3.1545 \(\int \frac{b+2 c x}{(d+e x) (a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=397 \[ -\frac{e \left (-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e+3 b^2 c d e-2 b^3 e^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}+\frac{e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{e^4 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

[Out]

-((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x)/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^
2) - (e*(3*b^2*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b*c*(c*d^2 - 7*a*e^2) - 2*c*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3
*a*e))*x))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)) - (e*(2*c^4*d^4 - b^4*e^4 - 4*c^3*d^2
*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c
]])/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^3) - (e^4*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)
^3 + (e^4*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.822982, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{e \left (-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e+3 b^2 c d e-2 b^3 e^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}+\frac{e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac{e^4 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/((d + e*x)*(a + b*x + c*x^2)^3),x]

[Out]

-((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x)/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^
2) - (e*(3*b^2*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b*c*(c*d^2 - 7*a*e^2) - 2*c*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3
*a*e))*x))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)) - (e*(2*c^4*d^4 - b^4*e^4 - 4*c^3*d^2
*e*(b*d - 3*a*e) - 6*a*c^2*e^3*(2*b*d + a*e) + 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c
]])/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)^3) - (e^4*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)
^3 + (e^4*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(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) \left (a+b x+c x^2\right )^3} \, dx &=-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{\left (b^2-4 a c\right ) e (c d-2 b e)-3 c \left (b^2-4 a c\right ) e^2 x}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx}{2 \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}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{\int \frac{2 \left (b^2-4 a c\right ) e \left (c^3 d^3+b^3 e^3-c^2 d e (b d-5 a e)-b c e^2 (b d+4 a e)\right )+2 c \left (b^2-4 a c\right ) e^2 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{\int \left (\frac{2 \left (b^2-4 a c\right )^2 e^5 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{2 \left (b^2-4 a c\right ) e \left (c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x\right )}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac{e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{e \int \frac{c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac{e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\left (e^4 (2 c d-b e)\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac{\left (e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac{e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac{\left (e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (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 (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac{e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}-\frac{e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 1.36692, size = 356, normalized size = 0.9 \[ \frac{1}{2} \left (\frac{e \left (b c \left (c d (d-2 e x)-7 a e^2\right )+2 c^2 \left (a e (4 d-3 e x)+c d^2 x\right )+b^2 c e (2 e x-3 d)+2 b^3 e^2\right )}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}-\frac{2 e \left (-2 b^2 c e^3 (3 a e+b d)+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (a e+2 b d)+b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2} \left (e (b d-a e)-c d^2\right )^3}+\frac{2 e^4 (b e-2 c d) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^3}+\frac{e^4 (2 c d-b e) \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^3}+\frac{b e-c d+c e x}{(a+x (b+c x))^2 \left (e (a e-b d)+c d^2\right )}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)/((d + e*x)*(a + b*x + c*x^2)^3),x]

[Out]

((-(c*d) + b*e + c*e*x)/((c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^2) + (e*(2*b^3*e^2 + b^2*c*e*(-3*d + 2*e
*x) + 2*c^2*(c*d^2*x + a*e*(4*d - 3*e*x)) + b*c*(-7*a*e^2 + c*d*(d - 2*e*x))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*
d) + a*e))^2*(a + x*(b + c*x))) - (2*e*(-2*c^4*d^4 + b^4*e^4 + 4*c^3*d^2*e*(b*d - 3*a*e) + 6*a*c^2*e^3*(2*b*d
+ a*e) - 2*b^2*c*e^3*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(3/2)*(-(c*d^2) +
e*(b*d - a*e))^3) + (2*e^4*(-2*c*d + b*e)*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 + (e^4*(2*c*d - b*e)*Log[
a + x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^3)/2

________________________________________________________________________________________

Maple [B]  time = 0.027, size = 3233, normalized size = 8.1 \begin{align*} \text{output 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)/(c*x^2+b*x+a)^3,x)

[Out]

-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^4/(4*a*c-b^2)*x^3*a*b*d+4/(a*e^2-b*d*e+c*d^2)^3*e^4/(4*a*c-b^2)
*c^2*ln(c*x^2+b*x+a)*a*d-1/(a*e^2-b*d*e+c*d^2)^3*e^4/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^2*d-6/(a*e^2-b*d*e+c*d^2)
^3*e^5/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*c-12/(a*e^2-b*d*e+c*d^2)^3*e^3/(4*a*c-b^2)^
(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^3*d^2+5/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*
a^3*c^2-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a*b^4+3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^
2*c^3*e^5/(4*a*c-b^2)*x^3*a^2-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^5*e/(4*a*c-b^2)*x^3*d^4-6/(a*e^2-b*d*e
+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^3*c^2*d*e^4-8/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*c^
3*d^3*e^2+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b^4*d*e^4+3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x
+a)^2/(4*a*c-b^2)*b^4*c*d^3*e^2-3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^3*c^2*d^4*e-2/(a*e^2-b
*d*e+c*d^2)^3*e^5/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*a*b-2/(a*e^2-b*d*e+c*d^2)^3*e^4/(4*a*c-b^2)^(3/2)*arctan((2*c*
x+b)/(4*a*c-b^2)^(1/2))*b^3*c*d+4/(a*e^2-b*d*e+c*d^2)^3*e^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/
2))*b*c^3*d^3+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x*b^5*d+11/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^
2+b*x+a)^2/(4*a*c-b^2)*a^3*b*c*e^5-3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b^3*e^5-2/(a*e^2-
b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*c^4*d^5-1/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^5
*d^2*e^3+1/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*b^2*c^3*d^5-2/(a*e^2-b*d*e+c*d^2)^3*e/(4*a*c-b^
2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^4*d^4+6/(a*e^2-b*d*e+c*d^2)^3*e^5/(4*a*c-b^2)^(3/2)*arctan((2*c
*x+b)/(4*a*c-b^2)^(1/2))*a^2*c^2-4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^4/(4*a*c-b^2)*x^2*a*b^2*d-6/(a*
e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x*a*b*c^3*d^3+9/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e
^3/(4*a*c-b^2)*x^2*a*b*d^2+6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x*a*b^2*c^2*d^2-10/(a*e^2-b
*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x*a^2*b*c^2*d+e^5/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b+1/(a*e^2-b*d
*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^4/(4*a*c-b^2)*x^3*b^3*d-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e/(4*a*c-b^2
)*x*b^2*c^3*d^4-2*e^4/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c*d+1/(a*e^2-b*d*e+c*d^2)^3*e^5/(4*a*c-b^2)^(3/2)*arctan
((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4+1/2/(a*e^2-b*d*e+c*d^2)^3*e^5/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^3-3/(a*e^2-b*d*e
+c*d^2)^3/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x*b^4*c*d^2+11/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a
*b*c^3*d^4*e+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e^3/(4*a*c-b^2)*x^3*a*d^2-9/2/(a*e^2-b*d*e+c*d^2)^3/(
c*x^2+b*x+a)^2*c^2*e^3/(4*a*c-b^2)*x^2*b^3*d^2+4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^2/(4*a*c-b^2)*x^2
*b^2*d^3-3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e/(4*a*c-b^2)*x^2*b*d^4+15/(a*e^2-b*d*e+c*d^2)^3/(c*x^2
+b*x+a)^2/(4*a*c-b^2)*a^2*b*c^2*d^2*e^3-5/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b^3*c*d^2*e^3+
12/(a*e^2-b*d*e+c*d^2)^3*e^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c^2*d+1/(a*e^2-b*d*e+c*
d^2)^3/(c*x^2+b*x+a)^2*e/(4*a*c-b^2)*x*a*c^4*d^4-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*b^2*c^2
*d^3*e^2-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^3/(4*a*c-b^2)*x^3*b^2*d^2+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^
2+b*x+a)^2*c^4*e^2/(4*a*c-b^2)*x^3*b*d^3+13/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^5/(4*a*c-b^2)*x^2*a^
2*b-4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3*e^4/(4*a*c-b^2)*x^2*a^2*d-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a
)^2*c*e^5/(4*a*c-b^2)*x^2*a*b^3-4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4*e^2/(4*a*c-b^2)*x^2*a*d^3+2/(a*e^2
-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c*e^4/(4*a*c-b^2)*x^2*b^4*d-11/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(4*a*c-
b^2)*a^2*b^2*c*d*e^4-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2*e^5/(4*a*c-b^2)*x^3*a*b^2+3/(a*e^2-b*d*e+c*d^
2)^3/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x*b^3*c^2*d^3+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a
^2*b^2*c+6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x*a^2*c^3*d^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)/(c*x^2+b*x+a)^3,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)/(c*x^2+b*x+a)^3,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)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.24145, size = 1508, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/2*(2*c*d*e^4 - b*e^5)*log(c*x^2 + b*x + a)/(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^5 - b*e^
6)*log(abs(x*e + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*
d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7) + (2*c^4*d^4*e - 4*b*c^3*d^3*e^2 + 12*a
*c^3*d^2*e^3 + 2*b^3*c*d*e^4 - 12*a*b*c^2*d*e^4 - b^4*e^5 + 6*a*b^2*c*e^5 - 6*a^2*c^2*e^5)*arctan((2*c*x + b)/
sqrt(-b^2 + 4*a*c))/((b^2*c^3*d^6 - 4*a*c^4*d^6 - 3*b^3*c^2*d^5*e + 12*a*b*c^3*d^5*e + 3*b^4*c*d^4*e^2 - 9*a*b
^2*c^2*d^4*e^2 - 12*a^2*c^3*d^4*e^2 - b^5*d^3*e^3 - 2*a*b^3*c*d^3*e^3 + 24*a^2*b*c^2*d^3*e^3 + 3*a*b^4*d^2*e^4
- 9*a^2*b^2*c*d^2*e^4 - 12*a^3*c^2*d^2*e^4 - 3*a^2*b^3*d*e^5 + 12*a^3*b*c*d*e^5 + a^3*b^2*e^6 - 4*a^4*c*e^6)*
sqrt(-b^2 + 4*a*c)) - 1/2*(b^2*c^3*d^5 - 4*a*c^4*d^5 - 3*b^3*c^2*d^4*e + 11*a*b*c^3*d^4*e + 3*b^4*c*d^3*e^2 -
6*a*b^2*c^2*d^3*e^2 - 16*a^2*c^3*d^3*e^2 - b^5*d^2*e^3 - 5*a*b^3*c*d^2*e^3 + 30*a^2*b*c^2*d^2*e^3 + 4*a*b^4*d*
e^4 - 11*a^2*b^2*c*d*e^4 - 12*a^3*c^2*d*e^4 - 3*a^2*b^3*e^5 + 11*a^3*b*c*e^5 - 2*(c^5*d^4*e - 2*b*c^4*d^3*e^2
+ 2*b^2*c^3*d^2*e^3 - 2*a*c^4*d^2*e^3 - b^3*c^2*d*e^4 + 2*a*b*c^3*d*e^4 + a*b^2*c^2*e^5 - 3*a^2*c^3*e^5)*x^3 -
(3*b*c^4*d^4*e - 8*b^2*c^3*d^3*e^2 + 8*a*c^4*d^3*e^2 + 9*b^3*c^2*d^2*e^3 - 18*a*b*c^3*d^2*e^3 - 4*b^4*c*d*e^4
+ 8*a*b^2*c^2*d*e^4 + 8*a^2*c^3*d*e^4 + 4*a*b^3*c*e^5 - 13*a^2*b*c^2*e^5)*x^2 - 2*(b^2*c^3*d^4*e - a*c^4*d^4*
e - 3*b^3*c^2*d^3*e^2 + 6*a*b*c^3*d^3*e^2 + 3*b^4*c*d^2*e^3 - 6*a*b^2*c^2*d^2*e^3 - 6*a^2*c^3*d^2*e^3 - b^5*d*
e^4 + 10*a^2*b*c^2*d*e^4 + a*b^4*e^5 - 2*a^2*b^2*c*e^5 - 5*a^3*c^2*e^5)*x)/((c*d^2 - b*d*e + a*e^2)^3*(c*x^2 +
b*x + a)^2*(b^2 - 4*a*c))