3.2376 \(\int \frac{f+g x}{(d+e x) (a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=666 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-6 c^2 e^2 \left (a^2 b e^2 (d g+5 e f)-2 a^3 e^3 g-6 a b^2 d^2 e g+2 b^3 d^3 g\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (d g+5 e f)+b^2 d^2 (4 d g+5 e f)\right )+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (d g+5 e f))+b^5 \left (-e^4\right ) (e f-d g)+12 c^5 d^5 f\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{c x \left (3 c e (b d-2 a e) (2 a e g-b (d g+e f)+2 c d f)-(2 c d-b e) \left (c (2 a e (4 e f-d g)-3 b d (d g+e f))-2 b^2 e (e f-d g)+6 c^2 d^2 f\right )\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (c (2 a e (4 e f-d g)-3 b d (d g+e f))-2 b^2 e (e f-d g)+6 c^2 d^2 f\right )+3 a c e (2 c d-b e) (2 a e g-b (d g+e f)+2 c d f)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^4 (e f-d g) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^4 (e f-d g) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

[Out]

-(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))*x)/(2*(b^2 - 4*a
*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (3*a*c*e*(2*c*d - b*e)*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))
- (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d*g) + c*(2*a*e*(4*e*f - d*g) - 3*b*d*(e*f + d*g)))
+ c*(3*c*e*(b*d - 2*a*e)*(2*c*d*f + 2*a*e*g - b*(e*f + d*g)) - (2*c*d - b*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d*g
) + c*(2*a*e*(4*e*f - d*g) - 3*b*d*(e*f + d*g))))*x)/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c
*x^2)) - ((12*c^5*d^5*f - b^5*e^4*(e*f - d*g) + 10*a*b^3*c*e^4*(e*f - d*g) + 2*c^4*d^3*(2*a*e*(10*e*f - d*g) -
3*b*d*(5*e*f + d*g)) - 6*c^2*e^2*(2*b^3*d^3*g - 6*a*b^2*d^2*e*g - 2*a^3*e^3*g + a^2*b*e^2*(5*e*f + d*g)) + 4*
c^3*d*e*(3*a^2*e^2*(5*e*f - 2*d*g) - 3*a*b*d*e*(5*e*f + d*g) + b^2*d^2*(5*e*f + 4*d*g)))*ArcTanh[(b + 2*c*x)/S
qrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2 - b*d*e + a*e^2)^3) + (e^4*(e*f - d*g)*Log[d + e*x])/(c*d^2 - b
*d*e + a*e^2)^3 - (e^4*(e*f - d*g)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^3)

________________________________________________________________________________________

Rubi [A]  time = 2.55086, antiderivative size = 664, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {822, 800, 634, 618, 206, 628} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-6 c^2 e^2 \left (a^2 b e^2 (d g+5 e f)-2 a^3 e^3 g-6 a b^2 d^2 e g+2 b^3 d^3 g\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (d g+5 e f)+b^2 d^2 (4 d g+5 e f)\right )+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (d g+5 e f))+b^5 \left (-e^4\right ) (e f-d g)+12 c^5 d^5 f\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{c x \left (3 c e (b d-2 a e) (2 a e g-b (d g+e f)+2 c d f)-(2 c d-b e) \left (2 a c e (4 e f-d g)-2 b^2 e (e f-d g)-3 b c d (d g+e f)+6 c^2 d^2 f\right )\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (2 a c e (4 e f-d g)-2 b^2 e (e f-d g)-3 b c d (d g+e f)+6 c^2 d^2 f\right )+3 a c e (2 c d-b e) (2 a e g-b (d g+e f)+2 c d f)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^4 (e f-d g) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^4 (e f-d g) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))*x)/(2*(b^2 - 4*a
*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (3*a*c*e*(2*c*d - b*e)*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))
- (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d*g) + 2*a*c*e*(4*e*f - d*g) - 3*b*c*d*(e*f + d*g))
+ c*(3*c*e*(b*d - 2*a*e)*(2*c*d*f + 2*a*e*g - b*(e*f + d*g)) - (2*c*d - b*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d*g
) + 2*a*c*e*(4*e*f - d*g) - 3*b*c*d*(e*f + d*g)))*x)/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c
*x^2)) - ((12*c^5*d^5*f - b^5*e^4*(e*f - d*g) + 10*a*b^3*c*e^4*(e*f - d*g) + 2*c^4*d^3*(2*a*e*(10*e*f - d*g) -
3*b*d*(5*e*f + d*g)) - 6*c^2*e^2*(2*b^3*d^3*g - 6*a*b^2*d^2*e*g - 2*a^3*e^3*g + a^2*b*e^2*(5*e*f + d*g)) + 4*
c^3*d*e*(3*a^2*e^2*(5*e*f - 2*d*g) - 3*a*b*d*e*(5*e*f + d*g) + b^2*d^2*(5*e*f + 4*d*g)))*ArcTanh[(b + 2*c*x)/S
qrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2 - b*d*e + a*e^2)^3) + (e^4*(e*f - d*g)*Log[d + e*x])/(c*d^2 - b
*d*e + a*e^2)^3 - (e^4*(e*f - d*g)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^3)

Rule 822

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + 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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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{f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)+3 c e (2 c d f+2 a e g-b (e f+d g)) x}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{\int \frac{-3 c d e \left (b c d-b^2 e+2 a c e\right ) (2 c d f+2 a e g-b (e f+d g))+\left (2 c^2 d^2-b^2 e^2-c e (b d-4 a e)\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )-c e \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{\int \left (\frac{2 \left (b^2-4 a c\right )^2 e^5 (e f-d g)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{2 \left (6 c^5 d^5 f-b^5 e^4 (e f-d g)+9 a b^3 c e^4 (e f-d g)-c^2 e^2 \left (6 b^3 d^3 g-18 a b^2 d^2 e g-6 a^3 e^3 g+a^2 b e^2 (23 e f-5 d g)\right )+c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))+2 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )-c \left (b^2-4 a c\right )^2 e^4 (e f-d g) x\right )}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{e^4 (e f-d g) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\int \frac{6 c^5 d^5 f-b^5 e^4 (e f-d g)+9 a b^3 c e^4 (e f-d g)-c^2 e^2 \left (6 b^3 d^3 g-18 a b^2 d^2 e g-6 a^3 e^3 g+a^2 b e^2 (23 e f-5 d g)\right )+c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))+2 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )-c \left (b^2-4 a c\right )^2 e^4 (e f-d g) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{e^4 (e f-d g) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{\left (e^4 (e f-d g)\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac{\left (12 c^5 d^5 f-b^5 e^4 (e f-d g)+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))-6 c^2 e^2 \left (2 b^3 d^3 g-6 a b^2 d^2 e g-2 a^3 e^3 g+a^2 b e^2 (5 e f+d g)\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac{e^4 (e f-d g) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{e^4 (e f-d g) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac{\left (12 c^5 d^5 f-b^5 e^4 (e f-d g)+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))-6 c^2 e^2 \left (2 b^3 d^3 g-6 a b^2 d^2 e g-2 a^3 e^3 g+a^2 b e^2 (5 e f+d g)\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 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 )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac{3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (12 c^5 d^5 f-b^5 e^4 (e f-d g)+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))-6 c^2 e^2 \left (2 b^3 d^3 g-6 a b^2 d^2 e g-2 a^3 e^3 g+a^2 b e^2 (5 e f+d g)\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^3}+\frac{e^4 (e f-d g) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{e^4 (e f-d g) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 3.57337, size = 668, normalized size = 1. \[ \frac{1}{2} \left (\frac{2 b c \left (3 a^2 e^3 g-a c e \left (d^2 g-7 d e f+5 d e g x+7 e^2 f x\right )+3 c^2 d^2 (d f-d g x-3 e f x)\right )-4 c^2 \left (a^2 e^2 (4 d g-4 e f-3 e g x)+a c d e x (d g-7 e f)-3 c^2 d^3 f x\right )+b^2 c \left (3 a e^2 (d g-5 e f)+c d \left (-3 d^2 g-9 d e f+10 d e g x+2 e^2 f x\right )\right )+b^3 c e \left (5 d^2 g+d e (f-2 g x)+2 e^2 f x\right )+2 b^4 e^2 (e f-d g)}{\left (b^2-4 a c\right )^2 (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}-\frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (6 c^2 e^2 \left (-a^2 b e^2 (d g+5 e f)+2 a^3 e^3 g+6 a b^2 d^2 e g-2 b^3 d^3 g\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (d g+5 e f)+b^2 d^2 (4 d g+5 e f)\right )+10 a b^3 c e^4 (e f-d g)-2 c^4 d^3 (2 a e (d g-10 e f)+3 b d (d g+5 e f))+b^5 e^4 (d g-e f)+12 c^5 d^5 f\right )}{\left (4 a c-b^2\right )^{5/2} \left (e (b d-a e)-c d^2\right )^3}+\frac{b (a e g+c d (f-g x)-c e f x)+2 c (-a d g+a e (f+g x)+c d f x)+b^2 (-e) f}{\left (b^2-4 a c\right ) (a+x (b+c x))^2 \left (e (b d-a e)-c d^2\right )}+\frac{2 e^4 (e f-d g) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^3}+\frac{e^4 (d g-e f) \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.034, size = 9360, normalized size = 14.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.24608, size = 3089, normalized size = 4.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(d*g*e^4 - f*e^5)*log(c*x^2 + b*x + a)/(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*
d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6) - (d*g*e^5 - f*e^6)*l
og(abs(x*e + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*
e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7) + (12*c^5*d^5*f - 6*b*c^4*d^5*g - 30*b*c^4*
d^4*f*e + 16*b^2*c^3*d^4*g*e - 4*a*c^4*d^4*g*e + 20*b^2*c^3*d^3*f*e^2 + 40*a*c^4*d^3*f*e^2 - 12*b^3*c^2*d^3*g*
e^2 - 12*a*b*c^3*d^3*g*e^2 - 60*a*b*c^3*d^2*f*e^3 + 36*a*b^2*c^2*d^2*g*e^3 - 24*a^2*c^3*d^2*g*e^3 + 60*a^2*c^3
*d*f*e^4 + b^5*d*g*e^4 - 10*a*b^3*c*d*g*e^4 - 6*a^2*b*c^2*d*g*e^4 - b^5*f*e^5 + 10*a*b^3*c*f*e^5 - 30*a^2*b*c^
2*f*e^5 + 12*a^3*c^2*g*e^5)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^3*d^6 - 8*a*b^2*c^4*d^6 + 16*a^2*c^
5*d^6 - 3*b^5*c^2*d^5*e + 24*a*b^3*c^3*d^5*e - 48*a^2*b*c^4*d^5*e + 3*b^6*c*d^4*e^2 - 21*a*b^4*c^2*d^4*e^2 + 2
4*a^2*b^2*c^3*d^4*e^2 + 48*a^3*c^4*d^4*e^2 - b^7*d^3*e^3 + 2*a*b^5*c*d^3*e^3 + 32*a^2*b^3*c^2*d^3*e^3 - 96*a^3
*b*c^3*d^3*e^3 + 3*a*b^6*d^2*e^4 - 21*a^2*b^4*c*d^2*e^4 + 24*a^3*b^2*c^2*d^2*e^4 + 48*a^4*c^3*d^2*e^4 - 3*a^2*
b^5*d*e^5 + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 + a^3*b^4*e^6 - 8*a^4*b^2*c*e^6 + 16*a^5*c^2*e^6)*sqrt(-b^
2 + 4*a*c)) - 1/2*(b^3*c^3*d^5*f - 10*a*b*c^4*d^5*f + a*b^2*c^3*d^5*g + 8*a^2*c^4*d^5*g - 3*b^4*c^2*d^4*f*e +
29*a*b^2*c^3*d^4*f*e - 8*a^2*c^4*d^4*f*e - 3*a*b^3*c^2*d^4*g*e - 18*a^2*b*c^3*d^4*g*e + 3*b^5*c*d^3*f*e^2 - 24
*a*b^3*c^2*d^3*f*e^2 - 12*a^2*b*c^3*d^3*f*e^2 + 3*a*b^4*c*d^3*g*e^2 + 10*a^2*b^2*c^2*d^3*g*e^2 + 32*a^3*c^3*d^
3*g*e^2 - b^6*d^2*f*e^3 + a*b^4*c*d^2*f*e^3 + 50*a^2*b^2*c^2*d^2*f*e^3 - 32*a^3*c^3*d^2*f*e^3 - a*b^5*d^2*g*e^
3 - 44*a^3*b*c^2*d^2*g*e^3 + 4*a*b^5*d*f*e^4 - 27*a^2*b^3*c*d*f*e^4 + 14*a^3*b*c^2*d*f*e^4 + 9*a^3*b^2*c*d*g*e
^4 + 24*a^4*c^2*d*g*e^4 - 3*a^2*b^4*f*e^5 + 21*a^3*b^2*c*f*e^5 - 24*a^4*c^2*f*e^5 + a^3*b^3*g*e^5 - 10*a^4*b*c
*g*e^5 - 2*(6*c^6*d^5*f - 3*b*c^5*d^5*g - 15*b*c^5*d^4*f*e + 8*b^2*c^4*d^4*g*e - 2*a*c^5*d^4*g*e + 10*b^2*c^4*
d^3*f*e^2 + 20*a*c^5*d^3*f*e^2 - 6*b^3*c^3*d^3*g*e^2 - 6*a*b*c^4*d^3*g*e^2 - 30*a*b*c^4*d^2*f*e^3 + b^4*c^2*d^
2*g*e^3 + 10*a*b^2*c^3*d^2*g*e^3 + 4*a^2*c^4*d^2*g*e^3 - b^4*c^2*d*f*e^4 + 8*a*b^2*c^3*d*f*e^4 + 14*a^2*c^4*d*
f*e^4 - a*b^3*c^2*d*g*e^4 - 11*a^2*b*c^3*d*g*e^4 + a*b^3*c^2*f*e^5 - 7*a^2*b*c^3*f*e^5 + 6*a^3*c^3*g*e^5)*x^3
- (18*b*c^5*d^5*f - 9*b^2*c^4*d^5*g - 45*b^2*c^4*d^4*f*e + 24*b^3*c^3*d^4*g*e - 6*a*b*c^4*d^4*g*e + 30*b^3*c^3
*d^3*f*e^2 + 60*a*b*c^4*d^3*f*e^2 - 19*b^4*c^2*d^3*g*e^2 - 10*a*b^2*c^3*d^3*g*e^2 - 16*a^2*c^4*d^3*g*e^2 + b^4
*c^2*d^2*f*e^3 - 98*a*b^2*c^3*d^2*f*e^3 + 16*a^2*c^4*d^2*f*e^3 + 4*b^5*c*d^2*g*e^3 + 22*a*b^3*c^2*d^2*g*e^3 +
28*a^2*b*c^3*d^2*g*e^3 - 4*b^5*c*d*f*e^4 + 32*a*b^3*c^2*d*f*e^4 + 26*a^2*b*c^3*d*f*e^4 - 4*a*b^4*c*d*g*e^4 - 2
5*a^2*b^2*c^2*d*g*e^4 - 16*a^3*c^3*d*g*e^4 + 4*a*b^4*c*f*e^5 - 29*a^2*b^2*c^2*f*e^5 + 16*a^3*c^3*f*e^5 + 18*a^
3*b*c^2*g*e^5)*x^2 - 2*(2*b^2*c^4*d^5*f + 10*a*c^5*d^5*f - b^3*c^3*d^5*g - 5*a*b*c^4*d^5*g - 5*b^3*c^3*d^4*f*e
- 25*a*b*c^4*d^4*f*e + 3*b^4*c^2*d^4*g*e + 10*a*b^2*c^3*d^4*g*e + 2*a^2*c^4*d^4*g*e + 3*b^4*c^2*d^3*f*e^2 + 2
6*a*b^2*c^3*d^3*f*e^2 + 28*a^2*c^4*d^3*f*e^2 - 3*b^5*c*d^3*g*e^2 - 4*a*b^3*c^2*d^3*g*e^2 - 26*a^2*b*c^3*d^3*g*
e^2 + b^5*c*d^2*f*e^3 - 18*a*b^3*c^2*d^2*f*e^3 - 34*a^2*b*c^3*d^2*f*e^3 + b^6*d^2*g*e^3 + 26*a^2*b^2*c^2*d^2*g
*e^3 + 12*a^3*c^3*d^2*g*e^3 - b^6*d*f*e^4 + 6*a*b^4*c*d*f*e^4 + 10*a^2*b^2*c^2*d*f*e^4 + 18*a^3*c^3*d*f*e^4 -
a*b^5*d*g*e^4 - 29*a^3*b*c^2*d*g*e^4 + a*b^5*f*e^5 - 6*a^2*b^3*c*f*e^5 - a^3*b*c^2*f*e^5 + 2*a^3*b^2*c*g*e^5 +
10*a^4*c^2*g*e^5)*x)/((c*d^2 - b*d*e + a*e^2)^3*(c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2)