3.1538 \(\int \frac{b+2 c x}{(d+e x)^2 (a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=327 \[ \frac{e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac{2 e^2 \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{2 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{2 e^2 (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.534025, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {822, 800, 634, 618, 206, 628} \[ \frac{e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{\left (a e^2-b d e+c d^2\right )^3}-\frac{2 e^2 \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{2 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{2 e^2 (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

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

Mathematica [A]  time = 0.683963, size = 275, normalized size = 0.84 \[ \frac{-\frac{\left (e (a e-b d)+c d^2\right ) \left (c e (-a e-2 b d+b e x)+b^2 e^2+c^2 d (d-2 e x)\right )}{a+x (b+c x)}-2 e^2 \log (d+e x) \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right )+e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log (a+x (b+c x))+\frac{2 e (b e-2 c d) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{e^2 (b e-2 c d) \left (e (a e-b d)+c d^2\right )}{d+e x}}{\left (e (a e-b d)+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.021, size = 1177, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*b^2*c*d^2*e^2+3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*b*d^3*c^2*e-3/(a*e^
2-b*d*e+c*d^2)^3*c*ln(c*x^2+b*x+a)*b*d*e^3-4/(a*e^2-b*d*e+c*d^2)^3*e/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c
-b^2)^(1/2))*c^3*d^3+6*e^3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b*c*d+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*e*d^3*x
*c^3+1/(a*e^2-b*d*e+c*d^2)^3*ln(c*x^2+b*x+a)*b^2*e^4-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^3*d^4-e^3/(a*e^2-
b*d*e+c*d^2)^2/(e*x+d)*b-2*e^4/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b^2+2*e^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)*c*d+2/(
a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e^4+1/(a*e^2-b*d*e+c*d^2)^3/(c*
x^2+b*x+a)*b^3*d*e^3+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c*e^4*a^2-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*a*b
^2*e^4+2*e^4/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*a*c-6*e^2/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c^2*d^2-1/(a*e^2-b*d*e+
c*d^2)^3*c*ln(c*x^2+b*x+a)*a*e^4+3/(a*e^2-b*d*e+c*d^2)^3*c^2*ln(c*x^2+b*x+a)*d^2*e^2-1/(a*e^2-b*d*e+c*d^2)^3/(
c*x^2+b*x+a)*x*a*b*c*e^4+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*x*a*c^2*d*e^3+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*
x+a)*x*b^2*c*d*e^3-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*x*b*c^2*d^2*e^2+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)
*a*b*c*d*e^3-6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c*e^4*a*b+12/(a*e^2
-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^2*d*e^3-6/(a*e^2-b*d*e+c*d^2)^3/(4*a
*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c*d*e^3+6/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)^(1/2)*arctan
((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^2*d^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((2*c*x+b)/(e*x+d)^2/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [B]  time = 1.27494, size = 1007, normalized size = 3.08 \begin{align*} \frac{2 \,{\left (2 \, c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} - 6 \, a c^{2} d e^{5} - b^{3} e^{6} + 3 \, a b c e^{6}\right )} \arctan \left (-\frac{{\left (2 \, c d - \frac{2 \, c d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4} - a c e^{4}\right )} \log \left (-c + \frac{2 \, c d}{x e + d} - \frac{c 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{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}} + \frac{\frac{2 \, c d e^{6}}{x e + d} - \frac{b e^{7}}{x e + d}}{c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6} + 2 \, a c d^{2} e^{6} - 2 \, a b d e^{7} + a^{2} e^{8}} + \frac{\frac{3 \, c^{3} d^{2} e^{2} - 3 \, b c^{2} d e^{3} + b^{2} c e^{4} - a c^{2} e^{4}}{c d^{2} - b d e + a e^{2}} - \frac{{\left (4 \, c^{3} d^{3} e^{3} - 6 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} - 4 \, a c^{2} d e^{5} - b^{3} e^{6} + 2 \, a b c e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e + a e^{2}\right )}{\left (x e + d\right )}}}{{\left (c d^{2} - b d e + a e^{2}\right )}^{2}{\left (c - \frac{2 \, c d}{x e + d} + \frac{c 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{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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