### 3.2201 $$\int \frac{(d+e x)^5}{(a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=388 $-\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{e^2 x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )^2}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{e^5 \log \left (a+b x+c x^2\right )}{2 c^3}$

[Out]

-((e^2*(2*c*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/(c^2*(b^2 - 4*a*c)^2)) - ((d + e*x)^4*(b*d
- 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - ((d + e*x)^2*(8*a*c*e*(c*d^2 + 2*a*e^2) -
6*b*c*d*(c*d^2 + 3*a*e^2) + b^2*(7*c*d^2*e - a*e^3) - (2*c*d - b*e)*(6*c^2*d^2 - b^2*e^2 - 2*c*e*(3*b*d - 5*a
*e))*x))/(2*c*(b^2 - 4*a*c)^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]])/(c^3*(b^2 - 4*a*c)^(5/2)) + (e^5*Log[a + b*x + c*x^2])/(2*c^3)

________________________________________________________________________________________

Rubi [A]  time = 1.11005, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.35, Rules used = {738, 818, 773, 634, 618, 206, 628} $-\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{e^2 x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )^2}-\frac{(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{e^5 \log \left (a+b x+c x^2\right )}{2 c^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5/(a + b*x + c*x^2)^3,x]

[Out]

-((e^2*(2*c*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/(c^2*(b^2 - 4*a*c)^2)) - ((d + e*x)^4*(b*d
- 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - ((d + e*x)^2*(8*a*c*e*(c*d^2 + 2*a*e^2) -
6*b*c*d*(c*d^2 + 3*a*e^2) + b^2*(7*c*d^2*e - a*e^3) - (2*c*d - b*e)*(6*c^2*d^2 - b^2*e^2 - 2*c*e*(3*b*d - 5*a
*e))*x))/(2*c*(b^2 - 4*a*c)^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]])/(c^3*(b^2 - 4*a*c)^(5/2)) + (e^5*Log[a + b*x + c*x^2])/(2*c^3)

Rule 738

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

Rule 818

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

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*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]

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)^5}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{(d+e x)^3 \left (6 c d^2-e (7 b d-8 a e)-e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\int \frac{(d+e x) \left (-2 \left (6 c^3 d^4-a b^2 e^4-c^2 d^2 e (15 b d-14 a e)+c e^2 \left (10 b^2 d^2-21 a b d e+16 a^2 e^2\right )\right )+2 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\right )}{a+b x+c x^2} \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=-\frac{e^2 (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}{c^2 \left (b^2-4 a c\right )^2}-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\int \frac{-2 a e^2 (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right )-2 c d \left (6 c^3 d^4-a b^2 e^4-c^2 d^2 e (15 b d-14 a e)+c e^2 \left (10 b^2 d^2-21 a b d e+16 a^2 e^2\right )\right )+\left (2 c d 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 )-2 b e^2 (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right )-2 c e \left (6 c^3 d^4-a b^2 e^4-c^2 d^2 e (15 b d-14 a e)+c e^2 \left (10 b^2 d^2-21 a b d e+16 a^2 e^2\right )\right )\right ) x}{a+b x+c x^2} \, dx}{2 c^2 \left (b^2-4 a c\right )^2}\\ &=-\frac{e^2 (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}{c^2 \left (b^2-4 a c\right )^2}-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{e^5 \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^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 c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac{e^2 (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}{c^2 \left (b^2-4 a c\right )^2}-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{e^5 \log \left (a+b x+c x^2\right )}{2 c^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 )}{c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac{e^2 (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}{c^2 \left (b^2-4 a c\right )^2}-\frac{(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\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 )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{e^5 \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end{align*}

Mathematica [A]  time = 1.20017, size = 628, normalized size = 1.62 $\frac{\frac{-2 b^2 c e^2 \left (2 a^2 e^3-5 a c d e (d+2 e x)+5 c^2 d^3 x\right )+b c^2 \left (5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)-c^2 d^4 (d-5 e x)\right )+2 c^2 \left (-5 a^2 c d e^3 (2 d+e x)+a^3 e^5+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right )-5 b^3 c e^3 \left (a e (d+e x)-2 c d^2 x\right )+b^4 e^4 (a e-5 c d x)+b^5 e^5 x}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{b^2 c^2 e \left (-39 a^2 e^4+10 a c d e^2 (5 d+8 e x)-5 c^2 d^3 (3 d-4 e x)\right )+2 b c^3 \left (5 a^2 e^4 (11 d+5 e x)+10 a c d^2 e^2 (d-3 e x)+3 c^2 d^4 (d-5 e x)\right )+4 c^3 \left (-5 a^2 c d e^3 (8 d+5 e x)+8 a^3 e^5+10 a c^2 d^3 e^2 x+3 c^3 d^5 x\right )+10 b^3 c^2 e^2 \left (c d^3-a e^2 (4 d+3 e x)\right )+b^4 c e^3 \left (11 a e^2-10 c d (d+e x)\right )+b^5 c e^4 (5 d+4 e x)+b^6 \left (-e^5\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{2 c (2 c d-b e) \left (2 c^2 e^2 \left (15 a^2 e^2-10 a b d e+2 b^2 d^2\right )+2 b^2 c e^3 (b d-5 a e)-4 c^3 d^2 e (3 b d-5 a e)+b^4 e^4+6 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+c e^5 \log (a+x (b+c x))}{2 c^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5/(a + b*x + c*x^2)^3,x]

[Out]

((b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*e^3*(-2*c*d^2*x + a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 + 5*
c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d
+ 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a^2*e^4*(3*d + e*x) - 10*a*c*d^2*e^2*(d + 3*e*x)))/((b^2 - 4*a*
c)*(a + x*(b + c*x))^2) + (-(b^6*e^5) + b^5*c*e^4*(5*d + 4*e*x) + b^4*c*e^3*(11*a*e^2 - 10*c*d*(d + e*x)) + 10
*b^3*c^2*e^2*(c*d^3 - a*e^2*(4*d + 3*e*x)) + 4*c^3*(8*a^3*e^5 + 3*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e
^3*(8*d + 5*e*x)) + 2*b*c^3*(3*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 5*a^2*e^4*(11*d + 5*e*x)) +
b^2*c^2*e*(-39*a^2*e^4 - 5*c^2*d^3*(3*d - 4*e*x) + 10*a*c*d*e^2*(5*d + 8*e*x)))/((b^2 - 4*a*c)^2*(a + x*(b + c
*x))) + (2*c*(2*c*d - b*e)*(6*c^4*d^4 + b^4*e^4 + 2*b^2*c*e^3*(b*d - 5*a*e) - 4*c^3*d^2*e*(3*b*d - 5*a*e) + 2*
c^2*e^2*(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)^(5/2) +
c*e^5*Log[a + x*(b + c*x)])/(2*c^4)

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Maple [B]  time = 0.168, size = 1444, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*x^2+b*x+a)^3,x)

[Out]

((25*a^2*b*c^2*e^5-50*a^2*c^3*d*e^4-15*a*b^3*c*e^5+40*a*b^2*c^2*d*e^4-30*a*b*c^3*d^2*e^3+20*a*c^4*d^3*e^2+2*b^
5*e^5-5*b^4*c*d*e^4+10*b^2*c^3*d^3*e^2-15*b*c^4*d^4*e+6*c^5*d^5)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*(32*a^
3*c^3*e^5+11*a^2*b^2*c^2*e^5+10*a^2*b*c^3*d*e^4-160*a^2*c^4*d^2*e^3-19*a*b^4*c*e^5+40*a*b^3*c^2*d*e^4-10*a*b^2
*c^3*d^2*e^3+60*a*b*c^4*d^3*e^2+3*b^6*e^5-5*b^5*c*d*e^4-10*b^4*c^2*d^2*e^3+30*b^3*c^3*d^3*e^2-45*b^2*c^4*d^4*e
+18*b*c^5*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x^2+(31*a^3*b*c^2*e^5-30*a^3*c^3*d*e^4-22*a^2*b^3*c*e^5+50*a^2*b
^2*c^2*d*e^4-50*a^2*b*c^3*d^2*e^3-20*a^2*c^4*d^3*e^2+3*a*b^5*e^5-5*a*b^4*c*d*e^4-10*a*b^3*c^2*d^2*e^3+50*a*b^2
*c^3*d^3*e^2-25*a*b*c^4*d^4*e+10*a*c^5*d^5-5*b^3*c^3*d^4*e+2*b^2*c^4*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x+1/2
/c^3*(24*a^4*c^2*e^5-21*a^3*b^2*c*e^5+50*a^3*b*c^2*d*e^4-80*a^3*c^3*d^2*e^3+3*a^2*b^4*e^5-5*a^2*b^3*c*d*e^4-10
*a^2*b^2*c^2*d^2*e^3+60*a^2*b*c^3*d^3*e^2-40*a^2*c^4*d^4*e-5*a*b^2*c^3*d^4*e+10*a*b*c^4*d^5-b^3*c^3*d^5)/(16*a
^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+8/c/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*a^2*e^5-4/c^2/(16*a^2*c^
2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*a*b^2*e^5+1/2/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*b^4*e^5-30/c/(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*e^5+60/(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))*d*a^2*e^4+10/c^2/(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*e^5-60/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*ar
ctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*d^2*e^3+40*c/(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))*d^3*a*e^2+20/(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*d^3*e^2-30*c/(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*d^4*
e+12*c^2/(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))*d^5-1/c^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

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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)^5/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.36605, size = 8014, normalized size = 20.65 \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)^5/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/2*((b^5*c^3 - 14*a*b^3*c^4 + 40*a^2*b*c^5)*d^5 + 5*(a*b^4*c^3 + 4*a^2*b^2*c^4 - 32*a^3*c^5)*d^4*e - 60*(a^
2*b^3*c^3 - 4*a^3*b*c^4)*d^3*e^2 + 10*(a^2*b^4*c^2 + 4*a^3*b^2*c^3 - 32*a^4*c^4)*d^2*e^3 + 5*(a^2*b^5*c - 14*a
^3*b^3*c^2 + 40*a^4*b*c^3)*d*e^4 - 3*(a^2*b^6 - 11*a^3*b^4*c + 36*a^4*b^2*c^2 - 32*a^5*c^3)*e^5 - 2*(6*(b^2*c^
6 - 4*a*c^7)*d^5 - 15*(b^3*c^5 - 4*a*b*c^6)*d^4*e + 10*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^3*e^2 - 30*(a*b^3
*c^4 - 4*a^2*b*c^5)*d^2*e^3 - 5*(b^6*c^2 - 12*a*b^4*c^3 + 42*a^2*b^2*c^4 - 40*a^3*c^5)*d*e^4 + (2*b^7*c - 23*a
*b^5*c^2 + 85*a^2*b^3*c^3 - 100*a^3*b*c^4)*e^5)*x^3 - (18*(b^3*c^5 - 4*a*b*c^6)*d^5 - 45*(b^4*c^4 - 4*a*b^2*c^
5)*d^4*e + 30*(b^5*c^3 - 2*a*b^3*c^4 - 8*a^2*b*c^5)*d^3*e^2 - 10*(b^6*c^2 - 3*a*b^4*c^3 + 12*a^2*b^2*c^4 - 64*
a^3*c^5)*d^2*e^3 - 5*(b^7*c - 12*a*b^5*c^2 + 30*a^2*b^3*c^3 + 8*a^3*b*c^4)*d*e^4 + (3*b^8 - 31*a*b^6*c + 87*a^
2*b^4*c^2 - 12*a^3*b^2*c^3 - 128*a^4*c^4)*e^5)*x^2 + (12*a^2*c^5*d^5 - 30*a^2*b*c^4*d^4*e - 60*a^3*b*c^3*d^2*e
^3 + 60*a^4*c^3*d*e^4 + 20*(a^2*b^2*c^3 + 2*a^3*c^4)*d^3*e^2 - (a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2)*e^5 + (
12*c^7*d^5 - 30*b*c^6*d^4*e - 60*a*b*c^5*d^2*e^3 + 60*a^2*c^5*d*e^4 + 20*(b^2*c^5 + 2*a*c^6)*d^3*e^2 - (b^5*c^
2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^5)*x^4 + 2*(12*b*c^6*d^5 - 30*b^2*c^5*d^4*e - 60*a*b^2*c^4*d^2*e^3 + 60*a^2
*b*c^4*d*e^4 + 20*(b^3*c^4 + 2*a*b*c^5)*d^3*e^2 - (b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*e^5)*x^3 + (12*(b^2*
c^5 + 2*a*c^6)*d^5 - 30*(b^3*c^4 + 2*a*b*c^5)*d^4*e + 20*(b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^3*e^2 - 60*(a*b
^3*c^3 + 2*a^2*b*c^4)*d^2*e^3 + 60*(a^2*b^2*c^3 + 2*a^3*c^4)*d*e^4 - (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^
3*b*c^3)*e^5)*x^2 + 2*(12*a*b*c^5*d^5 - 30*a*b^2*c^4*d^4*e - 60*a^2*b^2*c^3*d^2*e^3 + 60*a^3*b*c^3*d*e^4 + 20*
(a*b^3*c^3 + 2*a^2*b*c^4)*d^3*e^2 - (a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*e^5)*x)*sqrt(b^2 - 4*a*c)*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)) - 2*(2*(b^4*c^4 + a*b^2*c^5
- 20*a^2*c^6)*d^5 - 5*(b^5*c^3 + a*b^3*c^4 - 20*a^2*b*c^5)*d^4*e + 10*(5*a*b^4*c^3 - 22*a^2*b^2*c^4 + 8*a^3*c
^5)*d^3*e^2 - 10*(a*b^5*c^2 + a^2*b^3*c^3 - 20*a^3*b*c^4)*d^2*e^3 - 5*(a*b^6*c - 14*a^2*b^4*c^2 + 46*a^3*b^2*c
^3 - 24*a^4*c^4)*d*e^4 + (3*a*b^7 - 34*a^2*b^5*c + 119*a^3*b^3*c^2 - 124*a^4*b*c^3)*e^5)*x - ((b^6*c^2 - 12*a*
b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*e^5*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e^5*
x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*e^5*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 4
8*a^3*b^3*c^2 - 64*a^4*b*c^3)*e^5*x + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^5)*log(c*x^2 +
b*x + a))/(a^2*b^6*c^3 - 12*a^3*b^4*c^4 + 48*a^4*b^2*c^5 - 64*a^5*c^6 + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c
^7 - 64*a^3*c^8)*x^4 + 2*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*x^3 + (b^8*c^3 - 10*a*b^6*c^
4 + 24*a^2*b^4*c^5 + 32*a^3*b^2*c^6 - 128*a^4*c^7)*x^2 + 2*(a*b^7*c^3 - 12*a^2*b^5*c^4 + 48*a^3*b^3*c^5 - 64*a
^4*b*c^6)*x), -1/2*((b^5*c^3 - 14*a*b^3*c^4 + 40*a^2*b*c^5)*d^5 + 5*(a*b^4*c^3 + 4*a^2*b^2*c^4 - 32*a^3*c^5)*d
^4*e - 60*(a^2*b^3*c^3 - 4*a^3*b*c^4)*d^3*e^2 + 10*(a^2*b^4*c^2 + 4*a^3*b^2*c^3 - 32*a^4*c^4)*d^2*e^3 + 5*(a^2
*b^5*c - 14*a^3*b^3*c^2 + 40*a^4*b*c^3)*d*e^4 - 3*(a^2*b^6 - 11*a^3*b^4*c + 36*a^4*b^2*c^2 - 32*a^5*c^3)*e^5 -
2*(6*(b^2*c^6 - 4*a*c^7)*d^5 - 15*(b^3*c^5 - 4*a*b*c^6)*d^4*e + 10*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^3*e^
2 - 30*(a*b^3*c^4 - 4*a^2*b*c^5)*d^2*e^3 - 5*(b^6*c^2 - 12*a*b^4*c^3 + 42*a^2*b^2*c^4 - 40*a^3*c^5)*d*e^4 + (2
*b^7*c - 23*a*b^5*c^2 + 85*a^2*b^3*c^3 - 100*a^3*b*c^4)*e^5)*x^3 - (18*(b^3*c^5 - 4*a*b*c^6)*d^5 - 45*(b^4*c^4
- 4*a*b^2*c^5)*d^4*e + 30*(b^5*c^3 - 2*a*b^3*c^4 - 8*a^2*b*c^5)*d^3*e^2 - 10*(b^6*c^2 - 3*a*b^4*c^3 + 12*a^2*
b^2*c^4 - 64*a^3*c^5)*d^2*e^3 - 5*(b^7*c - 12*a*b^5*c^2 + 30*a^2*b^3*c^3 + 8*a^3*b*c^4)*d*e^4 + (3*b^8 - 31*a*
b^6*c + 87*a^2*b^4*c^2 - 12*a^3*b^2*c^3 - 128*a^4*c^4)*e^5)*x^2 + 2*(12*a^2*c^5*d^5 - 30*a^2*b*c^4*d^4*e - 60*
a^3*b*c^3*d^2*e^3 + 60*a^4*c^3*d*e^4 + 20*(a^2*b^2*c^3 + 2*a^3*c^4)*d^3*e^2 - (a^2*b^5 - 10*a^3*b^3*c + 30*a^4
*b*c^2)*e^5 + (12*c^7*d^5 - 30*b*c^6*d^4*e - 60*a*b*c^5*d^2*e^3 + 60*a^2*c^5*d*e^4 + 20*(b^2*c^5 + 2*a*c^6)*d^
3*e^2 - (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^5)*x^4 + 2*(12*b*c^6*d^5 - 30*b^2*c^5*d^4*e - 60*a*b^2*c^4*d
^2*e^3 + 60*a^2*b*c^4*d*e^4 + 20*(b^3*c^4 + 2*a*b*c^5)*d^3*e^2 - (b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*e^5)*
x^3 + (12*(b^2*c^5 + 2*a*c^6)*d^5 - 30*(b^3*c^4 + 2*a*b*c^5)*d^4*e + 20*(b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^
3*e^2 - 60*(a*b^3*c^3 + 2*a^2*b*c^4)*d^2*e^3 + 60*(a^2*b^2*c^3 + 2*a^3*c^4)*d*e^4 - (b^7 - 8*a*b^5*c + 10*a^2*
b^3*c^2 + 60*a^3*b*c^3)*e^5)*x^2 + 2*(12*a*b*c^5*d^5 - 30*a*b^2*c^4*d^4*e - 60*a^2*b^2*c^3*d^2*e^3 + 60*a^3*b*
c^3*d*e^4 + 20*(a*b^3*c^3 + 2*a^2*b*c^4)*d^3*e^2 - (a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*e^5)*x)*sqrt(-b^2 +
4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 2*(2*(b^4*c^4 + a*b^2*c^5 - 20*a^2*c^6)*d^5 -
5*(b^5*c^3 + a*b^3*c^4 - 20*a^2*b*c^5)*d^4*e + 10*(5*a*b^4*c^3 - 22*a^2*b^2*c^4 + 8*a^3*c^5)*d^3*e^2 - 10*(a*b
^5*c^2 + a^2*b^3*c^3 - 20*a^3*b*c^4)*d^2*e^3 - 5*(a*b^6*c - 14*a^2*b^4*c^2 + 46*a^3*b^2*c^3 - 24*a^4*c^4)*d*e^
4 + (3*a*b^7 - 34*a^2*b^5*c + 119*a^3*b^3*c^2 - 124*a^4*b*c^3)*e^5)*x - ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*
c^4 - 64*a^3*c^5)*e^5*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e^5*x^3 + (b^8 - 10*a*b^6
*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*e^5*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^
4*b*c^3)*e^5*x + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^5)*log(c*x^2 + b*x + a))/(a^2*b^6*c^
3 - 12*a^3*b^4*c^4 + 48*a^4*b^2*c^5 - 64*a^5*c^6 + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*x^4
+ 2*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*x^3 + (b^8*c^3 - 10*a*b^6*c^4 + 24*a^2*b^4*c^5 +
32*a^3*b^2*c^6 - 128*a^4*c^7)*x^2 + 2*(a*b^7*c^3 - 12*a^2*b^5*c^4 + 48*a^3*b^3*c^5 - 64*a^4*b*c^6)*x)]

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Sympy [B]  time = 138.706, size = 3403, normalized size = 8.77 \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)**5/(c*x**2+b*x+a)**3,x)

[Out]

(e**5/(2*c**3) - sqrt(-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 10*a*b**2*c*e**4 - 20*a*b*c**2*d*
e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 + 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d
**4)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b
**10)))*log(x + (-64*a**3*c**5*(e**5/(2*c**3) - sqrt(-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 10
*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 + 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**
2 - 12*b*c**3*d**3*e + 6*c**4*d**4)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a
**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 32*a**3*c**2*e**5 + 48*a**2*b**2*c**4*(e**5/(2*c**3) - sqrt(-(4*a*c -
b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 10*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b*
*4*e**4 + 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d**4)/(2*c**3*(1024*a**5*c**5 -
1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 9*a**2*b**2*c*e**5 -
30*a**2*b*c**2*d*e**4 - 12*a*b**4*c**3*(e**5/(2*c**3) - sqrt(-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e
**4 - 10*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 + 2*b**3*c*d*e**3 + 4*b**2*c**2*
d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d**4)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3
- 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + a*b**4*e**5 + 30*a*b**2*c**2*d**2*e**3 - 20*a*b*c**3*d**3*e**
2 + b**6*c**2*(e**5/(2*c**3) - sqrt(-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 10*a*b**2*c*e**4 -
20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 + 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3*d**
3*e + 6*c**4*d**4)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 2
0*a*b**8*c - b**10))) - 10*b**3*c**2*d**3*e**2 + 15*b**2*c**3*d**4*e - 6*b*c**4*d**5)/(30*a**2*b*c**2*e**5 - 6
0*a**2*c**3*d*e**4 - 10*a*b**3*c*e**5 + 60*a*b*c**3*d**2*e**3 - 40*a*c**4*d**3*e**2 + b**5*e**5 - 20*b**2*c**3
*d**3*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5)) + (e**5/(2*c**3) + sqrt(-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a
**2*c**2*e**4 - 10*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 + 2*b**3*c*d*e**3 + 4*
b**2*c**2*d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d**4)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3
*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)))*log(x + (-64*a**3*c**5*(e**5/(2*c**3) + sqrt(-(4*a*c
- b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 10*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b
**4*e**4 + 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d**4)/(2*c**3*(1024*a**5*c**5 -
1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + 32*a**3*c**2*e**5 +
48*a**2*b**2*c**4*(e**5/(2*c**3) + sqrt(-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 10*a*b**2*c*e**
4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 + 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3
*d**3*e + 6*c**4*d**4)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2
+ 20*a*b**8*c - b**10))) - 9*a**2*b**2*c*e**5 - 30*a**2*b*c**2*d*e**4 - 12*a*b**4*c**3*(e**5/(2*c**3) + sqrt(
-(4*a*c - b**2)**5)*(b*e - 2*c*d)*(30*a**2*c**2*e**4 - 10*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*
e**2 + b**4*e**4 + 2*b**3*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d**4)/(2*c**3*(1024*a**
5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) + a*b**4*e**5
+ 30*a*b**2*c**2*d**2*e**3 - 20*a*b*c**3*d**3*e**2 + b**6*c**2*(e**5/(2*c**3) + sqrt(-(4*a*c - b**2)**5)*(b*e
- 2*c*d)*(30*a**2*c**2*e**4 - 10*a*b**2*c*e**4 - 20*a*b*c**2*d*e**3 + 20*a*c**3*d**2*e**2 + b**4*e**4 + 2*b**3
*c*d*e**3 + 4*b**2*c**2*d**2*e**2 - 12*b*c**3*d**3*e + 6*c**4*d**4)/(2*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c
**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10))) - 10*b**3*c**2*d**3*e**2 + 15*b**2*c**3
*d**4*e - 6*b*c**4*d**5)/(30*a**2*b*c**2*e**5 - 60*a**2*c**3*d*e**4 - 10*a*b**3*c*e**5 + 60*a*b*c**3*d**2*e**3
- 40*a*c**4*d**3*e**2 + b**5*e**5 - 20*b**2*c**3*d**3*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5)) + (24*a**4*c**
2*e**5 - 21*a**3*b**2*c*e**5 + 50*a**3*b*c**2*d*e**4 - 80*a**3*c**3*d**2*e**3 + 3*a**2*b**4*e**5 - 5*a**2*b**3
*c*d*e**4 - 10*a**2*b**2*c**2*d**2*e**3 + 60*a**2*b*c**3*d**3*e**2 - 40*a**2*c**4*d**4*e - 5*a*b**2*c**3*d**4*
e + 10*a*b*c**4*d**5 - b**3*c**3*d**5 + x**3*(50*a**2*b*c**3*e**5 - 100*a**2*c**4*d*e**4 - 30*a*b**3*c**2*e**5
+ 80*a*b**2*c**3*d*e**4 - 60*a*b*c**4*d**2*e**3 + 40*a*c**5*d**3*e**2 + 4*b**5*c*e**5 - 10*b**4*c**2*d*e**4 +
20*b**2*c**4*d**3*e**2 - 30*b*c**5*d**4*e + 12*c**6*d**5) + x**2*(32*a**3*c**3*e**5 + 11*a**2*b**2*c**2*e**5
+ 10*a**2*b*c**3*d*e**4 - 160*a**2*c**4*d**2*e**3 - 19*a*b**4*c*e**5 + 40*a*b**3*c**2*d*e**4 - 10*a*b**2*c**3*
d**2*e**3 + 60*a*b*c**4*d**3*e**2 + 3*b**6*e**5 - 5*b**5*c*d*e**4 - 10*b**4*c**2*d**2*e**3 + 30*b**3*c**3*d**3
*e**2 - 45*b**2*c**4*d**4*e + 18*b*c**5*d**5) + x*(62*a**3*b*c**2*e**5 - 60*a**3*c**3*d*e**4 - 44*a**2*b**3*c*
e**5 + 100*a**2*b**2*c**2*d*e**4 - 100*a**2*b*c**3*d**2*e**3 - 40*a**2*c**4*d**3*e**2 + 6*a*b**5*e**5 - 10*a*b
**4*c*d*e**4 - 20*a*b**3*c**2*d**2*e**3 + 100*a*b**2*c**3*d**3*e**2 - 50*a*b*c**4*d**4*e + 20*a*c**5*d**5 - 10
*b**3*c**3*d**4*e + 4*b**2*c**4*d**5))/(32*a**4*c**5 - 16*a**3*b**2*c**4 + 2*a**2*b**4*c**3 + x**4*(32*a**2*c*
*7 - 16*a*b**2*c**6 + 2*b**4*c**5) + x**3*(64*a**2*b*c**6 - 32*a*b**3*c**5 + 4*b**5*c**4) + x**2*(64*a**3*c**6
- 12*a*b**4*c**4 + 2*b**6*c**3) + x*(64*a**3*b*c**5 - 32*a**2*b**3*c**4 + 4*a*b**5*c**3))

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Giac [B]  time = 1.13553, size = 1087, normalized size = 2.8 \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)^5/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

(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)/sqrt(-b^2 + 4*a*c))/((b^4*c^3 - 8*a*b^2*c^4 +
16*a^2*c^5)*sqrt(-b^2 + 4*a*c)) + 1/2*e^5*log(c*x^2 + b*x + a)/c^3 - 1/2*(b^3*c^3*d^5 - 10*a*b*c^4*d^5 + 5*a*
b^2*c^3*d^4*e + 40*a^2*c^4*d^4*e - 60*a^2*b*c^3*d^3*e^2 + 10*a^2*b^2*c^2*d^2*e^3 + 80*a^3*c^3*d^2*e^3 + 5*a^2*
b^3*c*d*e^4 - 50*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 - 5*b^4*c^2*d*e^4 + 40*a*b^2*c^3*d*e^4 - 5
0*a^2*c^4*d*e^4 + 2*b^5*c*e^5 - 15*a*b^3*c^2*e^5 + 25*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 - 10*b^4*c^2*d^2*e^3 - 10*a*b^2*c^3*d^2*e^3 - 160*a^2*c^4*d^2*e^3 - 5*
b^5*c*d*e^4 + 40*a*b^3*c^2*d*e^4 + 10*a^2*b*c^3*d*e^4 + 3*b^6*e^5 - 19*a*b^4*c*e^5 + 11*a^2*b^2*c^2*e^5 + 32*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 + 50*a*b^2*c^3*d^3*e^2
- 20*a^2*c^4*d^3*e^2 - 10*a*b^3*c^2*d^2*e^3 - 50*a^2*b*c^3*d^2*e^3 - 5*a*b^4*c*d*e^4 + 50*a^2*b^2*c^2*d*e^4 -
30*a^3*c^3*d*e^4 + 3*a*b^5*e^5 - 22*a^2*b^3*c*e^5 + 31*a^3*b*c^2*e^5)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*
c^3)