3.1.72 \(\int \frac {x}{(a+\frac {c}{x^2}+\frac {b}{x}) (d+e x)^2} \, dx\) [72]

Optimal. Leaf size=246 \[ \frac {d^3}{e^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {\left (b^3 d^2-2 b^2 c d e+4 a c^2 d e-b c \left (3 a d^2-c e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {d^2 \left (a d^2-e (2 b d-3 c e)\right ) \log (d+e x)}{e^2 \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (b^2 d^2-2 b c d e-c \left (a d^2-c e^2\right )\right ) \log \left (c+b x+a x^2\right )}{2 a \left (a d^2-e (b d-c e)\right )^2} \]

[Out]

d^3/e^2/(a*d^2-e*(b*d-c*e))/(e*x+d)+d^2*(a*d^2-e*(2*b*d-3*c*e))*ln(e*x+d)/e^2/(a*d^2-e*(b*d-c*e))^2+1/2*(b^2*d
^2-2*b*c*d*e-c*(a*d^2-c*e^2))*ln(a*x^2+b*x+c)/a/(a*d^2-e*(b*d-c*e))^2+(b^3*d^2-2*b^2*c*d*e+4*a*c^2*d*e-b*c*(3*
a*d^2-c*e^2))*arctanh((2*a*x+b)/(-4*a*c+b^2)^(1/2))/a/(a*d^2-e*(b*d-c*e))^2/(-4*a*c+b^2)^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1583, 1642, 648, 632, 212, 642} \begin {gather*} \frac {\left (-c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )^2}+\frac {\left (-b c \left (3 a d^2-c e^2\right )+4 a c^2 d e+b^3 d^2-2 b^2 c d e\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {d^2 \log (d+e x) \left (a d^2-e (2 b d-3 c e)\right )}{e^2 \left (a d^2-e (b d-c e)\right )^2}+\frac {d^3}{e^2 (d+e x) \left (a d^2-e (b d-c e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((a + c/x^2 + b/x)*(d + e*x)^2),x]

[Out]

d^3/(e^2*(a*d^2 - e*(b*d - c*e))*(d + e*x)) + ((b^3*d^2 - 2*b^2*c*d*e + 4*a*c^2*d*e - b*c*(3*a*d^2 - c*e^2))*A
rcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))^2) + (d^2*(a*d^2 - e*(2*b*
d - 3*c*e))*Log[d + e*x])/(e^2*(a*d^2 - e*(b*d - c*e))^2) + ((b^2*d^2 - 2*b*c*d*e - c*(a*d^2 - c*e^2))*Log[c +
b*x + a*x^2])/(2*a*(a*d^2 - e*(b*d - c*e))^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 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 1583

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbo
l] :> Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, m, n, q}, x] && E
qQ[mn, -n] && EqQ[mn2, 2*mn] && IntegerQ[p]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]
time = 0.16, size = 207, normalized size = 0.84 \begin {gather*} \frac {\frac {2 d^3 \left (a d^2+e (-b d+c e)\right )}{e^2 (d+e x)}-\frac {2 \left (b^3 d^2-2 b^2 c d e+4 a c^2 d e+b c \left (-3 a d^2+c e^2\right )\right ) \tan ^{-1}\left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{a \sqrt {-b^2+4 a c}}+\frac {2 \left (a d^4+d^2 e (-2 b d+3 c e)\right ) \log (d+e x)}{e^2}+\frac {\left (b^2 d^2-2 b c d e+c \left (-a d^2+c e^2\right )\right ) \log (c+x (b+a x))}{a}}{2 \left (a d^2+e (-b d+c e)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/((a + c/x^2 + b/x)*(d + e*x)^2),x]

[Out]

((2*d^3*(a*d^2 + e*(-(b*d) + c*e)))/(e^2*(d + e*x)) - (2*(b^3*d^2 - 2*b^2*c*d*e + 4*a*c^2*d*e + b*c*(-3*a*d^2
+ c*e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a*Sqrt[-b^2 + 4*a*c]) + (2*(a*d^4 + d^2*e*(-2*b*d + 3*c*e))
*Log[d + e*x])/e^2 + ((b^2*d^2 - 2*b*c*d*e + c*(-(a*d^2) + c*e^2))*Log[c + x*(b + a*x)])/a)/(2*(a*d^2 + e*(-(b
*d) + c*e))^2)

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Maple [A]
time = 0.27, size = 228, normalized size = 0.93

method	result	size
default	\(\frac {d^{2} \left (a \,d^{2}-2 d e b +3 c \,e^{2}\right ) \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2} e^{2}}+\frac {d^{3}}{e^{2} \left (a \,d^{2}-d e b +c \,e^{2}\right ) \left (e x +d \right )}+\frac {\frac {\left (-a c \,d^{2}+b^{2} d^{2}-2 b c d e +c^{2} e^{2}\right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (b c \,d^{2}-2 c^{2} d e -\frac {\left (-a c \,d^{2}+b^{2} d^{2}-2 b c d e +c^{2} e^{2}\right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2}}\)	\(228\)
risch	\(\text {Expression too large to display}\)	\(33066\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2*(a*d^2-2*b*d*e+3*c*e^2)/(a*d^2-b*d*e+c*e^2)^2/e^2*ln(e*x+d)+d^3/e^2/(a*d^2-b*d*e+c*e^2)/(e*x+d)+1/(a*d^2-b
*d*e+c*e^2)^2*(1/2*(-a*c*d^2+b^2*d^2-2*b*c*d*e+c^2*e^2)/a*ln(a*x^2+b*x+c)+2*(b*c*d^2-2*c^2*d*e-1/2*(-a*c*d^2+b
^2*d^2-2*b*c*d*e+c^2*e^2)*b/a)/(4*a*c-b^2)^(1/2)*arctan((2*a*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/(a+c/x^2+b/x)/(e*x+d)^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. 710 vs. \(2 (251) = 502\).
time = 11.04, size = 1439, normalized size = 5.85 \begin {gather*} \left [\frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{5} - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{4} e + 2 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{3} e^{2} + {\left (b c^{2} x e^{5} + {\left (b^{3} - 3 \, a b c\right )} d^{3} e^{2} + {\left (b c^{2} d - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d x\right )} e^{4} + {\left ({\left (b^{3} - 3 \, a b c\right )} d^{2} x - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d^{2}\right )} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, a^{2} x^{2} + 2 \, a b x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, a x + b\right )}}{a x^{2} + b x + c}\right ) + {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{3} e^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x e^{5} - {\left (2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d x - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d\right )} e^{4} + {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2}\right )} e^{3}\right )} \log \left (a x^{2} + b x + c\right ) + 2 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{5} + 3 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{2} x e^{3} - {\left (2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{3} x - 3 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{3}\right )} e^{2} + {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{4} x - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{4}\right )} e\right )} \log \left (x e + d\right )}{2 \, {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{5} e^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x e^{7} - {\left (2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x - {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d\right )} e^{6} + {\left ({\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2} x - 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{2}\right )} e^{5} - {\left (2 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d^{3} x - {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{3}\right )} e^{4} + {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{4} x - 2 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d^{4}\right )} e^{3}\right )}}, \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{5} - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{4} e + 2 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{3} e^{2} + 2 \, {\left (b c^{2} x e^{5} + {\left (b^{3} - 3 \, a b c\right )} d^{3} e^{2} + {\left (b c^{2} d - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d x\right )} e^{4} + {\left ({\left (b^{3} - 3 \, a b c\right )} d^{2} x - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d^{2}\right )} e^{3}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, a x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{3} e^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x e^{5} - {\left (2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d x - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d\right )} e^{4} + {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2}\right )} e^{3}\right )} \log \left (a x^{2} + b x + c\right ) + 2 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{5} + 3 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{2} x e^{3} - {\left (2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{3} x - 3 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{3}\right )} e^{2} + {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{4} x - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{4}\right )} e\right )} \log \left (x e + d\right )}{2 \, {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{5} e^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x e^{7} - {\left (2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x - {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d\right )} e^{6} + {\left ({\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{2} x - 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{2}\right )} e^{5} - {\left (2 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d^{3} x - {\left (a b^{4} - 2 \, a^{2} b^{2} c - 8 \, a^{3} c^{2}\right )} d^{3}\right )} e^{4} + {\left ({\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d^{4} x - 2 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d^{4}\right )} e^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*(a^2*b^2 - 4*a^3*c)*d^5 - 2*(a*b^3 - 4*a^2*b*c)*d^4*e + 2*(a*b^2*c - 4*a^2*c^2)*d^3*e^2 + (b*c^2*x*e^5
+ (b^3 - 3*a*b*c)*d^3*e^2 + (b*c^2*d - 2*(b^2*c - 2*a*c^2)*d*x)*e^4 + ((b^3 - 3*a*b*c)*d^2*x - 2*(b^2*c - 2*a
*c^2)*d^2)*e^3)*sqrt(b^2 - 4*a*c)*log((2*a^2*x^2 + 2*a*b*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*a*x + b))/(a*x
^2 + b*x + c)) + ((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d^3*e^2 + (b^2*c^2 - 4*a*c^3)*x*e^5 - (2*(b^3*c - 4*a*b*c^2)*d
*x - (b^2*c^2 - 4*a*c^3)*d)*e^4 + ((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d^2*x - 2*(b^3*c - 4*a*b*c^2)*d^2)*e^3)*log(a
*x^2 + b*x + c) + 2*((a^2*b^2 - 4*a^3*c)*d^5 + 3*(a*b^2*c - 4*a^2*c^2)*d^2*x*e^3 - (2*(a*b^3 - 4*a^2*b*c)*d^3*
x - 3*(a*b^2*c - 4*a^2*c^2)*d^3)*e^2 + ((a^2*b^2 - 4*a^3*c)*d^4*x - 2*(a*b^3 - 4*a^2*b*c)*d^4)*e)*log(x*e + d)
)/((a^3*b^2 - 4*a^4*c)*d^5*e^2 + (a*b^2*c^2 - 4*a^2*c^3)*x*e^7 - (2*(a*b^3*c - 4*a^2*b*c^2)*d*x - (a*b^2*c^2 -
4*a^2*c^3)*d)*e^6 + ((a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*x - 2*(a*b^3*c - 4*a^2*b*c^2)*d^2)*e^5 - (2*(a^2*b
^3 - 4*a^3*b*c)*d^3*x - (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3)*e^4 + ((a^3*b^2 - 4*a^4*c)*d^4*x - 2*(a^2*b^3 -
4*a^3*b*c)*d^4)*e^3), 1/2*(2*(a^2*b^2 - 4*a^3*c)*d^5 - 2*(a*b^3 - 4*a^2*b*c)*d^4*e + 2*(a*b^2*c - 4*a^2*c^2)*
d^3*e^2 + 2*(b*c^2*x*e^5 + (b^3 - 3*a*b*c)*d^3*e^2 + (b*c^2*d - 2*(b^2*c - 2*a*c^2)*d*x)*e^4 + ((b^3 - 3*a*b*c
)*d^2*x - 2*(b^2*c - 2*a*c^2)*d^2)*e^3)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*a*x + b)/(b^2 - 4*a*c
)) + ((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d^3*e^2 + (b^2*c^2 - 4*a*c^3)*x*e^5 - (2*(b^3*c - 4*a*b*c^2)*d*x - (b^2*c^
2 - 4*a*c^3)*d)*e^4 + ((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d^2*x - 2*(b^3*c - 4*a*b*c^2)*d^2)*e^3)*log(a*x^2 + b*x +
c) + 2*((a^2*b^2 - 4*a^3*c)*d^5 + 3*(a*b^2*c - 4*a^2*c^2)*d^2*x*e^3 - (2*(a*b^3 - 4*a^2*b*c)*d^3*x - 3*(a*b^2
*c - 4*a^2*c^2)*d^3)*e^2 + ((a^2*b^2 - 4*a^3*c)*d^4*x - 2*(a*b^3 - 4*a^2*b*c)*d^4)*e)*log(x*e + d))/((a^3*b^2
- 4*a^4*c)*d^5*e^2 + (a*b^2*c^2 - 4*a^2*c^3)*x*e^7 - (2*(a*b^3*c - 4*a^2*b*c^2)*d*x - (a*b^2*c^2 - 4*a^2*c^3)*
d)*e^6 + ((a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*x - 2*(a*b^3*c - 4*a^2*b*c^2)*d^2)*e^5 - (2*(a^2*b^3 - 4*a^3*b
*c)*d^3*x - (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3)*e^4 + ((a^3*b^2 - 4*a^4*c)*d^4*x - 2*(a^2*b^3 - 4*a^3*b*c)*
d^4)*e^3)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 2.93, size = 412, normalized size = 1.67 \begin {gather*} \frac {1}{2} \, {\left (\frac {2 \, d^{3} e^{2}}{{\left (a d^{2} e^{3} - b d e^{4} + c e^{5}\right )} {\left (x e + d\right )}} + \frac {2 \, {\left (b^{3} d^{2} e^{3} - 3 \, a b c d^{2} e^{3} - 2 \, b^{2} c d e^{4} + 4 \, a c^{2} d e^{4} + b c^{2} e^{5}\right )} \arctan \left (-\frac {{\left (2 \, a d - \frac {2 \, a d^{2}}{x e + d} - b e + \frac {2 \, b d e}{x e + d} - \frac {2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{3} d^{4} - 2 \, a^{2} b d^{3} e + a b^{2} d^{2} e^{2} + 2 \, a^{2} c d^{2} e^{2} - 2 \, a b c d e^{3} + a c^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {{\left (b^{2} d^{2} e - a c d^{2} e - 2 \, b c d e^{2} + c^{2} e^{3}\right )} \log \left (-a + \frac {2 \, a d}{x e + d} - \frac {a d^{2}}{{\left (x e + d\right )}^{2}} - \frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} - \frac {c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{a^{3} d^{4} - 2 \, a^{2} b d^{3} e + a b^{2} d^{2} e^{2} + 2 \, a^{2} c d^{2} e^{2} - 2 \, a b c d e^{3} + a c^{2} e^{4}} - \frac {2 \, e^{\left (-1\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right )}{a}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*d^3*e^2/((a*d^2*e^3 - b*d*e^4 + c*e^5)*(x*e + d)) + 2*(b^3*d^2*e^3 - 3*a*b*c*d^2*e^3 - 2*b^2*c*d*e^4 +
4*a*c^2*d*e^4 + b*c^2*e^5)*arctan(-(2*a*d - 2*a*d^2/(x*e + d) - b*e + 2*b*d*e/(x*e + d) - 2*c*e^2/(x*e + d))*e
^(-1)/sqrt(-b^2 + 4*a*c))*e^(-2)/((a^3*d^4 - 2*a^2*b*d^3*e + a*b^2*d^2*e^2 + 2*a^2*c*d^2*e^2 - 2*a*b*c*d*e^3 +
a*c^2*e^4)*sqrt(-b^2 + 4*a*c)) + (b^2*d^2*e - a*c*d^2*e - 2*b*c*d*e^2 + c^2*e^3)*log(-a + 2*a*d/(x*e + d) - a
*d^2/(x*e + d)^2 - b*e/(x*e + d) + b*d*e/(x*e + d)^2 - c*e^2/(x*e + d)^2)/(a^3*d^4 - 2*a^2*b*d^3*e + a*b^2*d^2
*e^2 + 2*a^2*c*d^2*e^2 - 2*a*b*c*d*e^3 + a*c^2*e^4) - 2*e^(-1)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2)/a)*e^(-1)

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Mupad [B]
time = 5.11, size = 2037, normalized size = 8.28 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (a\,d^4-2\,b\,d^3\,e+3\,c\,d^2\,e^2\right )}{a^2\,d^4\,e^2-2\,a\,b\,d^3\,e^3+2\,a\,c\,d^2\,e^4+b^2\,d^2\,e^4-2\,b\,c\,d\,e^5+c^2\,e^6}-\frac {\ln \left (a^2\,b^2\,d^6-4\,a^3\,c\,d^6-2\,c^4\,e^6-b^4\,d^4\,e^2+c^3\,e^6\,x\,\sqrt {b^2-4\,a\,c}+24\,a\,c^3\,d^2\,e^4+6\,b^3\,c\,d^3\,e^3+2\,b^4\,d^3\,e^3\,x-b^3\,d^4\,e^2\,\sqrt {b^2-4\,a\,c}-10\,a^2\,c^2\,d^4\,e^2-9\,b^2\,c^2\,d^2\,e^4-2\,a\,b^3\,d^5\,e+4\,b\,c^3\,d\,e^5-b\,c^3\,e^6\,x+a^2\,b\,d^6\,\sqrt {b^2-4\,a\,c}+4\,c^3\,d\,e^5\,\sqrt {b^2-4\,a\,c}+2\,a^3\,d^6\,x\,\sqrt {b^2-4\,a\,c}+8\,a^2\,b\,c\,d^5\,e+8\,a\,c^3\,d\,e^5\,x-8\,a^3\,c\,d^5\,e\,x-2\,a\,b^2\,d^5\,e\,\sqrt {b^2-4\,a\,c}-4\,a^2\,c\,d^5\,e\,\sqrt {b^2-4\,a\,c}-20\,a\,b\,c^2\,d^3\,e^3+6\,a\,b^2\,c\,d^4\,e^2-6\,a\,b^3\,d^4\,e^2\,x+2\,a^2\,b^2\,d^5\,e\,x-3\,b^3\,c\,d^2\,e^4\,x-16\,a\,c^2\,d^3\,e^3\,\sqrt {b^2-4\,a\,c}-3\,b\,c^2\,d^2\,e^4\,\sqrt {b^2-4\,a\,c}+2\,b^2\,c\,d^3\,e^3\,\sqrt {b^2-4\,a\,c}-2\,b^3\,d^3\,e^3\,x\,\sqrt {b^2-4\,a\,c}-32\,a^2\,c^2\,d^3\,e^3\,x+4\,a\,b^2\,d^4\,e^2\,x\,\sqrt {b^2-4\,a\,c}-12\,a\,c^2\,d^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}+5\,a^2\,c\,d^4\,e^2\,x\,\sqrt {b^2-4\,a\,c}+3\,b^2\,c\,d^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}+14\,a\,b\,c\,d^4\,e^2\,\sqrt {b^2-4\,a\,c}-6\,a^2\,b\,d^5\,e\,x\,\sqrt {b^2-4\,a\,c}+6\,a\,b\,c^2\,d^2\,e^4\,x+2\,a\,b^2\,c\,d^3\,e^3\,x+23\,a^2\,b\,c\,d^4\,e^2\,x+2\,a\,b\,c\,d^3\,e^3\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,d^2-4\,a\,c^3\,e^2+b^3\,d^2\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,d^2+b^2\,c^2\,e^2-2\,b^3\,c\,d\,e-5\,a\,b^2\,c\,d^2+b\,c^2\,e^2\,\sqrt {b^2-4\,a\,c}+8\,a\,b\,c^2\,d\,e-3\,a\,b\,c\,d^2\,\sqrt {b^2-4\,a\,c}+4\,a\,c^2\,d\,e\,\sqrt {b^2-4\,a\,c}-2\,b^2\,c\,d\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^4\,c\,d^4-a^3\,b^2\,d^4-8\,a^3\,b\,c\,d^3\,e+8\,a^3\,c^2\,d^2\,e^2+2\,a^2\,b^3\,d^3\,e+2\,a^2\,b^2\,c\,d^2\,e^2-8\,a^2\,b\,c^2\,d\,e^3+4\,a^2\,c^3\,e^4-a\,b^4\,d^2\,e^2+2\,a\,b^3\,c\,d\,e^3-a\,b^2\,c^2\,e^4\right )}-\frac {\ln \left (2\,c^4\,e^6+4\,a^3\,c\,d^6-a^2\,b^2\,d^6+b^4\,d^4\,e^2+c^3\,e^6\,x\,\sqrt {b^2-4\,a\,c}-24\,a\,c^3\,d^2\,e^4-6\,b^3\,c\,d^3\,e^3-2\,b^4\,d^3\,e^3\,x-b^3\,d^4\,e^2\,\sqrt {b^2-4\,a\,c}+10\,a^2\,c^2\,d^4\,e^2+9\,b^2\,c^2\,d^2\,e^4+2\,a\,b^3\,d^5\,e-4\,b\,c^3\,d\,e^5+b\,c^3\,e^6\,x+a^2\,b\,d^6\,\sqrt {b^2-4\,a\,c}+4\,c^3\,d\,e^5\,\sqrt {b^2-4\,a\,c}+2\,a^3\,d^6\,x\,\sqrt {b^2-4\,a\,c}-8\,a^2\,b\,c\,d^5\,e-8\,a\,c^3\,d\,e^5\,x+8\,a^3\,c\,d^5\,e\,x-2\,a\,b^2\,d^5\,e\,\sqrt {b^2-4\,a\,c}-4\,a^2\,c\,d^5\,e\,\sqrt {b^2-4\,a\,c}+20\,a\,b\,c^2\,d^3\,e^3-6\,a\,b^2\,c\,d^4\,e^2+6\,a\,b^3\,d^4\,e^2\,x-2\,a^2\,b^2\,d^5\,e\,x+3\,b^3\,c\,d^2\,e^4\,x-16\,a\,c^2\,d^3\,e^3\,\sqrt {b^2-4\,a\,c}-3\,b\,c^2\,d^2\,e^4\,\sqrt {b^2-4\,a\,c}+2\,b^2\,c\,d^3\,e^3\,\sqrt {b^2-4\,a\,c}-2\,b^3\,d^3\,e^3\,x\,\sqrt {b^2-4\,a\,c}+32\,a^2\,c^2\,d^3\,e^3\,x+4\,a\,b^2\,d^4\,e^2\,x\,\sqrt {b^2-4\,a\,c}-12\,a\,c^2\,d^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}+5\,a^2\,c\,d^4\,e^2\,x\,\sqrt {b^2-4\,a\,c}+3\,b^2\,c\,d^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}+14\,a\,b\,c\,d^4\,e^2\,\sqrt {b^2-4\,a\,c}-6\,a^2\,b\,d^5\,e\,x\,\sqrt {b^2-4\,a\,c}-6\,a\,b\,c^2\,d^2\,e^4\,x-2\,a\,b^2\,c\,d^3\,e^3\,x-23\,a^2\,b\,c\,d^4\,e^2\,x+2\,a\,b\,c\,d^3\,e^3\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,d^2-4\,a\,c^3\,e^2-b^3\,d^2\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,d^2+b^2\,c^2\,e^2-2\,b^3\,c\,d\,e-5\,a\,b^2\,c\,d^2-b\,c^2\,e^2\,\sqrt {b^2-4\,a\,c}+8\,a\,b\,c^2\,d\,e+3\,a\,b\,c\,d^2\,\sqrt {b^2-4\,a\,c}-4\,a\,c^2\,d\,e\,\sqrt {b^2-4\,a\,c}+2\,b^2\,c\,d\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^4\,c\,d^4-a^3\,b^2\,d^4-8\,a^3\,b\,c\,d^3\,e+8\,a^3\,c^2\,d^2\,e^2+2\,a^2\,b^3\,d^3\,e+2\,a^2\,b^2\,c\,d^2\,e^2-8\,a^2\,b\,c^2\,d\,e^3+4\,a^2\,c^3\,e^4-a\,b^4\,d^2\,e^2+2\,a\,b^3\,c\,d\,e^3-a\,b^2\,c^2\,e^4\right )}+\frac {d^3}{e^2\,\left (d+e\,x\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((d + e*x)^2*(a + b/x + c/x^2)),x)

[Out]

(log(d + e*x)*(a*d^4 + 3*c*d^2*e^2 - 2*b*d^3*e))/(c^2*e^6 + a^2*d^4*e^2 + b^2*d^2*e^4 - 2*b*c*d*e^5 - 2*a*b*d^
3*e^3 + 2*a*c*d^2*e^4) - (log(a^2*b^2*d^6 - 4*a^3*c*d^6 - 2*c^4*e^6 - b^4*d^4*e^2 + c^3*e^6*x*(b^2 - 4*a*c)^(1
/2) + 24*a*c^3*d^2*e^4 + 6*b^3*c*d^3*e^3 + 2*b^4*d^3*e^3*x - b^3*d^4*e^2*(b^2 - 4*a*c)^(1/2) - 10*a^2*c^2*d^4*
e^2 - 9*b^2*c^2*d^2*e^4 - 2*a*b^3*d^5*e + 4*b*c^3*d*e^5 - b*c^3*e^6*x + a^2*b*d^6*(b^2 - 4*a*c)^(1/2) + 4*c^3*
d*e^5*(b^2 - 4*a*c)^(1/2) + 2*a^3*d^6*x*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c*d^5*e + 8*a*c^3*d*e^5*x - 8*a^3*c*d^5*
e*x - 2*a*b^2*d^5*e*(b^2 - 4*a*c)^(1/2) - 4*a^2*c*d^5*e*(b^2 - 4*a*c)^(1/2) - 20*a*b*c^2*d^3*e^3 + 6*a*b^2*c*d
^4*e^2 - 6*a*b^3*d^4*e^2*x + 2*a^2*b^2*d^5*e*x - 3*b^3*c*d^2*e^4*x - 16*a*c^2*d^3*e^3*(b^2 - 4*a*c)^(1/2) - 3*
b*c^2*d^2*e^4*(b^2 - 4*a*c)^(1/2) + 2*b^2*c*d^3*e^3*(b^2 - 4*a*c)^(1/2) - 2*b^3*d^3*e^3*x*(b^2 - 4*a*c)^(1/2)
- 32*a^2*c^2*d^3*e^3*x + 4*a*b^2*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) - 12*a*c^2*d^2*e^4*x*(b^2 - 4*a*c)^(1/2) + 5*a^
2*c*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) + 3*b^2*c*d^2*e^4*x*(b^2 - 4*a*c)^(1/2) + 14*a*b*c*d^4*e^2*(b^2 - 4*a*c)^(1/
2) - 6*a^2*b*d^5*e*x*(b^2 - 4*a*c)^(1/2) + 6*a*b*c^2*d^2*e^4*x + 2*a*b^2*c*d^3*e^3*x + 23*a^2*b*c*d^4*e^2*x +
2*a*b*c*d^3*e^3*x*(b^2 - 4*a*c)^(1/2))*(b^4*d^2 - 4*a*c^3*e^2 + b^3*d^2*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*d^2 +
b^2*c^2*e^2 - 2*b^3*c*d*e - 5*a*b^2*c*d^2 + b*c^2*e^2*(b^2 - 4*a*c)^(1/2) + 8*a*b*c^2*d*e - 3*a*b*c*d^2*(b^2 -
4*a*c)^(1/2) + 4*a*c^2*d*e*(b^2 - 4*a*c)^(1/2) - 2*b^2*c*d*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a^4*c*d^4 - a^3*b^2*
d^4 + 4*a^2*c^3*e^4 - a*b^2*c^2*e^4 - a*b^4*d^2*e^2 + 2*a^2*b^3*d^3*e + 8*a^3*c^2*d^2*e^2 + 2*a*b^3*c*d*e^3 -
8*a^3*b*c*d^3*e - 8*a^2*b*c^2*d*e^3 + 2*a^2*b^2*c*d^2*e^2)) - (log(2*c^4*e^6 + 4*a^3*c*d^6 - a^2*b^2*d^6 + b^4
*d^4*e^2 + c^3*e^6*x*(b^2 - 4*a*c)^(1/2) - 24*a*c^3*d^2*e^4 - 6*b^3*c*d^3*e^3 - 2*b^4*d^3*e^3*x - b^3*d^4*e^2*
(b^2 - 4*a*c)^(1/2) + 10*a^2*c^2*d^4*e^2 + 9*b^2*c^2*d^2*e^4 + 2*a*b^3*d^5*e - 4*b*c^3*d*e^5 + b*c^3*e^6*x + a
^2*b*d^6*(b^2 - 4*a*c)^(1/2) + 4*c^3*d*e^5*(b^2 - 4*a*c)^(1/2) + 2*a^3*d^6*x*(b^2 - 4*a*c)^(1/2) - 8*a^2*b*c*d
^5*e - 8*a*c^3*d*e^5*x + 8*a^3*c*d^5*e*x - 2*a*b^2*d^5*e*(b^2 - 4*a*c)^(1/2) - 4*a^2*c*d^5*e*(b^2 - 4*a*c)^(1/
2) + 20*a*b*c^2*d^3*e^3 - 6*a*b^2*c*d^4*e^2 + 6*a*b^3*d^4*e^2*x - 2*a^2*b^2*d^5*e*x + 3*b^3*c*d^2*e^4*x - 16*a
*c^2*d^3*e^3*(b^2 - 4*a*c)^(1/2) - 3*b*c^2*d^2*e^4*(b^2 - 4*a*c)^(1/2) + 2*b^2*c*d^3*e^3*(b^2 - 4*a*c)^(1/2) -
2*b^3*d^3*e^3*x*(b^2 - 4*a*c)^(1/2) + 32*a^2*c^2*d^3*e^3*x + 4*a*b^2*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) - 12*a*c^2
*d^2*e^4*x*(b^2 - 4*a*c)^(1/2) + 5*a^2*c*d^4*e^2*x*(b^2 - 4*a*c)^(1/2) + 3*b^2*c*d^2*e^4*x*(b^2 - 4*a*c)^(1/2)
+ 14*a*b*c*d^4*e^2*(b^2 - 4*a*c)^(1/2) - 6*a^2*b*d^5*e*x*(b^2 - 4*a*c)^(1/2) - 6*a*b*c^2*d^2*e^4*x - 2*a*b^2*
c*d^3*e^3*x - 23*a^2*b*c*d^4*e^2*x + 2*a*b*c*d^3*e^3*x*(b^2 - 4*a*c)^(1/2))*(b^4*d^2 - 4*a*c^3*e^2 - b^3*d^2*(
b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*d^2 + b^2*c^2*e^2 - 2*b^3*c*d*e - 5*a*b^2*c*d^2 - b*c^2*e^2*(b^2 - 4*a*c)^(1/2)
+ 8*a*b*c^2*d*e + 3*a*b*c*d^2*(b^2 - 4*a*c)^(1/2) - 4*a*c^2*d*e*(b^2 - 4*a*c)^(1/2) + 2*b^2*c*d*e*(b^2 - 4*a*
c)^(1/2)))/(2*(4*a^4*c*d^4 - a^3*b^2*d^4 + 4*a^2*c^3*e^4 - a*b^2*c^2*e^4 - a*b^4*d^2*e^2 + 2*a^2*b^3*d^3*e + 8
*a^3*c^2*d^2*e^2 + 2*a*b^3*c*d*e^3 - 8*a^3*b*c*d^3*e - 8*a^2*b*c^2*d*e^3 + 2*a^2*b^2*c*d^2*e^2)) + d^3/(e^2*(d
+ e*x)*(a*d^2 + c*e^2 - b*d*e))

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