3.5.22 \(\int \frac {x}{(c+\frac {a}{x^2}+\frac {b}{x})^2} \, dx\) [422]

Optimal. Leaf size=196 \[ -\frac {b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \]

[Out]

-b*(-11*a*c+3*b^2)*x/c^3/(-4*a*c+b^2)+1/2*(-8*a*c+3*b^2)*x^2/c^2/(-4*a*c+b^2)-b*x^3/c/(-4*a*c+b^2)+x^4*(b*x+2*
a)/(-4*a*c+b^2)/(c*x^2+b*x+a)+b*(30*a^2*c^2-20*a*b^2*c+3*b^4)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^4/(-4*a*
c+b^2)^(3/2)+1/2*(-2*a*c+3*b^2)*ln(c*x^2+b*x+a)/c^4

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Rubi [A]
time = 0.15, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1368, 752, 814, 648, 632, 212, 642} \begin {gather*} \frac {b \left (30 a^2 c^2-20 a b^2 c+3 b^4\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 )^{3/2}}+\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {b x \left (3 b^2-11 a c\right )}{c^3 \left (b^2-4 a c\right )}+\frac {x^2 \left (3 b^2-8 a c\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(c + a/x^2 + b/x)^2,x]

[Out]

-((b*(3*b^2 - 11*a*c)*x)/(c^3*(b^2 - 4*a*c))) + ((3*b^2 - 8*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (b*x^3)/(c*(b^2
- 4*a*c)) + (x^4*(2*a + b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTanh
[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(3/2)) + ((3*b^2 - 2*a*c)*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 1368

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]

Rubi steps

\begin {align*} \int \frac {x}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^2} \, dx &=\int \frac {x^5}{\left (a+b x+c x^2\right )^2} \, dx\\ &=\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {x^3 (8 a+3 b x)}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {b \left (3 b^2-11 a c\right )}{c^3}-\frac {\left (3 b^2-8 a c\right ) x}{c^2}+\frac {3 b x^2}{c}-\frac {a b \left (3 b^2-11 a c\right )+\left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx}{-b^2+4 a c}\\ &=-\frac {b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {a b \left (3 b^2-11 a c\right )+\left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=-\frac {b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (3 b^2-2 a c\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}-\frac {\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4 \left (b^2-4 a c\right )}\\ &=-\frac {b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\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 )}\\ &=-\frac {b \left (3 b^2-11 a c\right ) x}{c^3 \left (b^2-4 a c\right )}+\frac {\left (3 b^2-8 a c\right ) x^2}{2 c^2 \left (b^2-4 a c\right )}-\frac {b x^3}{c \left (b^2-4 a c\right )}+\frac {x^4 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {b \left (3 b^4-20 a b^2 c+30 a^2 c^2\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 )^{3/2}}+\frac {\left (3 b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 163, normalized size = 0.83 \begin {gather*} \frac {-4 b c x+c^2 x^2+\frac {2 \left (2 a^3 c^2+b^5 x+a b^3 (b-5 c x)+a^2 b c (-4 b+5 c x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+\left (3 b^2-2 a c\right ) \log (a+x (b+c x))}{2 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(c + a/x^2 + b/x)^2,x]

[Out]

(-4*b*c*x + c^2*x^2 + (2*(2*a^3*c^2 + b^5*x + a*b^3*(b - 5*c*x) + a^2*b*c*(-4*b + 5*c*x)))/((b^2 - 4*a*c)*(a +
x*(b + c*x))) + (2*b*(3*b^4 - 20*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)
^(3/2) + (3*b^2 - 2*a*c)*Log[a + x*(b + c*x)])/(2*c^4)

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Maple [A]
time = 0.06, size = 238, normalized size = 1.21

method	result	size
default	\(-\frac {-\frac {1}{2} c \,x^{2}+2 b x}{c^{3}}+\frac {\frac {-\frac {b \left (5 a^{2} c^{2}-5 a \,b^{2} c +b^{4}\right ) x}{c \left (4 a c -b^{2}\right )}-\frac {a \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (-8 a^{2} c^{2}+14 a \,b^{2} c -3 b^{4}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (11 a^{2} b c -3 a \,b^{3}-\frac {\left (-8 a^{2} c^{2}+14 a \,b^{2} c -3 b^{4}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{3}}\)	\(238\)
risch	\(\text {Expression too large to display}\)	\(1596\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c^3*(-1/2*c*x^2+2*b*x)+1/c^3*((-b*(5*a^2*c^2-5*a*b^2*c+b^4)/c/(4*a*c-b^2)*x-a/c*(2*a^2*c^2-4*a*b^2*c+b^4)/(
4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b^2)*(1/2*(-8*a^2*c^2+14*a*b^2*c-3*b^4)/c*ln(c*x^2+b*x+a)+2*(11*a^2*b*c-3*a
*b^3-1/2*(-8*a^2*c^2+14*a*b^2*c-3*b^4)*b/c)/(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(x/(c+a/x^2+b/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. 504 vs. \(2 (188) = 376\).
time = 0.35, size = 1029, normalized size = 5.25 \begin {gather*} \left [\frac {2 \, a b^{6} - 16 \, a^{2} b^{4} c + 36 \, a^{3} b^{2} c^{2} - 16 \, a^{4} c^{3} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} - 3 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{3} - {\left (4 \, b^{6} c - 33 \, a b^{4} c^{2} + 72 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} x^{2} - {\left (3 \, a b^{5} - 20 \, a^{2} b^{3} c + 30 \, a^{3} b c^{2} + {\left (3 \, b^{5} c - 20 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x^{2} + {\left (3 \, b^{6} - 20 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2}\right )} x\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 (b^{7} - 11 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} - 52 \, a^{3} b c^{3}\right )} x + {\left (3 \, a b^{6} - 26 \, a^{2} b^{4} c + 64 \, a^{3} b^{2} c^{2} - 32 \, a^{4} c^{3} + {\left (3 \, b^{6} c - 26 \, a b^{4} c^{2} + 64 \, a^{2} b^{2} c^{3} - 32 \, a^{3} c^{4}\right )} x^{2} + {\left (3 \, b^{7} - 26 \, a b^{5} c + 64 \, a^{2} b^{3} c^{2} - 32 \, a^{3} b c^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{4} - 8 \, a^{2} b^{2} c^{5} + 16 \, a^{3} c^{6} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{2} + {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x\right )}}, \frac {2 \, a b^{6} - 16 \, a^{2} b^{4} c + 36 \, a^{3} b^{2} c^{2} - 16 \, a^{4} c^{3} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} - 3 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{3} - {\left (4 \, b^{6} c - 33 \, a b^{4} c^{2} + 72 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} x^{2} + 2 \, {\left (3 \, a b^{5} - 20 \, a^{2} b^{3} c + 30 \, a^{3} b c^{2} + {\left (3 \, b^{5} c - 20 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x^{2} + {\left (3 \, b^{6} - 20 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2}\right )} x\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 (b^{7} - 11 \, a b^{5} c + 41 \, a^{2} b^{3} c^{2} - 52 \, a^{3} b c^{3}\right )} x + {\left (3 \, a b^{6} - 26 \, a^{2} b^{4} c + 64 \, a^{3} b^{2} c^{2} - 32 \, a^{4} c^{3} + {\left (3 \, b^{6} c - 26 \, a b^{4} c^{2} + 64 \, a^{2} b^{2} c^{3} - 32 \, a^{3} c^{4}\right )} x^{2} + {\left (3 \, b^{7} - 26 \, a b^{5} c + 64 \, a^{2} b^{3} c^{2} - 32 \, a^{3} b c^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (a b^{4} c^{4} - 8 \, a^{2} b^{2} c^{5} + 16 \, a^{3} c^{6} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{2} + {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*a*b^6 - 16*a^2*b^4*c + 36*a^3*b^2*c^2 - 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 - 3*(b^5
*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 - (4*b^6*c - 33*a*b^4*c^2 + 72*a^2*b^2*c^3 - 16*a^3*c^4)*x^2 - (3*a*b^5
- 20*a^2*b^3*c + 30*a^3*b*c^2 + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*x^2 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^
2*c^2)*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*(b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*x + (3*a*b^6 - 26*a^2*b^4*c + 64*a^3*b^2*c^2 -
32*a^4*c^3 + (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^2 + (3*b^7 - 26*a*b^5*c + 64*a^2*b^3*c^
2 - 32*a^3*b*c^3)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6 + (b^4*c^5 - 8*a*b^2*c^6 +
16*a^2*c^7)*x^2 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x), 1/2*(2*a*b^6 - 16*a^2*b^4*c + 36*a^3*b^2*c^2 - 16
*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 - 3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 - (4*b^6*c
- 33*a*b^4*c^2 + 72*a^2*b^2*c^3 - 16*a^3*c^4)*x^2 + 2*(3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2 + (3*b^5*c - 20*a
*b^3*c^2 + 30*a^2*b*c^3)*x^2 + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 +
4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(b^7 - 11*a*b^5*c + 41*a^2*b^3*c^2 - 52*a^3*b*c^3)*x + (3*a*b^6 - 26*a^
2*b^4*c + 64*a^3*b^2*c^2 - 32*a^4*c^3 + (3*b^6*c - 26*a*b^4*c^2 + 64*a^2*b^2*c^3 - 32*a^3*c^4)*x^2 + (3*b^7 -
26*a*b^5*c + 64*a^2*b^3*c^2 - 32*a^3*b*c^3)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^4 - 8*a^2*b^2*c^5 + 16*a^3*c^6 +
(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^2 + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1012 vs. \(2 (180) = 360\).
time = 1.48, size = 1012, normalized size = 5.16 \begin {gather*} - \frac {2 b x}{c^{3}} + \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) \log {\left (x + \frac {16 a^{3} c^{2} - 17 a^{2} b^{2} c + 16 a^{2} c^{5} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) + 3 a b^{4} - 8 a b^{2} c^{4} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) + b^{4} c^{3} \left (- \frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right )}{30 a^{2} b c^{2} - 20 a b^{3} c + 3 b^{5}} \right )} + \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) \log {\left (x + \frac {16 a^{3} c^{2} - 17 a^{2} b^{2} c + 16 a^{2} c^{5} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) + 3 a b^{4} - 8 a b^{2} c^{4} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right ) + b^{4} c^{3} \left (\frac {b \sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (30 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4}\right )}{2 c^{4} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} - \frac {2 a c - 3 b^{2}}{2 c^{4}}\right )}{30 a^{2} b c^{2} - 20 a b^{3} c + 3 b^{5}} \right )} + \frac {- 2 a^{3} c^{2} + 4 a^{2} b^{2} c - a b^{4} + x \left (- 5 a^{2} b c^{2} + 5 a b^{3} c - b^{5}\right )}{4 a^{2} c^{5} - a b^{2} c^{4} + x^{2} \cdot \left (4 a c^{6} - b^{2} c^{5}\right ) + x \left (4 a b c^{5} - b^{3} c^{4}\right )} + \frac {x^{2}}{2 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b*x/c**3 + (-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a*
*2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4))*log(x + (16*a**3*c**2 - 17*a**2*b**2*c + 16*a
**2*c**5*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b*
*2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)) + 3*a*b**4 - 8*a*b**2*c**4*(-b*sqrt(-(4*a*c - b**2
)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) -
(2*a*c - 3*b**2)/(2*c**4)) + b**4*c**3*(-b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*
c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)))/(30*a**2*b*c**2 -
20*a*b**3*c + 3*b**5)) + (b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c*
*3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4))*log(x + (16*a**3*c**2 - 17*a**2*b**
2*c + 16*a**2*c**5*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 4
8*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)) + 3*a*b**4 - 8*a*b**2*c**4*(b*sqrt(-(4*a*
c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b**4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c -
b**6)) - (2*a*c - 3*b**2)/(2*c**4)) + b**4*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(30*a**2*c**2 - 20*a*b**2*c + 3*b*
*4)/(2*c**4*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*a*c - 3*b**2)/(2*c**4)))/(30*a**2*b*
c**2 - 20*a*b**3*c + 3*b**5)) + (-2*a**3*c**2 + 4*a**2*b**2*c - a*b**4 + x*(-5*a**2*b*c**2 + 5*a*b**3*c - b**5
))/(4*a**2*c**5 - a*b**2*c**4 + x**2*(4*a*c**6 - b**2*c**5) + x*(4*a*b*c**5 - b**3*c**4)) + x**2/(2*c**2)

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Giac [A]
time = 3.28, size = 188, normalized size = 0.96 \begin {gather*} -\frac {{\left (3 \, b^{5} - 20 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (3 \, b^{2} - 2 \, a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {c^{2} x^{2} - 4 \, b c x}{2 \, c^{4}} + \frac {a b^{4} - 4 \, a^{2} b^{2} c + 2 \, a^{3} c^{2} + {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*b^5 - 20*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^4 - 4*a*c^5)*sqrt(-b^2 + 4
*a*c)) + 1/2*(3*b^2 - 2*a*c)*log(c*x^2 + b*x + a)/c^4 + 1/2*(c^2*x^2 - 4*b*c*x)/c^4 + (a*b^4 - 4*a^2*b^2*c + 2
*a^3*c^2 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^4)

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Mupad [B]
time = 1.82, size = 382, normalized size = 1.95 \begin {gather*} \frac {x^2}{2\,c^2}-\frac {\frac {a\,\left (2\,a^2\,c^2-4\,a\,b^2\,c+b^4\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {b\,x\,\left (5\,a^2\,c^2-5\,a\,b^2\,c+b^4\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^4\,x^2+b\,c^3\,x+a\,c^3}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (128\,a^4\,c^4-288\,a^3\,b^2\,c^3+168\,a^2\,b^4\,c^2-38\,a\,b^6\,c+3\,b^8\right )}{2\,\left (64\,a^3\,c^7-48\,a^2\,b^2\,c^6+12\,a\,b^4\,c^5-b^6\,c^4\right )}-\frac {2\,b\,x}{c^3}+\frac {b\,\mathrm {atan}\left (\frac {c^4\,\left (\frac {2\,b\,x\,\left (30\,a^2\,c^2-20\,a\,b^2\,c+3\,b^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^3}-\frac {b\,\left (b^3\,c^3-4\,a\,b\,c^4\right )\,\left (30\,a^2\,c^2-20\,a\,b^2\,c+3\,b^4\right )}{c^7\,{\left (4\,a\,c-b^2\right )}^4}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}}{30\,a^2\,b\,c^2-20\,a\,b^3\,c+3\,b^5}\right )\,\left (30\,a^2\,c^2-20\,a\,b^2\,c+3\,b^4\right )}{c^4\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(c + a/x^2 + b/x)^2,x)

[Out]

x^2/(2*c^2) - ((a*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/(c*(4*a*c - b^2)) + (b*x*(b^4 + 5*a^2*c^2 - 5*a*b^2*c))/(c*(4
*a*c - b^2)))/(a*c^3 + c^4*x^2 + b*c^3*x) - (log(a + b*x + c*x^2)*(3*b^8 + 128*a^4*c^4 + 168*a^2*b^4*c^2 - 288
*a^3*b^2*c^3 - 38*a*b^6*c))/(2*(64*a^3*c^7 - b^6*c^4 + 12*a*b^4*c^5 - 48*a^2*b^2*c^6)) - (2*b*x)/c^3 + (b*atan
((c^4*((2*b*x*(3*b^4 + 30*a^2*c^2 - 20*a*b^2*c))/(c^3*(4*a*c - b^2)^3) - (b*(b^3*c^3 - 4*a*b*c^4)*(3*b^4 + 30*
a^2*c^2 - 20*a*b^2*c))/(c^7*(4*a*c - b^2)^4))*(4*a*c - b^2)^(5/2))/(3*b^5 + 30*a^2*b*c^2 - 20*a*b^3*c))*(3*b^4
+ 30*a^2*c^2 - 20*a*b^2*c))/(c^4*(4*a*c - b^2)^(3/2))

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