Integrand size = 18, antiderivative size = 239 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^8} \, dx=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 \left (b^2-4 a c\right )^2 x}+\frac {b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log \left (a+b x+c x^2\right )}{2 a^4} \] Output:
-3*(-5*a*c+b^2)*(-2*a*c+b^2)/a^3/(-4*a*c+b^2)^2/x+1/2*(b*c*x-2*a*c+b^2)/a/ (-4*a*c+b^2)/x/(c*x^2+b*x+a)^2+1/2*(3*b^4-20*a*b^2*c+20*c^2*a^2+3*b*c*(-6* a*c+b^2)*x)/a^2/(-4*a*c+b^2)^2/x/(c*x^2+b*x+a)-3*(-20*a^3*c^3+30*a^2*b^2*c ^2-10*a*b^4*c+b^6)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^4/(-4*a*c+b^2)^ (5/2)-3*b*ln(x)/a^4+3/2*b*ln(c*x^2+b*x+a)/a^4
Time = 0.40 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^8} \, dx=\frac {-\frac {2 a}{x}+\frac {a^2 \left (b^3-3 a b c+b^2 c x-2 a c^2 x\right )}{\left (-b^2+4 a c\right ) (a+x (b+c x))^2}-\frac {a \left (4 b^5-29 a b^3 c+46 a^2 b c^2+4 b^4 c x-26 a b^2 c^2 x+28 a^2 c^3 x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {6 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}-6 b \log (x)+3 b \log (a+x (b+c x))}{2 a^4} \] Input:
Integrate[1/((c + a/x^2 + b/x)^3*x^8),x]
Output:
((-2*a)/x + (a^2*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((-b^2 + 4*a*c)*(a + x*(b + c*x))^2) - (a*(4*b^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x - 2 6*a*b^2*c^2*x + 28*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (6*(b ^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^ 2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) - 6*b*Log[x] + 3*b*Log[a + x*(b + c*x)]) /(2*a^4)
Time = 0.57 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1692, 1165, 25, 1235, 27, 1200, 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.
\(\displaystyle \int \frac {1}{x^8 \left (\frac {a}{x^2}+\frac {b}{x}+c\right )^3} \, dx\) |
\(\Big \downarrow \) 1692 |
\(\displaystyle \int \frac {1}{x^2 \left (a+b x+c x^2\right )^3}dx\) |
\(\Big \downarrow \) 1165 |
\(\displaystyle \frac {-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int -\frac {3 b^2+4 c x b-10 a c}{x^2 \left (c x^2+b x+a\right )^2}dx}{2 a \left (b^2-4 a c\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {3 b^2+4 c x b-10 a c}{x^2 \left (c x^2+b x+a\right )^2}dx}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 1235 |
\(\displaystyle \frac {\frac {20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {6 \left (\left (b^2-5 a c\right ) \left (b^2-2 a c\right )+b c \left (b^2-6 a c\right ) x\right )}{x^2 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {6 \int \frac {\left (b^2-5 a c\right ) \left (b^2-2 a c\right )+b c \left (b^2-6 a c\right ) x}{x^2 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}+\frac {20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {\frac {6 \int \left (-\frac {b \left (4 a c-b^2\right )^2}{a^2 x}+\frac {b^6-9 a c b^4+23 a^2 c^2 b^2+c \left (b^2-4 a c\right )^2 x b-10 a^3 c^3}{a^2 \left (c x^2+b x+a\right )}+\frac {\left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a x^2}\right )dx}{a \left (b^2-4 a c\right )}+\frac {20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {6 \left (\frac {b \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )}{2 a^2}-\frac {b \log (x) \left (b^2-4 a c\right )^2}{a^2}-\frac {\left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c}}-\frac {\left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a x}\right )}{a \left (b^2-4 a c\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
Input:
Int[1/((c + a/x^2 + b/x)^3*x^8),x]
Output:
(b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)^2) + ((3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x)/(a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)) + (6*(-(((b^2 - 5*a*c)*(b^2 - 2*a*c))/(a*x)) - ((b^6 - 10*a *b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c ]])/(a^2*Sqrt[b^2 - 4*a*c]) - (b*(b^2 - 4*a*c)^2*Log[x])/a^2 + (b*(b^2 - 4 *a*c)^2*Log[a + b*x + c*x^2])/(2*a^2)))/(a*(b^2 - 4*a*c)))/(2*a*(b^2 - 4*a *c))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + 2*n*p)*(c + b/x^n + a/x^(2*n))^p, x] /; FreeQ[{a, b, c, m, n }, x] && EqQ[n2, 2*n] && ILtQ[p, 0] && NegQ[n]
Time = 0.09 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.69
method | result | size |
default | \(-\frac {\frac {\frac {a \,c^{2} \left (14 a^{2} c^{2}-13 a \,b^{2} c +2 b^{4}\right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {a b c \left (74 a^{2} c^{2}-55 a \,b^{2} c +8 b^{4}\right ) x^{2}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {a \left (18 a^{3} c^{3}+7 a^{2} b^{2} c^{2}-12 a \,b^{4} c +2 b^{6}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {a^{2} b \left (58 a^{2} c^{2}-36 a \,b^{2} c +5 b^{4}\right )}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {3 \left (-16 a^{2} b \,c^{3}+8 a \,b^{3} c^{2}-b^{5} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {6 \left (10 a^{3} c^{3}-23 a^{2} b^{2} c^{2}+9 a \,b^{4} c -b^{6}-\frac {\left (-16 a^{2} b \,c^{3}+8 a \,b^{3} c^{2}-b^{5} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{a^{4}}-\frac {1}{a^{3} x}-\frac {3 b \ln \left (x \right )}{a^{4}}\) | \(404\) |
risch | \(\frac {-\frac {3 c^{2} \left (10 a^{2} c^{2}-7 a \,b^{2} c +b^{4}\right ) x^{4}}{a^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 b c \left (46 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{3}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}-\frac {\left (50 a^{3} c^{3}+7 a^{2} b^{2} c^{2}-18 a \,b^{4} c +3 b^{6}\right ) x^{2}}{a^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \left (122 a^{2} c^{2}-68 a \,b^{2} c +9 b^{4}\right ) x}{2 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {1}{a}}{x \left (c \,x^{2}+b x +a \right )^{2}}-\frac {3 b \ln \left (x \right )}{a^{4}}+3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (1024 a^{9} c^{5}-1280 a^{8} b^{2} c^{4}+640 a^{7} b^{4} c^{3}-160 a^{6} b^{6} c^{2}+20 a^{5} b^{8} c -a^{4} b^{10}\right ) \textit {\_Z}^{2}+\left (-1024 a^{5} b \,c^{5}+1280 a^{4} b^{3} c^{4}-640 b^{5} c^{3} a^{3}+160 a^{2} b^{7} c^{2}-20 a \,b^{9} c +b^{11}\right ) \textit {\_Z} +100 a^{2} c^{6}-44 a \,b^{2} c^{5}+5 b^{4} c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (1536 a^{11} c^{5}-2048 a^{10} b^{2} c^{4}+1088 a^{9} b^{4} c^{3}-288 a^{8} b^{6} c^{2}+38 a^{7} b^{8} c -2 a^{6} b^{10}\right ) \textit {\_R}^{2}+\left (-608 a^{7} b \,c^{5}+576 a^{6} b^{3} c^{4}-206 a^{5} b^{5} c^{3}+33 a^{4} b^{7} c^{2}-2 a^{3} b^{9} c \right ) \textit {\_R} +100 a^{4} c^{6}-140 a^{3} b^{2} c^{5}+69 a^{2} b^{4} c^{4}-14 a \,b^{6} c^{3}+b^{8} c^{2}\right ) x +\left (-256 a^{11} b \,c^{4}+256 a^{10} b^{3} c^{3}-96 a^{9} b^{5} c^{2}+16 a^{8} b^{7} c -a^{7} b^{9}\right ) \textit {\_R}^{2}+\left (160 a^{8} c^{5}-448 a^{7} b^{2} c^{4}+338 a^{6} b^{4} c^{3}-111 a^{5} b^{6} c^{2}+17 a^{4} b^{8} c -a^{3} b^{10}\right ) \textit {\_R} +160 a^{4} b \,c^{5}-192 a^{3} b^{3} c^{4}+82 a^{2} b^{5} c^{3}-15 a \,b^{7} c^{2}+b^{9} c \right )\right )\) | \(724\) |
Input:
int(1/(c+a/x^2+b/x)^3/x^8,x,method=_RETURNVERBOSE)
Output:
-1/a^4*((a*c^2*(14*a^2*c^2-13*a*b^2*c+2*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^ 3+1/2*a*b*c*(74*a^2*c^2-55*a*b^2*c+8*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+a *(18*a^3*c^3+7*a^2*b^2*c^2-12*a*b^4*c+2*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x+ 1/2*a^2*b*(58*a^2*c^2-36*a*b^2*c+5*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2 +b*x+a)^2+3/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(-16*a^2*b*c^3+8*a*b^3*c^2-b^5 *c)/c*ln(c*x^2+b*x+a)+2*(10*a^3*c^3-23*a^2*b^2*c^2+9*a*b^4*c-b^6-1/2*(-16* a^2*b*c^3+8*a*b^3*c^2-b^5*c)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a* c-b^2)^(1/2))))-1/a^3/x-3*b*ln(x)/a^4
Leaf count of result is larger than twice the leaf count of optimal. 1130 vs. \(2 (229) = 458\).
Time = 0.46 (sec) , antiderivative size = 2280, normalized size of antiderivative = 9.54 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^8} \, dx=\text {Too large to display} \] Input:
integrate(1/(c+a/x^2+b/x)^3/x^8,x, algorithm="fricas")
Output:
[-1/2*(2*a^3*b^6 - 24*a^4*b^4*c + 96*a^5*b^2*c^2 - 128*a^6*c^3 + 6*(a*b^6* c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4*c^5)*x^4 + 3*(4*a*b^7*c - 4 5*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*x^3 + 2*(3*a*b^8 - 30*a^2 *b^6*c + 79*a^3*b^4*c^2 + 22*a^4*b^2*c^3 - 200*a^5*c^4)*x^2 + 3*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^5 + 2*(b^7*c - 10*a*b^5*c^ 2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^4 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^3 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3 *c^2 - 20*a^4*b*c^3)*x^2 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a ^5*c^3)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt (b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + (9*a^2*b^7 - 104*a^3*b^5*c + 394*a^4*b^3*c^2 - 488*a^5*b*c^3)*x - 3*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^ 2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^4 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^3 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4* b^2*c^3)*x^2 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x) *log(c*x^2 + b*x + a) + 6*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a ^3*b*c^5)*x^5 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4) *x^4 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4 )*x^3 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^2 + ( a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*log(x))/((a^...
Timed out. \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^8} \, dx=\text {Timed out} \] Input:
integrate(1/(c+a/x**2+b/x)**3/x**8,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^8} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(c+a/x^2+b/x)^3/x^8,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.12 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^8} \, dx=\frac {3 \, {\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, b \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {2 \, a^{3} b^{4} - 16 \, a^{4} b^{2} c + 32 \, a^{5} c^{2} + 6 \, {\left (a b^{4} c^{2} - 7 \, a^{2} b^{2} c^{3} + 10 \, a^{3} c^{4}\right )} x^{4} + 3 \, {\left (4 \, a b^{5} c - 29 \, a^{2} b^{3} c^{2} + 46 \, a^{3} b c^{3}\right )} x^{3} + 2 \, {\left (3 \, a b^{6} - 18 \, a^{2} b^{4} c + 7 \, a^{3} b^{2} c^{2} + 50 \, a^{4} c^{3}\right )} x^{2} + {\left (9 \, a^{2} b^{5} - 68 \, a^{3} b^{3} c + 122 \, a^{4} b c^{2}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} a^{4} x} \] Input:
integrate(1/(c+a/x^2+b/x)^3/x^8,x, algorithm="giac")
Output:
3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x + b)/sqrt (-b^2 + 4*a*c))/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a*c)) + 3/2*b*log(c*x^2 + b*x + a)/a^4 - 3*b*log(abs(x))/a^4 - 1/2*(2*a^3*b^4 - 16*a^4*b^2*c + 32*a^5*c^2 + 6*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*x^4 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*x^3 + 2*(3*a*b^6 - 18*a^2 *b^4*c + 7*a^3*b^2*c^2 + 50*a^4*c^3)*x^2 + (9*a^2*b^5 - 68*a^3*b^3*c + 122 *a^4*b*c^2)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*a^4*x)
Time = 20.48 (sec) , antiderivative size = 1255, normalized size of antiderivative = 5.25 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^8} \, dx =\text {Too large to display} \] Input:
int(1/(x^8*(c + a/x^2 + b/x)^3),x)
Output:
- (1/a + (x^2*(3*b^6 + 50*a^3*c^3 + 7*a^2*b^2*c^2 - 18*a*b^4*c))/(a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x*(9*b^5 + 122*a^2*b*c^2 - 68*a*b^3*c))/(2* a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*x^3*(4*b^5*c - 29*a*b^3*c^2 + 46* a^2*b*c^3))/(2*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*c^2*x^4*(b^4 + 10* a^2*c^2 - 7*a*b^2*c))/(a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^3*(2*a*c + b^2) + a^2*x + c^2*x^5 + 2*a*b*x^2 + 2*b*c*x^4) - (3*b*log(x))/a^4 - (3*lo g(2*a*b^11 + 2*b^12*x + 2*a*b^6*(-(4*a*c - b^2)^5)^(1/2) - 39*a^2*b^9*c - 1696*a^6*b*c^5 + 320*a^6*c^6*x + 2*b^7*x*(-(4*a*c - b^2)^5)^(1/2) + 303*a^ 3*b^7*c^2 - 1170*a^4*b^5*c^3 + 2240*a^5*b^3*c^4 - 10*a^4*c^3*(-(4*a*c - b^ 2)^5)^(1/2) - 17*a^2*b^4*c*(-(4*a*c - b^2)^5)^(1/2) + 321*a^2*b^8*c^2*x - 1296*a^3*b^6*c^3*x + 2660*a^4*b^4*c^4*x - 2336*a^5*b^2*c^5*x - 40*a*b^10*c *x + 39*a^3*b^2*c^2*(-(4*a*c - b^2)^5)^(1/2) - 20*a*b^5*c*x*(-(4*a*c - b^2 )^5)^(1/2) - 58*a^3*b*c^3*x*(-(4*a*c - b^2)^5)^(1/2) + 63*a^2*b^3*c^2*x*(- (4*a*c - b^2)^5)^(1/2))*(b^11 + b^6*(-(4*a*c - b^2)^5)^(1/2) - 1024*a^5*b* c^5 + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 20*a^3*c^3*(- (4*a*c - b^2)^5)^(1/2) - 20*a*b^9*c + 30*a^2*b^2*c^2*(-(4*a*c - b^2)^5)^(1 /2) - 10*a*b^4*c*(-(4*a*c - b^2)^5)^(1/2)))/(2*a^4*(4*a*c - b^2)^5) - (3*l og(2*a*b^11 + 2*b^12*x - 2*a*b^6*(-(4*a*c - b^2)^5)^(1/2) - 39*a^2*b^9*c - 1696*a^6*b*c^5 + 320*a^6*c^6*x - 2*b^7*x*(-(4*a*c - b^2)^5)^(1/2) + 303*a ^3*b^7*c^2 - 1170*a^4*b^5*c^3 + 2240*a^5*b^3*c^4 + 10*a^4*c^3*(-(4*a*c ...
Time = 0.17 (sec) , antiderivative size = 2201, normalized size of antiderivative = 9.21 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^8} \, dx =\text {Too large to display} \] Input:
int(1/(c+a/x^2+b/x)^3/x^8,x)
Output:
( - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b*c** 3*x + 180*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b** 3*c**2*x - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a** 4*b**2*c**3*x**2 - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b* *2))*a**4*b*c**4*x**3 - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**5*c*x + 360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a* c - b**2))*a**3*b**4*c**2*x**2 + 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/s qrt(4*a*c - b**2))*a**3*b**3*c**3*x**3 - 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**4*x**4 - 120*sqrt(4*a*c - b**2)*at an((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**5*x**5 + 6*sqrt(4*a*c - b**2) *atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**7*x - 120*sqrt(4*a*c - b**2) *atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**6*c*x**2 + 60*sqrt(4*a*c - b **2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**5*c**2*x**3 + 360*sqrt(4 *a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**4*c**3*x**4 + 18 0*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c**4*x **5 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**8*x* *2 - 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**7*c*x **3 - 120*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**6*c **2*x**4 - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b* *5*c**3*x**5 + 6*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)...