3.1.75 \(\int \frac {1}{(a+\frac {c}{x^2}+\frac {b}{x}) x^2 (d+e x)^2} \, dx\) [75]

Optimal. Leaf size=189 \[ -\frac {e}{\left (a d^2-b d e+c e^2\right ) (d+e x)}-\frac {\left (2 a^2 d^2+b^2 e^2-2 a e (b d+c e)\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac {e (2 a d-b e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2}-\frac {e (2 a d-b e) \log \left (c+b x+a x^2\right )}{2 \left (a d^2-e (b d-c e)\right )^2} \]

[Out]

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

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Rubi [A]
time = 0.21, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1583, 723, 814, 648, 632, 212, 642} \begin {gather*} -\frac {\left (2 a^2 d^2-2 a e (b d+c e)+b^2 e^2\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac {e}{(d+e x) \left (a d^2-b d e+c e^2\right )}-\frac {e (2 a d-b e) \log \left (a x^2+b x+c\right )}{2 \left (a d^2-e (b d-c e)\right )^2}+\frac {e (2 a d-b e) \log (d+e x)}{\left (a d^2-e (b d-c e)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + c/x^2 + b/x)*x^2*(d + e*x)^2),x]

[Out]

-(e/((a*d^2 - b*d*e + c*e^2)*(d + e*x))) - ((2*a^2*d^2 + b^2*e^2 - 2*a*e*(b*d + c*e))*ArcTanh[(b + 2*a*x)/Sqrt
[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))^2) + (e*(2*a*d - b*e)*Log[d + e*x])/(a*d^2 - e*(b*d
- c*e))^2 - (e*(2*a*d - b*e)*Log[c + b*x + a*x^2])/(2*(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 723

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m
+ 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c
*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, m}, 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[m, -1]

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 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]

Rubi steps

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

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Mathematica [A]
time = 0.15, size = 151, normalized size = 0.80 \begin {gather*} \frac {-\frac {2 e \left (a d^2+e (-b d+c e)\right )}{d+e x}+\frac {2 \left (2 a^2 d^2+b^2 e^2-2 a e (b d+c e)\right ) \tan ^{-1}\left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-2 e (-2 a d+b e) \log (d+e x)+e (-2 a d+b e) \log (c+x (b+a x))}{2 \left (a d^2+e (-b d+c e)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.30, size = 197, normalized size = 1.04

method	result	size
default	\(-\frac {e}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) \left (e x +d \right )}+\frac {e \left (2 a d -e b \right ) \ln \left (e x +d \right )}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2}}+\frac {\frac {\left (-2 d e \,a^{2}+a b \,e^{2}\right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (a^{2} d^{2}-2 a b d e -a c \,e^{2}+e^{2} b^{2}-\frac {\left (-2 d e \,a^{2}+a b \,e^{2}\right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-d e b +c \,e^{2}\right )^{2}}\)	\(197\)
risch	\(\text {Expression too large to display}\)	\(12760\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e/(a*d^2-b*d*e+c*e^2)/(e*x+d)+e*(2*a*d-b*e)/(a*d^2-b*d*e+c*e^2)^2*ln(e*x+d)+1/(a*d^2-b*d*e+c*e^2)^2*(1/2*(-2*
a^2*d*e+a*b*e^2)/a*ln(a*x^2+b*x+c)+2*(a^2*d^2-2*a*b*d*e-a*c*e^2+e^2*b^2-1/2*(-2*a^2*d*e+a*b*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(1/(a+c/x^2+b/x)/x^2/(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. 526 vs. \(2 (197) = 394\).
time = 1.94, size = 1071, normalized size = 5.67 \begin {gather*} \left [-\frac {2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} e - 2 \, {\left (b^{3} - 4 \, a b c\right )} d e^{2} + {\left (2 \, a^{2} d^{3} + {\left (b^{2} - 2 \, a c\right )} x e^{3} - {\left (2 \, a b d x - {\left (b^{2} - 2 \, a c\right )} d\right )} e^{2} + 2 \, {\left (a^{2} d^{2} x - a b d^{2}\right )} e\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 ) + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{3} + {\left (2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} e - {\left (b^{3} - 4 \, a b c\right )} x e^{3} + {\left (2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d x - {\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2}\right )} \log \left (a x^{2} + b x + c\right ) - 2 \, {\left (2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} e - {\left (b^{3} - 4 \, a b c\right )} x e^{3} + {\left (2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d x - {\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2}\right )} \log \left (x e + d\right )}{2 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{5} + {\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} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} d^{2} x - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2}\right )} e^{3} - {\left (2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{3} x - {\left (b^{4} - 2 \, a b^{2} c - 8 \, 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 )}}, -\frac {2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} e - 2 \, {\left (b^{3} - 4 \, a b c\right )} d e^{2} + 2 \, {\left (2 \, a^{2} d^{3} + {\left (b^{2} - 2 \, a c\right )} x e^{3} - {\left (2 \, a b d x - {\left (b^{2} - 2 \, a c\right )} d\right )} e^{2} + 2 \, {\left (a^{2} d^{2} x - a b d^{2}\right )} e\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 ) + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{3} + {\left (2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} e - {\left (b^{3} - 4 \, a b c\right )} x e^{3} + {\left (2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d x - {\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2}\right )} \log \left (a x^{2} + b x + c\right ) - 2 \, {\left (2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} e - {\left (b^{3} - 4 \, a b c\right )} x e^{3} + {\left (2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d x - {\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2}\right )} \log \left (x e + d\right )}{2 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{5} + {\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} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} d^{2} x - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{2}\right )} e^{3} - {\left (2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{3} x - {\left (b^{4} - 2 \, a b^{2} c - 8 \, 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 )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]
time = 4.13, size = 331, normalized size = 1.75 \begin {gather*} -\frac {{\left (2 \, a^{2} d^{2} e^{2} - 2 \, a b d e^{3} + b^{2} e^{4} - 2 \, a c e^{4}\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^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + c^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (2 \, a d e - b e^{2}\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 )}{2 \, {\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, b c d e^{3} + c^{2} e^{4}\right )}} - \frac {e^{3}}{{\left (a d^{2} e^{2} - b d e^{3} + c e^{4}\right )} {\left (x e + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 8.11, size = 1782, normalized size = 9.43 \begin {gather*} \frac {\ln \left (c\,e^4\,{\left (b^2-4\,a\,c\right )}^{5/2}-8\,b^5\,c\,e^4-8\,b^6\,e^4\,x-4\,a^3\,d^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-4\,a^3\,b^3\,d^4+4\,b^3\,e^4\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}+60\,a\,b^3\,c^2\,e^4-112\,a^2\,b\,c^3\,e^4+4\,a\,b^5\,d^2\,e^2-8\,a^2\,b^4\,d^3\,e+256\,a^3\,c^3\,d\,e^3-256\,a^4\,c^2\,d^3\,e-8\,a^4\,b^2\,d^4\,x+32\,a^3\,c^3\,e^4\,x+10\,b\,d\,e^3\,{\left (b^2-4\,a\,c\right )}^{5/2}+4\,b\,e^4\,x\,{\left (b^2-4\,a\,c\right )}^{5/2}+16\,a^4\,b\,c\,d^4+32\,a^5\,c\,d^4\,x-14\,a\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{5/2}+7\,b^2\,c\,e^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-10\,b^3\,d\,e^3\,{\left (b^2-4\,a\,c\right )}^{3/2}-8\,a\,d\,e^3\,x\,{\left (b^2-4\,a\,c\right )}^{5/2}+24\,a\,b^4\,c\,d\,e^3+64\,a\,b^4\,c\,e^4\,x+32\,a\,b^5\,d\,e^3\,x-8\,a^2\,b\,d^3\,e\,{\left (b^2-4\,a\,c\right )}^{3/2}-32\,a^3\,d^3\,e\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}+96\,a^3\,b^2\,c\,d^3\,e+16\,a^3\,b^3\,d^3\,e\,x+18\,a\,b^2\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{3/2}-160\,a^2\,b^2\,c^2\,d\,e^3-56\,a^2\,b^3\,c\,d^2\,e^2+160\,a^3\,b\,c^2\,d^2\,e^2-136\,a^2\,b^2\,c^2\,e^4\,x-40\,a^2\,b^4\,d^2\,e^2\,x-448\,a^4\,c^2\,d^2\,e^2\,x+48\,a^2\,b\,d^2\,e^2\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}+272\,a^3\,b^2\,c\,d^2\,e^2\,x-64\,a^4\,b\,c\,d^3\,e\,x-24\,a\,b^2\,d\,e^3\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}-240\,a^2\,b^3\,c\,d\,e^3\,x+448\,a^3\,b\,c^2\,d\,e^3\,x\right )\,\left (a\,\left (\left (2\,b\,c-c\,\sqrt {b^2-4\,a\,c}\right )\,e^2+\left (b^2\,d-b\,d\,\sqrt {b^2-4\,a\,c}\right )\,e\right )-e^2\,\left (\frac {b^3}{2}-\frac {b^2\,\sqrt {b^2-4\,a\,c}}{2}\right )+a^2\,\left (d^2\,\sqrt {b^2-4\,a\,c}-4\,c\,d\,e\right )\right )}{4\,a^3\,c\,d^4-a^2\,b^2\,d^4-8\,a^2\,b\,c\,d^3\,e+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d^3\,e+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d\,e^3+4\,a\,c^3\,e^4-b^4\,d^2\,e^2+2\,b^3\,c\,d\,e^3-b^2\,c^2\,e^4}-\frac {\ln \left (d+e\,x\right )\,\left (b\,e^2-2\,a\,d\,e\right )}{a^2\,d^4-2\,a\,b\,d^3\,e+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2-2\,b\,c\,d\,e^3+c^2\,e^4}-\frac {\ln \left (c\,e^4\,{\left (b^2-4\,a\,c\right )}^{5/2}+8\,b^5\,c\,e^4+8\,b^6\,e^4\,x-4\,a^3\,d^4\,{\left (b^2-4\,a\,c\right )}^{3/2}+4\,a^3\,b^3\,d^4+4\,b^3\,e^4\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}-60\,a\,b^3\,c^2\,e^4+112\,a^2\,b\,c^3\,e^4-4\,a\,b^5\,d^2\,e^2+8\,a^2\,b^4\,d^3\,e-256\,a^3\,c^3\,d\,e^3+256\,a^4\,c^2\,d^3\,e+8\,a^4\,b^2\,d^4\,x-32\,a^3\,c^3\,e^4\,x+10\,b\,d\,e^3\,{\left (b^2-4\,a\,c\right )}^{5/2}+4\,b\,e^4\,x\,{\left (b^2-4\,a\,c\right )}^{5/2}-16\,a^4\,b\,c\,d^4-32\,a^5\,c\,d^4\,x-14\,a\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{5/2}+7\,b^2\,c\,e^4\,{\left (b^2-4\,a\,c\right )}^{3/2}-10\,b^3\,d\,e^3\,{\left (b^2-4\,a\,c\right )}^{3/2}-8\,a\,d\,e^3\,x\,{\left (b^2-4\,a\,c\right )}^{5/2}-24\,a\,b^4\,c\,d\,e^3-64\,a\,b^4\,c\,e^4\,x-32\,a\,b^5\,d\,e^3\,x-8\,a^2\,b\,d^3\,e\,{\left (b^2-4\,a\,c\right )}^{3/2}-32\,a^3\,d^3\,e\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}-96\,a^3\,b^2\,c\,d^3\,e-16\,a^3\,b^3\,d^3\,e\,x+18\,a\,b^2\,d^2\,e^2\,{\left (b^2-4\,a\,c\right )}^{3/2}+160\,a^2\,b^2\,c^2\,d\,e^3+56\,a^2\,b^3\,c\,d^2\,e^2-160\,a^3\,b\,c^2\,d^2\,e^2+136\,a^2\,b^2\,c^2\,e^4\,x+40\,a^2\,b^4\,d^2\,e^2\,x+448\,a^4\,c^2\,d^2\,e^2\,x+48\,a^2\,b\,d^2\,e^2\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}-272\,a^3\,b^2\,c\,d^2\,e^2\,x+64\,a^4\,b\,c\,d^3\,e\,x-24\,a\,b^2\,d\,e^3\,x\,{\left (b^2-4\,a\,c\right )}^{3/2}+240\,a^2\,b^3\,c\,d\,e^3\,x-448\,a^3\,b\,c^2\,d\,e^3\,x\right )\,\left (e^2\,\left (\frac {b^3}{2}+\frac {b^2\,\sqrt {b^2-4\,a\,c}}{2}\right )-a\,\left (\left (2\,b\,c+c\,\sqrt {b^2-4\,a\,c}\right )\,e^2+\left (b^2\,d+b\,d\,\sqrt {b^2-4\,a\,c}\right )\,e\right )+a^2\,\left (d^2\,\sqrt {b^2-4\,a\,c}+4\,c\,d\,e\right )\right )}{4\,a^3\,c\,d^4-a^2\,b^2\,d^4-8\,a^2\,b\,c\,d^3\,e+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d^3\,e+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d\,e^3+4\,a\,c^3\,e^4-b^4\,d^2\,e^2+2\,b^3\,c\,d\,e^3-b^2\,c^2\,e^4}-\frac {e}{\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(1/(x^2*(d + e*x)^2*(a + b/x + c/x^2)),x)

[Out]

(log(c*e^4*(b^2 - 4*a*c)^(5/2) - 8*b^5*c*e^4 - 8*b^6*e^4*x - 4*a^3*d^4*(b^2 - 4*a*c)^(3/2) - 4*a^3*b^3*d^4 + 4
*b^3*e^4*x*(b^2 - 4*a*c)^(3/2) + 60*a*b^3*c^2*e^4 - 112*a^2*b*c^3*e^4 + 4*a*b^5*d^2*e^2 - 8*a^2*b^4*d^3*e + 25
6*a^3*c^3*d*e^3 - 256*a^4*c^2*d^3*e - 8*a^4*b^2*d^4*x + 32*a^3*c^3*e^4*x + 10*b*d*e^3*(b^2 - 4*a*c)^(5/2) + 4*
b*e^4*x*(b^2 - 4*a*c)^(5/2) + 16*a^4*b*c*d^4 + 32*a^5*c*d^4*x - 14*a*d^2*e^2*(b^2 - 4*a*c)^(5/2) + 7*b^2*c*e^4
*(b^2 - 4*a*c)^(3/2) - 10*b^3*d*e^3*(b^2 - 4*a*c)^(3/2) - 8*a*d*e^3*x*(b^2 - 4*a*c)^(5/2) + 24*a*b^4*c*d*e^3 +
64*a*b^4*c*e^4*x + 32*a*b^5*d*e^3*x - 8*a^2*b*d^3*e*(b^2 - 4*a*c)^(3/2) - 32*a^3*d^3*e*x*(b^2 - 4*a*c)^(3/2)
+ 96*a^3*b^2*c*d^3*e + 16*a^3*b^3*d^3*e*x + 18*a*b^2*d^2*e^2*(b^2 - 4*a*c)^(3/2) - 160*a^2*b^2*c^2*d*e^3 - 56*
a^2*b^3*c*d^2*e^2 + 160*a^3*b*c^2*d^2*e^2 - 136*a^2*b^2*c^2*e^4*x - 40*a^2*b^4*d^2*e^2*x - 448*a^4*c^2*d^2*e^2
*x + 48*a^2*b*d^2*e^2*x*(b^2 - 4*a*c)^(3/2) + 272*a^3*b^2*c*d^2*e^2*x - 64*a^4*b*c*d^3*e*x - 24*a*b^2*d*e^3*x*
(b^2 - 4*a*c)^(3/2) - 240*a^2*b^3*c*d*e^3*x + 448*a^3*b*c^2*d*e^3*x)*(a*(e^2*(2*b*c - c*(b^2 - 4*a*c)^(1/2)) +
e*(b^2*d - b*d*(b^2 - 4*a*c)^(1/2))) - e^2*(b^3/2 - (b^2*(b^2 - 4*a*c)^(1/2))/2) + a^2*(d^2*(b^2 - 4*a*c)^(1/
2) - 4*c*d*e)))/(4*a^3*c*d^4 + 4*a*c^3*e^4 - a^2*b^2*d^4 - b^2*c^2*e^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a
*b^3*d^3*e + 2*b^3*c*d*e^3 - 8*a*b*c^2*d*e^3 - 8*a^2*b*c*d^3*e + 2*a*b^2*c*d^2*e^2) - (log(d + e*x)*(b*e^2 - 2
*a*d*e))/(a^2*d^4 + c^2*e^4 + b^2*d^2*e^2 - 2*a*b*d^3*e - 2*b*c*d*e^3 + 2*a*c*d^2*e^2) - (log(c*e^4*(b^2 - 4*a
*c)^(5/2) + 8*b^5*c*e^4 + 8*b^6*e^4*x - 4*a^3*d^4*(b^2 - 4*a*c)^(3/2) + 4*a^3*b^3*d^4 + 4*b^3*e^4*x*(b^2 - 4*a
*c)^(3/2) - 60*a*b^3*c^2*e^4 + 112*a^2*b*c^3*e^4 - 4*a*b^5*d^2*e^2 + 8*a^2*b^4*d^3*e - 256*a^3*c^3*d*e^3 + 256
*a^4*c^2*d^3*e + 8*a^4*b^2*d^4*x - 32*a^3*c^3*e^4*x + 10*b*d*e^3*(b^2 - 4*a*c)^(5/2) + 4*b*e^4*x*(b^2 - 4*a*c)
^(5/2) - 16*a^4*b*c*d^4 - 32*a^5*c*d^4*x - 14*a*d^2*e^2*(b^2 - 4*a*c)^(5/2) + 7*b^2*c*e^4*(b^2 - 4*a*c)^(3/2)
- 10*b^3*d*e^3*(b^2 - 4*a*c)^(3/2) - 8*a*d*e^3*x*(b^2 - 4*a*c)^(5/2) - 24*a*b^4*c*d*e^3 - 64*a*b^4*c*e^4*x - 3
2*a*b^5*d*e^3*x - 8*a^2*b*d^3*e*(b^2 - 4*a*c)^(3/2) - 32*a^3*d^3*e*x*(b^2 - 4*a*c)^(3/2) - 96*a^3*b^2*c*d^3*e
- 16*a^3*b^3*d^3*e*x + 18*a*b^2*d^2*e^2*(b^2 - 4*a*c)^(3/2) + 160*a^2*b^2*c^2*d*e^3 + 56*a^2*b^3*c*d^2*e^2 - 1
60*a^3*b*c^2*d^2*e^2 + 136*a^2*b^2*c^2*e^4*x + 40*a^2*b^4*d^2*e^2*x + 448*a^4*c^2*d^2*e^2*x + 48*a^2*b*d^2*e^2
*x*(b^2 - 4*a*c)^(3/2) - 272*a^3*b^2*c*d^2*e^2*x + 64*a^4*b*c*d^3*e*x - 24*a*b^2*d*e^3*x*(b^2 - 4*a*c)^(3/2) +
240*a^2*b^3*c*d*e^3*x - 448*a^3*b*c^2*d*e^3*x)*(e^2*(b^3/2 + (b^2*(b^2 - 4*a*c)^(1/2))/2) - a*(e^2*(2*b*c + c
*(b^2 - 4*a*c)^(1/2)) + e*(b^2*d + b*d*(b^2 - 4*a*c)^(1/2))) + a^2*(d^2*(b^2 - 4*a*c)^(1/2) + 4*c*d*e)))/(4*a^
3*c*d^4 + 4*a*c^3*e^4 - a^2*b^2*d^4 - b^2*c^2*e^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d^3*e + 2*b^3*c*
d*e^3 - 8*a*b*c^2*d*e^3 - 8*a^2*b*c*d^3*e + 2*a*b^2*c*d^2*e^2) - e/((d + e*x)*(a*d^2 + c*e^2 - b*d*e))

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