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

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

[Out]

-e^2*(-b*e+2*c*d)*(3*c^2*d^2-b^2*e^2-c*e*(-7*a*e+3*b*d))*x/c^2/(-4*a*c+b^2)^2-1/2*(e*x+d)^4*(b*d-2*a*e+(-b*e+2
*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2-1/2*(e*x+d)^2*(8*a*c*e*(2*a*e^2+c*d^2)-6*b*c*d*(3*a*e^2+c*d^2)+b^2*(-a*e
^3+7*c*d^2*e)-(-b*e+2*c*d)*(6*c^2*d^2-b^2*e^2-2*c*e*(-5*a*e+3*b*d))*x)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)-(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*(-4*a*e+3*b*d)+20*c^3*d*e^2*(3*a^2*e^2-3*a*b*d*e+b^2*
d^2))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(5/2)+1/2*e^5*ln(c*x^2+b*x+a)/c^3

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Rubi [A]  time = 1.11, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, 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 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 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 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]

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 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 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 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])

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*}

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Mathematica [A]  time = 1.07, size = 628, normalized size = 1.62 \[ \frac {\frac {2 c (2 c d-b e) \left (2 c^2 e^2 \left (15 a^2 e^2-10 a b d e+2 b^2 d^2\right )+2 b^2 c e^3 (b d-5 a e)-4 c^3 d^2 e (3 b d-5 a e)+b^4 e^4+6 c^4 d^4\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac {-2 b^2 c e^2 \left (2 a^2 e^3-5 a c d e (d+2 e x)+5 c^2 d^3 x\right )+b c^2 \left (5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)-c^2 d^4 (d-5 e x)\right )+2 c^2 \left (a^3 e^5-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)-c^3 d^5 x\right )+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left (a e (d+e x)-2 c d^2 x\right )+b^5 e^5 x}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {b^2 c^2 e \left (-39 a^2 e^4+10 a c d e^2 (5 d+8 e x)-5 c^2 d^3 (3 d-4 e x)\right )+2 b c^3 \left (5 a^2 e^4 (11 d+5 e x)+10 a c d^2 e^2 (d-3 e x)+3 c^2 d^4 (d-5 e x)\right )+4 c^3 \left (8 a^3 e^5-5 a^2 c d e^3 (8 d+5 e x)+10 a c^2 d^3 e^2 x+3 c^3 d^5 x\right )+b^4 c e^3 \left (11 a e^2-10 c d (d+e x)\right )+10 b^3 c^2 e^2 \left (c d^3-a e^2 (4 d+3 e x)\right )+b^6 \left (-e^5\right )+b^5 c e^4 (5 d+4 e x)}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+c e^5 \log (a+x (b+c x))}{2 c^4} \]

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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fricas [B]  time = 1.47, size = 3865, normalized size = 9.96 \[ \text {result too large to display} \]

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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giac [B]  time = 0.19, size = 805, normalized size = 2.07 \[ \frac {{\left (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}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {e^{5} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac {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 \, {\left (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} - 50 \, 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}\right )} x^{3} - {\left (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}\right )} x^{2} - 2 \, {\left (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}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{3}} \]

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)

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maple [B]  time = 0.09, size = 1444, normalized size = 3.72 \[ \text {result too large to display} \]

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))*d^2*a*b*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))*d^3*b^2*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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 3.03, size = 1486, normalized size = 3.83 \[ \frac {\mathrm {atan}\left (\frac {\left (\frac {x\,\left (b\,e-2\,c\,d\right )\,\left (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\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^5}+\frac {\left (b\,e-2\,c\,d\right )\,\left (16\,a^2\,b\,c^4-8\,a\,b^3\,c^3+b^5\,c^2\right )\,\left (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\right )}{2\,c^5\,{\left (4\,a\,c-b^2\right )}^5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (32\,a^2\,c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,b^4\,c^3\,{\left (4\,a\,c-b^2\right )}^{5/2}-16\,a\,b^2\,c^4\,{\left (4\,a\,c-b^2\right )}^{5/2}\right )}{-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}\right )\,\left (b\,e-2\,c\,d\right )\,\left (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\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-1024\,a^5\,c^5\,e^5+1280\,a^4\,b^2\,c^4\,e^5-640\,a^3\,b^4\,c^3\,e^5+160\,a^2\,b^6\,c^2\,e^5-20\,a\,b^8\,c\,e^5+b^{10}\,e^5\right )}{2\,\left (1024\,a^5\,c^8-1280\,a^4\,b^2\,c^7+640\,a^3\,b^4\,c^6-160\,a^2\,b^6\,c^5+20\,a\,b^8\,c^4-b^{10}\,c^3\right )}-\frac {\frac {-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}{2\,c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (-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\right )}{c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {x^2\,\left (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\right )}{2\,c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {x^3\,\left (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\right )}{c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((x*(b*e - 2*c*d)*(b^4*e^4 + 6*c^4*d^4 + 30*a^2*c^2*e^4 + 20*a*c^3*d^2*e^2 + 4*b^2*c^2*d^2*e^2 - 10*a*b
^2*c*e^4 - 12*b*c^3*d^3*e + 2*b^3*c*d*e^3 - 20*a*b*c^2*d*e^3))/(c^2*(4*a*c - b^2)^5) + ((b*e - 2*c*d)*(b^5*c^2
- 8*a*b^3*c^3 + 16*a^2*b*c^4)*(b^4*e^4 + 6*c^4*d^4 + 30*a^2*c^2*e^4 + 20*a*c^3*d^2*e^2 + 4*b^2*c^2*d^2*e^2 -
10*a*b^2*c*e^4 - 12*b*c^3*d^3*e + 2*b^3*c*d*e^3 - 20*a*b*c^2*d*e^3))/(2*c^5*(4*a*c - b^2)^5*(b^4 + 16*a^2*c^2
- 8*a*b^2*c)))*(32*a^2*c^5*(4*a*c - b^2)^(5/2) + 2*b^4*c^3*(4*a*c - b^2)^(5/2) - 16*a*b^2*c^4*(4*a*c - b^2)^(5
/2)))/(12*c^5*d^5 - b^5*e^5 - 30*a^2*b*c^2*e^5 + 40*a*c^4*d^3*e^2 + 60*a^2*c^3*d*e^4 + 20*b^2*c^3*d^3*e^2 + 10
*a*b^3*c*e^5 - 30*b*c^4*d^4*e - 60*a*b*c^3*d^2*e^3))*(b*e - 2*c*d)*(b^4*e^4 + 6*c^4*d^4 + 30*a^2*c^2*e^4 + 20*
a*c^3*d^2*e^2 + 4*b^2*c^2*d^2*e^2 - 10*a*b^2*c*e^4 - 12*b*c^3*d^3*e + 2*b^3*c*d*e^3 - 20*a*b*c^2*d*e^3))/(c^3*
(4*a*c - b^2)^(5/2)) - (log(a + b*x + c*x^2)*(b^10*e^5 - 1024*a^5*c^5*e^5 + 160*a^2*b^6*c^2*e^5 - 640*a^3*b^4*
c^3*e^5 + 1280*a^4*b^2*c^4*e^5 - 20*a*b^8*c*e^5))/(2*(1024*a^5*c^8 - b^10*c^3 + 20*a*b^8*c^4 - 160*a^2*b^6*c^5
+ 640*a^3*b^4*c^6 - 1280*a^4*b^2*c^7)) - ((b^3*c^3*d^5 - 24*a^4*c^2*e^5 - 3*a^2*b^4*e^5 + 21*a^3*b^2*c*e^5 +
40*a^2*c^4*d^4*e + 80*a^3*c^3*d^2*e^3 - 10*a*b*c^4*d^5 + 10*a^2*b^2*c^2*d^2*e^3 + 5*a*b^2*c^3*d^4*e + 5*a^2*b^
3*c*d*e^4 - 50*a^3*b*c^2*d*e^4 - 60*a^2*b*c^3*d^3*e^2)/(2*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x*(22*a^2*b^3
*c*e^5 - 10*a*c^5*d^5 - 2*b^2*c^4*d^5 - 3*a*b^5*e^5 - 31*a^3*b*c^2*e^5 + 30*a^3*c^3*d*e^4 + 5*b^3*c^3*d^4*e +
20*a^2*c^4*d^3*e^2 + 25*a*b*c^4*d^4*e + 5*a*b^4*c*d*e^4 - 50*a*b^2*c^3*d^3*e^2 + 10*a*b^3*c^2*d^2*e^3 + 50*a^2
*b*c^3*d^2*e^3 - 50*a^2*b^2*c^2*d*e^4))/(c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^2*(3*b^6*e^5 + 18*b*c^5*d^5
+ 32*a^3*c^3*e^5 - 45*b^2*c^4*d^4*e + 11*a^2*b^2*c^2*e^5 - 160*a^2*c^4*d^2*e^3 + 30*b^3*c^3*d^3*e^2 - 10*b^4*c
^2*d^2*e^3 - 19*a*b^4*c*e^5 - 5*b^5*c*d*e^4 + 60*a*b*c^4*d^3*e^2 + 40*a*b^3*c^2*d*e^4 + 10*a^2*b*c^3*d*e^4 - 1
0*a*b^2*c^3*d^2*e^3))/(2*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^3*(2*b^5*e^5 + 6*c^5*d^5 + 25*a^2*b*c^2*e^5
+ 20*a*c^4*d^3*e^2 - 50*a^2*c^3*d*e^4 + 10*b^2*c^3*d^3*e^2 - 15*a*b^3*c*e^5 - 15*b*c^4*d^4*e - 5*b^4*c*d*e^4 -
30*a*b*c^3*d^2*e^3 + 40*a*b^2*c^2*d*e^4))/(c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^2*(2*a*c + b^2) + a^2 + c^
2*x^4 + 2*a*b*x + 2*b*c*x^3)

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sympy [B]  time = 119.90, size = 3403, normalized size = 8.77 \[ \text {result too large to display} \]

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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