\(\int \frac {(d+e x)^4 (f+g x)}{(a+b x+c x^2)^3} \, dx\) [2371]

Optimal result

Integrand size = 25, antiderivative size = 543 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {(d+e x)^3 \left (2 a c (e f+d g)-b (c d f+a e g)-\left (2 c^2 d f+b^2 e g-c (b e f+b d g+2 a e g)\right ) x\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {(d+e x) \left (b^3 e \left (c d^2-2 a e^2\right ) g-b^2 c d \left (a e^2 g+3 c d (2 e f+d g)\right )+2 b c \left (3 c^2 d^3 f+7 a^2 e^3 g+a c d e (9 e f+7 d g)\right )-4 a c^2 e \left (3 c d^2 f+a e (3 e f+8 d g)\right )+\left (12 c^4 d^3 f-2 b^4 e^3 g+b^2 c e^2 (b d+15 a e) g-2 c^3 d (3 b d (3 e f+d g)-2 a e (3 e f+4 d g))-c^2 e \left (16 a^2 e^2 g-b^2 d (6 e f+5 d g)+2 a b e (3 e f+11 d g)\right )\right ) x\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^4 f-b^5 e^4 g+10 a b^3 c e^4 g-30 a^2 b c^2 e^4 g-2 c^4 d^2 (3 b d (4 e f+d g)-4 a e (3 e f+2 d g))+4 c^3 e \left (b^2 d^2 (3 e f+2 d g)-3 a b d e (2 e f+3 d g)+3 a^2 e^2 (e f+4 d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {e^4 g \log \left (a+b x+c x^2\right )}{2 c^3} \]

[Out]

1/2*(e*x+d)^3*(2*a*c*(d*g+e*f)-b*(a*e*g+c*d*f)-(2*c^2*d*f+b^2*e*g-c*(2*a*e*g+b*d*g+b*e*f))*x)/c/(-4*a*c+b^2)/(
c*x^2+b*x+a)^2+1/2*(e*x+d)*(b^3*e*(-2*a*e^2+c*d^2)*g-b^2*c*d*(a*e^2*g+3*c*d*(d*g+2*e*f))+2*b*c*(3*c^2*d^3*f+7*
a^2*e^3*g+a*c*d*e*(7*d*g+9*e*f))-4*a*c^2*e*(3*c*d^2*f+a*e*(8*d*g+3*e*f))+(12*c^4*d^3*f-2*b^4*e^3*g+b^2*c*e^2*(
15*a*e+b*d)*g-2*c^3*d*(3*b*d*(d*g+3*e*f)-2*a*e*(4*d*g+3*e*f))-c^2*e*(16*a^2*e^2*g-b^2*d*(5*d*g+6*e*f)+2*a*b*e*
(11*d*g+3*e*f)))*x)/c^2/(-4*a*c+b^2)^2/(c*x^2+b*x+a)-(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*(d*g+4*e*f)-4*a*e*(2*d*g+3*e*f))+4*c^3*e*(b^2*d^2*(2*d*g+3*e*f)-3*a*b*d*e*(3*d*g+2*e*f)+3*
a^2*e^2*(4*d*g+e*f)))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(5/2)+1/2*e^4*g*ln(c*x^2+b*x+a)/c
^3

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {832, 648, 632, 212, 642} \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (4 c^3 e \left (3 a^2 e^2 (4 d g+e f)-3 a b d e (3 d g+2 e f)+b^2 d^2 (2 d g+3 e f)\right )-30 a^2 b c^2 e^4 g+10 a b^3 c e^4 g-2 c^4 d^2 (3 b d (d g+4 e f)-4 a e (2 d g+3 e f))+b^5 \left (-e^4\right ) g+12 c^5 d^4 f\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {(d+e x) \left (x \left (-c^2 e \left (16 a^2 e^2 g+2 a b e (11 d g+3 e f)+b^2 (-d) (5 d g+6 e f)\right )+b^2 c e^2 g (15 a e+b d)-2 c^3 d (3 b d (d g+3 e f)-2 a e (4 d g+3 e f))-2 b^4 e^3 g+12 c^4 d^3 f\right )+2 b c \left (7 a^2 e^3 g+a c d e (7 d g+9 e f)+3 c^2 d^3 f\right )+b^3 e g \left (c d^2-2 a e^2\right )-b^2 c d \left (a e^2 g+3 c d (d g+2 e f)\right )-4 a c^2 e \left (a e (8 d g+3 e f)+3 c d^2 f\right )\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {(d+e x)^3 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {e^4 g \log \left (a+b x+c x^2\right )}{2 c^3} \]

[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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

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 642

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 648

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 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(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] &&
NeQ[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*} \text {integral}& = \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 ) \text {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] (verified)

Time = 1.16 (sec) , antiderivative size = 897, normalized size of antiderivative = 1.65 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {b^5 e^4 g x+b^4 e^3 (a e g-c (e f+4 d g) x)-b^3 c e^2 (-2 c d (2 e f+3 d g) x+a e (e f+4 d g+5 e g x))+2 c^2 \left (a^3 e^4 g-c^3 d^4 f x-a^2 c e^2 \left (6 d^2 g+e^2 f x+4 d e (f+g x)\right )+a c^2 d^2 \left (d^2 g+6 e^2 f x+4 d e (f+g x)\right )\right )+b c^2 \left (c^2 d^3 (-d f+4 e f x+d g x)+a^2 e^3 (3 e f+12 d g+5 e g x)-2 a c d e \left (2 d^2 g+6 e^2 f x+3 d e (f+3 g x)\right )\right )+2 b^2 c e \left (-2 a^2 e^3 g-c^2 d^2 (3 e f+2 d g) x+a c e \left (3 d^2 g+2 e^2 f x+2 d e (f+4 g x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {-b^6 e^4 g+b^5 c e^3 (4 d g+e (f+4 g x))+2 b^3 c^2 e \left (c d^2 (3 e f+2 d g)-a e^2 (4 e f+16 d g+15 e g x)\right )+b^2 c^2 \left (-39 a^2 e^4 g+c^2 d^2 \left (-12 d e f-3 d^2 g+12 e^2 f x+8 d e g x\right )+2 a c e^2 \left (10 d e f+15 d^2 g+8 e^2 f x+32 d e g x\right )\right )+2 b c^3 \left (a^2 e^3 (11 e f+44 d g+25 e g x)+2 a c d e \left (2 d^2 g-6 e^2 f x+3 d e (f-3 g x)\right )+3 c^2 d^3 (-4 e f x+d (f-g x))\right )-b^4 c e^2 \left (-11 a e^2 g+2 c \left (3 d^2 g+e^2 f x+2 d e (f+2 g x)\right )\right )+4 c^3 \left (8 a^3 e^4 g+3 c^3 d^4 f x+2 a c^2 d^2 e (3 e f+2 d g) x-a^2 c e^2 \left (24 d^2 g+5 e^2 f x+4 d e (4 f+5 g x)\right )\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {2 c \left (12 c^5 d^4 f-b^5 e^4 g+10 a b^3 c e^4 g-30 a^2 b c^2 e^4 g+2 c^4 d^2 (-3 b d (4 e f+d g)+4 a e (3 e f+2 d g))+4 c^3 e \left (b^2 d^2 (3 e f+2 d g)-3 a b d e (2 e f+3 d g)+3 a^2 e^2 (e f+4 d g)\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}+c e^4 g \log (a+x (b+c x))}{2 c^4} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1500\) vs. \(2(533)=1066\).

Time = 0.56 (sec) , antiderivative size = 1501, normalized size of antiderivative = 2.76

method	result	size
default	\(\text {Expression too large to display}\)	\(1501\)
risch	\(\text {Expression too large to display}\)	\(8708\)

[In]

int((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[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+1/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(16*a^2*c^2*e^4*g-8*a*b^2*c*e^4*g+b
^4*e^4*g)/c*ln(c*x^2+b*x+a)+2*(-7*a^2*b*c*e^4*g+24*a^2*c^2*d*e^3*g+6*a^2*c^2*e^4*f+a*b^3*e^4*g-18*a*b*c^2*d^2*
e^2*g-12*a*b*c^2*d*e^3*f+8*a*c^3*d^3*e*g+12*a*c^3*d^2*e^2*f+4*b^2*c^2*d^3*e*g+6*b^2*c^2*d^2*e^2*f-3*b*c^3*d^4*
g-12*b*c^3*d^3*e*f+6*c^4*d^4*f-1/2*(16*a^2*c^2*e^4*g-8*a*b^2*c*e^4*g+b^4*e^4*g)*b/c)/(4*a*c-b^2)^(1/2)*arctan(
(2*c*x+b)/(4*a*c-b^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2828 vs. \(2 (533) = 1066\).

Time = 1.72 (sec) , antiderivative size = 5676, normalized size of antiderivative = 10.45 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**4*(g*x+f)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^4*(g*x+f)/(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 deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1368 vs. \(2 (533) = 1066\).

Time = 0.29 (sec) , antiderivative size = 1368, normalized size of antiderivative = 2.52 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 14.90 (sec) , antiderivative size = 1763, normalized size of antiderivative = 3.25 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan((x*(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)))/
(c^2*(4*a*c - b^2)^5) + ((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))*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4))/(2*c^5*(4*a*c - b^2)^5*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))
*(12*c^5*d^4*f - b^5*e^4*g + 12*a^2*c^3*e^4*f - 6*b*c^4*d^4*g + 10*a*b^3*c*e^4*g + 16*a*c^4*d^3*e*g - 24*b*c^4
*d^3*e*f - 30*a^2*b*c^2*e^4*g + 24*a*c^4*d^2*e^2*f + 48*a^2*c^3*d*e^3*g + 8*b^2*c^3*d^3*e*g + 12*b^2*c^3*d^2*e
^2*f - 24*a*b*c^3*d*e^3*f - 36*a*b*c^3*d^2*e^2*g))/(c^3*(4*a*c - b^2)^(5/2)) - (log(a + b*x + c*x^2)*(b^10*e^4
*g - 1024*a^5*c^5*e^4*g - 20*a*b^8*c*e^4*g + 160*a^2*b^6*c^2*e^4*g - 640*a^3*b^4*c^3*e^4*g + 1280*a^4*b^2*c^4*
e^4*g))/(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)) -
((8*a^2*c^4*d^4*g - 3*a^2*b^4*e^4*g + b^3*c^3*d^4*f - 24*a^4*c^2*e^4*g - 10*a*b*c^4*d^4*f + a*b^2*c^3*d^4*g +
a^2*b^3*c*e^4*f - 10*a^3*b*c^2*e^4*f + 21*a^3*b^2*c*e^4*g + 32*a^2*c^4*d^3*e*f + 32*a^3*c^3*d*e^3*f + 48*a^3*c
^3*d^2*e^2*g - 36*a^2*b*c^3*d^2*e^2*f + 4*a^2*b^2*c^2*d*e^3*f + 6*a^2*b^2*c^2*d^2*e^2*g + 4*a*b^2*c^3*d^3*e*f
- 24*a^2*b*c^3*d^3*e*g + 4*a^2*b^3*c*d*e^3*g - 40*a^3*b*c^2*d*e^3*g)/(2*c^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) -
(x^3*(6*c^5*d^4*f + 2*b^5*e^4*g - 10*a^2*c^3*e^4*f - 3*b*c^4*d^4*g - b^4*c*e^4*f - 15*a*b^3*c*e^4*g + 8*a*c^4*
d^3*e*g - 12*b*c^4*d^3*e*f - 4*b^4*c*d*e^3*g + 8*a*b^2*c^2*e^4*f + 25*a^2*b*c^2*e^4*g + 12*a*c^4*d^2*e^2*f - 4
0*a^2*c^3*d*e^3*g + 4*b^2*c^3*d^3*e*g + 6*b^2*c^3*d^2*e^2*f - 12*a*b*c^3*d*e^3*f - 18*a*b*c^3*d^2*e^2*g + 32*a
*b^2*c^2*d*e^3*g))/(c^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^2*(3*b^6*e^4*g + 32*a^3*c^3*e^4*g - 9*b^2*c^4*d^4
*g + 18*b*c^5*d^4*f - b^5*c*e^4*f - 19*a*b^4*c*e^4*g - 4*b^5*c*d*e^3*g + 8*a*b^3*c^2*e^4*f + 2*a^2*b*c^3*e^4*f
- 64*a^2*c^4*d*e^3*f - 36*b^2*c^4*d^3*e*f - 4*b^4*c^2*d*e^3*f + 12*b^3*c^3*d^3*e*g + 11*a^2*b^2*c^2*e^4*g - 9
6*a^2*c^4*d^2*e^2*g + 18*b^3*c^3*d^2*e^2*f - 6*b^4*c^2*d^2*e^2*g - 6*a*b^2*c^3*d^2*e^2*g + 24*a*b*c^4*d^3*e*g
+ 36*a*b*c^4*d^2*e^2*f - 4*a*b^2*c^3*d*e^3*f + 32*a*b^3*c^2*d*e^3*g + 8*a^2*b*c^3*d*e^3*g))/(2*c^3*(b^4 + 16*a
^2*c^2 - 8*a*b^2*c)) + (x*(6*a^3*c^3*e^4*f - 2*b^2*c^4*d^4*f + b^3*c^3*d^4*g - 10*a*c^5*d^4*f - 3*a*b^5*e^4*g
+ 5*a*b*c^4*d^4*g + a*b^4*c*e^4*f + 22*a^2*b^3*c*e^4*g - 31*a^3*b*c^2*e^4*g + 8*a^2*c^4*d^3*e*g + 24*a^3*c^3*d
*e^3*g + 4*b^3*c^3*d^3*e*f - 10*a^2*b^2*c^2*e^4*f + 12*a^2*c^4*d^2*e^2*f - 30*a*b^2*c^3*d^2*e^2*f + 6*a*b^3*c^
2*d^2*e^2*g + 30*a^2*b*c^3*d^2*e^2*g - 40*a^2*b^2*c^2*d*e^3*g + 20*a*b*c^4*d^3*e*f + 4*a*b^4*c*d*e^3*g + 4*a*b
^3*c^2*d*e^3*f + 20*a^2*b*c^3*d*e^3*f - 20*a*b^2*c^3*d^3*e*g))/(c^3*(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)