Integrand size = 14, antiderivative size = 232 \[ \int \frac {1}{\left (-b x^2+a x^8\right )^4} \, dx=\frac {-1296 b^4-32400 a b^3 x^6+114475 a^2 b^2 x^{12}-123500 a^3 b x^{18}+43225 a^4 x^{24}}{9072 b^5 x^7 \left (b-a x^6\right )^3}-\frac {6175 a^{7/6} \arctan \left (\frac {-\sqrt [3]{b}+\sqrt [3]{a} x^2}{\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x}\right )}{2592 \sqrt {3} b^{31/6}}+\frac {6175 a^{7/6} \text {arctanh}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3888 b^{31/6}}-\frac {6175 a^{7/6} \log \left (\sqrt [3]{b}-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{15552 b^{31/6}}+\frac {6175 a^{7/6} \log \left (\sqrt [3]{b}+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{15552 b^{31/6}} \] Output:
1/9072*(43225*a^4*x^24-123500*a^3*b*x^18+114475*a^2*b^2*x^12-32400*a*b^3*x ^6-1296*b^4)/b^5/x^7/(-a*x^6+b)^3-6175/7776*a^(7/6)*arctan(1/3*(-b^(1/3)+a ^(1/3)*x^2)*3^(1/2)/a^(1/6)/b^(1/6)/x)*3^(1/2)/b^(31/6)+6175/3888*a^(7/6)* arctanh(a^(1/6)*x/b^(1/6))/b^(31/6)-6175/15552*a^(7/6)*ln(b^(1/3)-a^(1/6)* b^(1/6)*x+a^(1/3)*x^2)/b^(31/6)+6175/15552*a^(7/6)*ln(b^(1/3)+a^(1/6)*b^(1 /6)*x+a^(1/3)*x^2)/b^(31/6)
Time = 0.17 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (-b x^2+a x^8\right )^4} \, dx=\frac {-\frac {15552 b^{7/6}}{x^7}-\frac {435456 a \sqrt [6]{b}}{x}+\frac {24696 a^2 b^{7/6} x^5}{\left (b-a x^6\right )^2}+\frac {83244 a^2 \sqrt [6]{b} x^5}{b-a x^6}-\frac {6048 a^2 b^{13/6} x^5}{\left (-b+a x^6\right )^3}+86450 \sqrt {3} a^{7/6} \arctan \left (\frac {1-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}}{\sqrt {3}}\right )-86450 \sqrt {3} a^{7/6} \arctan \left (\frac {1+\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}}{\sqrt {3}}\right )-86450 a^{7/6} \log \left (\sqrt [6]{b}-\sqrt [6]{a} x\right )+86450 a^{7/6} \log \left (\sqrt [6]{b}+\sqrt [6]{a} x\right )-43225 a^{7/6} \log \left (\sqrt [3]{b}-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )+43225 a^{7/6} \log \left (\sqrt [3]{b}+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{108864 b^{31/6}} \] Input:
Integrate[(-(b*x^2) + a*x^8)^(-4),x]
Output:
((-15552*b^(7/6))/x^7 - (435456*a*b^(1/6))/x + (24696*a^2*b^(7/6)*x^5)/(b - a*x^6)^2 + (83244*a^2*b^(1/6)*x^5)/(b - a*x^6) - (6048*a^2*b^(13/6)*x^5) /(-b + a*x^6)^3 + 86450*Sqrt[3]*a^(7/6)*ArcTan[(1 - (2*a^(1/6)*x)/b^(1/6)) /Sqrt[3]] - 86450*Sqrt[3]*a^(7/6)*ArcTan[(1 + (2*a^(1/6)*x)/b^(1/6))/Sqrt[ 3]] - 86450*a^(7/6)*Log[b^(1/6) - a^(1/6)*x] + 86450*a^(7/6)*Log[b^(1/6) + a^(1/6)*x] - 43225*a^(7/6)*Log[b^(1/3) - a^(1/6)*b^(1/6)*x + a^(1/3)*x^2] + 43225*a^(7/6)*Log[b^(1/3) + a^(1/6)*b^(1/6)*x + a^(1/3)*x^2])/(108864*b ^(31/6))
Time = 1.02 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.35, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {2026, 819, 25, 819, 819, 847, 847, 825, 27, 221, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\left (a x^8-b x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {1}{x^8 \left (a x^6-b\right )^4}dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {1}{18 b x^7 \left (b-a x^6\right )^3}-\frac {25 \int -\frac {1}{x^8 \left (b-a x^6\right )^3}dx}{18 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {25 \int \frac {1}{x^8 \left (b-a x^6\right )^3}dx}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {25 \left (\frac {19 \int \frac {1}{x^8 \left (b-a x^6\right )^2}dx}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \int \frac {1}{x^8 \left (b-a x^6\right )}dx}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \int \frac {1}{x^2 \left (b-a x^6\right )}dx}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \int \frac {x^4}{b-a x^6}dx}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \left (\frac {\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{a} x^2}dx}{3 a^{2/3}}+\frac {\int -\frac {\sqrt [6]{a} x+\sqrt [6]{b}}{2 \left (\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}\right )}dx}{3 a^{2/3} \sqrt [6]{b}}+\frac {\int -\frac {\sqrt [6]{b}-\sqrt [6]{a} x}{2 \left (\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}\right )}dx}{3 a^{2/3} \sqrt [6]{b}}\right )}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \left (\frac {\int \frac {1}{\sqrt [3]{b}-\sqrt [3]{a} x^2}dx}{3 a^{2/3}}-\frac {\int \frac {\sqrt [6]{a} x+\sqrt [6]{b}}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 a^{2/3} \sqrt [6]{b}}-\frac {\int \frac {\sqrt [6]{b}-\sqrt [6]{a} x}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 a^{2/3} \sqrt [6]{b}}\right )}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \left (-\frac {\int \frac {\sqrt [6]{a} x+\sqrt [6]{b}}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 a^{2/3} \sqrt [6]{b}}-\frac {\int \frac {\sqrt [6]{b}-\sqrt [6]{a} x}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 a^{2/3} \sqrt [6]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\int -\frac {\sqrt [6]{a} \left (\sqrt [6]{b}-2 \sqrt [6]{a} x\right )}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 a^{2/3} \sqrt [6]{b}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a} x+\sqrt [6]{b}\right )}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 a^{2/3} \sqrt [6]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\int \frac {\sqrt [6]{a} \left (\sqrt [6]{b}-2 \sqrt [6]{a} x\right )}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 a^{2/3} \sqrt [6]{b}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a} x+\sqrt [6]{b}\right )}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 a^{2/3} \sqrt [6]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \left (-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {1}{2} \int \frac {\sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 a^{2/3} \sqrt [6]{b}}-\frac {\frac {3}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {1}{2} \int \frac {2 \sqrt [6]{a} x+\sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 a^{2/3} \sqrt [6]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \left (-\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )^2-3}d\left (1-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{\sqrt [6]{a}}-\frac {1}{2} \int \frac {\sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 a^{2/3} \sqrt [6]{b}}-\frac {-\frac {1}{2} \int \frac {2 \sqrt [6]{a} x+\sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {3 \int \frac {1}{-\left (\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}+1\right )^2-3}d\left (\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}+1\right )}{\sqrt [6]{a}}}{6 a^{2/3} \sqrt [6]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \left (-\frac {-\frac {1}{2} \int \frac {\sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}}{\sqrt {3}}\right )}{\sqrt [6]{a}}}{6 a^{2/3} \sqrt [6]{b}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}+1}{\sqrt {3}}\right )}{\sqrt [6]{a}}-\frac {1}{2} \int \frac {2 \sqrt [6]{a} x+\sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 a^{2/3} \sqrt [6]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {25 \left (\frac {19 \left (\frac {13 \left (\frac {a \left (\frac {a \left (-\frac {\frac {\log \left (-\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{2 \sqrt [6]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}}{\sqrt {3}}\right )}{\sqrt [6]{a}}}{6 a^{2/3} \sqrt [6]{b}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}+1}{\sqrt {3}}\right )}{\sqrt [6]{a}}-\frac {\log \left (\sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{2 \sqrt [6]{a}}}{6 a^{2/3} \sqrt [6]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}-\frac {1}{b x}\right )}{b}-\frac {1}{7 b x^7}\right )}{6 b}+\frac {1}{6 b x^7 \left (b-a x^6\right )}\right )}{12 b}+\frac {1}{12 b x^7 \left (b-a x^6\right )^2}\right )}{18 b}+\frac {1}{18 b x^7 \left (b-a x^6\right )^3}\) |
Input:
Int[(-(b*x^2) + a*x^8)^(-4),x]
Output:
1/(18*b*x^7*(b - a*x^6)^3) + (25*(1/(12*b*x^7*(b - a*x^6)^2) + (19*(1/(6*b *x^7*(b - a*x^6)) + (13*(-1/7*1/(b*x^7) + (a*(-(1/(b*x)) + (a*(ArcTanh[(a^ (1/6)*x)/b^(1/6)]/(3*a^(5/6)*b^(1/6)) - (-((Sqrt[3]*ArcTan[(1 - (2*a^(1/6) *x)/b^(1/6))/Sqrt[3]])/a^(1/6)) + Log[b^(1/3) - a^(1/6)*b^(1/6)*x + a^(1/3 )*x^2]/(2*a^(1/6)))/(6*a^(2/3)*b^(1/6)) - ((Sqrt[3]*ArcTan[(1 + (2*a^(1/6) *x)/b^(1/6))/Sqrt[3]])/a^(1/6) - Log[b^(1/3) + a^(1/6)*b^(1/6)*x + a^(1/3) *x^2]/(2*a^(1/6)))/(6*a^(2/3)*b^(1/6))))/b))/b))/(6*b)))/(12*b)))/(18*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k *m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m)) Int[1/ (r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 ] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(\frac {\frac {25 a \,x^{6}}{7 b^{2}}-\frac {114475 a^{2} x^{12}}{9072 b^{3}}+\frac {30875 a^{3} x^{18}}{2268 b^{4}}-\frac {6175 a^{4} x^{24}}{1296 b^{5}}+\frac {1}{7 b}}{x^{7} \left (a \,x^{6}-b \right )^{3}}+\frac {6175 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{31} \textit {\_Z}^{6}-a^{7}\right )}{\sum }\textit {\_R} \ln \left (\left (-7 \textit {\_R}^{6} b^{31}+6 a^{7}\right ) x -a \,b^{26} \textit {\_R}^{5}\right )\right )}{7776}\) | \(113\) |
| default | \(-\frac {a^{2} \left (\frac {\frac {1357}{1296} b^{2} x^{5}-\frac {569}{324} a b \,x^{11}+\frac {991}{1296} a^{2} x^{17}}{\left (a \,x^{6}-b \right )^{3}}+\frac {6175 \ln \left (-x +\left (\frac {b}{a}\right )^{\frac {1}{6}}\right )}{7776 a \left (\frac {b}{a}\right )^{\frac {1}{6}}}+\frac {6175 \left (\frac {b}{a}\right )^{\frac {5}{6}} \ln \left (\left (\frac {b}{a}\right )^{\frac {1}{6}} x -x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{15552 b}+\frac {6175 \sqrt {3}\, \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 x \sqrt {3}}{3 \left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{7776 a \left (\frac {b}{a}\right )^{\frac {1}{6}}}-\frac {6175 \left (\frac {b}{a}\right )^{\frac {5}{6}} \ln \left (x^{2}+\left (\frac {b}{a}\right )^{\frac {1}{6}} x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{15552 b}+\frac {6175 \sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3 \left (\frac {b}{a}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{7776 a \left (\frac {b}{a}\right )^{\frac {1}{6}}}-\frac {6175 \ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{6}}\right )}{7776 a \left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{b^{5}}-\frac {1}{7 b^{4} x^{7}}-\frac {4 a}{b^{5} x}\) | \(250\) |
Input:
int(1/(a*x^8-b*x^2)^4,x,method=_RETURNVERBOSE)
Output:
(25/7*a/b^2*x^6-114475/9072/b^3*a^2*x^12+30875/2268*a^3/b^4*x^18-6175/1296 *a^4/b^5*x^24+1/7/b)/x^7/(a*x^6-b)^3+6175/7776*sum(_R*ln((-7*_R^6*b^31+6*a ^7)*x-a*b^26*_R^5),_R=RootOf(_Z^6*b^31-a^7))
Leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (176) = 352\).
Time = 0.08 (sec) , antiderivative size = 728, normalized size of antiderivative = 3.14 \[ \int \frac {1}{\left (-b x^2+a x^8\right )^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(a*x^8-b*x^2)^4,x, algorithm="fricas")
Output:
-1/108864*(518700*a^4*x^24 - 1482000*a^3*b*x^18 + 1373700*a^2*b^2*x^12 - 3 88800*a*b^3*x^6 - 15552*b^4 - 86450*(a^3*b^5*x^25 - 3*a^2*b^6*x^19 + 3*a*b ^7*x^13 - b^8*x^7)*(a^7/b^31)^(1/6)*log(8978107675849609375*b^26*(a^7/b^31 )^(5/6) + 8978107675849609375*a^6*x) + 86450*(a^3*b^5*x^25 - 3*a^2*b^6*x^1 9 + 3*a*b^7*x^13 - b^8*x^7)*(a^7/b^31)^(1/6)*log(-8978107675849609375*b^26 *(a^7/b^31)^(5/6) + 8978107675849609375*a^6*x) - 43225*(a^3*b^5*x^25 - 3*a ^2*b^6*x^19 + 3*a*b^7*x^13 - b^8*x^7 - sqrt(-3)*(a^3*b^5*x^25 - 3*a^2*b^6* x^19 + 3*a*b^7*x^13 - b^8*x^7))*(a^7/b^31)^(1/6)*log(8978107675849609375*a ^6*x + 8978107675849609375/2*(sqrt(-3)*b^26 + b^26)*(a^7/b^31)^(5/6)) + 43 225*(a^3*b^5*x^25 - 3*a^2*b^6*x^19 + 3*a*b^7*x^13 - b^8*x^7 - sqrt(-3)*(a^ 3*b^5*x^25 - 3*a^2*b^6*x^19 + 3*a*b^7*x^13 - b^8*x^7))*(a^7/b^31)^(1/6)*lo g(8978107675849609375*a^6*x - 8978107675849609375/2*(sqrt(-3)*b^26 + b^26) *(a^7/b^31)^(5/6)) + 43225*(a^3*b^5*x^25 - 3*a^2*b^6*x^19 + 3*a*b^7*x^13 - b^8*x^7 + sqrt(-3)*(a^3*b^5*x^25 - 3*a^2*b^6*x^19 + 3*a*b^7*x^13 - b^8*x^ 7))*(a^7/b^31)^(1/6)*log(8978107675849609375*a^6*x + 8978107675849609375/2 *(sqrt(-3)*b^26 - b^26)*(a^7/b^31)^(5/6)) - 43225*(a^3*b^5*x^25 - 3*a^2*b^ 6*x^19 + 3*a*b^7*x^13 - b^8*x^7 + sqrt(-3)*(a^3*b^5*x^25 - 3*a^2*b^6*x^19 + 3*a*b^7*x^13 - b^8*x^7))*(a^7/b^31)^(1/6)*log(8978107675849609375*a^6*x - 8978107675849609375/2*(sqrt(-3)*b^26 - b^26)*(a^7/b^31)^(5/6)))/(a^3*b^5 *x^25 - 3*a^2*b^6*x^19 + 3*a*b^7*x^13 - b^8*x^7)
Time = 0.53 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (-b x^2+a x^8\right )^4} \, dx=\operatorname {RootSum} {\left (221073919720733357899776 t^{6} b^{31} - 55439814898371337890625 a^{7}, \left ( t \mapsto t \log {\left (\frac {28430288029929701376 t^{5} b^{26}}{8978107675849609375 a^{6}} + x \right )} \right )\right )} + \frac {- 43225 a^{4} x^{24} + 123500 a^{3} b x^{18} - 114475 a^{2} b^{2} x^{12} + 32400 a b^{3} x^{6} + 1296 b^{4}}{9072 a^{3} b^{5} x^{25} - 27216 a^{2} b^{6} x^{19} + 27216 a b^{7} x^{13} - 9072 b^{8} x^{7}} \] Input:
integrate(1/(a*x**8-b*x**2)**4,x)
Output:
RootSum(221073919720733357899776*_t**6*b**31 - 55439814898371337890625*a** 7, Lambda(_t, _t*log(28430288029929701376*_t**5*b**26/(8978107675849609375 *a**6) + x))) + (-43225*a**4*x**24 + 123500*a**3*b*x**18 - 114475*a**2*b** 2*x**12 + 32400*a*b**3*x**6 + 1296*b**4)/(9072*a**3*b**5*x**25 - 27216*a** 2*b**6*x**19 + 27216*a*b**7*x**13 - 9072*b**8*x**7)
Time = 0.11 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (-b x^2+a x^8\right )^4} \, dx=-\frac {43225 \, a^{4} x^{24} - 123500 \, a^{3} b x^{18} + 114475 \, a^{2} b^{2} x^{12} - 32400 \, a b^{3} x^{6} - 1296 \, b^{4}}{9072 \, {\left (a^{3} b^{5} x^{25} - 3 \, a^{2} b^{6} x^{19} + 3 \, a b^{7} x^{13} - b^{8} x^{7}\right )}} - \frac {6175 \, a^{2} {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}}\right )}{a \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}}\right )}{a \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} + x \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {2}{3}}\right )}{a \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {2}{3}}\right )}{a \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x + \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}\right )}{a \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}} + \frac {2 \, \log \left (x - \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}\right )}{a \left (\frac {\sqrt {b}}{\sqrt {a}}\right )^{\frac {1}{3}}}\right )}}{15552 \, b^{5}} \] Input:
integrate(1/(a*x^8-b*x^2)^4,x, algorithm="maxima")
Output:
-1/9072*(43225*a^4*x^24 - 123500*a^3*b*x^18 + 114475*a^2*b^2*x^12 - 32400* a*b^3*x^6 - 1296*b^4)/(a^3*b^5*x^25 - 3*a^2*b^6*x^19 + 3*a*b^7*x^13 - b^8* x^7) - 6175/15552*a^2*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + (sqrt(b)/sqrt(a ))^(1/3))/(sqrt(b)/sqrt(a))^(1/3))/(a*(sqrt(b)/sqrt(a))^(1/3)) + 2*sqrt(3) *arctan(1/3*sqrt(3)*(2*x - (sqrt(b)/sqrt(a))^(1/3))/(sqrt(b)/sqrt(a))^(1/3 ))/(a*(sqrt(b)/sqrt(a))^(1/3)) - log(x^2 + x*(sqrt(b)/sqrt(a))^(1/3) + (sq rt(b)/sqrt(a))^(2/3))/(a*(sqrt(b)/sqrt(a))^(1/3)) + log(x^2 - x*(sqrt(b)/s qrt(a))^(1/3) + (sqrt(b)/sqrt(a))^(2/3))/(a*(sqrt(b)/sqrt(a))^(1/3)) - 2*l og(x + (sqrt(b)/sqrt(a))^(1/3))/(a*(sqrt(b)/sqrt(a))^(1/3)) + 2*log(x - (s qrt(b)/sqrt(a))^(1/3))/(a*(sqrt(b)/sqrt(a))^(1/3)))/b^5
Time = 0.17 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (-b x^2+a x^8\right )^4} \, dx=\frac {6175 \, a^{2} \left (-\frac {b}{a}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (-\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{3888 \, b^{6}} - \frac {6175 \, \sqrt {3} \left (-a^{5} b\right )^{\frac {5}{6}} \log \left (x^{2} + \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{6}} + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}{15552 \, a^{3} b^{6}} + \frac {6175 \, \sqrt {3} \left (-a^{5} b\right )^{\frac {5}{6}} \log \left (x^{2} - \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{6}} + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}{15552 \, a^{3} b^{6}} - \frac {991 \, a^{4} x^{17} - 2276 \, a^{3} b x^{11} + 1357 \, a^{2} b^{2} x^{5}}{1296 \, {\left (a x^{6} - b\right )}^{3} b^{5}} + \frac {6175 \, \left (-a^{5} b\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (-\frac {b}{a}\right )^{\frac {1}{6}}}{\left (-\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{7776 \, a^{3} b^{6}} + \frac {6175 \, \left (-a^{5} b\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (-\frac {b}{a}\right )^{\frac {1}{6}}}{\left (-\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{7776 \, a^{3} b^{6}} - \frac {28 \, a x^{6} + b}{7 \, b^{5} x^{7}} \] Input:
integrate(1/(a*x^8-b*x^2)^4,x, algorithm="giac")
Output:
6175/3888*a^2*(-b/a)^(5/6)*arctan(x/(-b/a)^(1/6))/b^6 - 6175/15552*sqrt(3) *(-a^5*b)^(5/6)*log(x^2 + sqrt(3)*x*(-b/a)^(1/6) + (-b/a)^(1/3))/(a^3*b^6) + 6175/15552*sqrt(3)*(-a^5*b)^(5/6)*log(x^2 - sqrt(3)*x*(-b/a)^(1/6) + (- b/a)^(1/3))/(a^3*b^6) - 1/1296*(991*a^4*x^17 - 2276*a^3*b*x^11 + 1357*a^2* b^2*x^5)/((a*x^6 - b)^3*b^5) + 6175/7776*(-a^5*b)^(5/6)*arctan((2*x + sqrt (3)*(-b/a)^(1/6))/(-b/a)^(1/6))/(a^3*b^6) + 6175/7776*(-a^5*b)^(5/6)*arcta n((2*x - sqrt(3)*(-b/a)^(1/6))/(-b/a)^(1/6))/(a^3*b^6) - 1/7*(28*a*x^6 + b )/(b^5*x^7)
Time = 0.38 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (-b x^2+a x^8\right )^4} \, dx=\frac {\frac {1}{7\,b}+\frac {25\,a\,x^6}{7\,b^2}-\frac {114475\,a^2\,x^{12}}{9072\,b^3}+\frac {30875\,a^3\,x^{18}}{2268\,b^4}-\frac {6175\,a^4\,x^{24}}{1296\,b^5}}{a^3\,x^{25}-3\,a^2\,b\,x^{19}+3\,a\,b^2\,x^{13}-b^3\,x^7}-\frac {a^{7/6}\,\mathrm {atan}\left (\frac {a^{1/6}\,x\,1{}\mathrm {i}}{b^{1/6}}\right )\,6175{}\mathrm {i}}{3888\,b^{31/6}}-\frac {a^{7/6}\,\mathrm {atan}\left (\frac {a^{23/2}\,b^{45/2}\,x\,50656457029092189500160000000000{}\mathrm {i}}{25328228514546094750080000000000\,a^{34/3}\,b^{68/3}-\sqrt {3}\,a^{34/3}\,b^{68/3}\,25328228514546094750080000000000{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,6175{}\mathrm {i}}{3888\,b^{31/6}}+\frac {a^{7/6}\,\mathrm {atan}\left (\frac {a^{23/2}\,b^{45/2}\,x\,50656457029092189500160000000000{}\mathrm {i}}{25328228514546094750080000000000\,a^{34/3}\,b^{68/3}+\sqrt {3}\,a^{34/3}\,b^{68/3}\,25328228514546094750080000000000{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,6175{}\mathrm {i}}{3888\,b^{31/6}} \] Input:
int(1/(a*x^8 - b*x^2)^4,x)
Output:
(1/(7*b) + (25*a*x^6)/(7*b^2) - (114475*a^2*x^12)/(9072*b^3) + (30875*a^3* x^18)/(2268*b^4) - (6175*a^4*x^24)/(1296*b^5))/(a^3*x^25 - b^3*x^7 + 3*a*b ^2*x^13 - 3*a^2*b*x^19) - (a^(7/6)*atan((a^(1/6)*x*1i)/b^(1/6))*6175i)/(38 88*b^(31/6)) - (a^(7/6)*atan((a^(23/2)*b^(45/2)*x*506564570290921895001600 00000000i)/(25328228514546094750080000000000*a^(34/3)*b^(68/3) - 3^(1/2)*a ^(34/3)*b^(68/3)*25328228514546094750080000000000i))*((3^(1/2)*1i)/2 + 1/2 )*6175i)/(3888*b^(31/6)) + (a^(7/6)*atan((a^(23/2)*b^(45/2)*x*506564570290 92189500160000000000i)/(25328228514546094750080000000000*a^(34/3)*b^(68/3) + 3^(1/2)*a^(34/3)*b^(68/3)*25328228514546094750080000000000i))*((3^(1/2) *1i)/2 - 1/2)*6175i)/(3888*b^(31/6))
Time = 0.30 (sec) , antiderivative size = 857, normalized size of antiderivative = 3.69 \[ \int \frac {1}{\left (-b x^2+a x^8\right )^4} \, dx =\text {Too large to display} \] Input:
int(1/(a*x^8-b*x^2)^4,x)
Output:
(86450*b**(1/6)*a**(1/6)*sqrt(3)*atan((b**(1/6)*a**(1/6) - 2*a**(1/3)*x)/( b**(1/6)*a**(1/6)*sqrt(3)))*a**4*x**25 - 259350*b**(1/6)*a**(1/6)*sqrt(3)* atan((b**(1/6)*a**(1/6) - 2*a**(1/3)*x)/(b**(1/6)*a**(1/6)*sqrt(3)))*a**3* b*x**19 + 259350*b**(1/6)*a**(1/6)*sqrt(3)*atan((b**(1/6)*a**(1/6) - 2*a** (1/3)*x)/(b**(1/6)*a**(1/6)*sqrt(3)))*a**2*b**2*x**13 - 86450*b**(1/6)*a** (1/6)*sqrt(3)*atan((b**(1/6)*a**(1/6) - 2*a**(1/3)*x)/(b**(1/6)*a**(1/6)*s qrt(3)))*a*b**3*x**7 - 86450*b**(1/6)*a**(1/6)*sqrt(3)*atan((b**(1/6)*a**( 1/6) + 2*a**(1/3)*x)/(b**(1/6)*a**(1/6)*sqrt(3)))*a**4*x**25 + 259350*b**( 1/6)*a**(1/6)*sqrt(3)*atan((b**(1/6)*a**(1/6) + 2*a**(1/3)*x)/(b**(1/6)*a* *(1/6)*sqrt(3)))*a**3*b*x**19 - 259350*b**(1/6)*a**(1/6)*sqrt(3)*atan((b** (1/6)*a**(1/6) + 2*a**(1/3)*x)/(b**(1/6)*a**(1/6)*sqrt(3)))*a**2*b**2*x**1 3 + 86450*b**(1/6)*a**(1/6)*sqrt(3)*atan((b**(1/6)*a**(1/6) + 2*a**(1/3)*x )/(b**(1/6)*a**(1/6)*sqrt(3)))*a*b**3*x**7 - 43225*b**(1/6)*a**(1/6)*log( - b**(1/6)*a**(1/6)*x + a**(1/3)*x**2 + b**(1/3))*a**4*x**25 + 129675*b**( 1/6)*a**(1/6)*log( - b**(1/6)*a**(1/6)*x + a**(1/3)*x**2 + b**(1/3))*a**3* b*x**19 - 129675*b**(1/6)*a**(1/6)*log( - b**(1/6)*a**(1/6)*x + a**(1/3)*x **2 + b**(1/3))*a**2*b**2*x**13 + 43225*b**(1/6)*a**(1/6)*log( - b**(1/6)* a**(1/6)*x + a**(1/3)*x**2 + b**(1/3))*a*b**3*x**7 - 86450*b**(1/6)*a**(1/ 6)*log( - b**(1/6)*a**(1/6) + a**(1/3)*x)*a**4*x**25 + 259350*b**(1/6)*a** (1/6)*log( - b**(1/6)*a**(1/6) + a**(1/3)*x)*a**3*b*x**19 - 259350*b**(...