3.2372 \(\int \frac{(d+e x)^3 (f+g x)}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=195 \[ -\frac{6 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (2 a e g-b d g-b e f+2 c d f)}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^3 (-2 a g+x (2 c f-b g)+b f)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 (d+e x) (-2 a e+x (2 c d-b e)+b d) (2 a e g-b d g-b e f+2 c d f)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

[Out]

-((d + e*x)^3*(b*f - 2*a*g + (2*c*f - b*g)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3*(2*c*d*f - b*e*f - b
*d*g + 2*a*e*g)*(d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (6*(c*d^2 -
b*d*e + a*e^2)*(2*c*d*f - b*e*f - b*d*g + 2*a*e*g)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2
)

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Rubi [A]  time = 0.161656, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {804, 722, 618, 206} \[ -\frac{6 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (2 a e g-b d g-b e f+2 c d f)}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x)^3 (-2 a g+x (2 c f-b g)+b f)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 (d+e x) (-2 a e+x (2 c d-b e)+b d) (2 a e g-b d g-b e f+2 c d f)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d + e*x)^3*(b*f - 2*a*g + (2*c*f - b*g)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3*(2*c*d*f - b*e*f - b
*d*g + 2*a*e*g)*(d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (6*(c*d^2 -
b*d*e + a*e^2)*(2*c*d*f - b*e*f - b*d*g + 2*a*e*g)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2
)

Rule 804

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

Rule 722

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

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

Rubi steps

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

Mathematica [B]  time = 1.19428, size = 550, normalized size = 2.82 \[ \frac{1}{2} \left (\frac{2 b c^2 \left (11 a^2 e^3 g+3 a c e \left (d^2 g+d e (f-3 g x)-e^2 f x\right )+3 c^2 d^2 (d (f-g x)-3 e f x)\right )-4 c^3 \left (a^2 e^2 (12 d g+4 e f+5 e g x)-3 a c d e x (d g+e f)-3 c^2 d^3 f x\right )+b^2 c^2 \left (a e^2 (15 d g+5 e f+16 e g x)-3 c d \left (d^2 g+3 d e f-2 d e g x-2 e^2 f x\right )\right )+b^3 c e \left (3 c d (d g+e f)-8 a e^2 g\right )-b^4 c e^2 (3 d g+e (f+2 g x))+b^5 e^3 g}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{b c \left (-3 a^2 e^3 g+3 a c e \left (d^2 g+d e (f+3 g x)+e^2 f x\right )+c^2 d^2 (d (f-g x)-3 e f x)\right )+2 c^2 \left (a^2 e^2 (3 d g+e (f+g x))-a c d \left (d^2 g+3 d e (f+g x)+3 e^2 f x\right )+c^2 d^3 f x\right )+b^3 e^2 (a e g-c x (3 d g+e f))-b^2 c e (a e (3 d g+e f+4 e g x)-3 c d x (d g+e f))+b^4 e^3 g x}{c^3 \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{12 \left (e (a e-b d)+c d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) (2 a e g-b (d g+e f)+2 c d f)}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^5*e^3*g + b^3*c*e*(-8*a*e^2*g + 3*c*d*(e*f + d*g)) - b^4*c*e^2*(3*d*g + e*(f + 2*g*x)) - 4*c^3*(-3*c^2*d^3
*f*x - 3*a*c*d*e*(e*f + d*g)*x + a^2*e^2*(4*e*f + 12*d*g + 5*e*g*x)) + b^2*c^2*(a*e^2*(5*e*f + 15*d*g + 16*e*g
*x) - 3*c*d*(3*d*e*f + d^2*g - 2*e^2*f*x - 2*d*e*g*x)) + 2*b*c^2*(11*a^2*e^3*g + 3*a*c*e*(d^2*g - e^2*f*x + d*
e*(f - 3*g*x)) + 3*c^2*d^2*(-3*e*f*x + d*(f - g*x))))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (b^4*e^3*g*x +
b^3*e^2*(a*e*g - c*(e*f + 3*d*g)*x) - b^2*c*e*(-3*c*d*(e*f + d*g)*x + a*e*(e*f + 3*d*g + 4*e*g*x)) + 2*c^2*(c
^2*d^3*f*x + a^2*e^2*(3*d*g + e*(f + g*x)) - a*c*d*(d^2*g + 3*e^2*f*x + 3*d*e*(f + g*x))) + b*c*(-3*a^2*e^3*g
+ c^2*d^2*(-3*e*f*x + d*(f - g*x)) + 3*a*c*e*(d^2*g + e^2*f*x + d*e*(f + 3*g*x))))/(c^3*(-b^2 + 4*a*c)*(a + x*
(b + c*x))^2) + (12*(c*d^2 + e*(-(b*d) + a*e))*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))*ArcTan[(b + 2*c*x)/Sqrt[-b^
2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/2

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Maple [B]  time = 0.011, size = 1458, normalized size = 7.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(10*a^2*c^2*e^3*g-8*a*b^2*c*e^3*g+9*a*b*c^2*d*e^2*g+3*a*b*c^2*e^3*f-6*a*c^3*d^2*e*g-6*a*c^3*d*e^2*f+b^4*e^3*
g-3*b^2*c^2*d^2*e*g-3*b^2*c^2*d*e^2*f+3*b*c^3*d^3*g+9*b*c^3*d^2*e*f-6*c^4*d^3*f)/c/(16*a^2*c^2-8*a*b^2*c+b^4)*
x^3+1/2*(2*a^2*b*c^2*e^3*g-48*a^2*c^3*d*e^2*g-16*a^2*c^3*e^3*f+8*a*b^3*c*e^3*g-3*a*b^2*c^2*d*e^2*g-a*b^2*c^2*e
^3*f+18*a*b*c^3*d^2*e*g+18*a*b*c^3*d*e^2*f-b^5*e^3*g-3*b^4*c*d*e^2*g-b^4*c*e^3*f+9*b^3*c^2*d^2*e*g+9*b^3*c^2*d
*e^2*f-9*b^2*c^3*d^3*g-27*b^2*c^3*d^2*e*f+18*b*c^4*d^3*f)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x^2-(6*a^3*c^2*e^3*g-
10*a^2*b^2*c*e^3*g+15*a^2*b*c^2*d*e^2*g+5*a^2*b*c^2*e^3*f+6*a^2*c^3*d^2*e*g+6*a^2*c^3*d*e^2*f+a*b^4*e^3*g+3*a*
b^3*c*d*e^2*g+a*b^3*c*e^3*f-15*a*b^2*c^2*d^2*e*g-15*a*b^2*c^2*d*e^2*f+5*a*b*c^3*d^3*g+15*a*b*c^3*d^2*e*f-10*a*
c^4*d^3*f+b^3*c^2*d^3*g+3*b^3*c^2*d^2*e*f-2*b^2*c^3*d^3*f)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x+1/2/c^2*(10*a^3*b*
c*e^3*g-24*a^3*c^2*d*e^2*g-8*a^3*c^2*e^3*f-a^2*b^3*e^3*g-3*a^2*b^2*c*d*e^2*g-a^2*b^2*c*e^3*f+18*a^2*b*c^2*d^2*
e*g+18*a^2*b*c^2*d*e^2*f-8*a^2*c^3*d^3*g-24*a^2*c^3*d^2*e*f-a*b^2*c^2*d^3*g-3*a*b^2*c^2*d^2*e*f+10*a*b*c^3*d^3
*f-b^3*c^2*d^3*f)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+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^3*g-18/(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^2*g-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*e^3*f+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*c*d^2*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))*a*c*d*e^2*f+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))*b^2*d^2*e*g+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))*b^2*d*e^2*f-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))*b*c*d^3*g-18/(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*c*d^2*e*f+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))*c^2*d^3*f

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.68122, size = 7891, normalized size = 40.47 \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)^3*(g*x+f)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[1/2*(2*(3*(2*(b^2*c^5 - 4*a*c^6)*d^3 - 3*(b^3*c^4 - 4*a*b*c^5)*d^2*e + (b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d*
e^2 - (a*b^3*c^3 - 4*a^2*b*c^4)*e^3)*f - (3*(b^3*c^4 - 4*a*b*c^5)*d^3 - 3*(b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*
d^2*e + 9*(a*b^3*c^3 - 4*a^2*b*c^4)*d*e^2 + (b^6*c - 12*a*b^4*c^2 + 42*a^2*b^2*c^3 - 40*a^3*c^4)*e^3)*g)*x^3 +
((18*(b^3*c^4 - 4*a*b*c^5)*d^3 - 27*(b^4*c^3 - 4*a*b^2*c^4)*d^2*e + 9*(b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d
*e^2 - (b^6*c - 3*a*b^4*c^2 + 12*a^2*b^2*c^3 - 64*a^3*c^4)*e^3)*f - (9*(b^4*c^3 - 4*a*b^2*c^4)*d^3 - 9*(b^5*c^
2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d^2*e + 3*(b^6*c - 3*a*b^4*c^2 + 12*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^2 + (b^7 - 12
*a*b^5*c + 30*a^2*b^3*c^2 + 8*a^3*b*c^3)*e^3)*g)*x^2 + 6*(((2*c^6*d^3 - 3*b*c^5*d^2*e - a*b*c^4*e^3 + (b^2*c^4
+ 2*a*c^5)*d*e^2)*f - (b*c^5*d^3 + 3*a*b*c^4*d*e^2 - 2*a^2*c^4*e^3 - (b^2*c^4 + 2*a*c^5)*d^2*e)*g)*x^4 + 2*((
2*b*c^5*d^3 - 3*b^2*c^4*d^2*e - a*b^2*c^3*e^3 + (b^3*c^3 + 2*a*b*c^4)*d*e^2)*f - (b^2*c^4*d^3 + 3*a*b^2*c^3*d*
e^2 - 2*a^2*b*c^3*e^3 - (b^3*c^3 + 2*a*b*c^4)*d^2*e)*g)*x^3 + ((2*(b^2*c^4 + 2*a*c^5)*d^3 - 3*(b^3*c^3 + 2*a*b
*c^4)*d^2*e + (b^4*c^2 + 4*a*b^2*c^3 + 4*a^2*c^4)*d*e^2 - (a*b^3*c^2 + 2*a^2*b*c^3)*e^3)*f - ((b^3*c^3 + 2*a*b
*c^4)*d^3 - (b^4*c^2 + 4*a*b^2*c^3 + 4*a^2*c^4)*d^2*e + 3*(a*b^3*c^2 + 2*a^2*b*c^3)*d*e^2 - 2*(a^2*b^2*c^2 + 2
*a^3*c^3)*e^3)*g)*x^2 + (2*a^2*c^4*d^3 - 3*a^2*b*c^3*d^2*e - a^3*b*c^2*e^3 + (a^2*b^2*c^2 + 2*a^3*c^3)*d*e^2)*
f - (a^2*b*c^3*d^3 + 3*a^3*b*c^2*d*e^2 - 2*a^4*c^2*e^3 - (a^2*b^2*c^2 + 2*a^3*c^3)*d^2*e)*g + 2*((2*a*b*c^4*d^
3 - 3*a*b^2*c^3*d^2*e - a^2*b^2*c^2*e^3 + (a*b^3*c^2 + 2*a^2*b*c^3)*d*e^2)*f - (a*b^2*c^3*d^3 + 3*a^2*b^2*c^2*
d*e^2 - 2*a^3*b*c^2*e^3 - (a*b^3*c^2 + 2*a^2*b*c^3)*d^2*e)*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^2 - 14*a*b^3*c^3 + 40*a^2*b*c^4)*d^3
+ 3*(a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*a^3*c^4)*d^2*e - 18*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d*e^2 + (a^2*b^4*c + 4*a^
3*b^2*c^2 - 32*a^4*c^3)*e^3)*f - ((a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*a^3*c^4)*d^3 - 18*(a^2*b^3*c^2 - 4*a^3*b*c^3
)*d^2*e + 3*(a^2*b^4*c + 4*a^3*b^2*c^2 - 32*a^4*c^3)*d*e^2 + (a^2*b^5 - 14*a^3*b^3*c + 40*a^4*b*c^2)*e^3)*g +
2*((2*(b^4*c^3 + a*b^2*c^4 - 20*a^2*c^5)*d^3 - 3*(b^5*c^2 + a*b^3*c^3 - 20*a^2*b*c^4)*d^2*e + 3*(5*a*b^4*c^2 -
22*a^2*b^2*c^3 + 8*a^3*c^4)*d*e^2 - (a*b^5*c + a^2*b^3*c^2 - 20*a^3*b*c^3)*e^3)*f - ((b^5*c^2 + a*b^3*c^3 - 2
0*a^2*b*c^4)*d^3 - 3*(5*a*b^4*c^2 - 22*a^2*b^2*c^3 + 8*a^3*c^4)*d^2*e + 3*(a*b^5*c + a^2*b^3*c^2 - 20*a^3*b*c^
3)*d*e^2 + (a*b^6 - 14*a^2*b^4*c + 46*a^3*b^2*c^2 - 24*a^4*c^3)*e^3)*g)*x)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*
a^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*x^4 + 2*(b^7*c^3 - 12*a*b^5*
c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^3 + (b^8*c^2 - 10*a*b^6*c^3 + 24*a^2*b^4*c^4 + 32*a^3*b^2*c^5 - 128*a^4
*c^6)*x^2 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 - 64*a^4*b*c^5)*x), 1/2*(2*(3*(2*(b^2*c^5 - 4*a*c^6
)*d^3 - 3*(b^3*c^4 - 4*a*b*c^5)*d^2*e + (b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d*e^2 - (a*b^3*c^3 - 4*a^2*b*c^4)*
e^3)*f - (3*(b^3*c^4 - 4*a*b*c^5)*d^3 - 3*(b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^2*e + 9*(a*b^3*c^3 - 4*a^2*b*c
^4)*d*e^2 + (b^6*c - 12*a*b^4*c^2 + 42*a^2*b^2*c^3 - 40*a^3*c^4)*e^3)*g)*x^3 + ((18*(b^3*c^4 - 4*a*b*c^5)*d^3
- 27*(b^4*c^3 - 4*a*b^2*c^4)*d^2*e + 9*(b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d*e^2 - (b^6*c - 3*a*b^4*c^2 + 12
*a^2*b^2*c^3 - 64*a^3*c^4)*e^3)*f - (9*(b^4*c^3 - 4*a*b^2*c^4)*d^3 - 9*(b^5*c^2 - 2*a*b^3*c^3 - 8*a^2*b*c^4)*d
^2*e + 3*(b^6*c - 3*a*b^4*c^2 + 12*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^2 + (b^7 - 12*a*b^5*c + 30*a^2*b^3*c^2 + 8*a^
3*b*c^3)*e^3)*g)*x^2 - 12*(((2*c^6*d^3 - 3*b*c^5*d^2*e - a*b*c^4*e^3 + (b^2*c^4 + 2*a*c^5)*d*e^2)*f - (b*c^5*d
^3 + 3*a*b*c^4*d*e^2 - 2*a^2*c^4*e^3 - (b^2*c^4 + 2*a*c^5)*d^2*e)*g)*x^4 + 2*((2*b*c^5*d^3 - 3*b^2*c^4*d^2*e -
a*b^2*c^3*e^3 + (b^3*c^3 + 2*a*b*c^4)*d*e^2)*f - (b^2*c^4*d^3 + 3*a*b^2*c^3*d*e^2 - 2*a^2*b*c^3*e^3 - (b^3*c^
3 + 2*a*b*c^4)*d^2*e)*g)*x^3 + ((2*(b^2*c^4 + 2*a*c^5)*d^3 - 3*(b^3*c^3 + 2*a*b*c^4)*d^2*e + (b^4*c^2 + 4*a*b^
2*c^3 + 4*a^2*c^4)*d*e^2 - (a*b^3*c^2 + 2*a^2*b*c^3)*e^3)*f - ((b^3*c^3 + 2*a*b*c^4)*d^3 - (b^4*c^2 + 4*a*b^2*
c^3 + 4*a^2*c^4)*d^2*e + 3*(a*b^3*c^2 + 2*a^2*b*c^3)*d*e^2 - 2*(a^2*b^2*c^2 + 2*a^3*c^3)*e^3)*g)*x^2 + (2*a^2*
c^4*d^3 - 3*a^2*b*c^3*d^2*e - a^3*b*c^2*e^3 + (a^2*b^2*c^2 + 2*a^3*c^3)*d*e^2)*f - (a^2*b*c^3*d^3 + 3*a^3*b*c^
2*d*e^2 - 2*a^4*c^2*e^3 - (a^2*b^2*c^2 + 2*a^3*c^3)*d^2*e)*g + 2*((2*a*b*c^4*d^3 - 3*a*b^2*c^3*d^2*e - a^2*b^2
*c^2*e^3 + (a*b^3*c^2 + 2*a^2*b*c^3)*d*e^2)*f - (a*b^2*c^3*d^3 + 3*a^2*b^2*c^2*d*e^2 - 2*a^3*b*c^2*e^3 - (a*b^
3*c^2 + 2*a^2*b*c^3)*d^2*e)*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^2 - 14*a*b^3*c^3 + 40*a^2*b*c^4)*d^3 + 3*(a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*a^3*c^4)*d^2*e - 18*(a^2*b^3*c
^2 - 4*a^3*b*c^3)*d*e^2 + (a^2*b^4*c + 4*a^3*b^2*c^2 - 32*a^4*c^3)*e^3)*f - ((a*b^4*c^2 + 4*a^2*b^2*c^3 - 32*a
^3*c^4)*d^3 - 18*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e + 3*(a^2*b^4*c + 4*a^3*b^2*c^2 - 32*a^4*c^3)*d*e^2 + (a^2*b
^5 - 14*a^3*b^3*c + 40*a^4*b*c^2)*e^3)*g + 2*((2*(b^4*c^3 + a*b^2*c^4 - 20*a^2*c^5)*d^3 - 3*(b^5*c^2 + a*b^3*c
^3 - 20*a^2*b*c^4)*d^2*e + 3*(5*a*b^4*c^2 - 22*a^2*b^2*c^3 + 8*a^3*c^4)*d*e^2 - (a*b^5*c + a^2*b^3*c^2 - 20*a^
3*b*c^3)*e^3)*f - ((b^5*c^2 + a*b^3*c^3 - 20*a^2*b*c^4)*d^3 - 3*(5*a*b^4*c^2 - 22*a^2*b^2*c^3 + 8*a^3*c^4)*d^2
*e + 3*(a*b^5*c + a^2*b^3*c^2 - 20*a^3*b*c^3)*d*e^2 + (a*b^6 - 14*a^2*b^4*c + 46*a^3*b^2*c^2 - 24*a^4*c^3)*e^3
)*g)*x)/(a^2*b^6*c^2 - 12*a^3*b^4*c^3 + 48*a^4*b^2*c^4 - 64*a^5*c^5 + (b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6
- 64*a^3*c^7)*x^4 + 2*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*x^3 + (b^8*c^2 - 10*a*b^6*c^3
+ 24*a^2*b^4*c^4 + 32*a^3*b^2*c^5 - 128*a^4*c^6)*x^2 + 2*(a*b^7*c^2 - 12*a^2*b^5*c^3 + 48*a^3*b^3*c^4 - 64*a^4
*b*c^5)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.17445, size = 1322, normalized size = 6.78 \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)^3*(g*x+f)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

6*(2*c^2*d^3*f - b*c*d^3*g - 3*b*c*d^2*f*e + b^2*d^2*g*e + 2*a*c*d^2*g*e + b^2*d*f*e^2 + 2*a*c*d*f*e^2 - 3*a*b
*d*g*e^2 - a*b*f*e^3 + 2*a^2*g*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqr
t(-b^2 + 4*a*c)) + 1/2*(12*c^5*d^3*f*x^3 - 6*b*c^4*d^3*g*x^3 - 18*b*c^4*d^2*f*x^3*e + 6*b^2*c^3*d^2*g*x^3*e +
12*a*c^4*d^2*g*x^3*e + 18*b*c^4*d^3*f*x^2 - 9*b^2*c^3*d^3*g*x^2 + 6*b^2*c^3*d*f*x^3*e^2 + 12*a*c^4*d*f*x^3*e^2
- 18*a*b*c^3*d*g*x^3*e^2 - 27*b^2*c^3*d^2*f*x^2*e + 9*b^3*c^2*d^2*g*x^2*e + 18*a*b*c^3*d^2*g*x^2*e + 4*b^2*c^
3*d^3*f*x + 20*a*c^4*d^3*f*x - 2*b^3*c^2*d^3*g*x - 10*a*b*c^3*d^3*g*x - 6*a*b*c^3*f*x^3*e^3 - 2*b^4*c*g*x^3*e^
3 + 16*a*b^2*c^2*g*x^3*e^3 - 20*a^2*c^3*g*x^3*e^3 + 9*b^3*c^2*d*f*x^2*e^2 + 18*a*b*c^3*d*f*x^2*e^2 - 3*b^4*c*d
*g*x^2*e^2 - 3*a*b^2*c^2*d*g*x^2*e^2 - 48*a^2*c^3*d*g*x^2*e^2 - 6*b^3*c^2*d^2*f*x*e - 30*a*b*c^3*d^2*f*x*e + 3
0*a*b^2*c^2*d^2*g*x*e - 12*a^2*c^3*d^2*g*x*e - b^3*c^2*d^3*f + 10*a*b*c^3*d^3*f - a*b^2*c^2*d^3*g - 8*a^2*c^3*
d^3*g - b^4*c*f*x^2*e^3 - a*b^2*c^2*f*x^2*e^3 - 16*a^2*c^3*f*x^2*e^3 - b^5*g*x^2*e^3 + 8*a*b^3*c*g*x^2*e^3 + 2
*a^2*b*c^2*g*x^2*e^3 + 30*a*b^2*c^2*d*f*x*e^2 - 12*a^2*c^3*d*f*x*e^2 - 6*a*b^3*c*d*g*x*e^2 - 30*a^2*b*c^2*d*g*
x*e^2 - 3*a*b^2*c^2*d^2*f*e - 24*a^2*c^3*d^2*f*e + 18*a^2*b*c^2*d^2*g*e - 2*a*b^3*c*f*x*e^3 - 10*a^2*b*c^2*f*x
*e^3 - 2*a*b^4*g*x*e^3 + 20*a^2*b^2*c*g*x*e^3 - 12*a^3*c^2*g*x*e^3 + 18*a^2*b*c^2*d*f*e^2 - 3*a^2*b^2*c*d*g*e^
2 - 24*a^3*c^2*d*g*e^2 - a^2*b^2*c*f*e^3 - 8*a^3*c^2*f*e^3 - a^2*b^3*g*e^3 + 10*a^3*b*c*g*e^3)/((b^4*c^2 - 8*a
*b^2*c^3 + 16*a^2*c^4)*(c*x^2 + b*x + a)^2)