\(\int \frac {1}{(c+\frac {a}{x^2}+\frac {b}{x})^3} \, dx\) [431]
Optimal result
Integrand size = 14, antiderivative size = 238 \[
\int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3} \, dx=\frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b \log \left (a+b x+c x^2\right )}{2 c^4}
\]
[Out]
3*(10*a^2*c^2-7*a*b^2*c+b^4)*x/c^3/(-4*a*c+b^2)^2-3/2*b*(-6*a*c+b^2)*x^2/c^2/(-4*a*c+b^2)^2+1/2*x^5*(b*x+2*a)/
(-4*a*c+b^2)/(c*x^2+b*x+a)^2+x^3*(a*(-10*a*c+b^2)+b*(-7*a*c+b^2)*x)/c/(-4*a*c+b^2)^2/(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))/c^4/(-4*a*c+b^2)^(5/2)-3/2*b*ln(c*x^2
+b*x+a)/c^4
Rubi [A] (verified)
Time = 0.21 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of
steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1354, 752, 832, 814, 648, 632,
212, 642} \[
\int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3} \, dx=\frac {3 x \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b x^2 \left (b^2-6 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (b x \left (b^2-7 a c\right )+a \left (b^2-10 a c\right )\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 b \log \left (a+b x+c x^2\right )}{2 c^4}
\]
[In]
Int[(c + a/x^2 + b/x)^(-3),x]
[Out]
(3*(b^4 - 7*a*b^2*c + 10*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^2) - (3*b*(b^2 - 6*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)^2) +
(x^5*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (x^3*(a*(b^2 - 10*a*c) + b*(b^2 - 7*a*c)*x))/(c*(b^
2 - 4*a*c)^2*(a + b*x + c*x^2)) - (3*(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]])/(c^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[a + b*x + c*x^2])/(2*c^4)
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 632
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 642
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]
Rule 648
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 752
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]
Rule 814
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]
Rule 832
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Rule 1354
Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(2*n*p)*(c + b/x^n + a/x^(2*n))^p,
x] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && LtQ[n, 0] && IntegerQ[p]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {x^6}{\left (a+b x+c x^2\right )^3} \, dx \\ & = \frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {x^4 (10 a+2 b x)}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )} \\ & = \frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {x^2 \left (6 a \left (b^2-10 a c\right )+6 b \left (b^2-6 a c\right ) x\right )}{a+b x+c x^2} \, dx}{2 c \left (b^2-4 a c\right )^2} \\ & = \frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \left (-\frac {6 \left (b^4-7 a b^2 c+10 a^2 c^2\right )}{c^2}+\frac {6 b \left (b^2-6 a c\right ) x}{c}+\frac {6 \left (a \left (b^4-7 a b^2 c+10 a^2 c^2\right )+b \left (b^2-4 a c\right )^2 x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{2 c \left (b^2-4 a c\right )^2} \\ & = \frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \int \frac {a \left (b^4-7 a b^2 c+10 a^2 c^2\right )+b \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )^2} \\ & = \frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {(3 b) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}+\frac {\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )^2} \\ & = \frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 b \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4 \left (b^2-4 a c\right )^2} \\ & = \frac {3 \left (b^4-7 a b^2 c+10 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right )^2}-\frac {3 b \left (b^2-6 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )^2}+\frac {x^5 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x^3 \left (a \left (b^2-10 a c\right )+b \left (b^2-7 a c\right ) x\right )}{c \left (b^2-4 a c\right )^2 \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 ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b \log \left (a+b x+c x^2\right )}{2 c^4} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.24 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.09
\[
\int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3} \, dx=\frac {2 c^2 x+\frac {b^7-14 a b^5 c+61 a^2 b^3 c^2-78 a^3 b c^3-6 b^6 c x+48 a b^4 c^2 x-102 a^2 b^2 c^3 x+36 a^3 c^4 x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {-b^6 x+a^2 b^2 c (5 b-9 c x)-a b^4 (b-6 c x)+a^3 c^2 (-5 b+2 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {6 c \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}-3 b c \log (a+x (b+c x))}{2 c^5}
\]
[In]
Integrate[(c + a/x^2 + b/x)^(-3),x]
[Out]
(2*c^2*x + (b^7 - 14*a*b^5*c + 61*a^2*b^3*c^2 - 78*a^3*b*c^3 - 6*b^6*c*x + 48*a*b^4*c^2*x - 102*a^2*b^2*c^3*x
+ 36*a^3*c^4*x)/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-(b^6*x) + a^2*b^2*c*(5*b - 9*c*x) - a*b^4*(b - 6*c*x)
+ a^3*c^2*(-5*b + 2*c*x))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (6*c*(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) - 3*b*c*Log[a + x*(b + c*x)])/(2*c^5)
Maple [A] (verified)
Time = 0.20 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.68
| | |
method | result | size |
| | |
default |
\(\frac {x}{c^{3}}-\frac {\frac {-\frac {3 \left (6 c^{3} a^{3}-17 a^{2} b^{2} c^{2}+8 a \,b^{4} c -b^{6}\right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {b \left (42 c^{3} a^{3}+41 a^{2} b^{2} c^{2}-34 a \,b^{4} c +5 b^{6}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c}-\frac {a \left (14 c^{3} a^{3}-71 a^{2} b^{2} c^{2}+38 a \,b^{4} c -5 b^{6}\right ) x}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \,a^{2} \left (58 a^{2} c^{2}-36 a \,b^{2} c +5 b^{4}\right )}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {3 \left (16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {6 \left (10 a^{3} c^{2}-7 a^{2} b^{2} c +b^{4} a -\frac {\left (16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{c^{3}}\) |
\(401\) |
risch |
\(\text {Expression too large to display}\) |
\(2019\) |
| | |
|
|
|
[In]
int(1/(c+a/x^2+b/x)^3,x,method=_RETURNVERBOSE)
[Out]
x/c^3-1/c^3*((-3*(6*a^3*c^3-17*a^2*b^2*c^2+8*a*b^4*c-b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*b*(42*a^3*c^3+41*
a^2*b^2*c^2-34*a*b^4*c+5*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)/c*x^2-a/c*(14*a^3*c^3-71*a^2*b^2*c^2+38*a*b^4*c-5*b^6
)/(16*a^2*c^2-8*a*b^2*c+b^4)*x+1/2*b*a^2/c*(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^2-8*a*b^3*c+b^5)/c*ln(c*x^2+b*x+a)+2*(10*a^3*c^2-7*a^2*b^
2*c+b^4*a-1/2*(16*a^2*b*c^2-8*a*b^3*c+b^5)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 953 vs. \(2 (228) = 456\).
Time = 0.29 (sec) , antiderivative size = 1926, normalized size of antiderivative = 8.09
\[
\int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(c+a/x^2+b/x)^3,x, algorithm="fricas")
[Out]
[-1/2*(5*a^2*b^7 - 56*a^3*b^5*c + 202*a^4*b^3*c^2 - 232*a^5*b*c^3 - 2*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5
- 64*a^3*c^6)*x^5 - 4*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^4 + 2*(2*b^8*c - 26*a*b^6*c^
2 + 123*a^2*b^4*c^3 - 254*a^3*b^2*c^4 + 200*a^4*c^5)*x^3 + (5*b^9 - 58*a*b^7*c + 225*a^2*b^5*c^2 - 314*a^3*b^3
*c^3 + 88*a^4*b*c^4)*x^2 + 3*(a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3 + (b^6*c^2 - 10*a*b^4*c^3 +
30*a^2*b^2*c^4 - 20*a^3*c^5)*x^4 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^3 + (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^2 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*
b*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)) + 2*(5*a*b^8 - 59*a^2*b^6*c + 235*a^3*b^4*c^2 - 346*a^4*b^2*c^3 + 120*a^5*c^4)*x + 3*(a^2*b^7 - 12*a^
3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^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^4 + 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^3 + (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^2 + 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)*log(c*x^2 + b*x + a))/(
a^2*b^6*c^4 - 12*a^3*b^4*c^5 + 48*a^4*b^2*c^6 - 64*a^5*c^7 + (b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3
*c^9)*x^4 + 2*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*x^3 + (b^8*c^4 - 10*a*b^6*c^5 + 24*a^2*
b^4*c^6 + 32*a^3*b^2*c^7 - 128*a^4*c^8)*x^2 + 2*(a*b^7*c^4 - 12*a^2*b^5*c^5 + 48*a^3*b^3*c^6 - 64*a^4*b*c^7)*x
), -1/2*(5*a^2*b^7 - 56*a^3*b^5*c + 202*a^4*b^3*c^2 - 232*a^5*b*c^3 - 2*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c
^5 - 64*a^3*c^6)*x^5 - 4*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^4 + 2*(2*b^8*c - 26*a*b^6*
c^2 + 123*a^2*b^4*c^3 - 254*a^3*b^2*c^4 + 200*a^4*c^5)*x^3 + (5*b^9 - 58*a*b^7*c + 225*a^2*b^5*c^2 - 314*a^3*b
^3*c^3 + 88*a^4*b*c^4)*x^2 + 6*(a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3 + (b^6*c^2 - 10*a*b^4*c^3
+ 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^4 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^3 + (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^2 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^
4*b*c^3)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(5*a*b^8 - 59*a^2*b^6
*c + 235*a^3*b^4*c^2 - 346*a^4*b^2*c^3 + 120*a^5*c^4)*x + 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*
b*c^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^4 + 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^3 + (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^2 + 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)*log(c*x^2 + b*x + a))/(a^2*b^6*c^4 - 12*a^3*b^4*c^5 + 48*
a^4*b^2*c^6 - 64*a^5*c^7 + (b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*x^4 + 2*(b^7*c^5 - 12*a*b^5*
c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*x^3 + (b^8*c^4 - 10*a*b^6*c^5 + 24*a^2*b^4*c^6 + 32*a^3*b^2*c^7 - 128*a^4
*c^8)*x^2 + 2*(a*b^7*c^4 - 12*a^2*b^5*c^5 + 48*a^3*b^3*c^6 - 64*a^4*b*c^7)*x)]
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1714 vs. \(2 (236) = 472\).
Time = 3.04 (sec) , antiderivative size = 1714, normalized size of antiderivative = 7.20
\[
\int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3} \, dx=\text {Too large to display}
\]
[In]
integrate(1/(c+a/x**2+b/x)**3,x)
[Out]
(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1
024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))*log(x +
(-66*a**3*b*c**2 - 64*a**3*c**6*(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2
+ 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2
+ 20*a*b**8*c - b**10))) + 27*a**2*b**3*c + 48*a**2*b**2*c**5*(-3*b/(2*c**4) - 3*sqrt(-(4*a*c - b**2)**5)*(20
*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*
b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 3*a*b**5 - 12*a*b**4*c**4*(-3*b/(2*c**4) - 3*sqrt(-(
4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*
b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + b**6*c**3*(-3*b/(2*c**4) - 3*sq
rt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*
a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))))/(60*a**3*c**3 - 90*a**2*b**
2*c**2 + 30*a*b**4*c - 3*b**6)) + (-3*b/(2*c**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**
2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**
2 + 20*a*b**8*c - b**10)))*log(x + (-66*a**3*b*c**2 - 64*a**3*c**6*(-3*b/(2*c**4) + 3*sqrt(-(4*a*c - b**2)**5)
*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a
**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 27*a**2*b**3*c + 48*a**2*b**2*c**5*(-3*b/(2*c**4
) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2*c**4*(1024*a**5*c**5
- 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 3*a*b**5 - 12*a*b*
*4*c**4*(-3*b/(2*c**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**6)/(2
*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))
+ b**6*c**3*(-3*b/(2*c**4) + 3*sqrt(-(4*a*c - b**2)**5)*(20*a**3*c**3 - 30*a**2*b**2*c**2 + 10*a*b**4*c - b**
6)/(2*c**4*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**
10))))/(60*a**3*c**3 - 90*a**2*b**2*c**2 + 30*a*b**4*c - 3*b**6)) + (-58*a**4*b*c**2 + 36*a**3*b**3*c - 5*a**2
*b**5 + x**3*(36*a**3*c**4 - 102*a**2*b**2*c**3 + 48*a*b**4*c**2 - 6*b**6*c) + x**2*(-42*a**3*b*c**3 - 41*a**2
*b**3*c**2 + 34*a*b**5*c - 5*b**7) + x*(28*a**4*c**3 - 142*a**3*b**2*c**2 + 76*a**2*b**4*c - 10*a*b**6))/(32*a
**4*c**6 - 16*a**3*b**2*c**5 + 2*a**2*b**4*c**4 + x**4*(32*a**2*c**8 - 16*a*b**2*c**7 + 2*b**4*c**6) + x**3*(6
4*a**2*b*c**7 - 32*a*b**3*c**6 + 4*b**5*c**5) + x**2*(64*a**3*c**7 - 12*a*b**4*c**5 + 2*b**6*c**4) + x*(64*a**
3*b*c**6 - 32*a**2*b**3*c**5 + 4*a*b**5*c**4)) + x/c**3
Maxima [F(-2)]
Exception generated. \[
\int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3} \, dx=\text {Exception raised: ValueError}
\]
[In]
integrate(1/(c+a/x^2+b/x)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta
Giac [A] (verification not implemented)
none
Time = 0.33 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.18
\[
\int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3} \, dx=\frac {3 \, {\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x}{c^{3}} - \frac {3 \, b \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac {5 \, a^{2} b^{5} - 36 \, a^{3} b^{3} c + 58 \, a^{4} b c^{2} + 6 \, {\left (b^{6} c - 8 \, a b^{4} c^{2} + 17 \, a^{2} b^{2} c^{3} - 6 \, a^{3} c^{4}\right )} x^{3} + {\left (5 \, b^{7} - 34 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} + 42 \, a^{3} b c^{3}\right )} x^{2} + 2 \, {\left (5 \, a b^{6} - 38 \, a^{2} b^{4} c + 71 \, a^{3} b^{2} c^{2} - 14 \, a^{4} c^{3}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{4}}
\]
[In]
integrate(1/(c+a/x^2+b/x)^3,x, algorithm="giac")
[Out]
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))/((b^4*c^4 - 8*a*b^2*
c^5 + 16*a^2*c^6)*sqrt(-b^2 + 4*a*c)) + x/c^3 - 3/2*b*log(c*x^2 + b*x + a)/c^4 - 1/2*(5*a^2*b^5 - 36*a^3*b^3*c
+ 58*a^4*b*c^2 + 6*(b^6*c - 8*a*b^4*c^2 + 17*a^2*b^2*c^3 - 6*a^3*c^4)*x^3 + (5*b^7 - 34*a*b^5*c + 41*a^2*b^3*
c^2 + 42*a^3*b*c^3)*x^2 + 2*(5*a*b^6 - 38*a^2*b^4*c + 71*a^3*b^2*c^2 - 14*a^4*c^3)*x)/((c*x^2 + b*x + a)^2*(b^
2 - 4*a*c)^2*c^4)
Mupad [B] (verification not implemented)
Time = 8.59 (sec) , antiderivative size = 705, normalized size of antiderivative = 2.96
\[
\int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3} \, dx=\frac {x}{c^3}-\frac {\frac {3\,x^3\,\left (-6\,a^3\,c^3+17\,a^2\,b^2\,c^2-8\,a\,b^4\,c+b^6\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {x^2\,\left (42\,a^3\,b\,c^3+41\,a^2\,b^3\,c^2-34\,a\,b^5\,c+5\,b^7\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a^2\,\left (58\,a^2\,b\,c^2-36\,a\,b^3\,c+5\,b^5\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,x\,\left (-14\,a^3\,c^3+71\,a^2\,b^2\,c^2-38\,a\,b^4\,c+5\,b^6\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{a^2\,c^3+c^5\,x^4+x^2\,\left (b^2\,c^3+2\,a\,c^4\right )+2\,b\,c^4\,x^3+2\,a\,b\,c^3\,x}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-3072\,a^5\,b\,c^5+3840\,a^4\,b^3\,c^4-1920\,a^3\,b^5\,c^3+480\,a^2\,b^7\,c^2-60\,a\,b^9\,c+3\,b^{11}\right )}{2\,\left (1024\,a^5\,c^9-1280\,a^4\,b^2\,c^8+640\,a^3\,b^4\,c^7-160\,a^2\,b^6\,c^6+20\,a\,b^8\,c^5-b^{10}\,c^4\right )}+\frac {3\,\mathrm {atan}\left (\frac {\left (\frac {3\,x\,\left (-20\,a^3\,c^3+30\,a^2\,b^2\,c^2-10\,a\,b^4\,c+b^6\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^5}+\frac {3\,\left (16\,a^2\,b\,c^5-8\,a\,b^3\,c^4+b^5\,c^3\right )\,\left (-20\,a^3\,c^3+30\,a^2\,b^2\,c^2-10\,a\,b^4\,c+b^6\right )}{2\,c^7\,{\left (4\,a\,c-b^2\right )}^5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (32\,a^2\,c^6\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,b^4\,c^4\,{\left (4\,a\,c-b^2\right )}^{5/2}-16\,a\,b^2\,c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}\right )}{-60\,a^3\,c^3+90\,a^2\,b^2\,c^2-30\,a\,b^4\,c+3\,b^6}\right )\,\left (-20\,a^3\,c^3+30\,a^2\,b^2\,c^2-10\,a\,b^4\,c+b^6\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^{5/2}}
\]
[In]
int(1/(c + a/x^2 + b/x)^3,x)
[Out]
x/c^3 - ((3*x^3*(b^6 - 6*a^3*c^3 + 17*a^2*b^2*c^2 - 8*a*b^4*c))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (x^2*(5*b^7 +
42*a^3*b*c^3 + 41*a^2*b^3*c^2 - 34*a*b^5*c))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a^2*(5*b^5 + 58*a^2*b*c^
2 - 36*a*b^3*c))/(2*c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a*x*(5*b^6 - 14*a^3*c^3 + 71*a^2*b^2*c^2 - 38*a*b^4*c
))/(c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(a^2*c^3 + c^5*x^4 + x^2*(2*a*c^4 + b^2*c^3) + 2*b*c^4*x^3 + 2*a*b*c^3*
x) + (log(a + b*x + c*x^2)*(3*b^11 - 3072*a^5*b*c^5 + 480*a^2*b^7*c^2 - 1920*a^3*b^5*c^3 + 3840*a^4*b^3*c^4 -
60*a*b^9*c))/(2*(1024*a^5*c^9 - b^10*c^4 + 20*a*b^8*c^5 - 160*a^2*b^6*c^6 + 640*a^3*b^4*c^7 - 1280*a^4*b^2*c^8
)) + (3*atan((((3*x*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c))/(c^3*(4*a*c - b^2)^5) + (3*(b^5*c^3 - 8*
a*b^3*c^4 + 16*a^2*b*c^5)*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4*c))/(2*c^7*(4*a*c - b^2)^5*(b^4 + 16*a
^2*c^2 - 8*a*b^2*c)))*(32*a^2*c^6*(4*a*c - b^2)^(5/2) + 2*b^4*c^4*(4*a*c - b^2)^(5/2) - 16*a*b^2*c^5*(4*a*c -
b^2)^(5/2)))/(3*b^6 - 60*a^3*c^3 + 90*a^2*b^2*c^2 - 30*a*b^4*c))*(b^6 - 20*a^3*c^3 + 30*a^2*b^2*c^2 - 10*a*b^4
*c))/(c^4*(4*a*c - b^2)^(5/2))