3.887 \(\int \frac{d+e x}{x^2 (a+b x+c x^2)} \, dx\)
Optimal. Leaf size=104 \[ -\frac{\left (-a b e-2 a c d+b^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c}}+\frac{(b d-a e) \log \left (a+b x+c x^2\right )}{2 a^2}-\frac{\log (x) (b d-a e)}{a^2}-\frac{d}{a x} \]
[Out]
-(d/(a*x)) - ((b^2*d - 2*a*c*d - a*b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) - ((b*
d - a*e)*Log[x])/a^2 + ((b*d - a*e)*Log[a + b*x + c*x^2])/(2*a^2)
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Rubi [A] time = 0.1474, antiderivative size = 104, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used =
{800, 634, 618, 206, 628} \[ -\frac{\left (-a b e-2 a c d+b^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c}}+\frac{(b d-a e) \log \left (a+b x+c x^2\right )}{2 a^2}-\frac{\log (x) (b d-a e)}{a^2}-\frac{d}{a x} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)/(x^2*(a + b*x + c*x^2)),x]
[Out]
-(d/(a*x)) - ((b^2*d - 2*a*c*d - a*b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) - ((b*
d - a*e)*Log[x])/a^2 + ((b*d - a*e)*Log[a + b*x + c*x^2])/(2*a^2)
Rule 800
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 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{d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac{d}{a x^2}+\frac{-b d+a e}{a^2 x}+\frac{b^2 d-a c d-a b e+c (b d-a e) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac{d}{a x}-\frac{(b d-a e) \log (x)}{a^2}+\frac{\int \frac{b^2 d-a c d-a b e+c (b d-a e) x}{a+b x+c x^2} \, dx}{a^2}\\ &=-\frac{d}{a x}-\frac{(b d-a e) \log (x)}{a^2}+\frac{(b d-a e) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 a^2}+\frac{\left (b^2 d-2 a c d-a b e\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 a^2}\\ &=-\frac{d}{a x}-\frac{(b d-a e) \log (x)}{a^2}+\frac{(b d-a e) \log \left (a+b x+c x^2\right )}{2 a^2}-\frac{\left (b^2 d-2 a c d-a b e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^2}\\ &=-\frac{d}{a x}-\frac{\left (b^2 d-2 a c d-a b e\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{b^2-4 a c}}-\frac{(b d-a e) \log (x)}{a^2}+\frac{(b d-a e) \log \left (a+b x+c x^2\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0859868, size = 100, normalized size = 0.96 \[ \frac{\frac{2 \left (-a b e-2 a c d+b^2 d\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+(b d-a e) \log (a+x (b+c x))+2 \log (x) (a e-b d)-\frac{2 a d}{x}}{2 a^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)/(x^2*(a + b*x + c*x^2)),x]
[Out]
((-2*a*d)/x + (2*(b^2*d - 2*a*c*d - a*b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*(-(b
*d) + a*e)*Log[x] + (b*d - a*e)*Log[a + x*(b + c*x)])/(2*a^2)
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Maple [A] time = 0.007, size = 180, normalized size = 1.7 \begin{align*} -{\frac{d}{ax}}+{\frac{e\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( x \right ) bd}{{a}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) e}{2\,a}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) bd}{2\,{a}^{2}}}-{\frac{be}{a}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{cd}{a\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}d}{{a}^{2}}\arctan \left ({(2\,cx+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((e*x+d)/x^2/(c*x^2+b*x+a),x)
[Out]
-d/a/x+e*ln(x)/a-1/a^2*ln(x)*b*d-1/2/a*ln(c*x^2+b*x+a)*e+1/2/a^2*ln(c*x^2+b*x+a)*b*d-1/a/(4*a*c-b^2)^(1/2)*arc
tan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*e-2/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c*d+1/a^2/(4*a*
c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d
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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((e*x+d)/x^2/(c*x^2+b*x+a),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.39309, size = 807, normalized size = 7.76 \begin{align*} \left [\frac{{\left (a b e -{\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt{b^{2} - 4 \, a c} x \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 ) +{\left ({\left (b^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x \log \left (c x^{2} + b x + a\right ) - 2 \,{\left ({\left (b^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x \log \left (x\right ) - 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} d}{2 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x}, \frac{2 \,{\left (a b e -{\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt{-b^{2} + 4 \, a c} x \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (b^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x \log \left (c x^{2} + b x + a\right ) - 2 \,{\left ({\left (b^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x \log \left (x\right ) - 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} d}{2 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/x^2/(c*x^2+b*x+a),x, algorithm="fricas")
[Out]
[1/2*((a*b*e - (b^2 - 2*a*c)*d)*sqrt(b^2 - 4*a*c)*x*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)) + ((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x*log(c*x^2 + b*x + a) - 2*((b^3
- 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x*log(x) - 2*(a*b^2 - 4*a^2*c)*d)/((a^2*b^2 - 4*a^3*c)*x), 1/2*(2*(a*b*e -
(b^2 - 2*a*c)*d)*sqrt(-b^2 + 4*a*c)*x*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + ((b^3 - 4*a*b*c
)*d - (a*b^2 - 4*a^2*c)*e)*x*log(c*x^2 + b*x + a) - 2*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x*log(x) - 2*(
a*b^2 - 4*a^2*c)*d)/((a^2*b^2 - 4*a^3*c)*x)]
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Sympy [B] time = 162.757, size = 3232, normalized size = 31.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/x**2/(c*x**2+b*x+a),x)
[Out]
(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2))*log(x + (24*a
**7*c**2*e*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2))**2
- 14*a**6*b**2*c*e*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*
a**2))**2 - 28*a**6*b*c**2*d*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e -
b*d)/(2*a**2))**2 - 12*a**6*c**2*e**2*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)
) - (a*e - b*d)/(2*a**2)) + 2*a**5*b**4*e*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b*
*2)) - (a*e - b*d)/(2*a**2))**2 + 15*a**5*b**3*c*d*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4
*a*c - b**2)) - (a*e - b*d)/(2*a**2))**2 + 3*a**5*b**2*c*e**2*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)
/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) + 16*a**5*b*c**2*d*e*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d -
b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) - 4*a**5*c**3*d**2*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a
*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) - 12*a**5*c**2*e**3 - 2*a**4*b**5*d*(-sqrt(-4*a
*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2))**2 - 4*a**4*b**3*c*d*e*(
-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) - 3*a**4*b**2*
c**2*d**2*(-sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) + 1
1*a**4*b**2*c*e**3 + 48*a**4*b*c**2*d*e**2 + 4*a**4*c**3*d**2*e + a**3*b**4*c*d**2*(-sqrt(-4*a*c + b**2)*(a*b*
e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) - 2*a**3*b**4*e**3 - 36*a**3*b**3*c*d*e*
*2 - 61*a**3*b**2*c**2*d**2*e - 4*a**3*b*c**3*d**3 + 6*a**2*b**5*d*e**2 + 39*a**2*b**4*c*d**2*e + 25*a**2*b**3
*c**2*d**3 - 6*a*b**6*d**2*e - 14*a*b**5*c*d**3 + 2*b**7*d**3)/(9*a**4*b*c**2*e**3 + 18*a**4*c**3*d*e**2 - 2*a
**3*b**3*c*e**3 - 30*a**3*b**2*c**2*d*e**2 - 33*a**3*b*c**3*d**2*e + 2*a**3*c**4*d**3 + 6*a**2*b**4*c*d*e**2 +
33*a**2*b**3*c**2*d**2*e + 15*a**2*b**2*c**3*d**3 - 6*a*b**5*c*d**2*e - 12*a*b**4*c**2*d**3 + 2*b**6*c*d**3))
+ (sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2))*log(x + (24
*a**7*c**2*e*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2))**
2 - 14*a**6*b**2*c*e*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*
a**2))**2 - 28*a**6*b*c**2*d*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e -
b*d)/(2*a**2))**2 - 12*a**6*c**2*e**2*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2))
- (a*e - b*d)/(2*a**2)) + 2*a**5*b**4*e*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)
) - (a*e - b*d)/(2*a**2))**2 + 15*a**5*b**3*c*d*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c
- b**2)) - (a*e - b*d)/(2*a**2))**2 + 3*a**5*b**2*c*e**2*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a
**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) + 16*a**5*b*c**2*d*e*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*
d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) - 4*a**5*c**3*d**2*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d -
b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) - 12*a**5*c**2*e**3 - 2*a**4*b**5*d*(sqrt(-4*a*c + b**
2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2))**2 - 4*a**4*b**3*c*d*e*(sqrt(-4*
a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) - 3*a**4*b**2*c**2*d**2
*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d - b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) + 11*a**4*b**
2*c*e**3 + 48*a**4*b*c**2*d*e**2 + 4*a**4*c**3*d**2*e + a**3*b**4*c*d**2*(sqrt(-4*a*c + b**2)*(a*b*e + 2*a*c*d
- b**2*d)/(2*a**2*(4*a*c - b**2)) - (a*e - b*d)/(2*a**2)) - 2*a**3*b**4*e**3 - 36*a**3*b**3*c*d*e**2 - 61*a**
3*b**2*c**2*d**2*e - 4*a**3*b*c**3*d**3 + 6*a**2*b**5*d*e**2 + 39*a**2*b**4*c*d**2*e + 25*a**2*b**3*c**2*d**3
- 6*a*b**6*d**2*e - 14*a*b**5*c*d**3 + 2*b**7*d**3)/(9*a**4*b*c**2*e**3 + 18*a**4*c**3*d*e**2 - 2*a**3*b**3*c*
e**3 - 30*a**3*b**2*c**2*d*e**2 - 33*a**3*b*c**3*d**2*e + 2*a**3*c**4*d**3 + 6*a**2*b**4*c*d*e**2 + 33*a**2*b*
*3*c**2*d**2*e + 15*a**2*b**2*c**3*d**3 - 6*a*b**5*c*d**2*e - 12*a*b**4*c**2*d**3 + 2*b**6*c*d**3)) - d/(a*x)
+ (a*e - b*d)*log(x + (-12*a**5*c**2*e**3 + 11*a**4*b**2*c*e**3 + 48*a**4*b*c**2*d*e**2 + 4*a**4*c**3*d**2*e -
12*a**4*c**2*e**2*(a*e - b*d) - 2*a**3*b**4*e**3 - 36*a**3*b**3*c*d*e**2 - 61*a**3*b**2*c**2*d**2*e + 3*a**3*
b**2*c*e**2*(a*e - b*d) - 4*a**3*b*c**3*d**3 + 16*a**3*b*c**2*d*e*(a*e - b*d) - 4*a**3*c**3*d**2*(a*e - b*d) +
24*a**3*c**2*e*(a*e - b*d)**2 + 6*a**2*b**5*d*e**2 + 39*a**2*b**4*c*d**2*e + 25*a**2*b**3*c**2*d**3 - 4*a**2*
b**3*c*d*e*(a*e - b*d) - 3*a**2*b**2*c**2*d**2*(a*e - b*d) - 14*a**2*b**2*c*e*(a*e - b*d)**2 - 28*a**2*b*c**2*
d*(a*e - b*d)**2 - 6*a*b**6*d**2*e - 14*a*b**5*c*d**3 + a*b**4*c*d**2*(a*e - b*d) + 2*a*b**4*e*(a*e - b*d)**2
+ 15*a*b**3*c*d*(a*e - b*d)**2 + 2*b**7*d**3 - 2*b**5*d*(a*e - b*d)**2)/(9*a**4*b*c**2*e**3 + 18*a**4*c**3*d*e
**2 - 2*a**3*b**3*c*e**3 - 30*a**3*b**2*c**2*d*e**2 - 33*a**3*b*c**3*d**2*e + 2*a**3*c**4*d**3 + 6*a**2*b**4*c
*d*e**2 + 33*a**2*b**3*c**2*d**2*e + 15*a**2*b**2*c**3*d**3 - 6*a*b**5*c*d**2*e - 12*a*b**4*c**2*d**3 + 2*b**6
*c*d**3))/a**2
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Giac [A] time = 1.3955, size = 142, normalized size = 1.37 \begin{align*} \frac{{\left (b d - a e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{2}} - \frac{{\left (b d - a e\right )} \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (b^{2} d - 2 \, a c d - a b e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a^{2}} - \frac{d}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)/x^2/(c*x^2+b*x+a),x, algorithm="giac")
[Out]
1/2*(b*d - a*e)*log(c*x^2 + b*x + a)/a^2 - (b*d - a*e)*log(abs(x))/a^2 + (b^2*d - 2*a*c*d - a*b*e)*arctan((2*c
*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^2) - d/(a*x)