} x^{14}-15 c \,d^{2} x^{11}+48 c^{2} d \,x^{8}+64 c^{3} x^{5}}d x \right ) c^{4} d \,x^{7}-312 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{3} x^{14}-15 c \,d^{2} x^{11}+48 c^{2} d \,x^{8}+64 c^{3} x^{5}}d x \right ) c^{3} d^{2} x^{10}-624 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{3} x^{11}-15 c \,d^{2} x^{8}+48 c^{2} d \,x^{5}+64 c^{3} x^{2}}d x \right ) c^{3} d^{2} x^{7}+78 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{3} x^{11}-15 c \,d^{2} x^{8}+48 c^{2} d \,x^{5}+64 c^{3} x^{2}}d x \right ) c^{2} d^{3} x^{10}+840 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c \,d^{4} x^{7}-105 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) d^{5} x^{10}+2688 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{2} d^{3} x^{7}-336 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c \,d^{4} x^{10}}{224 c^{2} x^{7} \left (-d \,x^{3}+8 c \right )} \] Input:
int((d*x^3+c)^(3/2)/x^8/(-d*x^3+8*c)^2,x)
Output:
( - 4*sqrt(c + d*x**3)*c**2 - 42*sqrt(c + d*x**3)*d**2*x**6 + 2496*int(sqr t(c + d*x**3)/(64*c**3*x**5 + 48*c**2*d*x**8 - 15*c*d**2*x**11 + d**3*x**1 4),x)*c**4*d*x**7 - 312*int(sqrt(c + d*x**3)/(64*c**3*x**5 + 48*c**2*d*x** 8 - 15*c*d**2*x**11 + d**3*x**14),x)*c**3*d**2*x**10 - 624*int(sqrt(c + d* x**3)/(64*c**3*x**2 + 48*c**2*d*x**5 - 15*c*d**2*x**8 + d**3*x**11),x)*c** 3*d**2*x**7 + 78*int(sqrt(c + d*x**3)/(64*c**3*x**2 + 48*c**2*d*x**5 - 15* c*d**2*x**8 + d**3*x**11),x)*c**2*d**3*x**10 + 840*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**4*x** 7 - 105*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**5*x**10 + 2688*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**3*x**7 - 336*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**4*x**10)/(224*c**2*x**7*(8*c - d*x**3))