Integrand size = 27, antiderivative size = 732 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {5}{648 c^3 x^7 \sqrt {c+d x^3}}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {191 \sqrt {c+d x^3}}{18144 c^4 x^7}+\frac {8257 d \sqrt {c+d x^3}}{580608 c^5 x^4}-\frac {5179 d^2 \sqrt {c+d x^3}}{145152 c^6 x}+\frac {5179 d^{7/3} \sqrt {c+d x^3}}{145152 c^6 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {7 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 )}{331776 \sqrt {3} c^{35/6}}+\frac {7 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 )}{995328 c^{35/6}}-\frac {7 d^{7/3} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{995328 c^{35/6}}-\frac {5179 \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 )}{96768\ 3^{3/4} c^{17/3} \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 {5179 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 )}{72576 \sqrt {2} \sqrt [4]{3} c^{17/3} \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^7/(d*x^3+c)^(1/2)+1/216/c^2/x^7/(-d*x^3+8*c)/(d*x^3+c)^(1/2)-1 91/18144*(d*x^3+c)^(1/2)/c^4/x^7+8257/580608*d*(d*x^3+c)^(1/2)/c^5/x^4-517 9/145152*d^2*(d*x^3+c)^(1/2)/c^6/x+5179/145152*d^(7/3)*(d*x^3+c)^(1/2)/c^6 /((1+3^(1/2))*c^(1/3)+d^(1/3)*x)-7/995328*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^(35/6)+7/995328*d^(7/3)*arct anh(1/3*(c^(1/3)+d^(1/3)*x)^2/c^(1/6)/(d*x^3+c)^(1/2))/c^(35/6)-7/995328*d ^(7/3)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(35/6)-5179/290304*(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)*3^(1/4 )/c^(17/3)/(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)+5179/435456*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)*Ell ipticF(((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^(17/3)/(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 = 10.21 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.29 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {829375 c d^3 x^9 \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )-8 \left (20 c \left (10368 c^4-18792 c^3 d x^3+101817 c^2 d^2 x^6+153269 c d^3 x^9-20716 d^4 x^{12}\right )+5179 d^4 x^{12} \left (8 c-d x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},\frac {d x^3}{8 c}\right )\right )}{92897280 c^7 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \] Input:
Integrate[1/(x^8*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
(829375*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*(10368*c^4 - 18792*c^3*d*x^3 + 101817*c^2*d^2*x^6 + 153269*c*d^3*x^9 - 20716*d^4*x^12) + 5179*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)]))/(92897280*c^7*x^7*(8*c - d*x^3)*Sqrt[c + d*x^3])
Time = 2.19 (sec) , antiderivative size = 757, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {972, 27, 1049, 27, 1053, 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.
| \(\displaystyle \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {\int \frac {d \left (23 d x^3+68 c\right )}{2 x^8 \left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx}{216 c^2 d}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {23 d x^3+68 c}{x^8 \left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle \frac {\frac {10}{3 c x^7 \sqrt {c+d x^3}}-\frac {2 \int -\frac {9 c d \left (764 c-85 d x^3\right )}{2 x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{27 c^2 d}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {764 c-85 d x^3}{x^8 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{3 c}+\frac {10}{3 c x^7 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {2 c d \left (16514 c-2101 d x^3\right )}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{56 c^2}-\frac {191 \sqrt {c+d x^3}}{14 c x^7}}{3 c}+\frac {10}{3 c x^7 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {d \int \frac {16514 c-2101 d x^3}{x^5 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{28 c}-\frac {191 \sqrt {c+d x^3}}{14 c x^7}}{3 c}+\frac {10}{3 c x^7 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {-\frac {d \left (-\frac {\int \frac {c d \left (331456 c-41285 d x^3\right )}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c^2}-\frac {8257 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {191 \sqrt {c+d x^3}}{14 c x^7}}{3 c}+\frac {10}{3 c x^7 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {d \left (-\frac {d \int \frac {331456 c-41285 d x^3}{x^2 \left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{32 c}-\frac {8257 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {191 \sqrt {c+d x^3}}{14 c x^7}}{3 c}+\frac {10}{3 c x^7 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {-\frac {d \left (-\frac {d \left (-\frac {\int -\frac {8 c d x \left (165875 c-20716 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{8 c^2}-\frac {41432 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {8257 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {191 \sqrt {c+d x^3}}{14 c x^7}}{3 c}+\frac {10}{3 c x^7 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {d \left (-\frac {d \left (\frac {d \int \frac {x \left (165875 c-20716 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx}{c}-\frac {41432 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {8257 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {191 \sqrt {c+d x^3}}{14 c x^7}}{3 c}+\frac {10}{3 c x^7 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {\frac {-\frac {d \left (-\frac {d \left (\frac {d \int \left (\frac {147 c x}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}+\frac {20716 x}{\sqrt {d x^3+c}}\right )dx}{c}-\frac {41432 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {8257 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {191 \sqrt {c+d x^3}}{14 c x^7}}{3 c}+\frac {10}{3 c x^7 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {d \left (-\frac {d \left (\frac {d \left (\frac {41432 \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 {20716 \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 {49 \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 {49 \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 {49 \sqrt [6]{c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{6 d^{2/3}}+\frac {41432 \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 {41432 \sqrt {c+d x^3}}{c x}\right )}{32 c}-\frac {8257 \sqrt {c+d x^3}}{16 c x^4}\right )}{28 c}-\frac {191 \sqrt {c+d x^3}}{14 c x^7}}{3 c}+\frac {10}{3 c x^7 \sqrt {c+d x^3}}}{432 c^2}+\frac {1}{216 c^2 x^7 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}\) |
Input:
Int[1/(x^8*(8*c - d*x^3)^2*(c + d*x^3)^(3/2)),x]
Output:
1/(216*c^2*x^7*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (10/(3*c*x^7*Sqrt[c + d*x^ 3]) + ((-191*Sqrt[c + d*x^3])/(14*c*x^7) - (d*((-8257*Sqrt[c + d*x^3])/(16 *c*x^4) - (d*((-41432*Sqrt[c + d*x^3])/(c*x) + (d*((41432*Sqrt[c + d*x^3]) /(d^(2/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (49*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)) + (49*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)) - (49*c^(1/6)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(6*d^(2/3) ) - (20716*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]) + (41 432*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)) /(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 = 5.80 (sec) , antiderivative size = 962, normalized size of antiderivative = 1.31
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(962\) |
| risch | \(\text {Expression too large to display}\) | \(2243\) |
| default | \(\text {Expression too large to display}\) | \(3300\) |
Input:
int(1/x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/448*(d*x^3+c)^(1/2)/c^4/x^7+43/7168*d*(d*x^3+c)^(1/2)/c^5/x^4-787/28672 *d^2*(d*x^3+c)^(1/2)/c^6/x-2/243*d^3/c^6*x^2/((x^3+c/d)*d)^(1/2)+1/995328* d^3*x^2/c^6*(d*x^3+c)^(1/2)/(-d*x^3+8*c)-5179/435456*I/c^6*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)))-7/1492992*I/c^6*2^(1/2)*sum(1/_al pha*(-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)^(...
Leaf count of result is larger than twice the leaf count of optimal. 2725 vs. \(2 (527) = 1054\).
Time = 5.80 (sec) , antiderivative size = 2725, normalized size of antiderivative = 3.72 \[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x, algorithm="fricas")
Output:
-1/83607552*(2983104*(d^4*x^13 - 7*c*d^3*x^10 - 8*c^2*d^2*x^7)*sqrt(d)*wei erstrassZeta(0, -4*c/d, weierstrassPInverse(0, -4*c/d, x)) - 49*(c^6*d^2*x ^13 - 7*c^7*d*x^10 - 8*c^8*x^7 + sqrt(-3)*(c^6*d^2*x^13 - 7*c^7*d*x^10 - 8 *c^8*x^7))*(d^14/c^35)^(1/6)*log(16807*(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^24*d^4*x^7 + 64*c^25*d^3*x^4 + 32*c^26 *d^2*x + sqrt(-3)*(5*c^24*d^4*x^7 + 64*c^25*d^3*x^4 + 32*c^26*d^2*x))*(d^1 4/c^35)^(2/3) + 3*sqrt(d*x^3 + c)*(6*(5*c^30*d*x^5 + 32*c^31*x^2 - sqrt(-3 )*(5*c^30*d*x^5 + 32*c^31*x^2))*(d^14/c^35)^(5/6) - 2*(7*c^18*d^6*x^6 + 15 2*c^19*d^5*x^3 + 64*c^20*d^4)*sqrt(d^14/c^35) + (c^6*d^11*x^7 + 80*c^7*d^1 0*x^4 + 160*c^8*d^9*x + sqrt(-3)*(c^6*d^11*x^7 + 80*c^7*d^10*x^4 + 160*c^8 *d^9*x))*(d^14/c^35)^(1/6)) - 9*(c^12*d^9*x^8 + 38*c^13*d^8*x^5 + 64*c^14* d^7*x^2 - sqrt(-3)*(c^12*d^9*x^8 + 38*c^13*d^8*x^5 + 64*c^14*d^7*x^2))*(d^ 14/c^35)^(1/3))/(d^3*x^9 - 24*c*d^2*x^6 + 192*c^2*d*x^3 - 512*c^3)) + 49*( c^6*d^2*x^13 - 7*c^7*d*x^10 - 8*c^8*x^7 + sqrt(-3)*(c^6*d^2*x^13 - 7*c^7*d *x^10 - 8*c^8*x^7))*(d^14/c^35)^(1/6)*log(16807*(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^24*d^4*x^7 + 64*c^25*d^3*x^4 + 32*c^26*d^2*x + sqrt(-3)*(5*c^24*d^4*x^7 + 64*c^25*d^3*x^4 + 32*c^26*d^2 *x))*(d^14/c^35)^(2/3) - 3*sqrt(d*x^3 + c)*(6*(5*c^30*d*x^5 + 32*c^31*x^2 - sqrt(-3)*(5*c^30*d*x^5 + 32*c^31*x^2))*(d^14/c^35)^(5/6) - 2*(7*c^18*d^6 *x^6 + 152*c^19*d^5*x^3 + 64*c^20*d^4)*sqrt(d^14/c^35) + (c^6*d^11*x^7 ...
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^{8} \left (- 8 c + d x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**8/(-d*x**3+8*c)**2/(d*x**3+c)**(3/2),x)
Output:
Integral(1/(x**8*(-8*c + d*x**3)**2*(c + d*x**3)**(3/2)), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{8}} \,d x } \] Input:
integrate(1/x^8/(-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^8), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} {\left (d x^{3} - 8 \, c\right )}^{2} x^{8}} \,d x } \] Input:
integrate(1/x^8/(-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^8), x)
Timed out. \[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{x^8\,{\left (d\,x^3+c\right )}^{3/2}\,{\left (8\,c-d\,x^3\right )}^2} \,d x \] Input:
int(1/(x^8*(c + d*x^3)^(3/2)*(8*c - d*x^3)^2),x)
Output:
int(1/(x^8*(c + d*x^3)^(3/2)*(8*c - d*x^3)^2), x)
\[ \int \frac {1}{x^8 \left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {-32 \sqrt {d \,x^{3}+c}\, c^{2}+58 \sqrt {d \,x^{3}+c}\, c d \,x^{3}-46 \sqrt {d \,x^{3}+c}\, d^{2} x^{6}+17168 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{4} x^{14}-14 c \,d^{3} x^{11}+33 c^{2} d^{2} x^{8}+112 c^{3} d \,x^{5}+64 c^{4} x^{2}}d x \right ) c^{4} d^{2} x^{7}+15022 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{4} x^{14}-14 c \,d^{3} x^{11}+33 c^{2} d^{2} x^{8}+112 c^{3} d \,x^{5}+64 c^{4} x^{2}}d x \right ) c^{3} d^{3} x^{10}-2146 \left (\int \frac {\sqrt {d \,x^{3}+c}}{d^{4} x^{14}-14 c \,d^{3} x^{11}+33 c^{2} d^{2} x^{8}+112 c^{3} d \,x^{5}+64 c^{4} x^{2}}d x \right ) c^{2} d^{4} x^{13}+2024 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{4} x^{7}+1771 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{5} x^{10}-253 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) d^{6} x^{13}-9832 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{3} d^{3} x^{7}-8603 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c^{2} d^{4} x^{10}+1229 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{d^{4} x^{12}-14 c \,d^{3} x^{9}+33 c^{2} d^{2} x^{6}+112 c^{3} d \,x^{3}+64 c^{4}}d x \right ) c \,d^{5} x^{13}}{1792 c^{4} x^{7} \left (-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}\right )} \] Input:
int(1/x^8/(-d*x^3+8*c)^2/(d*x^3+c)^(3/2),x)
Output:
( - 32*sqrt(c + d*x**3)*c**2 + 58*sqrt(c + d*x**3)*c*d*x**3 - 46*sqrt(c + d*x**3)*d**2*x**6 + 17168*int(sqrt(c + d*x**3)/(64*c**4*x**2 + 112*c**3*d* x**5 + 33*c**2*d**2*x**8 - 14*c*d**3*x**11 + d**4*x**14),x)*c**4*d**2*x**7 + 15022*int(sqrt(c + d*x**3)/(64*c**4*x**2 + 112*c**3*d*x**5 + 33*c**2*d* *2*x**8 - 14*c*d**3*x**11 + d**4*x**14),x)*c**3*d**3*x**10 - 2146*int(sqrt (c + d*x**3)/(64*c**4*x**2 + 112*c**3*d*x**5 + 33*c**2*d**2*x**8 - 14*c*d* *3*x**11 + d**4*x**14),x)*c**2*d**4*x**13 + 2024*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**4*x**7 + 1771*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*d**5* x**10 - 253*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**6*x**13 - 9832*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**3*x**7 - 8603*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**4*x**10 + 1229*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**5*x**13)/(1 792*c**4*x**7*(8*c**2 + 7*c*d*x**3 - d**2*x**6))