3.2198 \(\int \frac{1}{(d+e x)^3 (a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=485 \[ \frac{(2 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^4}-\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}-\frac{e (2 c d-b e) \left (-c e (11 a e+b d)+3 b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{e^3 \log (d+e x) \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^4}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

[Out]

-(e*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e)))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (
e*(2*c*d - b*e)*(c^2*d^2 + 3*b^2*e^2 - c*e*(b*d + 11*a*e)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)
) - (b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*(a + b*x
+ c*x^2)) + ((2*c*d - b*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d - 5*a*e) + 4*b^2*c*e^3*(b*d + 5*a*e) - 2*
c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*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)^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[d + e*x])/(c*d^2 - b*d*e + a*
e^2)^4 - (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^
4)

________________________________________________________________________________________

Rubi [A]  time = 1.05679, antiderivative size = 485, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {740, 800, 634, 618, 206, 628} \[ \frac{(2 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^4}-\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}-\frac{e (2 c d-b e) \left (-c e (11 a e+b d)+3 b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{e^3 \log (d+e x) \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^4}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + b*x + c*x^2)^2),x]

[Out]

-(e*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e)))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (
e*(2*c*d - b*e)*(c^2*d^2 + 3*b^2*e^2 - c*e*(b*d + 11*a*e)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)
) - (b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*(a + b*x
+ c*x^2)) + ((2*c*d - b*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d - 5*a*e) + 4*b^2*c*e^3*(b*d + 5*a*e) - 2*
c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*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)^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[d + e*x])/(c*d^2 - b*d*e + a*
e^2)^4 - (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^
4)

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), 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[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

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{1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}-\frac{\int \frac{2 c^2 d^2-3 b^2 e^2+c e (b d+8 a e)+3 c e (2 c d-b e) x}{(d+e x)^3 \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{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}-\frac{\int \left (\frac{e^2 \left (-4 c^2 d^2-3 b^2 e^2+4 c e (b d+2 a e)\right )}{\left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{e^2 (2 c d-b e) \left (-c^2 d^2-3 b^2 e^2+c e (b d+11 a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\left (b^2-4 a c\right ) e^4 \left (-10 c^2 d^2-3 b^2 e^2+2 c e (5 b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{2 c^5 d^5+3 b^5 e^5-5 c^4 d^3 e (b d-4 a e)-10 a c^3 d e^3 (5 b d+3 a e)-b^3 c e^4 (10 b d+17 a e)+b c^2 e^3 \left (10 b^2 d^2+50 a b d e+19 a^2 e^2\right )+c \left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) x}{\left (c d^2-b d e+a e^2\right )^3 \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{e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac{e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{\int \frac{2 c^5 d^5+3 b^5 e^5-5 c^4 d^3 e (b d-4 a e)-10 a c^3 d e^3 (5 b d+3 a e)-b^3 c e^4 (10 b d+17 a e)+b c^2 e^3 \left (10 b^2 d^2+50 a b d e+19 a^2 e^2\right )+c \left (b^2-4 a c\right ) e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) 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 )^4}\\ &=-\frac{e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac{e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{\left (e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right )\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 )^4}-\frac{\left ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\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 )^4}\\ &=-\frac{e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac{e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}+\frac{\left ((2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\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 )^4}\\ &=-\frac{e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (c^2 d^2+3 b^2 e^2-c e (b d+11 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )}+\frac{(2 c d-b e) \left (2 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (b d-5 a e)+4 b^2 c e^3 (b d+5 a e)-2 c^2 e^2 \left (b^2 d^2+10 a b d e+15 a^2 e^2\right )\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 )^4}+\frac{e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac{e^3 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}\\ \end{align*}

Mathematica [A]  time = 1.28918, size = 489, normalized size = 1.01 \[ \frac{2 c^2 \left (-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right )+b^2 c e \left (4 a e^2-3 c d (d-e x)\right )+b c^2 \left (3 a e^2 (e x-3 d)+c d^2 (d-3 e x)\right )+b^3 c e^2 (3 d-e x)+b^4 \left (-e^3\right )}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (b d-a e)-c d^2\right )^3}+\frac{(b e-2 c d) \left (2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )-4 b^2 c e^3 (5 a e+b d)+4 c^3 d^2 e (b d-5 a e)+3 b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2} \left (e (a e-b d)+c d^2\right )^4}+\frac{e^3 \log (d+e x) \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^4}-\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log (a+x (b+c x))}{2 \left (e (a e-b d)+c d^2\right )^4}+\frac{2 e^3 (b e-2 c d)}{(d+e x) \left (e (a e-b d)+c d^2\right )^3}-\frac{e^3}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^2),x]

[Out]

-e^3/(2*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^2) + (2*e^3*(-2*c*d + b*e))/((c*d^2 + e*(-(b*d) + a*e))^3*(d +
e*x)) + (-(b^4*e^3) + b^3*c*e^2*(3*d - e*x) + b^2*c*e*(4*a*e^2 - 3*c*d*(d - e*x)) + 2*c^2*(-(a^2*e^3) + c^2*d^
3*x + 3*a*c*d*e*(d - e*x)) + b*c^2*(c*d^2*(d - 3*e*x) + 3*a*e^2*(-3*d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b
*d - a*e))^3*(a + x*(b + c*x))) + ((-2*c*d + b*e)*(-2*c^4*d^4 + 3*b^4*e^4 + 4*c^3*d^2*e*(b*d - 5*a*e) - 4*b^2*
c*e^3*(b*d + 5*a*e) + 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e^2))*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))^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[d
+ e*x])/(c*d^2 + e*(-(b*d) + a*e))^4 - (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + x*(b + c*x
)])/(2*(c*d^2 + e*(-(b*d) + a*e))^4)

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Maple [B]  time = 0.177, size = 2554, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+b*x+a)^2,x)

[Out]

6/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*c^4*d^4*e-20/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)*c^3*ln(c*x^
2+b*x+a)*a*d^2*e^3-5/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)*c*ln(c*x^2+b*x+a)*b^3*d*e^4+5/(a*e^2-b*d*e+c*d^2)^4/(4*
a*c-b^2)*c^2*ln(c*x^2+b*x+a)*b^2*d^2*e^3+10/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^
2)^(1/2))*b^3*c^2*d^2*e^3-10/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^4
*d^4*e-10/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*c*d*e^4-60/(a*e^2-b*
d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*c^3*d*e^4-20/(a*e^2-b*d*e+c*d^2)^4/(4*a
*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*c*e^5+40/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arcta
n((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^4*d^3*e^2-4/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b^2*c^3*d^4*e-4
/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b^4*c*d^2*e^3+6/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^
2)*b^3*c^2*d^3*e^2+4/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*c^3*d^2*e^3-2*e^5/(a*e^2-b*d*e+c*d^2)
^4*ln(e*x+d)*a*c+10*e^3/(a*e^2-b*d*e+c*d^2)^4*ln(e*x+d)*c^2*d^2+3/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*e^5+4/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)
^(1/2))*c^5*d^5-4*e^3/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)*c*d+6/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c^3/(4*a*c-b^2)*
x*a*b*d^2*e^3+3/2/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^4*e^5-6/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x
+a)*c^3/(4*a*c-b^2)*x*d*a^2*e^4+3/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*a^2*b*e^5-4/(a*e^2-b*d
*e+c*d^2)^4/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*d^2*b^3*e^3+6/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c^3/(4*a*c-b^2)*
x*b^2*d^3*e^2-1/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*a*b^3*e^5-4/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b
*x+a)*c^4/(4*a*c-b^2)*x*d^3*a*e^2+1/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*b^4*d*e^4-5/(a*e^2-b*d
*e+c*d^2)^4/(c*x^2+b*x+a)*c^4/(4*a*c-b^2)*x*b*d^4*e-7/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b*c^
2*d*e^4-1/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^3*c*d*e^4-1/2*e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^
2+4/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b^2*c*e^5-7/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)*c*ln(c*x
^2+b*x+a)*a*b^2*e^5+30/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*c^2*e
^5+10/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^2*c^2*d^2*e^3-14/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)
/(4*a*c-b^2)*a*b*c^3*d^3*e^2+60/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*
b^2*c^2*d*e^4-60/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c^3*d^2*e^3+2
0/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)*c^2*ln(c*x^2+b*x+a)*a*b*d*e^4+2*e^4/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)*b+3*e^5/
(a*e^2-b*d*e+c*d^2)^4*ln(e*x+d)*b^2+2/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c^5/(4*a*c-b^2)*x*d^5-2/(a*e^2-b*d*e
+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a^3*c^2*e^5-1/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^4*e^5+1/
(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b^5*d*e^4+1/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b*
c^4*d^5+4/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)*c^2*ln(c*x^2+b*x+a)*a^2*e^5-10*e^4/(a*e^2-b*d*e+c*d^2)^4*ln(e*x+d)
*b*c*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(1/(e*x+d)^3/(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(1/(e*x+d)^3/(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(1/(e*x+d)**3/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.17698, size = 2176, normalized size = 4.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

-1/2*(10*c^2*d^2*e^3 - 10*b*c*d*e^4 + 3*b^2*e^5 - 2*a*c*e^5)*log(c*x^2 + b*x + a)/(c^4*d^8 - 4*b*c^3*d^7*e + 6
*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 +
6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^
7 + a^4*e^8) + (10*c^2*d^2*e^4 - 10*b*c*d*e^5 + 3*b^2*e^6 - 2*a*c*e^6)*log(abs(x*e + d))/(c^4*d^8*e - 4*b*c^3*
d^7*e^2 + 6*b^2*c^2*d^6*e^3 + 4*a*c^3*d^6*e^3 - 4*b^3*c*d^5*e^4 - 12*a*b*c^2*d^5*e^4 + b^4*d^4*e^5 + 12*a*b^2*
c*d^4*e^5 + 6*a^2*c^2*d^4*e^5 - 4*a*b^3*d^3*e^6 - 12*a^2*b*c*d^3*e^6 + 6*a^2*b^2*d^2*e^7 + 4*a^3*c*d^2*e^7 - 4
*a^3*b*d*e^8 + a^4*e^9) - (4*c^5*d^5 - 10*b*c^4*d^4*e + 40*a*c^4*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 60*a*b*c^3*d^2
*e^3 - 10*b^4*c*d*e^4 + 60*a*b^2*c^2*d*e^4 - 60*a^2*c^3*d*e^4 + 3*b^5*e^5 - 20*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5)
*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^4*d^8 - 4*a*c^5*d^8 - 4*b^3*c^3*d^7*e + 16*a*b*c^4*d^7*e + 6*b
^4*c^2*d^6*e^2 - 20*a*b^2*c^3*d^6*e^2 - 16*a^2*c^4*d^6*e^2 - 4*b^5*c*d^5*e^3 + 4*a*b^3*c^2*d^5*e^3 + 48*a^2*b*
c^3*d^5*e^3 + b^6*d^4*e^4 + 8*a*b^4*c*d^4*e^4 - 42*a^2*b^2*c^2*d^4*e^4 - 24*a^3*c^3*d^4*e^4 - 4*a*b^5*d^3*e^5
+ 4*a^2*b^3*c*d^3*e^5 + 48*a^3*b*c^2*d^3*e^5 + 6*a^2*b^4*d^2*e^6 - 20*a^3*b^2*c*d^2*e^6 - 16*a^4*c^2*d^2*e^6 -
4*a^3*b^3*d*e^7 + 16*a^4*b*c*d*e^7 + a^4*b^2*e^8 - 4*a^5*c*e^8)*sqrt(-b^2 + 4*a*c)) - 1/2*(2*b*c^4*d^7 - 8*b^
2*c^3*d^6*e + 12*a*c^4*d^6*e + 12*b^3*c^2*d^5*e^2 - 28*a*b*c^3*d^5*e^2 - 8*b^4*c*d^4*e^3 + 29*a*b^2*c^2*d^4*e^
3 - 28*a^2*c^3*d^4*e^3 + 2*b^5*d^3*e^4 - 16*a*b^3*c*d^3*e^4 + 42*a^2*b*c^2*d^3*e^4 + 3*a*b^4*d^2*e^5 - 2*a^2*b
^2*c*d^2*e^5 - 44*a^3*c^2*d^2*e^5 - 6*a^2*b^3*d*e^6 + 24*a^3*b*c*d*e^6 + a^3*b^2*e^7 - 4*a^4*c*e^7 + 2*(2*c^5*
d^5*e^2 - 5*b*c^4*d^4*e^3 + 10*b^2*c^3*d^3*e^4 - 20*a*c^4*d^3*e^4 - 10*b^3*c^2*d^2*e^5 + 30*a*b*c^3*d^2*e^5 +
3*b^4*c*d*e^6 - 4*a*b^2*c^2*d*e^6 - 22*a^2*c^3*d*e^6 - 3*a*b^3*c*e^7 + 11*a^2*b*c^2*e^7)*x^3 + (8*c^5*d^6*e -
18*b*c^4*d^5*e^2 + 25*b^2*c^3*d^4*e^3 - 40*a*c^4*d^4*e^3 - 10*b^3*c^2*d^3*e^4 + 20*a*b*c^3*d^3*e^4 - 11*b^4*c*
d^2*e^5 + 58*a*b^2*c^2*d^2*e^5 - 56*a^2*c^3*d^2*e^5 + 6*b^5*d*e^6 - 20*a*b^3*c*d*e^6 - 10*a^2*b*c^2*d*e^6 - 6*
a*b^4*e^7 + 25*a^2*b^2*c*e^7 - 8*a^3*c^2*e^7)*x^2 + (4*c^5*d^7 - 6*b*c^4*d^6*e - 4*b^2*c^3*d^5*e^2 + 16*a*c^4*
d^5*e^2 + 25*b^3*c^2*d^4*e^3 - 80*a*b*c^3*d^4*e^3 - 28*b^4*c*d^3*e^4 + 104*a*b^2*c^2*d^3*e^4 - 28*a^2*c^3*d^3*
e^4 + 9*b^5*d^2*e^5 - 28*a*b^3*c*d^2*e^5 - 14*a^2*b*c^2*d^2*e^5 - 6*a*b^4*d*e^6 + 32*a^2*b^2*c*d*e^6 - 40*a^3*
c^2*d*e^6 - 3*a^2*b^3*e^7 + 12*a^3*b*c*e^7)*x)/((c*d^2 - b*d*e + a*e^2)^4*(c*x^2 + b*x + a)*(b^2 - 4*a*c)*(x*e
+ d)^2)