### 3.2207 $$\int \frac{1}{(d+e x) (a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=429 $-\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^5 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3}$

[Out]

-(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) -
(3*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2 - 2*b^2*e^2 - c*e*(3*b*d - 8*a*e)) - 2*c*(2*c
*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x
+ c*x^2)) - ((12*c^5*d^5 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5 - 10*c^4*d^3*e*(3*b*d - 4*a*e) + 20*c^
3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2
- b*d*e + a*e^2)^3) + (e^5*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 - (e^5*Log[a + b*x + c*x^2])/(2*(c*d^2 - b
*d*e + a*e^2)^3)

________________________________________________________________________________________

Rubi [A]  time = 1.01571, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.35, Rules used = {740, 822, 800, 634, 618, 206, 628} $-\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^5 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) -
(3*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2 - 2*b^2*e^2 - c*e*(3*b*d - 8*a*e)) - 2*c*(2*c
*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x
+ c*x^2)) - ((12*c^5*d^5 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5 - 10*c^4*d^3*e*(3*b*d - 4*a*e) + 20*c^
3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2
- b*d*e + a*e^2)^3) + (e^5*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 - (e^5*Log[a + b*x + c*x^2])/(2*(c*d^2 - b
*d*e + a*e^2)^3)

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 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{1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b 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{6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)+3 c e (2 c d-b e) 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{b c d-b^2 e+2 a c e+c (2 c d-b 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{3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \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 (6 c^4 d^4+b^4 e^4-c^3 d^2 e (9 b d-14 a e)+b^2 c e^3 (b d-8 a e)+c^2 e^2 \left (b^2 d^2-7 a b d e+16 a^2 e^2\right )\right )+2 c e (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 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{b c d-b^2 e+2 a c e+c (2 c d-b 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{3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \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^6}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{2 \left (6 c^5 d^5-b^5 e^5+9 a b^3 c e^5-23 a^2 b c^2 e^5-5 c^4 d^3 e (3 b d-4 a e)+10 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )-c \left (b^2-4 a c\right )^2 e^5 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{b c d-b^2 e+2 a c e+c (2 c d-b 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{3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\int \frac{6 c^5 d^5-b^5 e^5+9 a b^3 c e^5-23 a^2 b c^2 e^5-5 c^4 d^3 e (3 b d-4 a e)+10 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )-c \left (b^2-4 a c\right )^2 e^5 x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b 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{3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{e^5 \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 (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b 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{3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{e^5 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac{\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\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 )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b 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{3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 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 )^{5/2} \left (c d^2-b d e+a e^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{e^5 \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 = 1.30148, size = 429, normalized size = 1. $\frac{1}{2} \left (\frac{4 c^2 \left (4 a^2 e^3+7 a c d e^2 x+3 c^2 d^3 x\right )+b^2 c e \left (c d (2 e x-9 d)-15 a e^2\right )+2 b c^2 \left (7 a e^2 (d-e x)+3 c d^2 (d-3 e x)\right )+b^3 c e^2 (d+2 e x)+2 b^4 e^3}{\left (b^2-4 a c\right )^2 (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}+\frac{2 \left (-20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+30 a^2 b c^2 e^5-10 a b^3 c e^5+10 c^4 d^3 e (3 b d-4 a e)+b^5 e^5-12 c^5 d^5\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2} \left (e (b d-a e)-c d^2\right )^3}+\frac{2 c (a e+c d x)+b^2 (-e)+b c (d-e x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2 \left (e (b d-a e)-c d^2\right )}+\frac{2 e^5 \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^3}-\frac{e^5 \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^3}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-(b^2*e) + 2*c*(a*e + c*d*x) + b*c*(d - e*x))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))^2)
+ (2*b^4*e^3 + b^3*c*e^2*(d + 2*e*x) + 4*c^2*(4*a^2*e^3 + 3*c^2*d^3*x + 7*a*c*d*e^2*x) + 2*b*c^2*(3*c*d^2*(d
- 3*e*x) + 7*a*e^2*(d - e*x)) + b^2*c*e*(-15*a*e^2 + c*d*(-9*d + 2*e*x)))/((b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*d)
+ a*e))^2*(a + x*(b + c*x))) + (2*(-12*c^5*d^5 + b^5*e^5 - 10*a*b^3*c*e^5 + 30*a^2*b*c^2*e^5 + 10*c^4*d^3*e*(3
*b*d - 4*a*e) - 20*c^3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2
+ 4*a*c)^(5/2)*(-(c*d^2) + e*(b*d - a*e))^3) + (2*e^5*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 - (e^5*Log[a
+ x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^3)/2

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Maple [B]  time = 0.178, size = 4701, normalized size = 11. \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^3*d^3*e^2-2/(a*e^2-b*d*e+c*d^2)^
3/(c*x^2+b*x+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^5*d*e^4+1/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2/(16
*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^4*d^2*e^3+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x
^2*a*b^4*e^5+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^5*c*d^2*e^3+3/(a*e^2-b*d*e
+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^4*c^2*d^3*e^2-5/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2
/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^3*c^3*d^4*e-7/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)
*a^3*b*c^2*d*e^4-15/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b*d^4*e-29/2/(a*e
^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a*b^2*c^3*d^4*e+28/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b
*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a^2*c^4*d^3*e^2+6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b
^2*c+b^4)*a^2*b*c^3*d^3*e^2-1/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a*b^4*c*d^2*e
^3+12/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a*b^3*c^2*d^3*e^2+14/(a*e^2-b*d*e+c*d^2
)^3/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*d*a^2*e^4+10/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4/
(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^2*d^3*e^2-6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*
x*a^2*b^3*c*e^5+20/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*d^3*a*e^2-1/(a*e^2
-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^4*d*e^4-29/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2
+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^2*b^2*e^5+8/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4/(16*a^2*c
^2-8*a*b^2*c+b^4)*x^2*a^2*d^2*e^3-25/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a^2*b^2*
c^2*d^2*e^3+27/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a^2*b^3*c*d*e^4-1/(a*e^2-b*d
*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a^3*b*c^2*e^5+18/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a^3*c^3*d*e^4+60/(a*e^2-b*d*e+c*d^2)^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(
1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*c^3*d*e^4+10/(a*e^2-b*d*e+c*d^2)^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*
a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*c*e^5-30/(a*e^2-b*d*e+c*d^2)^3/(16*a^2*c^2-8*a*b^2*c+
b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^4*d^4*e-30/(a*e^2-b*d*e+c*d^2)^3/(16*a^2*c^2-8*
a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*c^2*e^5+20/(a*e^2-b*d*e+c*d^2)^3/(16*
a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c^3*d^3*e^2+40/(a*e^2-b*d*e+c
*d^2)^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^4*d^3*e^2+1/(a*e^
2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a*b^3*e^5-7/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b
*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a^2*b*e^5+13/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-
8*a*b^2*c+b^4)*x^2*a^2*b*d*e^4-49/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a*b
^2*d^2*e^3-30/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a*b*d^2*e^3+8/(a*e^2-b*
d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a*b^2*d*e^4+26/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*
x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a*b^2*c^3*d^3*e^2-18/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a
*b^2*c+b^4)*x*a*b^3*c^2*d^2*e^3-60/(a*e^2-b*d*e+c*d^2)^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((
2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c^3*d^2*e^3+16/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c
+b^4)*x^2*a*b^3*d*e^4-34/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a^2*b*c^3*d^2*e^3+
6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a*b^4*c*d*e^4+30/(a*e^2-b*d*e+c*d^2)^3/(c
*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a*b*d^3*e^2-25/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*
c^2-8*a*b^2*c+b^4)*x*a*b*c^4*d^4*e+10/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a^2*b
^2*c^2*d*e^4+4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a^2*c^4*d^4*e-45/2/(a*e^2-b*d*
e+c*d^2)^3/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^2*d^4*e-1/2/(a*e^2-b*d*e+c*d^2)^3/(16*a^2*c^2-
8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*b^4*e^5-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a*b^
5*d*e^4+5/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a*b*c^4*d^5-3/2/(a*e^2-b*d*e+c*d^2)
^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^5*c*d^3*e^2+3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*
c^2-8*a*b^2*c+b^4)*b^4*c^2*d^4*e+4/(a*e^2-b*d*e+c*d^2)^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln(c*x^2+b*x+a)*a*b^2*e^
5+8/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^3*e^5+9/(a*e^2-b*d*e+c*d^2)^3/(
c*x^2+b*x+a)^2*c^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b*d^5+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*
a*b^2*c+b^4)*x*a*b^5*e^5+10/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*a*c^5*d^5+16/(a
*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a^3*c^3*d^2*e^3-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+
b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^6*d*e^4+2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+
b^4)*x*b^2*c^4*d^5-21/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a^3*b^2*c*e^5+12/(a*e
^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a^4*c^2*e^5+3/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+
a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*a^2*b^4*e^5+1/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^
4)*b^6*d^2*e^3-1/2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^3*c^3*d^5-1/(a*e^2-b*d*e
+c*d^2)^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*e^5+12/(a*e^2-b
*d*e+c*d^2)^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^5*d^5-8/(a*e^
2-b*d*e+c*d^2)^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*ln(c*x^2+b*x+a)*a^2*e^5+6/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^
2*c^6/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*d^5+e^5*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^3

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

[Out]

Timed out

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Giac [B]  time = 1.16121, size = 1871, normalized size = 4.36 \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)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-1/2*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) + 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) + (12*c^5*d^5 - 30*b*c^4*d^4*e + 20*b^2*c^3*d^3*e^2 + 40*a*c^4*d^3*e^2
- 60*a*b*c^3*d^2*e^3 + 60*a^2*c^3*d*e^4 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5)*arctan((2*c*x + b)/sqr
t(-b^2 + 4*a*c))/((b^4*c^3*d^6 - 8*a*b^2*c^4*d^6 + 16*a^2*c^5*d^6 - 3*b^5*c^2*d^5*e + 24*a*b^3*c^3*d^5*e - 48*
a^2*b*c^4*d^5*e + 3*b^6*c*d^4*e^2 - 21*a*b^4*c^2*d^4*e^2 + 24*a^2*b^2*c^3*d^4*e^2 + 48*a^3*c^4*d^4*e^2 - b^7*d
^3*e^3 + 2*a*b^5*c*d^3*e^3 + 32*a^2*b^3*c^2*d^3*e^3 - 96*a^3*b*c^3*d^3*e^3 + 3*a*b^6*d^2*e^4 - 21*a^2*b^4*c*d^
2*e^4 + 24*a^3*b^2*c^2*d^2*e^4 + 48*a^4*c^3*d^2*e^4 - 3*a^2*b^5*d*e^5 + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^
5 + a^3*b^4*e^6 - 8*a^4*b^2*c*e^6 + 16*a^5*c^2*e^6)*sqrt(-b^2 + 4*a*c)) - 1/2*(b^3*c^3*d^5 - 10*a*b*c^4*d^5 -
3*b^4*c^2*d^4*e + 29*a*b^2*c^3*d^4*e - 8*a^2*c^4*d^4*e + 3*b^5*c*d^3*e^2 - 24*a*b^3*c^2*d^3*e^2 - 12*a^2*b*c^3
*d^3*e^2 - b^6*d^2*e^3 + a*b^4*c*d^2*e^3 + 50*a^2*b^2*c^2*d^2*e^3 - 32*a^3*c^3*d^2*e^3 + 4*a*b^5*d*e^4 - 27*a^
2*b^3*c*d*e^4 + 14*a^3*b*c^2*d*e^4 - 3*a^2*b^4*e^5 + 21*a^3*b^2*c*e^5 - 24*a^4*c^2*e^5 - 2*(6*c^6*d^5 - 15*b*c
^5*d^4*e + 10*b^2*c^4*d^3*e^2 + 20*a*c^5*d^3*e^2 - 30*a*b*c^4*d^2*e^3 - b^4*c^2*d*e^4 + 8*a*b^2*c^3*d*e^4 + 14
*a^2*c^4*d*e^4 + a*b^3*c^2*e^5 - 7*a^2*b*c^3*e^5)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2
+ 60*a*b*c^4*d^3*e^2 + b^4*c^2*d^2*e^3 - 98*a*b^2*c^3*d^2*e^3 + 16*a^2*c^4*d^2*e^3 - 4*b^5*c*d*e^4 + 32*a*b^3*
c^2*d*e^4 + 26*a^2*b*c^3*d*e^4 + 4*a*b^4*c*e^5 - 29*a^2*b^2*c^2*e^5 + 16*a^3*c^3*e^5)*x^2 - 2*(2*b^2*c^4*d^5 +
10*a*c^5*d^5 - 5*b^3*c^3*d^4*e - 25*a*b*c^4*d^4*e + 3*b^4*c^2*d^3*e^2 + 26*a*b^2*c^3*d^3*e^2 + 28*a^2*c^4*d^3
*e^2 + b^5*c*d^2*e^3 - 18*a*b^3*c^2*d^2*e^3 - 34*a^2*b*c^3*d^2*e^3 - b^6*d*e^4 + 6*a*b^4*c*d*e^4 + 10*a^2*b^2*
c^2*d*e^4 + 18*a^3*c^3*d*e^4 + a*b^5*e^5 - 6*a^2*b^3*c*e^5 - a^3*b*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)^2)