3.5.33 \(\int \frac {1}{(c+\frac {a}{x^2}+\frac {b}{x})^3 x^2} \, dx\) [433]

Optimal. Leaf size=111 \[ \frac {b+\frac {2 a}{x}}{2 \left (b^2-4 a c\right ) \left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2}-\frac {3 a \left (b+\frac {2 a}{x}\right )}{\left (b^2-4 a c\right )^2 \left (c+\frac {a}{x^2}+\frac {b}{x}\right )}+\frac {12 a^2 \tanh ^{-1}\left (\frac {b+\frac {2 a}{x}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]

[Out]

1/2*(b+2*a/x)/(-4*a*c+b^2)/(c+a/x^2+b/x)^2-3*a*(b+2*a/x)/(-4*a*c+b^2)^2/(c+a/x^2+b/x)+12*a^2*arctanh((b+2*a/x)
/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(5/2)

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Rubi [A]
time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1366, 628, 632, 212} \begin {gather*} \frac {12 a^2 \tanh ^{-1}\left (\frac {\frac {2 a}{x}+b}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac {3 a \left (\frac {2 a}{x}+b\right )}{\left (b^2-4 a c\right )^2 \left (\frac {a}{x^2}+\frac {b}{x}+c\right )}+\frac {\frac {2 a}{x}+b}{2 \left (b^2-4 a c\right ) \left (\frac {a}{x^2}+\frac {b}{x}+c\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c + a/x^2 + b/x)^3*x^2),x]

[Out]

(b + (2*a)/x)/(2*(b^2 - 4*a*c)*(c + a/x^2 + b/x)^2) - (3*a*(b + (2*a)/x))/((b^2 - 4*a*c)^2*(c + a/x^2 + b/x))
+ (12*a^2*ArcTanh[(b + (2*a)/x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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 1366

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^2} \, dx &=-\text {Subst}\left (\int \frac {1}{\left (c+b x+a x^2\right )^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {b+\frac {2 a}{x}}{2 \left (b^2-4 a c\right ) \left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{\left (c+b x+a x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{b^2-4 a c}\\ &=\frac {b+\frac {2 a}{x}}{2 \left (b^2-4 a c\right ) \left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2}-\frac {3 a \left (b+\frac {2 a}{x}\right )}{\left (b^2-4 a c\right )^2 \left (c+\frac {a}{x^2}+\frac {b}{x}\right )}-\frac {\left (6 a^2\right ) \text {Subst}\left (\int \frac {1}{c+b x+a x^2} \, dx,x,\frac {1}{x}\right )}{\left (b^2-4 a c\right )^2}\\ &=\frac {b+\frac {2 a}{x}}{2 \left (b^2-4 a c\right ) \left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2}-\frac {3 a \left (b+\frac {2 a}{x}\right )}{\left (b^2-4 a c\right )^2 \left (c+\frac {a}{x^2}+\frac {b}{x}\right )}+\frac {\left (12 a^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+\frac {2 a}{x}\right )}{\left (b^2-4 a c\right )^2}\\ &=\frac {b+\frac {2 a}{x}}{2 \left (b^2-4 a c\right ) \left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2}-\frac {3 a \left (b+\frac {2 a}{x}\right )}{\left (b^2-4 a c\right )^2 \left (c+\frac {a}{x^2}+\frac {b}{x}\right )}+\frac {12 a^2 \tanh ^{-1}\left (\frac {b+\frac {2 a}{x}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 174, normalized size = 1.57 \begin {gather*} \frac {1}{2} \left (\frac {b^5-8 a b^3 c+22 a^2 b c^2-2 b^4 c x+16 a b^2 c^2 x-20 a^2 c^3 x}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {b^4 x+a b^2 (b-4 c x)+a^2 c (-3 b+2 c x)}{c^3 \left (-b^2+4 a c\right ) (a+x (b+c x))^2}+\frac {24 a^2 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c + a/x^2 + b/x)^3*x^2),x]

[Out]

((b^5 - 8*a*b^3*c + 22*a^2*b*c^2 - 2*b^4*c*x + 16*a*b^2*c^2*x - 20*a^2*c^3*x)/(c^3*(b^2 - 4*a*c)^2*(a + x*(b +
c*x))) + (b^4*x + a*b^2*(b - 4*c*x) + a^2*c*(-3*b + 2*c*x))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (24*a^
2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(259\) vs. \(2(105)=210\).
time = 0.05, size = 260, normalized size = 2.34

method	result	size
default	\(\frac {-\frac {\left (10 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) x^{3}}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \left (2 a^{2} c^{2}+8 a \,b^{2} c -b^{4}\right ) x^{2}}{2 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (6 a^{2} c^{2}-10 a \,b^{2} c +b^{4}\right ) x}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}+\frac {a^{2} b \left (10 a c -b^{2}\right )}{2 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {12 a^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}\)	\(260\)
risch	\(\frac {-\frac {\left (10 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) x^{3}}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \left (2 a^{2} c^{2}+8 a \,b^{2} c -b^{4}\right ) x^{2}}{2 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (6 a^{2} c^{2}-10 a \,b^{2} c +b^{4}\right ) x}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{2}}+\frac {a^{2} b \left (10 a c -b^{2}\right )}{2 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {6 a^{2} \ln \left (\left (32 a^{2} c^{3}-16 a \,b^{2} c^{2}+2 b^{4} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {6 a^{2} \ln \left (\left (-32 a^{2} c^{3}+16 a \,b^{2} c^{2}-2 b^{4} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\)	\(348\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c+a/x^2+b/x)^3/x^2,x,method=_RETURNVERBOSE)

[Out]

(-1/c*(10*a^2*c^2-8*a*b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*b*(2*a^2*c^2+8*a*b^2*c-b^4)/c^2/(16*a^2*c^
2-8*a*b^2*c+b^4)*x^2-a*(6*a^2*c^2-10*a*b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x+1/2*a^2*b*(10*a*c-b^2)/c^2/
(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+12*a^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+
b)/(4*a*c-b^2)^(1/2))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x^2+b/x)^3/x^2,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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (105) = 210\).
time = 0.37, size = 953, normalized size = 8.59 \begin {gather*} \left [-\frac {a^{2} b^{5} - 14 \, a^{3} b^{3} c + 40 \, a^{4} b c^{2} + 2 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 42 \, a^{2} b^{2} c^{3} - 40 \, a^{3} c^{4}\right )} x^{3} + {\left (b^{7} - 12 \, a b^{5} c + 30 \, a^{2} b^{3} c^{2} + 8 \, a^{3} b c^{3}\right )} x^{2} - 12 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{2} b c^{3} x^{3} + 2 \, a^{3} b c^{2} x + a^{4} c^{2} + {\left (a^{2} b^{2} c^{2} + 2 \, a^{3} c^{3}\right )} x^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left (a b^{6} - 14 \, a^{2} b^{4} c + 46 \, a^{3} b^{2} c^{2} - 24 \, a^{4} c^{3}\right )} x}{2 \, {\left (a^{2} b^{6} c^{2} - 12 \, a^{3} b^{4} c^{3} + 48 \, a^{4} b^{2} c^{4} - 64 \, a^{5} c^{5} + {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} x^{3} + {\left (b^{8} c^{2} - 10 \, a b^{6} c^{3} + 24 \, a^{2} b^{4} c^{4} + 32 \, a^{3} b^{2} c^{5} - 128 \, a^{4} c^{6}\right )} x^{2} + 2 \, {\left (a b^{7} c^{2} - 12 \, a^{2} b^{5} c^{3} + 48 \, a^{3} b^{3} c^{4} - 64 \, a^{4} b c^{5}\right )} x\right )}}, -\frac {a^{2} b^{5} - 14 \, a^{3} b^{3} c + 40 \, a^{4} b c^{2} + 2 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 42 \, a^{2} b^{2} c^{3} - 40 \, a^{3} c^{4}\right )} x^{3} + {\left (b^{7} - 12 \, a b^{5} c + 30 \, a^{2} b^{3} c^{2} + 8 \, a^{3} b c^{3}\right )} x^{2} + 24 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{2} b c^{3} x^{3} + 2 \, a^{3} b c^{2} x + a^{4} c^{2} + {\left (a^{2} b^{2} c^{2} + 2 \, a^{3} c^{3}\right )} x^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (a b^{6} - 14 \, a^{2} b^{4} c + 46 \, a^{3} b^{2} c^{2} - 24 \, a^{4} c^{3}\right )} x}{2 \, {\left (a^{2} b^{6} c^{2} - 12 \, a^{3} b^{4} c^{3} + 48 \, a^{4} b^{2} c^{4} - 64 \, a^{5} c^{5} + {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} x^{3} + {\left (b^{8} c^{2} - 10 \, a b^{6} c^{3} + 24 \, a^{2} b^{4} c^{4} + 32 \, a^{3} b^{2} c^{5} - 128 \, a^{4} c^{6}\right )} x^{2} + 2 \, {\left (a b^{7} c^{2} - 12 \, a^{2} b^{5} c^{3} + 48 \, a^{3} b^{3} c^{4} - 64 \, a^{4} b c^{5}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x^2+b/x)^3/x^2,x, algorithm="fricas")

[Out]

[-1/2*(a^2*b^5 - 14*a^3*b^3*c + 40*a^4*b*c^2 + 2*(b^6*c - 12*a*b^4*c^2 + 42*a^2*b^2*c^3 - 40*a^3*c^4)*x^3 + (b
^7 - 12*a*b^5*c + 30*a^2*b^3*c^2 + 8*a^3*b*c^3)*x^2 - 12*(a^2*c^4*x^4 + 2*a^2*b*c^3*x^3 + 2*a^3*b*c^2*x + a^4*
c^2 + (a^2*b^2*c^2 + 2*a^3*c^3)*x^2)*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*(a*b^6 - 14*a^2*b^4*c + 46*a^3*b^2*c^2 - 24*a^4*c^3)*x)/(a^2*b^6*c^2 -
12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^4 + 2
*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^3 + (b^8*c^2 - 10*a*b^6*c^3 + 24*a^2*b^4*c^4 + 32*
a^3*b^2*c^5 - 128*a^4*c^6)*x^2 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 - 64*a^4*b*c^5)*x), -1/2*(a^2*
b^5 - 14*a^3*b^3*c + 40*a^4*b*c^2 + 2*(b^6*c - 12*a*b^4*c^2 + 42*a^2*b^2*c^3 - 40*a^3*c^4)*x^3 + (b^7 - 12*a*b
^5*c + 30*a^2*b^3*c^2 + 8*a^3*b*c^3)*x^2 + 24*(a^2*c^4*x^4 + 2*a^2*b*c^3*x^3 + 2*a^3*b*c^2*x + a^4*c^2 + (a^2*
b^2*c^2 + 2*a^3*c^3)*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(a*b^6
- 14*a^2*b^4*c + 46*a^3*b^2*c^2 - 24*a^4*c^3)*x)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5 +
(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^4 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*
a^3*b*c^6)*x^3 + (b^8*c^2 - 10*a*b^6*c^3 + 24*a^2*b^4*c^4 + 32*a^3*b^2*c^5 - 128*a^4*c^6)*x^2 + 2*(a*b^7*c^2 -
12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 - 64*a^4*b*c^5)*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (94) = 188\).
time = 0.80, size = 547, normalized size = 4.93 \begin {gather*} - 6 a^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {- 384 a^{5} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 288 a^{4} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 72 a^{3} b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 a^{2} b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 a^{2} b}{12 a^{2} c} \right )} + 6 a^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {384 a^{5} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 288 a^{4} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 72 a^{3} b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 6 a^{2} b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 6 a^{2} b}{12 a^{2} c} \right )} + \frac {10 a^{3} b c - a^{2} b^{3} + x^{3} \left (- 20 a^{2} c^{3} + 16 a b^{2} c^{2} - 2 b^{4} c\right ) + x^{2} \cdot \left (2 a^{2} b c^{2} + 8 a b^{3} c - b^{5}\right ) + x \left (- 12 a^{3} c^{2} + 20 a^{2} b^{2} c - 2 a b^{4}\right )}{32 a^{4} c^{4} - 16 a^{3} b^{2} c^{3} + 2 a^{2} b^{4} c^{2} + x^{4} \cdot \left (32 a^{2} c^{6} - 16 a b^{2} c^{5} + 2 b^{4} c^{4}\right ) + x^{3} \cdot \left (64 a^{2} b c^{5} - 32 a b^{3} c^{4} + 4 b^{5} c^{3}\right ) + x^{2} \cdot \left (64 a^{3} c^{5} - 12 a b^{4} c^{3} + 2 b^{6} c^{2}\right ) + x \left (64 a^{3} b c^{4} - 32 a^{2} b^{3} c^{3} + 4 a b^{5} c^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x**2+b/x)**3/x**2,x)

[Out]

-6*a**2*sqrt(-1/(4*a*c - b**2)**5)*log(x + (-384*a**5*c**3*sqrt(-1/(4*a*c - b**2)**5) + 288*a**4*b**2*c**2*sqr
t(-1/(4*a*c - b**2)**5) - 72*a**3*b**4*c*sqrt(-1/(4*a*c - b**2)**5) + 6*a**2*b**6*sqrt(-1/(4*a*c - b**2)**5) +
6*a**2*b)/(12*a**2*c)) + 6*a**2*sqrt(-1/(4*a*c - b**2)**5)*log(x + (384*a**5*c**3*sqrt(-1/(4*a*c - b**2)**5)
- 288*a**4*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) + 72*a**3*b**4*c*sqrt(-1/(4*a*c - b**2)**5) - 6*a**2*b**6*sqrt
(-1/(4*a*c - b**2)**5) + 6*a**2*b)/(12*a**2*c)) + (10*a**3*b*c - a**2*b**3 + x**3*(-20*a**2*c**3 + 16*a*b**2*c
**2 - 2*b**4*c) + x**2*(2*a**2*b*c**2 + 8*a*b**3*c - b**5) + x*(-12*a**3*c**2 + 20*a**2*b**2*c - 2*a*b**4))/(3
2*a**4*c**4 - 16*a**3*b**2*c**3 + 2*a**2*b**4*c**2 + x**4*(32*a**2*c**6 - 16*a*b**2*c**5 + 2*b**4*c**4) + x**3
*(64*a**2*b*c**5 - 32*a*b**3*c**4 + 4*b**5*c**3) + x**2*(64*a**3*c**5 - 12*a*b**4*c**3 + 2*b**6*c**2) + x*(64*
a**3*b*c**4 - 32*a**2*b**3*c**3 + 4*a*b**5*c**2))

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Giac [A]
time = 3.68, size = 202, normalized size = 1.82 \begin {gather*} \frac {12 \, a^{2} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{4} c x^{3} - 16 \, a b^{2} c^{2} x^{3} + 20 \, a^{2} c^{3} x^{3} + b^{5} x^{2} - 8 \, a b^{3} c x^{2} - 2 \, a^{2} b c^{2} x^{2} + 2 \, a b^{4} x - 20 \, a^{2} b^{2} c x + 12 \, a^{3} c^{2} x + a^{2} b^{3} - 10 \, a^{3} b c}{2 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c+a/x^2+b/x)^3/x^2,x, algorithm="giac")

[Out]

12*a^2*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)) - 1/2*(2*b^4
*c*x^3 - 16*a*b^2*c^2*x^3 + 20*a^2*c^3*x^3 + b^5*x^2 - 8*a*b^3*c*x^2 - 2*a^2*b*c^2*x^2 + 2*a*b^4*x - 20*a^2*b^
2*c*x + 12*a^3*c^2*x + a^2*b^3 - 10*a^3*b*c)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*(c*x^2 + b*x + a)^2)

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Mupad [B]
time = 0.19, size = 343, normalized size = 3.09 \begin {gather*} \frac {12\,a^2\,\mathrm {atan}\left (\frac {\left (\frac {6\,a^2\,\left (16\,a^2\,b\,c^2-8\,a\,b^3\,c+b^5\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {12\,a^2\,c\,x}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{6\,a^2}\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {x^3\,\left (10\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a^2\,\left (b^3-10\,a\,b\,c\right )}{2\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {x^2\,\left (2\,a^2\,b\,c^2+8\,a\,b^3\,c-b^5\right )}{2\,c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a\,x\,\left (6\,a^2\,c^2-10\,a\,b^2\,c+b^4\right )}{c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(c + a/x^2 + b/x)^3),x)

[Out]

(12*a^2*atan((((6*a^2*(b^5 + 16*a^2*b*c^2 - 8*a*b^3*c))/((4*a*c - b^2)^(5/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) +
(12*a^2*c*x)/(4*a*c - b^2)^(5/2))*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(6*a^2)))/(4*a*c - b^2)^(5/2) - ((x^3*(b^4
+ 10*a^2*c^2 - 8*a*b^2*c))/(c*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a^2*(b^3 - 10*a*b*c))/(2*c^2*(b^4 + 16*a^2*c^
2 - 8*a*b^2*c)) - (x^2*(2*a^2*b*c^2 - b^5 + 8*a*b^3*c))/(2*c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (a*x*(b^4 + 6
*a^2*c^2 - 10*a*b^2*c))/(c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2
*b*c*x^3)

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