3.2371 \(\int \frac{(d+e x)^4 (f+g x)}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=543 \[ \frac{(d+e x) \left (x \left (-c^2 e \left (16 a^2 e^2 g+2 a b e (11 d g+3 e f)+b^2 (-d) (5 d g+6 e f)\right )+b^2 c e^2 g (15 a e+b d)-2 c^3 d (3 b d (d g+3 e f)-2 a e (4 d g+3 e f))-2 b^4 e^3 g+12 c^4 d^3 f\right )+2 b c \left (7 a^2 e^3 g+a c d e (7 d g+9 e f)+3 c^2 d^3 f\right )+b^3 e g \left (c d^2-2 a e^2\right )-b^2 c d \left (a e^2 g+3 c d (d g+2 e f)\right )-4 a c^2 e \left (a e (8 d g+3 e f)+3 c d^2 f\right )\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (4 c^3 e \left (3 a^2 e^2 (4 d g+e f)-3 a b d e (3 d g+2 e f)+b^2 d^2 (2 d g+3 e f)\right )-30 a^2 b c^2 e^4 g+10 a b^3 c e^4 g-2 c^4 d^2 (3 b d (d g+4 e f)-4 a e (2 d g+3 e f))+b^5 \left (-e^4\right ) g+12 c^5 d^4 f\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{(d+e x)^3 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{e^4 g \log \left (a+b x+c x^2\right )}{2 c^3} \]

[Out]

((d + e*x)^3*(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))/
(2*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + ((d + e*x)*(b^3*e*(c*d^2 - 2*a*e^2)*g - b^2*c*d*(a*e^2*g + 3*c*d*(2*
e*f + d*g)) + 2*b*c*(3*c^2*d^3*f + 7*a^2*e^3*g + a*c*d*e*(9*e*f + 7*d*g)) - 4*a*c^2*e*(3*c*d^2*f + a*e*(3*e*f
+ 8*d*g)) + (12*c^4*d^3*f - 2*b^4*e^3*g + b^2*c*e^2*(b*d + 15*a*e)*g - 2*c^3*d*(3*b*d*(3*e*f + d*g) - 2*a*e*(3
*e*f + 4*d*g)) - c^2*e*(16*a^2*e^2*g - b^2*d*(6*e*f + 5*d*g) + 2*a*b*e*(3*e*f + 11*d*g)))*x))/(2*c^2*(b^2 - 4*
a*c)^2*(a + b*x + c*x^2)) - ((12*c^5*d^4*f - b^5*e^4*g + 10*a*b^3*c*e^4*g - 30*a^2*b*c^2*e^4*g - 2*c^4*d^2*(3*
b*d*(4*e*f + d*g) - 4*a*e*(3*e*f + 2*d*g)) + 4*c^3*e*(b^2*d^2*(3*e*f + 2*d*g) - 3*a*b*d*e*(2*e*f + 3*d*g) + 3*
a^2*e^2*(e*f + 4*d*g)))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(5/2)) + (e^4*g*Log[a + b*x
+ c*x^2])/(2*c^3)

________________________________________________________________________________________

Rubi [A]  time = 1.21391, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {818, 634, 618, 206, 628} \[ \frac{(d+e x) \left (x \left (-c^2 e \left (16 a^2 e^2 g+2 a b e (11 d g+3 e f)+b^2 (-d) (5 d g+6 e f)\right )+b^2 c e^2 g (15 a e+b d)-2 c^3 d (3 b d (d g+3 e f)-2 a e (4 d g+3 e f))-2 b^4 e^3 g+12 c^4 d^3 f\right )+2 b c \left (7 a^2 e^3 g+a c d e (7 d g+9 e f)+3 c^2 d^3 f\right )+b^3 e g \left (c d^2-2 a e^2\right )-b^2 c d \left (a e^2 g+3 c d (d g+2 e f)\right )-4 a c^2 e \left (a e (8 d g+3 e f)+3 c d^2 f\right )\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (4 c^3 e \left (3 a^2 e^2 (4 d g+e f)-3 a b d e (3 d g+2 e f)+b^2 d^2 (2 d g+3 e f)\right )-30 a^2 b c^2 e^4 g+10 a b^3 c e^4 g-2 c^4 d^2 (3 b d (d g+4 e f)-4 a e (2 d g+3 e f))+b^5 \left (-e^4\right ) g+12 c^5 d^4 f\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{(d+e x)^3 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{e^4 g \log \left (a+b x+c x^2\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^4*(f + g*x))/(a + b*x + c*x^2)^3,x]

[Out]

((d + e*x)^3*(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))/
(2*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + ((d + e*x)*(b^3*e*(c*d^2 - 2*a*e^2)*g - b^2*c*d*(a*e^2*g + 3*c*d*(2*
e*f + d*g)) + 2*b*c*(3*c^2*d^3*f + 7*a^2*e^3*g + a*c*d*e*(9*e*f + 7*d*g)) - 4*a*c^2*e*(3*c*d^2*f + a*e*(3*e*f
+ 8*d*g)) + (12*c^4*d^3*f - 2*b^4*e^3*g + b^2*c*e^2*(b*d + 15*a*e)*g - 2*c^3*d*(3*b*d*(3*e*f + d*g) - 2*a*e*(3
*e*f + 4*d*g)) - c^2*e*(16*a^2*e^2*g - b^2*d*(6*e*f + 5*d*g) + 2*a*b*e*(3*e*f + 11*d*g)))*x))/(2*c^2*(b^2 - 4*
a*c)^2*(a + b*x + c*x^2)) - ((12*c^5*d^4*f - b^5*e^4*g + 10*a*b^3*c*e^4*g - 30*a^2*b*c^2*e^4*g - 2*c^4*d^2*(3*
b*d*(4*e*f + d*g) - 4*a*e*(3*e*f + 2*d*g)) + 4*c^3*e*(b^2*d^2*(3*e*f + 2*d*g) - 3*a*b*d*e*(2*e*f + 3*d*g) + 3*
a^2*e^2*(e*f + 4*d*g)))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(5/2)) + (e^4*g*Log[a + b*x
+ c*x^2])/(2*c^3)

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 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)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx &=\frac{(d+e x)^3 \left (2 a c (e f+d g)-b (c d f+a e g)-\left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) x\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\int \frac{(d+e x)^2 \left (-6 c^2 d^2 f-b e (b d-3 a e) g+3 b c d (2 e f+d g)-2 a c e (3 e f+4 d g)+2 \left (b^2-4 a c\right ) e^2 g x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{2 c \left (b^2-4 a c\right )}\\ &=\frac{(d+e x)^3 \left (2 a c (e f+d g)-b (c d f+a e g)-\left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) x\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{(d+e x) \left (b^3 e \left (c d^2-2 a e^2\right ) g-b^2 c d \left (a e^2 g+3 c d (2 e f+d g)\right )+2 b c \left (3 c^2 d^3 f+7 a^2 e^3 g+a c d e (9 e f+7 d g)\right )-4 a c^2 e \left (3 c d^2 f+a e (3 e f+8 d g)\right )+\left (12 c^4 d^3 f-2 b^4 e^3 g+b^2 c e^2 (b d+15 a e) g-2 c^3 d (3 b d (3 e f+d g)-2 a e (3 e f+4 d g))-c^2 e \left (16 a^2 e^2 g-b^2 d (6 e f+5 d g)+2 a b e (3 e f+11 d g)\right )\right ) x\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{\int \frac{2 \left (6 c^4 d^4 f+a b^3 e^4 g-7 a^2 b c e^4 g-c^3 d^2 (3 b d (4 e f+d g)-4 a e (3 e f+2 d g))+2 c^2 e \left (b^2 d^2 (3 e f+2 d g)-3 a b d e (2 e f+3 d g)+3 a^2 e^2 (e f+4 d g)\right )\right )+2 \left (b^2-4 a c\right )^2 e^4 g x}{a+b x+c x^2} \, dx}{2 c^2 \left (b^2-4 a c\right )^2}\\ &=\frac{(d+e x)^3 \left (2 a c (e f+d g)-b (c d f+a e g)-\left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) x\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{(d+e x) \left (b^3 e \left (c d^2-2 a e^2\right ) g-b^2 c d \left (a e^2 g+3 c d (2 e f+d g)\right )+2 b c \left (3 c^2 d^3 f+7 a^2 e^3 g+a c d e (9 e f+7 d g)\right )-4 a c^2 e \left (3 c d^2 f+a e (3 e f+8 d g)\right )+\left (12 c^4 d^3 f-2 b^4 e^3 g+b^2 c e^2 (b d+15 a e) g-2 c^3 d (3 b d (3 e f+d g)-2 a e (3 e f+4 d g))-c^2 e \left (16 a^2 e^2 g-b^2 d (6 e f+5 d g)+2 a b e (3 e f+11 d g)\right )\right ) x\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{\left (e^4 g\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^3}+\frac{\left (12 c^5 d^4 f-b^5 e^4 g+10 a b^3 c e^4 g-30 a^2 b c^2 e^4 g-2 c^4 d^2 (3 b d (4 e f+d g)-4 a e (3 e f+2 d g))+4 c^3 e \left (b^2 d^2 (3 e f+2 d g)-3 a b d e (2 e f+3 d g)+3 a^2 e^2 (e f+4 d g)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^3 \left (b^2-4 a c\right )^2}\\ &=\frac{(d+e x)^3 \left (2 a c (e f+d g)-b (c d f+a e g)-\left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) x\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{(d+e x) \left (b^3 e \left (c d^2-2 a e^2\right ) g-b^2 c d \left (a e^2 g+3 c d (2 e f+d g)\right )+2 b c \left (3 c^2 d^3 f+7 a^2 e^3 g+a c d e (9 e f+7 d g)\right )-4 a c^2 e \left (3 c d^2 f+a e (3 e f+8 d g)\right )+\left (12 c^4 d^3 f-2 b^4 e^3 g+b^2 c e^2 (b d+15 a e) g-2 c^3 d (3 b d (3 e f+d g)-2 a e (3 e f+4 d g))-c^2 e \left (16 a^2 e^2 g-b^2 d (6 e f+5 d g)+2 a b e (3 e f+11 d g)\right )\right ) x\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{e^4 g \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{\left (12 c^5 d^4 f-b^5 e^4 g+10 a b^3 c e^4 g-30 a^2 b c^2 e^4 g-2 c^4 d^2 (3 b d (4 e f+d g)-4 a e (3 e f+2 d g))+4 c^3 e \left (b^2 d^2 (3 e f+2 d g)-3 a b d e (2 e f+3 d g)+3 a^2 e^2 (e f+4 d g)\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{(d+e x)^3 \left (2 a c (e f+d g)-b (c d f+a e g)-\left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) x\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{(d+e x) \left (b^3 e \left (c d^2-2 a e^2\right ) g-b^2 c d \left (a e^2 g+3 c d (2 e f+d g)\right )+2 b c \left (3 c^2 d^3 f+7 a^2 e^3 g+a c d e (9 e f+7 d g)\right )-4 a c^2 e \left (3 c d^2 f+a e (3 e f+8 d g)\right )+\left (12 c^4 d^3 f-2 b^4 e^3 g+b^2 c e^2 (b d+15 a e) g-2 c^3 d (3 b d (3 e f+d g)-2 a e (3 e f+4 d g))-c^2 e \left (16 a^2 e^2 g-b^2 d (6 e f+5 d g)+2 a b e (3 e f+11 d g)\right )\right ) x\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (12 c^5 d^4 f-b^5 e^4 g+10 a b^3 c e^4 g-30 a^2 b c^2 e^4 g-2 c^4 d^2 (3 b d (4 e f+d g)-4 a e (3 e f+2 d g))+4 c^3 e \left (b^2 d^2 (3 e f+2 d g)-3 a b d e (2 e f+3 d g)+3 a^2 e^2 (e f+4 d g)\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^4 g \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end{align*}

Mathematica [A]  time = 2.35471, size = 897, normalized size = 1.65 \[ \frac{c g \log (a+x (b+c x)) e^4+\frac{e^4 g x b^5+e^3 (a e g-c (e f+4 d g) x) b^4-c e^2 (a e (e f+4 d g+5 e g x)-2 c d (2 e f+3 d g) x) b^3+2 c e \left (-2 a^2 g e^3+a c \left (3 g d^2+2 e (f+4 g x) d+2 e^2 f x\right ) e-c^2 d^2 (3 e f+2 d g) x\right ) b^2+c^2 \left (c^2 (-d f+4 e x f+d g x) d^3-2 a c e \left (2 g d^2+3 e (f+3 g x) d+6 e^2 f x\right ) d+a^2 e^3 (3 e f+12 d g+5 e g x)\right ) b+2 c^2 \left (-c^3 f x d^4+a c^2 \left (g d^2+4 e (f+g x) d+6 e^2 f x\right ) d^2+a^3 e^4 g-a^2 c e^2 \left (6 g d^2+4 e (f+g x) d+e^2 f x\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{-e^4 g b^6+c e^3 (4 d g+e (f+4 g x)) b^5-c e^2 \left (2 c \left (3 g d^2+2 e (f+2 g x) d+e^2 f x\right )-11 a e^2 g\right ) b^4+2 c^2 e \left (c d^2 (3 e f+2 d g)-a e^2 (4 e f+16 d g+15 e g x)\right ) b^3+c^2 \left (-39 a^2 g e^4+2 a c \left (15 g d^2+10 e f d+32 e g x d+8 e^2 f x\right ) e^2+c^2 d^2 \left (-3 g d^2-12 e f d+8 e g x d+12 e^2 f x\right )\right ) b^2+2 c^3 \left (3 c^2 (d (f-g x)-4 e f x) d^3+2 a c e \left (2 g d^2+3 e (f-3 g x) d-6 e^2 f x\right ) d+a^2 e^3 (11 e f+44 d g+25 e g x)\right ) b+4 c^3 \left (3 c^3 f x d^4+2 a c^2 e (3 e f+2 d g) x d^2+8 a^3 e^4 g-a^2 c e^2 \left (24 g d^2+4 e (4 f+5 g x) d+5 e^2 f x\right )\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{2 c \left (-e^4 g b^5+10 a c e^4 g b^3-30 a^2 c^2 e^4 g b+12 c^5 d^4 f+2 c^4 d^2 (4 a e (3 e f+2 d g)-3 b d (4 e f+d g))+4 c^3 e \left (b^2 (3 e f+2 d g) d^2-3 a b e (2 e f+3 d g) d+3 a^2 e^2 (e f+4 d g)\right )\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}}{2 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^4*(f + g*x))/(a + b*x + c*x^2)^3,x]

[Out]

((b^5*e^4*g*x + b^4*e^3*(a*e*g - c*(e*f + 4*d*g)*x) - b^3*c*e^2*(-2*c*d*(2*e*f + 3*d*g)*x + a*e*(e*f + 4*d*g +
5*e*g*x)) + 2*c^2*(a^3*e^4*g - c^3*d^4*f*x - a^2*c*e^2*(6*d^2*g + e^2*f*x + 4*d*e*(f + g*x)) + a*c^2*d^2*(d^2
*g + 6*e^2*f*x + 4*d*e*(f + g*x))) + b*c^2*(c^2*d^3*(-(d*f) + 4*e*f*x + d*g*x) + a^2*e^3*(3*e*f + 12*d*g + 5*e
*g*x) - 2*a*c*d*e*(2*d^2*g + 6*e^2*f*x + 3*d*e*(f + 3*g*x))) + 2*b^2*c*e*(-2*a^2*e^3*g - c^2*d^2*(3*e*f + 2*d*
g)*x + a*c*e*(3*d^2*g + 2*e^2*f*x + 2*d*e*(f + 4*g*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (-(b^6*e^4*g) +
b^5*c*e^3*(4*d*g + e*(f + 4*g*x)) + 2*b^3*c^2*e*(c*d^2*(3*e*f + 2*d*g) - a*e^2*(4*e*f + 16*d*g + 15*e*g*x)) +
b^2*c^2*(-39*a^2*e^4*g + c^2*d^2*(-12*d*e*f - 3*d^2*g + 12*e^2*f*x + 8*d*e*g*x) + 2*a*c*e^2*(10*d*e*f + 15*d^
2*g + 8*e^2*f*x + 32*d*e*g*x)) + 2*b*c^3*(a^2*e^3*(11*e*f + 44*d*g + 25*e*g*x) + 2*a*c*d*e*(2*d^2*g - 6*e^2*f*
x + 3*d*e*(f - 3*g*x)) + 3*c^2*d^3*(-4*e*f*x + d*(f - g*x))) - b^4*c*e^2*(-11*a*e^2*g + 2*c*(3*d^2*g + e^2*f*x
+ 2*d*e*(f + 2*g*x))) + 4*c^3*(8*a^3*e^4*g + 3*c^3*d^4*f*x + 2*a*c^2*d^2*e*(3*e*f + 2*d*g)*x - a^2*c*e^2*(24*
d^2*g + 5*e^2*f*x + 4*d*e*(4*f + 5*g*x))))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (2*c*(12*c^5*d^4*f - b^5*e^4*
g + 10*a*b^3*c*e^4*g - 30*a^2*b*c^2*e^4*g + 2*c^4*d^2*(-3*b*d*(4*e*f + d*g) + 4*a*e*(3*e*f + 2*d*g)) + 4*c^3*e
*(b^2*d^2*(3*e*f + 2*d*g) - 3*a*b*d*e*(2*e*f + 3*d*g) + 3*a^2*e^2*(e*f + 4*d*g)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2
+ 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + c*e^4*g*Log[a + x*(b + c*x)])/(2*c^4)

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Maple [B]  time = 0.019, size = 2221, normalized size = 4.1 \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*(g*x+f)/(c*x^2+b*x+a)^3,x)

[Out]

((25*a^2*b*c^2*e^4*g-40*a^2*c^3*d*e^3*g-10*a^2*c^3*e^4*f-15*a*b^3*c*e^4*g+32*a*b^2*c^2*d*e^3*g+8*a*b^2*c^2*e^4
*f-18*a*b*c^3*d^2*e^2*g-12*a*b*c^3*d*e^3*f+8*a*c^4*d^3*e*g+12*a*c^4*d^2*e^2*f+2*b^5*e^4*g-4*b^4*c*d*e^3*g-b^4*
c*e^4*f+4*b^2*c^3*d^3*e*g+6*b^2*c^3*d^2*e^2*f-3*b*c^4*d^4*g-12*b*c^4*d^3*e*f+6*c^5*d^4*f)/c^2/(16*a^2*c^2-8*a*
b^2*c+b^4)*x^3+1/2*(32*a^3*c^3*e^4*g+11*a^2*b^2*c^2*e^4*g+8*a^2*b*c^3*d*e^3*g+2*a^2*b*c^3*e^4*f-96*a^2*c^4*d^2
*e^2*g-64*a^2*c^4*d*e^3*f-19*a*b^4*c*e^4*g+32*a*b^3*c^2*d*e^3*g+8*a*b^3*c^2*e^4*f-6*a*b^2*c^3*d^2*e^2*g-4*a*b^
2*c^3*d*e^3*f+24*a*b*c^4*d^3*e*g+36*a*b*c^4*d^2*e^2*f+3*b^6*e^4*g-4*b^5*c*d*e^3*g-b^5*c*e^4*f-6*b^4*c^2*d^2*e^
2*g-4*b^4*c^2*d*e^3*f+12*b^3*c^3*d^3*e*g+18*b^3*c^3*d^2*e^2*f-9*b^2*c^4*d^4*g-36*b^2*c^4*d^3*e*f+18*b*c^5*d^4*
f)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x^2+(31*a^3*b*c^2*e^4*g-24*a^3*c^3*d*e^3*g-6*a^3*c^3*e^4*f-22*a^2*b^3*c*e^4*
g+40*a^2*b^2*c^2*d*e^3*g+10*a^2*b^2*c^2*e^4*f-30*a^2*b*c^3*d^2*e^2*g-20*a^2*b*c^3*d*e^3*f-8*a^2*c^4*d^3*e*g-12
*a^2*c^4*d^2*e^2*f+3*a*b^5*e^4*g-4*a*b^4*c*d*e^3*g-a*b^4*c*e^4*f-6*a*b^3*c^2*d^2*e^2*g-4*a*b^3*c^2*d*e^3*f+20*
a*b^2*c^3*d^3*e*g+30*a*b^2*c^3*d^2*e^2*f-5*a*b*c^4*d^4*g-20*a*b*c^4*d^3*e*f+10*a*c^5*d^4*f-b^3*c^3*d^4*g-4*b^3
*c^3*d^3*e*f+2*b^2*c^4*d^4*f)/(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^4*g-21*a^3*b^2*c*e^4*g+40
*a^3*b*c^2*d*e^3*g+10*a^3*b*c^2*e^4*f-48*a^3*c^3*d^2*e^2*g-32*a^3*c^3*d*e^3*f+3*a^2*b^4*e^4*g-4*a^2*b^3*c*d*e^
3*g-a^2*b^3*c*e^4*f-6*a^2*b^2*c^2*d^2*e^2*g-4*a^2*b^2*c^2*d*e^3*f+24*a^2*b*c^3*d^3*e*g+36*a^2*b*c^3*d^2*e^2*f-
8*a^2*c^4*d^4*g-32*a^2*c^4*d^3*e*f-a*b^2*c^3*d^4*g-4*a*b^2*c^3*d^3*e*f+10*a*b*c^4*d^4*f-b^3*c^3*d^4*f)/(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^4*g-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^4*g+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^4*g-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^4*g+48/(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*d*e^3*g+12/(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*e^4*f+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^4*g-36/(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*d^2*e^2*g-24/(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*d*e^3*f+16*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*d^3*e*g+24*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*d^2*e^2*
f+8/(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*g+12/(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^2*e^2*f-6*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*g-24*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^3*e*f+12*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^4*f-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^4*g

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 8.111, size = 11572, normalized size = 21.31 \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*(g*x+f)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[1/2*(2*((6*(b^2*c^6 - 4*a*c^7)*d^4 - 12*(b^3*c^5 - 4*a*b*c^6)*d^3*e + 6*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d
^2*e^2 - 12*(a*b^3*c^4 - 4*a^2*b*c^5)*d*e^3 - (b^6*c^2 - 12*a*b^4*c^3 + 42*a^2*b^2*c^4 - 40*a^3*c^5)*e^4)*f -
(3*(b^3*c^5 - 4*a*b*c^6)*d^4 - 4*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^3*e + 18*(a*b^3*c^4 - 4*a^2*b*c^5)*d^2*
e^2 + 4*(b^6*c^2 - 12*a*b^4*c^3 + 42*a^2*b^2*c^4 - 40*a^3*c^5)*d*e^3 - (2*b^7*c - 23*a*b^5*c^2 + 85*a^2*b^3*c^
3 - 100*a^3*b*c^4)*e^4)*g)*x^3 + ((18*(b^3*c^5 - 4*a*b*c^6)*d^4 - 36*(b^4*c^4 - 4*a*b^2*c^5)*d^3*e + 18*(b^5*c
^3 - 2*a*b^3*c^4 - 8*a^2*b*c^5)*d^2*e^2 - 4*(b^6*c^2 - 3*a*b^4*c^3 + 12*a^2*b^2*c^4 - 64*a^3*c^5)*d*e^3 - (b^7
*c - 12*a*b^5*c^2 + 30*a^2*b^3*c^3 + 8*a^3*b*c^4)*e^4)*f - (9*(b^4*c^4 - 4*a*b^2*c^5)*d^4 - 12*(b^5*c^3 - 2*a*
b^3*c^4 - 8*a^2*b*c^5)*d^3*e + 6*(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^2 + 4*(b^7*c - 12
*a*b^5*c^2 + 30*a^2*b^3*c^3 + 8*a^3*b*c^4)*d*e^3 - (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^4)*g)*x^2 - ((12*(c^7*d^4 - 2*b*c^6*d^3*e - 2*a*b*c^5*d*e^3 + a^2*c^5*e^4 + (b^2*c^5 + 2*a*c^6)*d^
2*e^2)*f - (6*b*c^6*d^4 + 36*a*b*c^5*d^2*e^2 - 48*a^2*c^5*d*e^3 - 8*(b^2*c^5 + 2*a*c^6)*d^3*e + (b^5*c^2 - 10*
a*b^3*c^3 + 30*a^2*b*c^4)*e^4)*g)*x^4 + 2*(12*(b*c^6*d^4 - 2*b^2*c^5*d^3*e - 2*a*b^2*c^4*d*e^3 + a^2*b*c^4*e^4
+ (b^3*c^4 + 2*a*b*c^5)*d^2*e^2)*f - (6*b^2*c^5*d^4 + 36*a*b^2*c^4*d^2*e^2 - 48*a^2*b*c^4*d*e^3 - 8*(b^3*c^4
+ 2*a*b*c^5)*d^3*e + (b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*e^4)*g)*x^3 + (12*((b^2*c^5 + 2*a*c^6)*d^4 - 2*(b
^3*c^4 + 2*a*b*c^5)*d^3*e + (b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^2*e^2 - 2*(a*b^3*c^3 + 2*a^2*b*c^4)*d*e^3 +
(a^2*b^2*c^3 + 2*a^3*c^4)*e^4)*f - (6*(b^3*c^4 + 2*a*b*c^5)*d^4 - 8*(b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^3*e
+ 36*(a*b^3*c^3 + 2*a^2*b*c^4)*d^2*e^2 - 48*(a^2*b^2*c^3 + 2*a^3*c^4)*d*e^3 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^
2 + 60*a^3*b*c^3)*e^4)*g)*x^2 + 12*(a^2*c^5*d^4 - 2*a^2*b*c^4*d^3*e - 2*a^3*b*c^3*d*e^3 + a^4*c^3*e^4 + (a^2*b
^2*c^3 + 2*a^3*c^4)*d^2*e^2)*f - (6*a^2*b*c^4*d^4 + 36*a^3*b*c^3*d^2*e^2 - 48*a^4*c^3*d*e^3 - 8*(a^2*b^2*c^3 +
2*a^3*c^4)*d^3*e + (a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2)*e^4)*g + 2*(12*(a*b*c^5*d^4 - 2*a*b^2*c^4*d^3*e -
2*a^2*b^2*c^3*d*e^3 + a^3*b*c^3*e^4 + (a*b^3*c^3 + 2*a^2*b*c^4)*d^2*e^2)*f - (6*a*b^2*c^4*d^4 + 36*a^2*b^2*c^3
*d^2*e^2 - 48*a^3*b*c^3*d*e^3 - 8*(a*b^3*c^3 + 2*a^2*b*c^4)*d^3*e + (a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*e^
4)*g)*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)) - ((b^5*c^3 - 14*a*b^3*c^4 + 40*a^2*b*c^5)*d^4 + 4*(a*b^4*c^3 + 4*a^2*b^2*c^4 - 32*a^3*c^5)*d^3*e - 36
*(a^2*b^3*c^3 - 4*a^3*b*c^4)*d^2*e^2 + 4*(a^2*b^4*c^2 + 4*a^3*b^2*c^3 - 32*a^4*c^4)*d*e^3 + (a^2*b^5*c - 14*a^
3*b^3*c^2 + 40*a^4*b*c^3)*e^4)*f - ((a*b^4*c^3 + 4*a^2*b^2*c^4 - 32*a^3*c^5)*d^4 - 24*(a^2*b^3*c^3 - 4*a^3*b*c
^4)*d^3*e + 6*(a^2*b^4*c^2 + 4*a^3*b^2*c^3 - 32*a^4*c^4)*d^2*e^2 + 4*(a^2*b^5*c - 14*a^3*b^3*c^2 + 40*a^4*b*c^
3)*d*e^3 - 3*(a^2*b^6 - 11*a^3*b^4*c + 36*a^4*b^2*c^2 - 32*a^5*c^3)*e^4)*g + 2*((2*(b^4*c^4 + a*b^2*c^5 - 20*a
^2*c^6)*d^4 - 4*(b^5*c^3 + a*b^3*c^4 - 20*a^2*b*c^5)*d^3*e + 6*(5*a*b^4*c^3 - 22*a^2*b^2*c^4 + 8*a^3*c^5)*d^2*
e^2 - 4*(a*b^5*c^2 + a^2*b^3*c^3 - 20*a^3*b*c^4)*d*e^3 - (a*b^6*c - 14*a^2*b^4*c^2 + 46*a^3*b^2*c^3 - 24*a^4*c
^4)*e^4)*f - ((b^5*c^3 + a*b^3*c^4 - 20*a^2*b*c^5)*d^4 - 4*(5*a*b^4*c^3 - 22*a^2*b^2*c^4 + 8*a^3*c^5)*d^3*e +
6*(a*b^5*c^2 + a^2*b^3*c^3 - 20*a^3*b*c^4)*d^2*e^2 + 4*(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^3 - (3*a*b^7 - 34*a^2*b^5*c + 119*a^3*b^3*c^2 - 124*a^4*b*c^3)*e^4)*g)*x + ((b^6*c^2 - 12*a*b^4*c^3 + 48
*a^2*b^2*c^4 - 64*a^3*c^5)*e^4*g*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^4*g*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^4*g*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^4*g*x + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4*g)*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*(2*((6*(b^2*c^6 - 4*a*c^7)*d^4 - 12*(b^3*c^5 - 4*a*b*c^6)*d^3*e + 6*(b^4*c^4 - 2*a*b^2*c^5 - 8
*a^2*c^6)*d^2*e^2 - 12*(a*b^3*c^4 - 4*a^2*b*c^5)*d*e^3 - (b^6*c^2 - 12*a*b^4*c^3 + 42*a^2*b^2*c^4 - 40*a^3*c^5
)*e^4)*f - (3*(b^3*c^5 - 4*a*b*c^6)*d^4 - 4*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^3*e + 18*(a*b^3*c^4 - 4*a^2*
b*c^5)*d^2*e^2 + 4*(b^6*c^2 - 12*a*b^4*c^3 + 42*a^2*b^2*c^4 - 40*a^3*c^5)*d*e^3 - (2*b^7*c - 23*a*b^5*c^2 + 85
*a^2*b^3*c^3 - 100*a^3*b*c^4)*e^4)*g)*x^3 + ((18*(b^3*c^5 - 4*a*b*c^6)*d^4 - 36*(b^4*c^4 - 4*a*b^2*c^5)*d^3*e
+ 18*(b^5*c^3 - 2*a*b^3*c^4 - 8*a^2*b*c^5)*d^2*e^2 - 4*(b^6*c^2 - 3*a*b^4*c^3 + 12*a^2*b^2*c^4 - 64*a^3*c^5)*d
*e^3 - (b^7*c - 12*a*b^5*c^2 + 30*a^2*b^3*c^3 + 8*a^3*b*c^4)*e^4)*f - (9*(b^4*c^4 - 4*a*b^2*c^5)*d^4 - 12*(b^5
*c^3 - 2*a*b^3*c^4 - 8*a^2*b*c^5)*d^3*e + 6*(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^2 + 4*
(b^7*c - 12*a*b^5*c^2 + 30*a^2*b^3*c^3 + 8*a^3*b*c^4)*d*e^3 - (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^4)*g)*x^2 - 2*((12*(c^7*d^4 - 2*b*c^6*d^3*e - 2*a*b*c^5*d*e^3 + a^2*c^5*e^4 + (b^2*c^5
+ 2*a*c^6)*d^2*e^2)*f - (6*b*c^6*d^4 + 36*a*b*c^5*d^2*e^2 - 48*a^2*c^5*d*e^3 - 8*(b^2*c^5 + 2*a*c^6)*d^3*e + (
b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^4)*g)*x^4 + 2*(12*(b*c^6*d^4 - 2*b^2*c^5*d^3*e - 2*a*b^2*c^4*d*e^3 +
a^2*b*c^4*e^4 + (b^3*c^4 + 2*a*b*c^5)*d^2*e^2)*f - (6*b^2*c^5*d^4 + 36*a*b^2*c^4*d^2*e^2 - 48*a^2*b*c^4*d*e^3
- 8*(b^3*c^4 + 2*a*b*c^5)*d^3*e + (b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*e^4)*g)*x^3 + (12*((b^2*c^5 + 2*a*c^
6)*d^4 - 2*(b^3*c^4 + 2*a*b*c^5)*d^3*e + (b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^2*e^2 - 2*(a*b^3*c^3 + 2*a^2*b*
c^4)*d*e^3 + (a^2*b^2*c^3 + 2*a^3*c^4)*e^4)*f - (6*(b^3*c^4 + 2*a*b*c^5)*d^4 - 8*(b^4*c^3 + 4*a*b^2*c^4 + 4*a^
2*c^5)*d^3*e + 36*(a*b^3*c^3 + 2*a^2*b*c^4)*d^2*e^2 - 48*(a^2*b^2*c^3 + 2*a^3*c^4)*d*e^3 + (b^7 - 8*a*b^5*c +
10*a^2*b^3*c^2 + 60*a^3*b*c^3)*e^4)*g)*x^2 + 12*(a^2*c^5*d^4 - 2*a^2*b*c^4*d^3*e - 2*a^3*b*c^3*d*e^3 + a^4*c^3
*e^4 + (a^2*b^2*c^3 + 2*a^3*c^4)*d^2*e^2)*f - (6*a^2*b*c^4*d^4 + 36*a^3*b*c^3*d^2*e^2 - 48*a^4*c^3*d*e^3 - 8*(
a^2*b^2*c^3 + 2*a^3*c^4)*d^3*e + (a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2)*e^4)*g + 2*(12*(a*b*c^5*d^4 - 2*a*b^2
*c^4*d^3*e - 2*a^2*b^2*c^3*d*e^3 + a^3*b*c^3*e^4 + (a*b^3*c^3 + 2*a^2*b*c^4)*d^2*e^2)*f - (6*a*b^2*c^4*d^4 + 3
6*a^2*b^2*c^3*d^2*e^2 - 48*a^3*b*c^3*d*e^3 - 8*(a*b^3*c^3 + 2*a^2*b*c^4)*d^3*e + (a*b^6 - 10*a^2*b^4*c + 30*a^
3*b^2*c^2)*e^4)*g)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - ((b^5*c^3 - 1
4*a*b^3*c^4 + 40*a^2*b*c^5)*d^4 + 4*(a*b^4*c^3 + 4*a^2*b^2*c^4 - 32*a^3*c^5)*d^3*e - 36*(a^2*b^3*c^3 - 4*a^3*b
*c^4)*d^2*e^2 + 4*(a^2*b^4*c^2 + 4*a^3*b^2*c^3 - 32*a^4*c^4)*d*e^3 + (a^2*b^5*c - 14*a^3*b^3*c^2 + 40*a^4*b*c^
3)*e^4)*f - ((a*b^4*c^3 + 4*a^2*b^2*c^4 - 32*a^3*c^5)*d^4 - 24*(a^2*b^3*c^3 - 4*a^3*b*c^4)*d^3*e + 6*(a^2*b^4*
c^2 + 4*a^3*b^2*c^3 - 32*a^4*c^4)*d^2*e^2 + 4*(a^2*b^5*c - 14*a^3*b^3*c^2 + 40*a^4*b*c^3)*d*e^3 - 3*(a^2*b^6 -
11*a^3*b^4*c + 36*a^4*b^2*c^2 - 32*a^5*c^3)*e^4)*g + 2*((2*(b^4*c^4 + a*b^2*c^5 - 20*a^2*c^6)*d^4 - 4*(b^5*c^
3 + a*b^3*c^4 - 20*a^2*b*c^5)*d^3*e + 6*(5*a*b^4*c^3 - 22*a^2*b^2*c^4 + 8*a^3*c^5)*d^2*e^2 - 4*(a*b^5*c^2 + a^
2*b^3*c^3 - 20*a^3*b*c^4)*d*e^3 - (a*b^6*c - 14*a^2*b^4*c^2 + 46*a^3*b^2*c^3 - 24*a^4*c^4)*e^4)*f - ((b^5*c^3
+ a*b^3*c^4 - 20*a^2*b*c^5)*d^4 - 4*(5*a*b^4*c^3 - 22*a^2*b^2*c^4 + 8*a^3*c^5)*d^3*e + 6*(a*b^5*c^2 + a^2*b^3*
c^3 - 20*a^3*b*c^4)*d^2*e^2 + 4*(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^3 - (3*a*b^7 - 34
*a^2*b^5*c + 119*a^3*b^3*c^2 - 124*a^4*b*c^3)*e^4)*g)*x + ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c
^5)*e^4*g*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^4*g*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^4*g*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^4*g*x + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*e^4*g)*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 [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*(g*x+f)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.14738, size = 1777, normalized size = 3.27 \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*(g*x+f)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

(12*c^5*d^4*f - 6*b*c^4*d^4*g - 24*b*c^4*d^3*f*e + 8*b^2*c^3*d^3*g*e + 16*a*c^4*d^3*g*e + 12*b^2*c^3*d^2*f*e^2
+ 24*a*c^4*d^2*f*e^2 - 36*a*b*c^3*d^2*g*e^2 - 24*a*b*c^3*d*f*e^3 + 48*a^2*c^3*d*g*e^3 + 12*a^2*c^3*f*e^4 - b^
5*g*e^4 + 10*a*b^3*c*g*e^4 - 30*a^2*b*c^2*g*e^4)*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*g*e^4*log(c*x^2 + b*x + a)/c^3 - 1/2*(b^3*c^3*d^4*f - 10*a*b*c^4*d^4
*f + a*b^2*c^3*d^4*g + 8*a^2*c^4*d^4*g + 4*a*b^2*c^3*d^3*f*e + 32*a^2*c^4*d^3*f*e - 24*a^2*b*c^3*d^3*g*e - 36*
a^2*b*c^3*d^2*f*e^2 + 6*a^2*b^2*c^2*d^2*g*e^2 + 48*a^3*c^3*d^2*g*e^2 + 4*a^2*b^2*c^2*d*f*e^3 + 32*a^3*c^3*d*f*
e^3 + 4*a^2*b^3*c*d*g*e^3 - 40*a^3*b*c^2*d*g*e^3 + a^2*b^3*c*f*e^4 - 10*a^3*b*c^2*f*e^4 - 3*a^2*b^4*g*e^4 + 21
*a^3*b^2*c*g*e^4 - 24*a^4*c^2*g*e^4 - 2*(6*c^6*d^4*f - 3*b*c^5*d^4*g - 12*b*c^5*d^3*f*e + 4*b^2*c^4*d^3*g*e +
8*a*c^5*d^3*g*e + 6*b^2*c^4*d^2*f*e^2 + 12*a*c^5*d^2*f*e^2 - 18*a*b*c^4*d^2*g*e^2 - 12*a*b*c^4*d*f*e^3 - 4*b^4
*c^2*d*g*e^3 + 32*a*b^2*c^3*d*g*e^3 - 40*a^2*c^4*d*g*e^3 - b^4*c^2*f*e^4 + 8*a*b^2*c^3*f*e^4 - 10*a^2*c^4*f*e^
4 + 2*b^5*c*g*e^4 - 15*a*b^3*c^2*g*e^4 + 25*a^2*b*c^3*g*e^4)*x^3 - (18*b*c^5*d^4*f - 9*b^2*c^4*d^4*g - 36*b^2*
c^4*d^3*f*e + 12*b^3*c^3*d^3*g*e + 24*a*b*c^4*d^3*g*e + 18*b^3*c^3*d^2*f*e^2 + 36*a*b*c^4*d^2*f*e^2 - 6*b^4*c^
2*d^2*g*e^2 - 6*a*b^2*c^3*d^2*g*e^2 - 96*a^2*c^4*d^2*g*e^2 - 4*b^4*c^2*d*f*e^3 - 4*a*b^2*c^3*d*f*e^3 - 64*a^2*
c^4*d*f*e^3 - 4*b^5*c*d*g*e^3 + 32*a*b^3*c^2*d*g*e^3 + 8*a^2*b*c^3*d*g*e^3 - b^5*c*f*e^4 + 8*a*b^3*c^2*f*e^4 +
2*a^2*b*c^3*f*e^4 + 3*b^6*g*e^4 - 19*a*b^4*c*g*e^4 + 11*a^2*b^2*c^2*g*e^4 + 32*a^3*c^3*g*e^4)*x^2 - 2*(2*b^2*
c^4*d^4*f + 10*a*c^5*d^4*f - b^3*c^3*d^4*g - 5*a*b*c^4*d^4*g - 4*b^3*c^3*d^3*f*e - 20*a*b*c^4*d^3*f*e + 20*a*b
^2*c^3*d^3*g*e - 8*a^2*c^4*d^3*g*e + 30*a*b^2*c^3*d^2*f*e^2 - 12*a^2*c^4*d^2*f*e^2 - 6*a*b^3*c^2*d^2*g*e^2 - 3
0*a^2*b*c^3*d^2*g*e^2 - 4*a*b^3*c^2*d*f*e^3 - 20*a^2*b*c^3*d*f*e^3 - 4*a*b^4*c*d*g*e^3 + 40*a^2*b^2*c^2*d*g*e^
3 - 24*a^3*c^3*d*g*e^3 - a*b^4*c*f*e^4 + 10*a^2*b^2*c^2*f*e^4 - 6*a^3*c^3*f*e^4 + 3*a*b^5*g*e^4 - 22*a^2*b^3*c
*g*e^4 + 31*a^3*b*c^2*g*e^4)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*c^3)