3.2222 \(\int \frac{(d+e x)^4}{(a+b x+c x^2)^5} \, dx\)

Optimal. Leaf size=545 \[ -\frac{-x \left (2 c^2 e^2 \left (-18 a^2 e^2-40 a b d e+305 b^2 d^2\right )-38 b^2 c e^3 (5 b d-a e)-40 c^3 d^2 e (21 b d-2 a e)+19 b^4 e^4+420 c^4 d^4\right )-10 b c \left (11 a^2 e^4+88 a c d^2 e^2+21 c^2 d^4\right )-5 b^3 \left (5 a e^4+19 c d^2 e^2\right )+4 b^2 c d e \left (83 a e^2+70 c d^2\right )+16 a c^2 d e \left (16 a e^2+35 c d^2\right )+6 b^4 d e^3}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac{(d+e x)^2 \left (-c x \left (-2 c e (35 b d-9 a e)+13 b^2 e^2+70 c^2 d^2\right )-b c \left (23 a e^2+35 c d^2\right )+28 a c^2 d e+28 b^2 c d e-3 b^3 e^2\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^3 \left (-6 a c e-2 b^2 e+7 c x (2 c d-b e)+7 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3} \]

[Out]

-((b + 2*c*x)*(d + e*x)^4)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^3*(7*b*c*d - 2*b^2*e - 6*a*c*e +
7*c*(2*c*d - b*e)*x))/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) + ((d + e*x)^2*(28*b^2*c*d*e + 28*a*c^2*d*e - 3
*b^3*e^2 - b*c*(35*c*d^2 + 23*a*e^2) - c*(70*c^2*d^2 + 13*b^2*e^2 - 2*c*e*(35*b*d - 9*a*e))*x))/(6*(b^2 - 4*a*
c)^3*(a + b*x + c*x^2)^2) - (6*b^4*d*e^3 + 16*a*c^2*d*e*(35*c*d^2 + 16*a*e^2) + 4*b^2*c*d*e*(70*c*d^2 + 83*a*e
^2) - 5*b^3*(19*c*d^2*e^2 + 5*a*e^4) - 10*b*c*(21*c^2*d^4 + 88*a*c*d^2*e^2 + 11*a^2*e^4) - (420*c^4*d^4 + 19*b
^4*e^4 - 40*c^3*d^2*e*(21*b*d - 2*a*e) - 38*b^2*c*e^3*(5*b*d - a*e) + 2*c^2*e^2*(305*b^2*d^2 - 40*a*b*d*e - 18
*a^2*e^2))*x)/(6*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) -
20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 -
4*a*c]])/(b^2 - 4*a*c)^(9/2)

________________________________________________________________________________________

Rubi [A]  time = 0.879508, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {736, 820, 777, 618, 206} \[ -\frac{-x \left (2 c^2 e^2 \left (-18 a^2 e^2-40 a b d e+305 b^2 d^2\right )-38 b^2 c e^3 (5 b d-a e)-40 c^3 d^2 e (21 b d-2 a e)+19 b^4 e^4+420 c^4 d^4\right )-10 b c \left (11 a^2 e^4+88 a c d^2 e^2+21 c^2 d^4\right )-5 b^3 \left (5 a e^4+19 c d^2 e^2\right )+4 b^2 c d e \left (83 a e^2+70 c d^2\right )+16 a c^2 d e \left (16 a e^2+35 c d^2\right )+6 b^4 d e^3}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac{(d+e x)^2 \left (-c x \left (-2 c e (35 b d-9 a e)+13 b^2 e^2+70 c^2 d^2\right )-b c \left (23 a e^2+35 c d^2\right )+28 a c^2 d e+28 b^2 c d e-3 b^3 e^2\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^3 \left (-6 a c e-2 b^2 e+7 c x (2 c d-b e)+7 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4/(a + b*x + c*x^2)^5,x]

[Out]

-((b + 2*c*x)*(d + e*x)^4)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^3*(7*b*c*d - 2*b^2*e - 6*a*c*e +
7*c*(2*c*d - b*e)*x))/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) + ((d + e*x)^2*(28*b^2*c*d*e + 28*a*c^2*d*e - 3
*b^3*e^2 - b*c*(35*c*d^2 + 23*a*e^2) - c*(70*c^2*d^2 + 13*b^2*e^2 - 2*c*e*(35*b*d - 9*a*e))*x))/(6*(b^2 - 4*a*
c)^3*(a + b*x + c*x^2)^2) - (6*b^4*d*e^3 + 16*a*c^2*d*e*(35*c*d^2 + 16*a*e^2) + 4*b^2*c*d*e*(70*c*d^2 + 83*a*e
^2) - 5*b^3*(19*c*d^2*e^2 + 5*a*e^4) - 10*b*c*(21*c^2*d^4 + 88*a*c*d^2*e^2 + 11*a^2*e^4) - (420*c^4*d^4 + 19*b
^4*e^4 - 40*c^3*d^2*e*(21*b*d - 2*a*e) - 38*b^2*c*e^3*(5*b*d - a*e) + 2*c^2*e^2*(305*b^2*d^2 - 40*a*b*d*e - 18
*a^2*e^2))*x)/(6*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) -
20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 -
4*a*c]])/(b^2 - 4*a*c)^(9/2)

Rule 736

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*(b + 2*
c*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
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*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
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], 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] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

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])

Rubi steps

\begin{align*} \int \frac{(d+e x)^4}{\left (a+b x+c x^2\right )^5} \, dx &=-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{\int \frac{(d+e x)^3 (-14 c d+4 b e-6 c e x)}{\left (a+b x+c x^2\right )^4} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{\int \frac{(d+e x)^2 \left (-4 \left (35 c^2 d^2+3 b^2 e^2-c e (28 b d-9 a e)\right )-28 c e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^3} \, dx}{12 \left (b^2-4 a c\right )^2}\\ &=-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (28 b^2 c d e+28 a c^2 d e-3 b^3 e^2-b c \left (35 c d^2+23 a e^2\right )-c \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{\int \frac{(d+e x) \left (-4 \left (210 c^3 d^3-6 b^3 e^3+b c e^2 (95 b d-46 a e)-10 c^2 d e (28 b d-11 a e)\right )-4 c e \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{24 \left (b^2-4 a c\right )^3}\\ &=-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (28 b^2 c d e+28 a c^2 d e-3 b^3 e^2-b c \left (35 c d^2+23 a e^2\right )-c \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{6 b^4 d e^3+16 a c^2 d e \left (35 c d^2+16 a e^2\right )+4 b^2 c d e \left (70 c d^2+83 a e^2\right )-5 b^3 \left (19 c d^2 e^2+5 a e^4\right )-10 b c \left (21 c^2 d^4+88 a c d^2 e^2+11 a^2 e^4\right )-\left (420 c^4 d^4+19 b^4 e^4-40 c^3 d^2 e (21 b d-2 a e)-38 b^2 c e^3 (5 b d-a e)+2 c^2 e^2 \left (305 b^2 d^2-40 a b d e-18 a^2 e^2\right )\right ) x}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4}\\ &=-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (28 b^2 c d e+28 a c^2 d e-3 b^3 e^2-b c \left (35 c d^2+23 a e^2\right )-c \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{6 b^4 d e^3+16 a c^2 d e \left (35 c d^2+16 a e^2\right )+4 b^2 c d e \left (70 c d^2+83 a e^2\right )-5 b^3 \left (19 c d^2 e^2+5 a e^4\right )-10 b c \left (21 c^2 d^4+88 a c d^2 e^2+11 a^2 e^4\right )-\left (420 c^4 d^4+19 b^4 e^4-40 c^3 d^2 e (21 b d-2 a e)-38 b^2 c e^3 (5 b d-a e)+2 c^2 e^2 \left (305 b^2 d^2-40 a b d e-18 a^2 e^2\right )\right ) x}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{\left (2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+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 )^4}\\ &=-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (28 b^2 c d e+28 a c^2 d e-3 b^3 e^2-b c \left (35 c d^2+23 a e^2\right )-c \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{6 b^4 d e^3+16 a c^2 d e \left (35 c d^2+16 a e^2\right )+4 b^2 c d e \left (70 c d^2+83 a e^2\right )-5 b^3 \left (19 c d^2 e^2+5 a e^4\right )-10 b c \left (21 c^2 d^4+88 a c d^2 e^2+11 a^2 e^4\right )-\left (420 c^4 d^4+19 b^4 e^4-40 c^3 d^2 e (21 b d-2 a e)-38 b^2 c e^3 (5 b d-a e)+2 c^2 e^2 \left (305 b^2 d^2-40 a b d e-18 a^2 e^2\right )\right ) x}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+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 )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.88643, size = 713, normalized size = 1.31 \[ \frac{1}{12} \left (\frac{6 (b+2 c x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{c \left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac{(b+2 c x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{c^2 \left (4 a c-b^2\right )^3 (a+x (b+c x))^2}+\frac{3 \left (b c \left (-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right )+2 c^2 \left (a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right )+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b^3 e^3 (a e-4 c d x)+b^4 e^4 x\right )}{c^3 \left (4 a c-b^2\right ) (a+x (b+c x))^4}+\frac{2 b c^2 \left (23 a^2 e^4+6 a c d e^2 (d-2 e x)+7 c^2 d^3 (d-4 e x)\right )-4 c^3 \left (a^2 e^3 (32 d+9 e x)-6 a c d^2 e^2 x-7 c^2 d^4 x\right )+4 b^2 c^2 e \left (a e^2 (13 d+6 e x)+c d^2 (9 e x-7 d)\right )+2 b^3 c e^2 \left (c d (9 d-4 e x)-10 a e^2\right )-2 b^4 c e^3 (6 d+e x)+3 b^5 e^4}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac{24 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4/(a + b*x + c*x^2)^5,x]

[Out]

(((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 -
10*a*b*d*e + a^2*e^2))*(b + 2*c*x))/(c^2*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))^2) + (6*(70*c^4*d^4 + b^4*e^4 - 4
*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(b
+ 2*c*x))/(c*(b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (3*(b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d
^2*x - 2*a*e*(d + e*x)) + b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*c^2*(c^2*d^4*x
+ a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x))))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (3*b^5*e^4 - 2
*b^4*c*e^3*(6*d + e*x) + 2*b^3*c*e^2*(-10*a*e^2 + c*d*(9*d - 4*e*x)) + 2*b*c^2*(23*a^2*e^4 + 7*c^2*d^3*(d - 4*
e*x) + 6*a*c*d*e^2*(d - 2*e*x)) + 4*b^2*c^2*e*(a*e^2*(13*d + 6*e*x) + c*d^2*(-7*d + 9*e*x)) - 4*c^3*(-7*c^2*d^
4*x - 6*a*c*d^2*e^2*x + a^2*e^3*(32*d + 9*e*x)))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) + (24*(70*c^4*d^4 +
b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a
^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12

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Maple [B]  time = 0.169, size = 2430, normalized size = 4.5 \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)^4/(c*x^2+b*x+a)^5,x)

[Out]

((6*a^2*c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140
*b*c^3*d^3*e+70*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*c^3*x^7+7/2*(6*a^2*c^2*e^
4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3*d^3*e+7
0*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*b*c^2*x^6+1/3*c*(11*a*c+13*b^2)*(6*a^2*
c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3*d
^3*e+70*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^5+5/12*b*(22*a*c+5*b^2)*(6*a^2*
c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3*d
^3*e+70*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^4-1/3*(66*a^4*c^3*e^4-450*a^3*b
^2*c^2*e^4+876*a^3*b*c^3*d*e^3-876*a^3*c^4*d^2*e^2-203*a^2*b^4*c*e^4+1504*a^2*b^3*c^2*d*e^3-2526*a^2*b^2*c^3*d
^2*e^2+2044*a^2*b*c^4*d^3*e-1022*a^2*c^5*d^4-37*a*b^6*e^4+440*a*b^5*c*d*e^3-1854*a*b^4*c^2*d^2*e^2+2828*a*b^3*
c^3*d^3*e-1414*a*b^2*c^4*d^4+12*b^7*d*e^3-54*b^6*c*d^2*e^2+84*b^5*c^2*d^3*e-42*b^4*c^3*d^4)/(256*a^4*c^4-256*a
^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^3+1/6*(314*a^4*b*c^2*e^4-1024*a^4*c^3*d*e^3+508*a^3*b^3*c*e^4-1604
*a^3*b^2*c^2*d*e^3+2628*a^3*b*c^3*d^2*e^2+129*a^2*b^5*e^4-1596*a^2*b^4*c*d*e^3+4278*a^2*b^3*c^2*d^2*e^2-6132*a
^2*b^2*c^3*d^3*e+3066*a^2*b*c^4*d^4-36*a*b^6*d*e^3+492*a*b^5*c*d^2*e^2-784*a*b^4*c^2*d^3*e+392*a*b^3*c^3*d^4-1
8*b^7*d^2*e^2+28*b^6*c*d^3*e-14*b^5*c^2*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^2-1
/3*(18*a^5*c^2*e^4-184*a^4*b^2*c*e^4+332*a^4*b*c^2*d*e^3+180*a^4*c^3*d^2*e^2-47*a^3*b^4*e^4+604*a^3*b^3*c*d*e^
3-1674*a^3*b^2*c^2*d^2*e^2+1116*a^3*b*c^3*d^3*e-558*a^3*c^4*d^4+12*a^2*b^5*d*e^3-168*a^2*b^4*c*d^2*e^2+696*a^2
*b^3*c^2*d^3*e-348*a^2*b^2*c^3*d^4+6*a*b^6*d^2*e^2-76*a*b^5*c*d^3*e+38*a*b^4*c^2*d^4+4*b^7*d^3*e-2*b^6*c*d^4)/
(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x+1/12*(220*a^5*b*c*e^4-512*a^5*c^2*d*e^3+50*a^4*b
^3*e^4-664*a^4*b^2*c*d*e^3+1944*a^4*b*c^2*d^2*e^2-1536*a^4*c^3*d^3*e-12*a^3*b^4*d*e^3+168*a^3*b^3*c*d^2*e^2-69
6*a^3*b^2*c^2*d^3*e+1116*a^3*b*c^3*d^4-6*a^2*b^5*d^2*e^2+76*a^2*b^4*c*d^3*e-326*a^2*b^3*c^2*d^4-4*a*b^6*d^3*e+
50*a*b^5*c*d^4-3*b^7*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8))/(c*x^2+b*x+a)^4+12/(256
*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*
c^2*a^2*e^4+24/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/
(4*a*c-b^2)^(1/2))*a*b^2*c*e^4-120/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/
2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c^2*d*e^3+120/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*
c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^3*a*d^2*e^2+2/(256*a^4*c^4-256*a^3*b^2*c^3+96*a
^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*e^4-40/(256*a^4*c^4-256*a
^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*c*d*e^3+18
0/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(
1/2))*b^2*c^2*d^2*e^2-280/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan
((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c^3*d^3*e+140/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a
*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^4*d^4

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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)^4/(c*x^2+b*x+a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.11802, size = 15663, normalized size = 28.74 \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)^4/(c*x^2+b*x+a)^5,x, algorithm="fricas")

[Out]

[1/12*(12*(70*(b^2*c^7 - 4*a*c^8)*d^4 - 140*(b^3*c^6 - 4*a*b*c^7)*d^3*e + 30*(3*b^4*c^5 - 10*a*b^2*c^6 - 8*a^2
*c^7)*d^2*e^2 - 20*(b^5*c^4 - a*b^3*c^5 - 12*a^2*b*c^6)*d*e^3 + (b^6*c^3 + 8*a*b^4*c^4 - 42*a^2*b^2*c^5 - 24*a
^3*c^6)*e^4)*x^7 + 42*(70*(b^3*c^6 - 4*a*b*c^7)*d^4 - 140*(b^4*c^5 - 4*a*b^2*c^6)*d^3*e + 30*(3*b^5*c^4 - 10*a
*b^3*c^5 - 8*a^2*b*c^6)*d^2*e^2 - 20*(b^6*c^3 - a*b^4*c^4 - 12*a^2*b^2*c^5)*d*e^3 + (b^7*c^2 + 8*a*b^5*c^3 - 4
2*a^2*b^3*c^4 - 24*a^3*b*c^5)*e^4)*x^6 + 4*(70*(13*b^4*c^5 - 41*a*b^2*c^6 - 44*a^2*c^7)*d^4 - 140*(13*b^5*c^4
- 41*a*b^3*c^5 - 44*a^2*b*c^6)*d^3*e + 30*(39*b^6*c^3 - 97*a*b^4*c^4 - 214*a^2*b^2*c^5 - 88*a^3*c^6)*d^2*e^2 -
20*(13*b^7*c^2 - 2*a*b^5*c^3 - 167*a^2*b^3*c^4 - 132*a^3*b*c^5)*d*e^3 + (13*b^8*c + 115*a*b^6*c^2 - 458*a^2*b
^4*c^3 - 774*a^3*b^2*c^4 - 264*a^4*c^5)*e^4)*x^5 - (3*b^9 - 62*a*b^7*c + 526*a^2*b^5*c^2 - 2420*a^3*b^3*c^3 +
4464*a^4*b*c^4)*d^4 - 4*(a*b^8 - 23*a^2*b^6*c + 250*a^3*b^4*c^2 - 312*a^4*b^2*c^3 - 1536*a^5*c^4)*d^3*e - 6*(a
^2*b^7 - 32*a^3*b^5*c - 212*a^4*b^3*c^2 + 1296*a^5*b*c^3)*d^2*e^2 - 4*(3*a^3*b^6 + 154*a^4*b^4*c - 536*a^5*b^2
*c^2 - 512*a^6*c^3)*d*e^3 + 10*(5*a^4*b^5 + 2*a^5*b^3*c - 88*a^6*b*c^2)*e^4 + 5*(70*(5*b^5*c^4 + 2*a*b^3*c^5 -
88*a^2*b*c^6)*d^4 - 140*(5*b^6*c^3 + 2*a*b^4*c^4 - 88*a^2*b^2*c^5)*d^3*e + 30*(15*b^7*c^2 + 16*a*b^5*c^3 - 26
0*a^2*b^3*c^4 - 176*a^3*b*c^5)*d^2*e^2 - 20*(5*b^8*c + 17*a*b^6*c^2 - 82*a^2*b^4*c^3 - 264*a^3*b^2*c^4)*d*e^3
+ (5*b^9 + 62*a*b^7*c - 34*a^2*b^5*c^2 - 1044*a^3*b^3*c^3 - 528*a^4*b*c^4)*e^4)*x^4 + 4*(14*(3*b^6*c^3 + 89*a*
b^4*c^4 - 331*a^2*b^2*c^5 - 292*a^3*c^6)*d^4 - 28*(3*b^7*c^2 + 89*a*b^5*c^3 - 331*a^2*b^3*c^4 - 292*a^3*b*c^5)
*d^3*e + 6*(9*b^8*c + 273*a*b^6*c^2 - 815*a^2*b^4*c^3 - 1538*a^3*b^2*c^4 - 584*a^4*c^5)*d^2*e^2 - 4*(3*b^9 + 9
8*a*b^7*c - 64*a^2*b^5*c^2 - 1285*a^3*b^3*c^3 - 876*a^4*b*c^4)*d*e^3 + (37*a*b^8 + 55*a^2*b^6*c - 362*a^3*b^4*
c^2 - 1866*a^4*b^2*c^3 + 264*a^5*c^4)*e^4)*x^3 - 2*(14*(b^7*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3*c^4 + 876*a^3*b*c
^5)*d^4 - 28*(b^8*c - 32*a*b^6*c^2 - 107*a^2*b^4*c^3 + 876*a^3*b^2*c^4)*d^3*e + 6*(3*b^9 - 94*a*b^7*c - 385*a^
2*b^5*c^2 + 2414*a^3*b^3*c^3 + 1752*a^4*b*c^4)*d^2*e^2 + 4*(9*a*b^8 + 363*a^2*b^6*c - 1195*a^3*b^4*c^2 - 1348*
a^4*b^2*c^3 - 1024*a^5*c^4)*d*e^3 - (129*a^2*b^7 - 8*a^3*b^5*c - 1718*a^4*b^3*c^2 - 1256*a^5*b*c^3)*e^4)*x^2 +
12*(70*a^4*c^4*d^4 - 140*a^4*b*c^3*d^3*e + (70*c^8*d^4 - 140*b*c^7*d^3*e + 30*(3*b^2*c^6 + 2*a*c^7)*d^2*e^2 -
20*(b^3*c^5 + 3*a*b*c^6)*d*e^3 + (b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*e^4)*x^8 + 4*(70*b*c^7*d^4 - 140*b^2*c^
6*d^3*e + 30*(3*b^3*c^5 + 2*a*b*c^6)*d^2*e^2 - 20*(b^4*c^4 + 3*a*b^2*c^5)*d*e^3 + (b^5*c^3 + 12*a*b^3*c^4 + 6*
a^2*b*c^5)*e^4)*x^7 + 2*(70*(3*b^2*c^6 + 2*a*c^7)*d^4 - 140*(3*b^3*c^5 + 2*a*b*c^6)*d^3*e + 30*(9*b^4*c^4 + 12
*a*b^2*c^5 + 4*a^2*c^6)*d^2*e^2 - 20*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d*e^3 + (3*b^6*c^2 + 38*a*b^4*c^
3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)*e^4)*x^6 + 4*(70*(b^3*c^5 + 3*a*b*c^6)*d^4 - 140*(b^4*c^4 + 3*a*b^2*c^5)*d^3*
e + 30*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d^2*e^2 - 20*(b^6*c^2 + 6*a*b^4*c^3 + 9*a^2*b^2*c^4)*d*e^3 + (
b^7*c + 15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*e^4)*x^5 + 30*(3*a^4*b^2*c^2 + 2*a^5*c^3)*d^2*e^2 - 20*(
a^4*b^3*c + 3*a^5*b*c^2)*d*e^3 + (a^4*b^4 + 12*a^5*b^2*c + 6*a^6*c^2)*e^4 + (70*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^
2*c^6)*d^4 - 140*(b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*d^3*e + 30*(3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4
+ 12*a^3*c^5)*d^2*e^2 - 20*(b^7*c + 15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*d*e^3 + (b^8 + 24*a*b^6*c +
156*a^2*b^4*c^2 + 144*a^3*b^2*c^3 + 36*a^4*c^4)*e^4)*x^4 + 4*(70*(a*b^3*c^4 + 3*a^2*b*c^5)*d^4 - 140*(a*b^4*c^
3 + 3*a^2*b^2*c^4)*d^3*e + 30*(3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*d^2*e^2 - 20*(a*b^6*c + 6*a^2*b^4*c
^2 + 9*a^3*b^2*c^3)*d*e^3 + (a*b^7 + 15*a^2*b^5*c + 42*a^3*b^3*c^2 + 18*a^4*b*c^3)*e^4)*x^3 + 2*(70*(3*a^2*b^2
*c^4 + 2*a^3*c^5)*d^4 - 140*(3*a^2*b^3*c^3 + 2*a^3*b*c^4)*d^3*e + 30*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c
^4)*d^2*e^2 - 20*(3*a^2*b^5*c + 11*a^3*b^3*c^2 + 6*a^4*b*c^3)*d*e^3 + (3*a^2*b^6 + 38*a^3*b^4*c + 42*a^4*b^2*c
^2 + 12*a^5*c^3)*e^4)*x^2 + 4*(70*a^3*b*c^4*d^4 - 140*a^3*b^2*c^3*d^3*e + 30*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d^2
*e^2 - 20*(a^3*b^4*c + 3*a^4*b^2*c^2)*d*e^3 + (a^3*b^5 + 12*a^4*b^3*c + 6*a^5*b*c^2)*e^4)*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)) + 4*(2*(b^8*c - 23
*a*b^6*c^2 + 250*a^2*b^4*c^3 - 417*a^3*b^2*c^4 - 1116*a^4*c^5)*d^4 - 4*(b^9 - 23*a*b^7*c + 250*a^2*b^5*c^2 - 4
17*a^3*b^3*c^3 - 1116*a^4*b*c^4)*d^3*e - 6*(a*b^8 - 32*a^2*b^6*c - 167*a^3*b^4*c^2 + 1146*a^4*b^2*c^3 - 120*a^
5*c^4)*d^2*e^2 - 4*(3*a^2*b^7 + 139*a^3*b^5*c - 521*a^4*b^3*c^2 - 332*a^5*b*c^3)*d*e^3 + (47*a^3*b^6 - 4*a^4*b
^4*c - 754*a^5*b^2*c^2 + 72*a^6*c^3)*e^4)*x)/(a^4*b^10 - 20*a^5*b^8*c + 160*a^6*b^6*c^2 - 640*a^7*b^4*c^3 + 12
80*a^8*b^2*c^4 - 1024*a^9*c^5 + (b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^
8 - 1024*a^5*c^9)*x^8 + 4*(b^11*c^3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 10
24*a^5*b*c^8)*x^7 + 2*(3*b^12*c^2 - 58*a*b^10*c^3 + 440*a^2*b^8*c^4 - 1600*a^3*b^6*c^5 + 2560*a^4*b^4*c^6 - 51
2*a^5*b^2*c^7 - 2048*a^6*c^8)*x^6 + 4*(b^13*c - 17*a*b^11*c^2 + 100*a^2*b^9*c^3 - 160*a^3*b^7*c^4 - 640*a^4*b^
5*c^5 + 2816*a^5*b^3*c^6 - 3072*a^6*b*c^7)*x^5 + (b^14 - 8*a*b^12*c - 74*a^2*b^10*c^2 + 1160*a^3*b^8*c^3 - 544
0*a^4*b^6*c^4 + 10496*a^5*b^4*c^5 - 4608*a^6*b^2*c^6 - 6144*a^7*c^7)*x^4 + 4*(a*b^13 - 17*a^2*b^11*c + 100*a^3
*b^9*c^2 - 160*a^4*b^7*c^3 - 640*a^5*b^5*c^4 + 2816*a^6*b^3*c^5 - 3072*a^7*b*c^6)*x^3 + 2*(3*a^2*b^12 - 58*a^3
*b^10*c + 440*a^4*b^8*c^2 - 1600*a^5*b^6*c^3 + 2560*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 2048*a^8*c^6)*x^2 + 4*(a^3
*b^11 - 20*a^4*b^9*c + 160*a^5*b^7*c^2 - 640*a^6*b^5*c^3 + 1280*a^7*b^3*c^4 - 1024*a^8*b*c^5)*x), 1/12*(12*(70
*(b^2*c^7 - 4*a*c^8)*d^4 - 140*(b^3*c^6 - 4*a*b*c^7)*d^3*e + 30*(3*b^4*c^5 - 10*a*b^2*c^6 - 8*a^2*c^7)*d^2*e^2
- 20*(b^5*c^4 - a*b^3*c^5 - 12*a^2*b*c^6)*d*e^3 + (b^6*c^3 + 8*a*b^4*c^4 - 42*a^2*b^2*c^5 - 24*a^3*c^6)*e^4)*
x^7 + 42*(70*(b^3*c^6 - 4*a*b*c^7)*d^4 - 140*(b^4*c^5 - 4*a*b^2*c^6)*d^3*e + 30*(3*b^5*c^4 - 10*a*b^3*c^5 - 8*
a^2*b*c^6)*d^2*e^2 - 20*(b^6*c^3 - a*b^4*c^4 - 12*a^2*b^2*c^5)*d*e^3 + (b^7*c^2 + 8*a*b^5*c^3 - 42*a^2*b^3*c^4
- 24*a^3*b*c^5)*e^4)*x^6 + 4*(70*(13*b^4*c^5 - 41*a*b^2*c^6 - 44*a^2*c^7)*d^4 - 140*(13*b^5*c^4 - 41*a*b^3*c^
5 - 44*a^2*b*c^6)*d^3*e + 30*(39*b^6*c^3 - 97*a*b^4*c^4 - 214*a^2*b^2*c^5 - 88*a^3*c^6)*d^2*e^2 - 20*(13*b^7*c
^2 - 2*a*b^5*c^3 - 167*a^2*b^3*c^4 - 132*a^3*b*c^5)*d*e^3 + (13*b^8*c + 115*a*b^6*c^2 - 458*a^2*b^4*c^3 - 774*
a^3*b^2*c^4 - 264*a^4*c^5)*e^4)*x^5 - (3*b^9 - 62*a*b^7*c + 526*a^2*b^5*c^2 - 2420*a^3*b^3*c^3 + 4464*a^4*b*c^
4)*d^4 - 4*(a*b^8 - 23*a^2*b^6*c + 250*a^3*b^4*c^2 - 312*a^4*b^2*c^3 - 1536*a^5*c^4)*d^3*e - 6*(a^2*b^7 - 32*a
^3*b^5*c - 212*a^4*b^3*c^2 + 1296*a^5*b*c^3)*d^2*e^2 - 4*(3*a^3*b^6 + 154*a^4*b^4*c - 536*a^5*b^2*c^2 - 512*a^
6*c^3)*d*e^3 + 10*(5*a^4*b^5 + 2*a^5*b^3*c - 88*a^6*b*c^2)*e^4 + 5*(70*(5*b^5*c^4 + 2*a*b^3*c^5 - 88*a^2*b*c^6
)*d^4 - 140*(5*b^6*c^3 + 2*a*b^4*c^4 - 88*a^2*b^2*c^5)*d^3*e + 30*(15*b^7*c^2 + 16*a*b^5*c^3 - 260*a^2*b^3*c^4
- 176*a^3*b*c^5)*d^2*e^2 - 20*(5*b^8*c + 17*a*b^6*c^2 - 82*a^2*b^4*c^3 - 264*a^3*b^2*c^4)*d*e^3 + (5*b^9 + 62
*a*b^7*c - 34*a^2*b^5*c^2 - 1044*a^3*b^3*c^3 - 528*a^4*b*c^4)*e^4)*x^4 + 4*(14*(3*b^6*c^3 + 89*a*b^4*c^4 - 331
*a^2*b^2*c^5 - 292*a^3*c^6)*d^4 - 28*(3*b^7*c^2 + 89*a*b^5*c^3 - 331*a^2*b^3*c^4 - 292*a^3*b*c^5)*d^3*e + 6*(9
*b^8*c + 273*a*b^6*c^2 - 815*a^2*b^4*c^3 - 1538*a^3*b^2*c^4 - 584*a^4*c^5)*d^2*e^2 - 4*(3*b^9 + 98*a*b^7*c - 6
4*a^2*b^5*c^2 - 1285*a^3*b^3*c^3 - 876*a^4*b*c^4)*d*e^3 + (37*a*b^8 + 55*a^2*b^6*c - 362*a^3*b^4*c^2 - 1866*a^
4*b^2*c^3 + 264*a^5*c^4)*e^4)*x^3 - 2*(14*(b^7*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3*c^4 + 876*a^3*b*c^5)*d^4 - 28*
(b^8*c - 32*a*b^6*c^2 - 107*a^2*b^4*c^3 + 876*a^3*b^2*c^4)*d^3*e + 6*(3*b^9 - 94*a*b^7*c - 385*a^2*b^5*c^2 + 2
414*a^3*b^3*c^3 + 1752*a^4*b*c^4)*d^2*e^2 + 4*(9*a*b^8 + 363*a^2*b^6*c - 1195*a^3*b^4*c^2 - 1348*a^4*b^2*c^3 -
1024*a^5*c^4)*d*e^3 - (129*a^2*b^7 - 8*a^3*b^5*c - 1718*a^4*b^3*c^2 - 1256*a^5*b*c^3)*e^4)*x^2 - 24*(70*a^4*c
^4*d^4 - 140*a^4*b*c^3*d^3*e + (70*c^8*d^4 - 140*b*c^7*d^3*e + 30*(3*b^2*c^6 + 2*a*c^7)*d^2*e^2 - 20*(b^3*c^5
+ 3*a*b*c^6)*d*e^3 + (b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*e^4)*x^8 + 4*(70*b*c^7*d^4 - 140*b^2*c^6*d^3*e + 30*
(3*b^3*c^5 + 2*a*b*c^6)*d^2*e^2 - 20*(b^4*c^4 + 3*a*b^2*c^5)*d*e^3 + (b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*e^
4)*x^7 + 2*(70*(3*b^2*c^6 + 2*a*c^7)*d^4 - 140*(3*b^3*c^5 + 2*a*b*c^6)*d^3*e + 30*(9*b^4*c^4 + 12*a*b^2*c^5 +
4*a^2*c^6)*d^2*e^2 - 20*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d*e^3 + (3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^
2*c^4 + 12*a^3*c^5)*e^4)*x^6 + 4*(70*(b^3*c^5 + 3*a*b*c^6)*d^4 - 140*(b^4*c^4 + 3*a*b^2*c^5)*d^3*e + 30*(3*b^5
*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d^2*e^2 - 20*(b^6*c^2 + 6*a*b^4*c^3 + 9*a^2*b^2*c^4)*d*e^3 + (b^7*c + 15*a*
b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*e^4)*x^5 + 30*(3*a^4*b^2*c^2 + 2*a^5*c^3)*d^2*e^2 - 20*(a^4*b^3*c + 3
*a^5*b*c^2)*d*e^3 + (a^4*b^4 + 12*a^5*b^2*c + 6*a^6*c^2)*e^4 + (70*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^4 -
140*(b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*d^3*e + 30*(3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)
*d^2*e^2 - 20*(b^7*c + 15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*d*e^3 + (b^8 + 24*a*b^6*c + 156*a^2*b^4*c
^2 + 144*a^3*b^2*c^3 + 36*a^4*c^4)*e^4)*x^4 + 4*(70*(a*b^3*c^4 + 3*a^2*b*c^5)*d^4 - 140*(a*b^4*c^3 + 3*a^2*b^2
*c^4)*d^3*e + 30*(3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*d^2*e^2 - 20*(a*b^6*c + 6*a^2*b^4*c^2 + 9*a^3*b^
2*c^3)*d*e^3 + (a*b^7 + 15*a^2*b^5*c + 42*a^3*b^3*c^2 + 18*a^4*b*c^3)*e^4)*x^3 + 2*(70*(3*a^2*b^2*c^4 + 2*a^3*
c^5)*d^4 - 140*(3*a^2*b^3*c^3 + 2*a^3*b*c^4)*d^3*e + 30*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)*d^2*e^2 -
20*(3*a^2*b^5*c + 11*a^3*b^3*c^2 + 6*a^4*b*c^3)*d*e^3 + (3*a^2*b^6 + 38*a^3*b^4*c + 42*a^4*b^2*c^2 + 12*a^5*c
^3)*e^4)*x^2 + 4*(70*a^3*b*c^4*d^4 - 140*a^3*b^2*c^3*d^3*e + 30*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d^2*e^2 - 20*(a^
3*b^4*c + 3*a^4*b^2*c^2)*d*e^3 + (a^3*b^5 + 12*a^4*b^3*c + 6*a^5*b*c^2)*e^4)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqr
t(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 4*(2*(b^8*c - 23*a*b^6*c^2 + 250*a^2*b^4*c^3 - 417*a^3*b^2*c^4 -
1116*a^4*c^5)*d^4 - 4*(b^9 - 23*a*b^7*c + 250*a^2*b^5*c^2 - 417*a^3*b^3*c^3 - 1116*a^4*b*c^4)*d^3*e - 6*(a*b^8
- 32*a^2*b^6*c - 167*a^3*b^4*c^2 + 1146*a^4*b^2*c^3 - 120*a^5*c^4)*d^2*e^2 - 4*(3*a^2*b^7 + 139*a^3*b^5*c - 5
21*a^4*b^3*c^2 - 332*a^5*b*c^3)*d*e^3 + (47*a^3*b^6 - 4*a^4*b^4*c - 754*a^5*b^2*c^2 + 72*a^6*c^3)*e^4)*x)/(a^4
*b^10 - 20*a^5*b^8*c + 160*a^6*b^6*c^2 - 640*a^7*b^4*c^3 + 1280*a^8*b^2*c^4 - 1024*a^9*c^5 + (b^10*c^4 - 20*a*
b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*x^8 + 4*(b^11*c^3 - 20*a*b^9*c^
4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024*a^5*b*c^8)*x^7 + 2*(3*b^12*c^2 - 58*a*b^10*c^3
+ 440*a^2*b^8*c^4 - 1600*a^3*b^6*c^5 + 2560*a^4*b^4*c^6 - 512*a^5*b^2*c^7 - 2048*a^6*c^8)*x^6 + 4*(b^13*c - 1
7*a*b^11*c^2 + 100*a^2*b^9*c^3 - 160*a^3*b^7*c^4 - 640*a^4*b^5*c^5 + 2816*a^5*b^3*c^6 - 3072*a^6*b*c^7)*x^5 +
(b^14 - 8*a*b^12*c - 74*a^2*b^10*c^2 + 1160*a^3*b^8*c^3 - 5440*a^4*b^6*c^4 + 10496*a^5*b^4*c^5 - 4608*a^6*b^2*
c^6 - 6144*a^7*c^7)*x^4 + 4*(a*b^13 - 17*a^2*b^11*c + 100*a^3*b^9*c^2 - 160*a^4*b^7*c^3 - 640*a^5*b^5*c^4 + 28
16*a^6*b^3*c^5 - 3072*a^7*b*c^6)*x^3 + 2*(3*a^2*b^12 - 58*a^3*b^10*c + 440*a^4*b^8*c^2 - 1600*a^5*b^6*c^3 + 25
60*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 2048*a^8*c^6)*x^2 + 4*(a^3*b^11 - 20*a^4*b^9*c + 160*a^5*b^7*c^2 - 640*a^6*
b^5*c^3 + 1280*a^7*b^3*c^4 - 1024*a^8*b*c^5)*x)]

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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((e*x+d)**4/(c*x**2+b*x+a)**5,x)

[Out]

Timed out

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Giac [B]  time = 1.17875, size = 2483, normalized size = 4.56 \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)^4/(c*x^2+b*x+a)^5,x, algorithm="giac")

[Out]

2*(70*c^4*d^4 - 140*b*c^3*d^3*e + 90*b^2*c^2*d^2*e^2 + 60*a*c^3*d^2*e^2 - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3 +
b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^8 - 16*a*b^6*c + 96*a^2*b
^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-b^2 + 4*a*c)) + 1/12*(840*c^7*d^4*x^7 - 1680*b*c^6*d^3*x^7*e + 2
940*b*c^6*d^4*x^6 + 1080*b^2*c^5*d^2*x^7*e^2 + 720*a*c^6*d^2*x^7*e^2 - 5880*b^2*c^5*d^3*x^6*e + 3640*b^2*c^5*d
^4*x^5 + 3080*a*c^6*d^4*x^5 - 240*b^3*c^4*d*x^7*e^3 - 720*a*b*c^5*d*x^7*e^3 + 3780*b^3*c^4*d^2*x^6*e^2 + 2520*
a*b*c^5*d^2*x^6*e^2 - 7280*b^3*c^4*d^3*x^5*e - 6160*a*b*c^5*d^3*x^5*e + 1750*b^3*c^4*d^4*x^4 + 7700*a*b*c^5*d^
4*x^4 + 12*b^4*c^3*x^7*e^4 + 144*a*b^2*c^4*x^7*e^4 + 72*a^2*c^5*x^7*e^4 - 840*b^4*c^3*d*x^6*e^3 - 2520*a*b^2*c
^4*d*x^6*e^3 + 4680*b^4*c^3*d^2*x^5*e^2 + 7080*a*b^2*c^4*d^2*x^5*e^2 + 2640*a^2*c^5*d^2*x^5*e^2 - 3500*b^4*c^3
*d^3*x^4*e - 15400*a*b^2*c^4*d^3*x^4*e + 168*b^4*c^3*d^4*x^3 + 5656*a*b^2*c^4*d^4*x^3 + 4088*a^2*c^5*d^4*x^3 +
42*b^5*c^2*x^6*e^4 + 504*a*b^3*c^3*x^6*e^4 + 252*a^2*b*c^4*x^6*e^4 - 1040*b^5*c^2*d*x^5*e^3 - 4000*a*b^3*c^3*
d*x^5*e^3 - 2640*a^2*b*c^4*d*x^5*e^3 + 2250*b^5*c^2*d^2*x^4*e^2 + 11400*a*b^3*c^3*d^2*x^4*e^2 + 6600*a^2*b*c^4
*d^2*x^4*e^2 - 336*b^5*c^2*d^3*x^3*e - 11312*a*b^3*c^3*d^3*x^3*e - 8176*a^2*b*c^4*d^3*x^3*e - 28*b^5*c^2*d^4*x
^2 + 784*a*b^3*c^3*d^4*x^2 + 6132*a^2*b*c^4*d^4*x^2 + 52*b^6*c*x^5*e^4 + 668*a*b^4*c^2*x^5*e^4 + 840*a^2*b^2*c
^3*x^5*e^4 + 264*a^3*c^4*x^5*e^4 - 500*b^6*c*d*x^4*e^3 - 3700*a*b^4*c^2*d*x^4*e^3 - 6600*a^2*b^2*c^3*d*x^4*e^3
+ 216*b^6*c*d^2*x^3*e^2 + 7416*a*b^4*c^2*d^2*x^3*e^2 + 10104*a^2*b^2*c^3*d^2*x^3*e^2 + 3504*a^3*c^4*d^2*x^3*e
^2 + 56*b^6*c*d^3*x^2*e - 1568*a*b^4*c^2*d^3*x^2*e - 12264*a^2*b^2*c^3*d^3*x^2*e + 8*b^6*c*d^4*x - 152*a*b^4*c
^2*d^4*x + 1392*a^2*b^2*c^3*d^4*x + 2232*a^3*c^4*d^4*x + 25*b^7*x^4*e^4 + 410*a*b^5*c*x^4*e^4 + 1470*a^2*b^3*c
^2*x^4*e^4 + 660*a^3*b*c^3*x^4*e^4 - 48*b^7*d*x^3*e^3 - 1760*a*b^5*c*d*x^3*e^3 - 6016*a^2*b^3*c^2*d*x^3*e^3 -
3504*a^3*b*c^3*d*x^3*e^3 - 36*b^7*d^2*x^2*e^2 + 984*a*b^5*c*d^2*x^2*e^2 + 8556*a^2*b^3*c^2*d^2*x^2*e^2 + 5256*
a^3*b*c^3*d^2*x^2*e^2 - 16*b^7*d^3*x*e + 304*a*b^5*c*d^3*x*e - 2784*a^2*b^3*c^2*d^3*x*e - 4464*a^3*b*c^3*d^3*x
*e - 3*b^7*d^4 + 50*a*b^5*c*d^4 - 326*a^2*b^3*c^2*d^4 + 1116*a^3*b*c^3*d^4 + 148*a*b^6*x^3*e^4 + 812*a^2*b^4*c
*x^3*e^4 + 1800*a^3*b^2*c^2*x^3*e^4 - 264*a^4*c^3*x^3*e^4 - 72*a*b^6*d*x^2*e^3 - 3192*a^2*b^4*c*d*x^2*e^3 - 32
08*a^3*b^2*c^2*d*x^2*e^3 - 2048*a^4*c^3*d*x^2*e^3 - 24*a*b^6*d^2*x*e^2 + 672*a^2*b^4*c*d^2*x*e^2 + 6696*a^3*b^
2*c^2*d^2*x*e^2 - 720*a^4*c^3*d^2*x*e^2 - 4*a*b^6*d^3*e + 76*a^2*b^4*c*d^3*e - 696*a^3*b^2*c^2*d^3*e - 1536*a^
4*c^3*d^3*e + 258*a^2*b^5*x^2*e^4 + 1016*a^3*b^3*c*x^2*e^4 + 628*a^4*b*c^2*x^2*e^4 - 48*a^2*b^5*d*x*e^3 - 2416
*a^3*b^3*c*d*x*e^3 - 1328*a^4*b*c^2*d*x*e^3 - 6*a^2*b^5*d^2*e^2 + 168*a^3*b^3*c*d^2*e^2 + 1944*a^4*b*c^2*d^2*e
^2 + 188*a^3*b^4*x*e^4 + 736*a^4*b^2*c*x*e^4 - 72*a^5*c^2*x*e^4 - 12*a^3*b^4*d*e^3 - 664*a^4*b^2*c*d*e^3 - 512
*a^5*c^2*d*e^3 + 50*a^4*b^3*e^4 + 220*a^5*b*c*e^4)/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256
*a^4*c^4)*(c*x^2 + b*x + a)^4)