### 3.72 $$\int \frac{x}{(a+\frac{c}{x^2}+\frac{b}{x}) (d+e x)^2} \, dx$$

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

[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)

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Rubi [A]  time = 0.395346, antiderivative size = 246, normalized size of antiderivative = 1., 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 = {1569, 1628, 634, 618, 206, 628} $\frac{\left (-b c \left (3 a d^2-c e^2\right )+4 a c^2 d e-2 b^2 c d e+b^3 d^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (-c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )^2}+\frac{d^3}{e^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac{d^2 \log (d+e x) \left (a d^2-e (2 b d-3 c e)\right )}{e^2 \left (a d^2-e (b d-c e)\right )^2}$

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 1569

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 1628

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]

Rule 634

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 618

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

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]

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 ) \operatorname{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*}

Mathematica [A]  time = 0.248042, size = 207, normalized size = 0.84 $\frac{-\frac{2 \left (b c \left (c e^2-3 a d^2\right )+4 a c^2 d e-2 b^2 c d e+b^3 d^2\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a \sqrt{4 a c-b^2}}+\frac{\left (c \left (c e^2-a d^2\right )+b^2 d^2-2 b c d e\right ) \log (x (a x+b)+c)}{a}+\frac{2 d^3 \left (a d^2+e (c e-b d)\right )}{e^2 (d+e x)}+\frac{2 \log (d+e x) \left (a d^4+d^2 e (3 c e-2 b d)\right )}{e^2}}{2 \left (a d^2+e (c e-b d)\right )^2}$

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 [B]  time = 0.007, size = 580, normalized size = 2.4 \begin{align*}{\frac{{d}^{4}\ln \left ( ex+d \right ) a}{{e}^{2} \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-2\,{\frac{{d}^{3}\ln \left ( ex+d \right ) b}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}e}}+3\,{\frac{{d}^{2}\ln \left ( ex+d \right ) c}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{{d}^{3}}{{e}^{2} \left ( a{d}^{2}-bde+{e}^{2}c \right ) \left ( ex+d \right ) }}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) c{d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}{d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) bcde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){c}^{2}{e}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}}+3\,{\frac{bc{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{c}^{2}de}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{{b}^{2}cde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}a}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{b{c}^{2}{e}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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Fricas [B]  time = 110.084, size = 3005, normalized size = 12.22 \begin{align*} \text{result too large to display} \end{align*}

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*d*e^4
+ (b^3 - 3*a*b*c)*d^3*e^2 - 2*(b^2*c - 2*a*c^2)*d^2*e^3 + (b*c^2*e^5 + (b^3 - 3*a*b*c)*d^2*e^3 - 2*(b^2*c - 2
*a*c^2)*d*e^4)*x)*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 - 2*(b^3*c - 4*a*b*c^2)*d^2*e^3 + (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*e^3 - 2*(b^3*c - 4*a*b*c^2)*d*e^4 + (b^2*c^2 - 4*a*c^3)*e^5)*x)*l
og(a*x^2 + b*x + c) + 2*((a^2*b^2 - 4*a^3*c)*d^5 - 2*(a*b^3 - 4*a^2*b*c)*d^4*e + 3*(a*b^2*c - 4*a^2*c^2)*d^3*e
^2 + ((a^2*b^2 - 4*a^3*c)*d^4*e - 2*(a*b^3 - 4*a^2*b*c)*d^3*e^2 + 3*(a*b^2*c - 4*a^2*c^2)*d^2*e^3)*x)*log(e*x
+ d))/((a^3*b^2 - 4*a^4*c)*d^5*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d^4*e^3 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3*e
^4 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^2*e^5 + (a*b^2*c^2 - 4*a^2*c^3)*d*e^6 + ((a^3*b^2 - 4*a^4*c)*d^4*e^3 - 2*(a^2
*b^3 - 4*a^3*b*c)*d^3*e^4 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^5 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^6 + (a*b
^2*c^2 - 4*a^2*c^3)*e^7)*x), 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*d*e^4 + (b^3 - 3*a*b*c)*d^3*e^2 - 2*(b^2*c - 2*a*c^2)*d^2*e^3 + (b*c^2*e^5 + (b^3 - 3
*a*b*c)*d^2*e^3 - 2*(b^2*c - 2*a*c^2)*d*e^4)*x)*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 - 2*(b^3*c - 4*a*b*c^2)*d^2*e^3 + (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*e^3 - 2*(b^3*c - 4*a*b*c^2)*d*e^4 + (b^2*c^2 - 4*a*c^3)*e^5)*x)*log(a*x
^2 + b*x + c) + 2*((a^2*b^2 - 4*a^3*c)*d^5 - 2*(a*b^3 - 4*a^2*b*c)*d^4*e + 3*(a*b^2*c - 4*a^2*c^2)*d^3*e^2 + (
(a^2*b^2 - 4*a^3*c)*d^4*e - 2*(a*b^3 - 4*a^2*b*c)*d^3*e^2 + 3*(a*b^2*c - 4*a^2*c^2)*d^2*e^3)*x)*log(e*x + d))/
((a^3*b^2 - 4*a^4*c)*d^5*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d^4*e^3 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3*e^4 - 2
*(a*b^3*c - 4*a^2*b*c^2)*d^2*e^5 + (a*b^2*c^2 - 4*a^2*c^3)*d*e^6 + ((a^3*b^2 - 4*a^4*c)*d^4*e^3 - 2*(a^2*b^3 -
4*a^3*b*c)*d^3*e^4 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^5 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^6 + (a*b^2*c^2
- 4*a^2*c^3)*e^7)*x)]

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

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 = 1.11668, size = 556, normalized size = 2.26 \begin{align*} \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{align*}

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)