3.372 \(\int \frac{d+e x+f x^2+g x^3+h x^4+i x^5}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=528 \[ \frac{2 c x \left (c^3 \left (-10 a^2 h-3 a b g+b^2 f\right )-b^3 c (15 a i+b h)-c^4 (3 b e-2 a f)+a b c^2 (25 a i+8 b h)+2 b^5 i+6 c^5 d\right )-b^2 c^2 \left (39 a^2 i-5 a c g+3 c^2 e\right )+2 b c^3 \left (11 a^2 h+a c f+3 c^2 d\right )-16 a^2 c^3 (c g-2 a i)+b^3 c^2 (c f-8 a h)-b^4 c (c g-11 a i)+b^5 c h+b^6 (-i)}{2 c^4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{x \left (c^3 \left (2 a^2 h+3 a b g+b^2 f\right )-b c^2 \left (5 a^2 i+4 a b h+b^2 g\right )+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right )+b c^2 \left (-3 a^2 h+a c f+c^2 d\right )-2 a c^2 \left (a^2 i-a c g+c^2 e\right )-a b^2 c (c g-4 a i)+a b^3 c h-a b^4 i}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c^3 \left (6 a^2 h-3 a b g+b^2 f\right )-30 a^2 b c^2 i+10 a b^3 c i-c^4 (6 b e-4 a f)+b^5 (-i)+12 c^5 d\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{i \log \left (a+b x+c x^2\right )}{2 c^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.31158, antiderivative size = 528, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {1660, 634, 618, 206, 628} \[ \frac{2 c x \left (c^3 \left (-10 a^2 h-3 a b g+b^2 f\right )-b^3 c (15 a i+b h)-c^4 (3 b e-2 a f)+a b c^2 (25 a i+8 b h)+2 b^5 i+6 c^5 d\right )-b^2 c^2 \left (39 a^2 i-5 a c g+3 c^2 e\right )+2 b c^3 \left (11 a^2 h+a c f+3 c^2 d\right )-16 a^2 c^3 (c g-2 a i)+b^3 c^2 (c f-8 a h)-b^4 c (c g-11 a i)+b^5 c h+b^6 (-i)}{2 c^4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{x \left (c^3 \left (2 a^2 h+3 a b g+b^2 f\right )-b c^2 \left (5 a^2 i+4 a b h+b^2 g\right )+b^3 c (5 a i+b h)-c^4 (2 a f+b e)+b^5 (-i)+2 c^5 d\right )+b c^2 \left (-3 a^2 h+a c f+c^2 d\right )-2 a c^2 \left (a^2 i-a c g+c^2 e\right )-a b^2 c (c g-4 a i)+a b^3 c h-a b^4 i}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c^3 \left (6 a^2 h-3 a b g+b^2 f\right )-30 a^2 b c^2 i+10 a b^3 c i-c^4 (6 b e-4 a f)+b^5 (-i)+12 c^5 d\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{i \log \left (a+b x+c x^2\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x + c*x^2)^3,x]

[Out]

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

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*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[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

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+f x^2+g x^3+h x^4+372 x^5}{\left (a+b x+c x^2\right )^3} \, dx &=\frac{744 a^3 c^2-b c^4 d-a^2 c \left (1488 b^2+2 c^2 g-3 b c h\right )+a \left (372 b^4+2 c^4 e-b c^3 f+b^2 c^2 g-b^3 c h\right )+\left (372 b^5-b^3 c (1860 a-c g)+b c^2 \left (1860 a^2+c^2 e-3 a c g\right )-b^4 c h-b^2 c^2 (c f-4 a h)-2 c^3 \left (c^2 d-a c f+a^2 h\right )\right ) x}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{-\frac{372 b^5-b^3 c (1116 a-c g)-b c^2 \left (372 a^2-3 c^2 e+a c g\right )-b^4 c h-b^2 c^2 (c f-2 a h)-2 c^3 \left (3 c^2 d+a c f-a^2 h\right )}{c^4}-\frac{2 \left (b^2-4 a c\right ) \left (372 b^2-c (372 a-c g)-b c h\right ) x}{c^3}+\frac{2 \left (b^2-4 a c\right ) (372 b-c h) x^2}{c^2}+744 \left (4 a-\frac{b^2}{c}\right ) x^3}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=\frac{744 a^3 c^2-b c^4 d-a^2 c \left (1488 b^2+2 c^2 g-3 b c h\right )+a \left (372 b^4+2 c^4 e-b c^3 f+b^2 c^2 g-b^3 c h\right )+\left (372 b^5-b^3 c (1860 a-c g)+b c^2 \left (1860 a^2+c^2 e-3 a c g\right )-b^4 c h-b^2 c^2 (c f-4 a h)-2 c^3 \left (c^2 d-a c f+a^2 h\right )\right ) x}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{372 b^6-16 a^2 c^3 (744 a-c g)-b^4 c (4092 a-c g)+b^2 c^2 \left (14508 a^2+3 c^2 e-5 a c g\right )-b^5 c h-b^3 c^2 (c f-8 a h)-2 b c^3 \left (3 c^2 d+a c f+11 a^2 h\right )-2 c \left (744 b^5-5580 a b^3 c+3 b c^2 \left (3100 a^2-c^2 e-a c g\right )-b^4 c h+b^2 c^2 (c f+8 a h)+2 c^3 \left (3 c^2 d+a c f-5 a^2 h\right )\right ) x}{2 c^4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{\int \frac{2 \left (6 c^2 d-3 b c e+b^2 f+a \left (\frac{372 b^3}{c^2}+2 c f-3 b g\right )-6 a^2 \left (\frac{434 b}{c}-h\right )\right )+\frac{744 \left (b^2-4 a c\right )^2 x}{c^2}}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^2}\\ &=\frac{744 a^3 c^2-b c^4 d-a^2 c \left (1488 b^2+2 c^2 g-3 b c h\right )+a \left (372 b^4+2 c^4 e-b c^3 f+b^2 c^2 g-b^3 c h\right )+\left (372 b^5-b^3 c (1860 a-c g)+b c^2 \left (1860 a^2+c^2 e-3 a c g\right )-b^4 c h-b^2 c^2 (c f-4 a h)-2 c^3 \left (c^2 d-a c f+a^2 h\right )\right ) x}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{372 b^6-16 a^2 c^3 (744 a-c g)-b^4 c (4092 a-c g)+b^2 c^2 \left (14508 a^2+3 c^2 e-5 a c g\right )-b^5 c h-b^3 c^2 (c f-8 a h)-2 b c^3 \left (3 c^2 d+a c f+11 a^2 h\right )-2 c \left (744 b^5-5580 a b^3 c+3 b c^2 \left (3100 a^2-c^2 e-a c g\right )-b^4 c h+b^2 c^2 (c f+8 a h)+2 c^3 \left (3 c^2 d+a c f-5 a^2 h\right )\right ) x}{2 c^4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{186 \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{c^3}-\frac{\left (186 b^5-1860 a b^3 c-b^2 c^3 f+3 b c^2 \left (1860 a^2+c^2 e+a c g\right )-2 c^3 \left (3 c^2 d+a c f+3 a^2 h\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )^2}\\ &=\frac{744 a^3 c^2-b c^4 d-a^2 c \left (1488 b^2+2 c^2 g-3 b c h\right )+a \left (372 b^4+2 c^4 e-b c^3 f+b^2 c^2 g-b^3 c h\right )+\left (372 b^5-b^3 c (1860 a-c g)+b c^2 \left (1860 a^2+c^2 e-3 a c g\right )-b^4 c h-b^2 c^2 (c f-4 a h)-2 c^3 \left (c^2 d-a c f+a^2 h\right )\right ) x}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{372 b^6-16 a^2 c^3 (744 a-c g)-b^4 c (4092 a-c g)+b^2 c^2 \left (14508 a^2+3 c^2 e-5 a c g\right )-b^5 c h-b^3 c^2 (c f-8 a h)-2 b c^3 \left (3 c^2 d+a c f+11 a^2 h\right )-2 c \left (744 b^5-5580 a b^3 c+3 b c^2 \left (3100 a^2-c^2 e-a c g\right )-b^4 c h+b^2 c^2 (c f+8 a h)+2 c^3 \left (3 c^2 d+a c f-5 a^2 h\right )\right ) x}{2 c^4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{186 \log \left (a+b x+c x^2\right )}{c^3}+\frac{\left (2 \left (186 b^5-1860 a b^3 c-b^2 c^3 f+3 b c^2 \left (1860 a^2+c^2 e+a c g\right )-2 c^3 \left (3 c^2 d+a c f+3 a^2 h\right )\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{744 a^3 c^2-b c^4 d-a^2 c \left (1488 b^2+2 c^2 g-3 b c h\right )+a \left (372 b^4+2 c^4 e-b c^3 f+b^2 c^2 g-b^3 c h\right )+\left (372 b^5-b^3 c (1860 a-c g)+b c^2 \left (1860 a^2+c^2 e-3 a c g\right )-b^4 c h-b^2 c^2 (c f-4 a h)-2 c^3 \left (c^2 d-a c f+a^2 h\right )\right ) x}{2 c^4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{372 b^6-16 a^2 c^3 (744 a-c g)-b^4 c (4092 a-c g)+b^2 c^2 \left (14508 a^2+3 c^2 e-5 a c g\right )-b^5 c h-b^3 c^2 (c f-8 a h)-2 b c^3 \left (3 c^2 d+a c f+11 a^2 h\right )-2 c \left (744 b^5-5580 a b^3 c+3 b c^2 \left (3100 a^2-c^2 e-a c g\right )-b^4 c h+b^2 c^2 (c f+8 a h)+2 c^3 \left (3 c^2 d+a c f-5 a^2 h\right )\right ) x}{2 c^4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{2 \left (186 b^5-1860 a b^3 c-b^2 c^3 f+3 b c^2 \left (1860 a^2+c^2 e+a c g\right )-2 c^3 \left (3 c^2 d+a c f+3 a^2 h\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{186 \log \left (a+b x+c x^2\right )}{c^3}\\ \end{align*}

Mathematica [A]  time = 1.31935, size = 488, normalized size = 0.92 \[ \frac{\frac{b^2 c \left (-4 a^2 i+a c (g+4 h x)-c^2 f x\right )+b c^2 \left (a^2 (3 h+5 i x)-a c (f+3 g x)+c^2 (e x-d)\right )+2 c^2 \left (-a^2 c (g+h x)+a^3 i+a c^2 (e+f x)-c^3 d x\right )+b^3 c (c g x-a (h+5 i x))+b^4 (a i-c h x)+b^5 i x}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{b^2 c^2 \left (-39 a^2 i+a c (5 g+16 h x)+c^2 (2 f x-3 e)\right )+2 b c^3 \left (a^2 (11 h+25 i x)+a c (f-3 g x)+3 c^2 (d-e x)\right )+4 c^3 \left (-a^2 c (4 g+5 h x)+8 a^3 i+a c^2 f x+3 c^3 d x\right )+b^3 c^2 (-8 a h-30 a i x+c f)-b^4 c (c (g+2 h x)-11 a i)+b^5 c (h+4 i x)+b^6 (-i)}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{2 c \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (2 c^3 \left (6 a^2 h-3 a b g+b^2 f\right )-30 a^2 b c^2 i+10 a b^3 c i+c^4 (4 a f-6 b e)+b^5 (-i)+12 c^5 d\right )}{\left (4 a c-b^2\right )^{5/2}}+c i \log (a+x (b+c x))}{2 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5)/(a + b*x + c*x^2)^3,x]

[Out]

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

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Maple [B]  time = 0.191, size = 1244, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^3,x)

[Out]

((25*a^2*b*c^2*i-10*a^2*c^3*h-15*a*b^3*c*i+8*a*b^2*c^2*h-3*a*b*c^3*g+2*a*c^4*f+2*b^5*i-b^4*c*h+b^2*c^3*f-3*b*c
^4*e+6*c^5*d)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*(32*a^3*c^3*i+11*a^2*b^2*c^2*i+2*a^2*b*c^3*h-16*a^2*c^4*g
-19*a*b^4*c*i+8*a*b^3*c^2*h-a*b^2*c^3*g+6*a*b*c^4*f+3*b^6*i-b^5*c*h-b^4*c^2*g+3*b^3*c^3*f-9*b^2*c^4*e+18*b*c^5
*d)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x^2+(31*a^3*b*c^2*i-6*a^3*c^3*h-22*a^2*b^3*c*i+10*a^2*b^2*c^2*h-5*a^2*b*c^3
*g-2*a^2*c^4*f+3*a*b^5*i-a*b^4*c*h-a*b^3*c^2*g+5*a*b^2*c^3*f-5*a*b*c^4*e+10*a*c^5*d-b^3*c^3*e+2*b^2*c^4*d)/(16
*a^2*c^2-8*a*b^2*c+b^4)/c^3*x+1/2/c^3*(24*a^4*c^2*i-21*a^3*b^2*c*i+10*a^3*b*c^2*h-8*a^3*c^3*g+3*a^2*b^4*i-a^2*
b^3*c*h-a^2*b^2*c^2*g+6*a^2*b*c^3*f-8*a^2*c^4*e-a*b^2*c^3*e+10*a*b*c^4*d-b^3*c^3*d)/(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*i-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*i+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*i-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*i+12/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arc
tan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*h+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*i-6/(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*g+4*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*f+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))*b^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*e+12*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan(
(2*c*x+b)/(4*a*c-b^2)^(1/2))*d-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*i

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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((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.84246, size = 7274, normalized size = 13.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.15391, size = 887, normalized size = 1.68 \begin{align*} \frac{{\left (12 \, c^{5} d i + 2 \, b^{2} c^{3} f i + 4 \, a c^{4} f i - 6 \, a b c^{3} g i + 12 \, a^{2} c^{3} h i - 6 \, b c^{4} i e + b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{3} i - 8 \, a b^{2} c^{4} i + 16 \, a^{2} c^{5} i\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{i \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac{b^{3} c^{3} d - 10 \, a b c^{4} d - 6 \, a^{2} b c^{3} f + a^{2} b^{2} c^{2} g + 8 \, a^{3} c^{3} g + a^{2} b^{3} c h - 10 \, a^{3} b c^{2} h - 3 \, a^{2} b^{4} i + 21 \, a^{3} b^{2} c i - 24 \, a^{4} c^{2} i + a b^{2} c^{3} e + 8 \, a^{2} c^{4} e - 2 \,{\left (6 \, c^{6} d + b^{2} c^{4} f + 2 \, a c^{5} f - 3 \, a b c^{4} g - b^{4} c^{2} h + 8 \, a b^{2} c^{3} h - 10 \, a^{2} c^{4} h + 2 \, b^{5} c i - 15 \, a b^{3} c^{2} i + 25 \, a^{2} b c^{3} i - 3 \, b c^{5} e\right )} x^{3} -{\left (18 \, b c^{5} d + 3 \, b^{3} c^{3} f + 6 \, a b c^{4} f - b^{4} c^{2} g - a b^{2} c^{3} g - 16 \, a^{2} c^{4} g - b^{5} c h + 8 \, a b^{3} c^{2} h + 2 \, a^{2} b c^{3} h + 3 \, b^{6} i - 19 \, a b^{4} c i + 11 \, a^{2} b^{2} c^{2} i + 32 \, a^{3} c^{3} i - 9 \, b^{2} c^{4} e\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{4} d + 10 \, a c^{5} d + 5 \, a b^{2} c^{3} f - 2 \, a^{2} c^{4} f - a b^{3} c^{2} g - 5 \, a^{2} b c^{3} g - a b^{4} c h + 10 \, a^{2} b^{2} c^{2} h - 6 \, a^{3} c^{3} h + 3 \, a b^{5} i - 22 \, a^{2} b^{3} c i + 31 \, a^{3} b c^{2} i - b^{3} c^{3} e - 5 \, a b c^{4} e\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )}^{2} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

(12*c^5*d*i + 2*b^2*c^3*f*i + 4*a*c^4*f*i - 6*a*b*c^3*g*i + 12*a^2*c^3*h*i - 6*b*c^4*i*e + b^5 - 10*a*b^3*c +
30*a^2*b*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^3*i - 8*a*b^2*c^4*i + 16*a^2*c^5*i)*sqrt(-b^2 + 4
*a*c)) + 1/2*i*log(c*x^2 + b*x + a)/c^3 - 1/2*(b^3*c^3*d - 10*a*b*c^4*d - 6*a^2*b*c^3*f + a^2*b^2*c^2*g + 8*a^
3*c^3*g + a^2*b^3*c*h - 10*a^3*b*c^2*h - 3*a^2*b^4*i + 21*a^3*b^2*c*i - 24*a^4*c^2*i + a*b^2*c^3*e + 8*a^2*c^4
*e - 2*(6*c^6*d + b^2*c^4*f + 2*a*c^5*f - 3*a*b*c^4*g - b^4*c^2*h + 8*a*b^2*c^3*h - 10*a^2*c^4*h + 2*b^5*c*i -
15*a*b^3*c^2*i + 25*a^2*b*c^3*i - 3*b*c^5*e)*x^3 - (18*b*c^5*d + 3*b^3*c^3*f + 6*a*b*c^4*f - b^4*c^2*g - a*b^
2*c^3*g - 16*a^2*c^4*g - b^5*c*h + 8*a*b^3*c^2*h + 2*a^2*b*c^3*h + 3*b^6*i - 19*a*b^4*c*i + 11*a^2*b^2*c^2*i +
32*a^3*c^3*i - 9*b^2*c^4*e)*x^2 - 2*(2*b^2*c^4*d + 10*a*c^5*d + 5*a*b^2*c^3*f - 2*a^2*c^4*f - a*b^3*c^2*g - 5
*a^2*b*c^3*g - a*b^4*c*h + 10*a^2*b^2*c^2*h - 6*a^3*c^3*h + 3*a*b^5*i - 22*a^2*b^3*c*i + 31*a^3*b*c^2*i - b^3*
c^3*e - 5*a*b*c^4*e)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*c^3)