### 3.2209 $$\int \frac{1}{x^3 (a+b x+c x^2)^3} \, dx$$

Optimal. Leaf size=306 $\frac{24 a^2 c^2+2 b c x \left (2 b^2-11 a c\right )-25 a b^2 c+4 b^4}{2 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{3 b \left (70 a^2 b^2 c^2-70 a^3 c^3-21 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{5/2}}-\frac{3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5}+\frac{3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 x \left (b^2-4 a c\right )^2}+\frac{3 \log (x) \left (2 b^2-a c\right )}{a^5}+\frac{-2 a c+b^2+b c x}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}$

[Out]

(-3*(2*b^4 - 13*a*b^2*c + 16*a^2*c^2))/(2*a^3*(b^2 - 4*a*c)^2*x^2) + (3*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c))/(a^4*
(b^2 - 4*a*c)^2*x) + (b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)*x^2*(a + b*x + c*x^2)^2) + (4*b^4 - 25*a*b^2*c +
24*a^2*c^2 + 2*b*c*(2*b^2 - 11*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*x^2*(a + b*x + c*x^2)) + (3*b*(2*b^6 - 21*a*b^4
*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^5*(b^2 - 4*a*c)^(5/2)) + (3*(2*b^
2 - a*c)*Log[x])/a^5 - (3*(2*b^2 - a*c)*Log[a + b*x + c*x^2])/(2*a^5)

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Rubi [A]  time = 0.459514, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.438, Rules used = {740, 822, 800, 634, 618, 206, 628} $\frac{24 a^2 c^2+2 b c x \left (2 b^2-11 a c\right )-25 a b^2 c+4 b^4}{2 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{3 b \left (70 a^2 b^2 c^2-70 a^3 c^3-21 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{5/2}}-\frac{3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5}+\frac{3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 x \left (b^2-4 a c\right )^2}+\frac{3 \log (x) \left (2 b^2-a c\right )}{a^5}+\frac{-2 a c+b^2+b c x}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x^3*(a + b*x + c*x^2)^3),x]

[Out]

(-3*(2*b^4 - 13*a*b^2*c + 16*a^2*c^2))/(2*a^3*(b^2 - 4*a*c)^2*x^2) + (3*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c))/(a^4*
(b^2 - 4*a*c)^2*x) + (b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)*x^2*(a + b*x + c*x^2)^2) + (4*b^4 - 25*a*b^2*c +
24*a^2*c^2 + 2*b*c*(2*b^2 - 11*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*x^2*(a + b*x + c*x^2)) + (3*b*(2*b^6 - 21*a*b^4
*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^5*(b^2 - 4*a*c)^(5/2)) + (3*(2*b^
2 - a*c)*Log[x])/a^5 - (3*(2*b^2 - a*c)*Log[a + b*x + c*x^2])/(2*a^5)

Rule 740

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

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{1}{x^3 \left (a+b x+c x^2\right )^3} \, dx &=\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}-\frac{\int \frac{-4 \left (b^2-3 a c\right )-5 b c x}{x^3 \left (a+b x+c x^2\right )^2} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac{4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac{\int \frac{6 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )+6 b c \left (2 b^2-11 a c\right ) x}{x^3 \left (a+b x+c x^2\right )} \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac{4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac{\int \left (\frac{6 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{a x^3}+\frac{6 b \left (2 b^2-9 a c\right ) \left (-b^2+3 a c\right )}{a^2 x^2}-\frac{6 \left (-2 b^2+a c\right ) \left (-b^2+4 a c\right )^2}{a^3 x}+\frac{6 \left (-b \left (2 b^6-19 a b^4 c+55 a^2 b^2 c^2-43 a^3 c^3\right )-c \left (b^2-4 a c\right )^2 \left (2 b^2-a c\right ) x\right )}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac{3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 \left (b^2-4 a c\right )^2 x}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac{4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac{3 \left (2 b^2-a c\right ) \log (x)}{a^5}+\frac{3 \int \frac{-b \left (2 b^6-19 a b^4 c+55 a^2 b^2 c^2-43 a^3 c^3\right )-c \left (b^2-4 a c\right )^2 \left (2 b^2-a c\right ) x}{a+b x+c x^2} \, dx}{a^5 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac{3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 \left (b^2-4 a c\right )^2 x}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac{4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac{3 \left (2 b^2-a c\right ) \log (x)}{a^5}-\frac{\left (3 \left (2 b^2-a c\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 a^5}-\frac{\left (3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 a^5 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac{3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 \left (b^2-4 a c\right )^2 x}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac{4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac{3 \left (2 b^2-a c\right ) \log (x)}{a^5}-\frac{3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5}+\frac{\left (3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^5 \left (b^2-4 a c\right )^2}\\ &=-\frac{3 \left (2 b^4-13 a b^2 c+16 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}+\frac{3 b \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{a^4 \left (b^2-4 a c\right )^2 x}+\frac{b^2-2 a c+b c x}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x+c x^2\right )^2}+\frac{4 b^4-25 a b^2 c+24 a^2 c^2+2 b c \left (2 b^2-11 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x+c x^2\right )}+\frac{3 b \left (2 b^6-21 a b^4 c+70 a^2 b^2 c^2-70 a^3 c^3\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{5/2}}+\frac{3 \left (2 b^2-a c\right ) \log (x)}{a^5}-\frac{3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^5}\\ \end{align*}

Mathematica [A]  time = 0.526325, size = 269, normalized size = 0.88 $\frac{\frac{a^2 \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^3 c x+b^4\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{a \left (97 a^2 b^2 c^2+66 a^2 b c^3 x-32 a^3 c^3-42 a b^3 c^2 x-47 a b^4 c+6 b^5 c x+6 b^6\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-\frac{6 b \left (70 a^2 b^2 c^2-70 a^3 c^3-21 a b^4 c+2 b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}-\frac{a^2}{x^2}+6 \log (x) \left (2 b^2-a c\right )+3 \left (a c-2 b^2\right ) \log (a+x (b+c x))+\frac{6 a b}{x}}{2 a^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x^3*(a + b*x + c*x^2)^3),x]

[Out]

(-(a^2/x^2) + (6*a*b)/x + (a^2*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*c*x - 3*a*b*c^2*x))/((b^2 - 4*a*c)*(a + x*(b
+ c*x))^2) + (a*(6*b^6 - 47*a*b^4*c + 97*a^2*b^2*c^2 - 32*a^3*c^3 + 6*b^5*c*x - 42*a*b^3*c^2*x + 66*a^2*b*c^3
*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) - (6*b*(2*b^6 - 21*a*b^4*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTan[(b +
2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + 6*(2*b^2 - a*c)*Log[x] + 3*(-2*b^2 + a*c)*Log[a + x*(b + c
*x)])/(2*a^5)

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Maple [B]  time = 0.171, size = 1110, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(c*x^2+b*x+a)^3,x)

[Out]

115/2/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^2*c^2-55/2/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)
*b^4*c-60/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*ln(c*x^2+b*x+a)*b^2+51/2/a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln(c*x^
2+b*x+a)*b^4-6/a^5/(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^7-1/2/a^
3/x^2-16/a/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+3/a^4/(c*x^2+b*x+a)^2*b^7/(16*a^2*c^2-8*a*b^2*c+
b^4)*x+7/2/a^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*b^6+24/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*ln(c*x^2+b
*x+a)-3/a^5/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a)*b^6-20/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3+3
/a^4*b/x-3/a^4*ln(x)*c+6/a^5*ln(x)*b^2+33/a^2/(c*x^2+b*x+a)^2*b*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-21/a^3/(c*x
^2+b*x+a)^2*b^3*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+3/a^4/(c*x^2+b*x+a)^2*b^5*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^
3+163/2/a^2/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^2-89/2/a^3/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*
a*b^2*c+b^4)*x^2*b^4+6/a^4/(c*x^2+b*x+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^6+23/a/(c*x^2+b*x+a)^2*b/(16*a^2
*c^2-8*a*b^2*c+b^4)*x*c^3+24/a^2/(c*x^2+b*x+a)^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c^2-20/a^3/(c*x^2+b*x+a)^2*b
^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c+210/a^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*c^3-210/a^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^
3*c^2+63/a^4/(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*c

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 16.8099, size = 5724, normalized size = 18.71 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/2*(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3 - 6*(2*a*b^7*c^2 - 23*a^2*b^5*c^3 + 87*a^3*b^3*c^4
- 108*a^4*b*c^5)*x^5 - 3*(8*a*b^8*c - 94*a^2*b^6*c^2 + 369*a^3*b^4*c^3 - 500*a^4*b^2*c^4 + 64*a^5*c^5)*x^4 -
2*(6*a*b^9 - 63*a^2*b^7*c + 188*a^3*b^5*c^2 - 25*a^4*b^3*c^3 - 412*a^5*b*c^4)*x^3 - (18*a^2*b^8 - 217*a^3*b^6*
c + 887*a^4*b^4*c^2 - 1300*a^5*b^2*c^3 + 288*a^6*c^4)*x^2 + 3*((2*b^7*c^2 - 21*a*b^5*c^3 + 70*a^2*b^3*c^4 - 70
*a^3*b*c^5)*x^6 + 2*(2*b^8*c - 21*a*b^6*c^2 + 70*a^2*b^4*c^3 - 70*a^3*b^2*c^4)*x^5 + (2*b^9 - 17*a*b^7*c + 28*
a^2*b^5*c^2 + 70*a^3*b^3*c^3 - 140*a^4*b*c^4)*x^4 + 2*(2*a*b^8 - 21*a^2*b^6*c + 70*a^3*b^4*c^2 - 70*a^4*b^2*c^
3)*x^3 + (2*a^2*b^7 - 21*a^3*b^5*c + 70*a^4*b^3*c^2 - 70*a^5*b*c^3)*x^2)*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)) - 4*(a^3*b^7 - 12*a^4*b^5*c + 48*a^5*b
^3*c^2 - 64*a^6*b*c^3)*x + 3*((2*b^8*c^2 - 25*a*b^6*c^3 + 108*a^2*b^4*c^4 - 176*a^3*b^2*c^5 + 64*a^4*c^6)*x^6
+ 2*(2*b^9*c - 25*a*b^7*c^2 + 108*a^2*b^5*c^3 - 176*a^3*b^3*c^4 + 64*a^4*b*c^5)*x^5 + (2*b^10 - 21*a*b^8*c + 5
8*a^2*b^6*c^2 + 40*a^3*b^4*c^3 - 288*a^4*b^2*c^4 + 128*a^5*c^5)*x^4 + 2*(2*a*b^9 - 25*a^2*b^7*c + 108*a^3*b^5*
c^2 - 176*a^4*b^3*c^3 + 64*a^5*b*c^4)*x^3 + (2*a^2*b^8 - 25*a^3*b^6*c + 108*a^4*b^4*c^2 - 176*a^5*b^2*c^3 + 64
*a^6*c^4)*x^2)*log(c*x^2 + b*x + a) - 6*((2*b^8*c^2 - 25*a*b^6*c^3 + 108*a^2*b^4*c^4 - 176*a^3*b^2*c^5 + 64*a^
4*c^6)*x^6 + 2*(2*b^9*c - 25*a*b^7*c^2 + 108*a^2*b^5*c^3 - 176*a^3*b^3*c^4 + 64*a^4*b*c^5)*x^5 + (2*b^10 - 21*
a*b^8*c + 58*a^2*b^6*c^2 + 40*a^3*b^4*c^3 - 288*a^4*b^2*c^4 + 128*a^5*c^5)*x^4 + 2*(2*a*b^9 - 25*a^2*b^7*c + 1
08*a^3*b^5*c^2 - 176*a^4*b^3*c^3 + 64*a^5*b*c^4)*x^3 + (2*a^2*b^8 - 25*a^3*b^6*c + 108*a^4*b^4*c^2 - 176*a^5*b
^2*c^3 + 64*a^6*c^4)*x^2)*log(x))/((a^5*b^6*c^2 - 12*a^6*b^4*c^3 + 48*a^7*b^2*c^4 - 64*a^8*c^5)*x^6 + 2*(a^5*b
^7*c - 12*a^6*b^5*c^2 + 48*a^7*b^3*c^3 - 64*a^8*b*c^4)*x^5 + (a^5*b^8 - 10*a^6*b^6*c + 24*a^7*b^4*c^2 + 32*a^8
*b^2*c^3 - 128*a^9*c^4)*x^4 + 2*(a^6*b^7 - 12*a^7*b^5*c + 48*a^8*b^3*c^2 - 64*a^9*b*c^3)*x^3 + (a^7*b^6 - 12*a
^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*x^2), -1/2*(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3 - 6*
(2*a*b^7*c^2 - 23*a^2*b^5*c^3 + 87*a^3*b^3*c^4 - 108*a^4*b*c^5)*x^5 - 3*(8*a*b^8*c - 94*a^2*b^6*c^2 + 369*a^3*
b^4*c^3 - 500*a^4*b^2*c^4 + 64*a^5*c^5)*x^4 - 2*(6*a*b^9 - 63*a^2*b^7*c + 188*a^3*b^5*c^2 - 25*a^4*b^3*c^3 - 4
12*a^5*b*c^4)*x^3 - (18*a^2*b^8 - 217*a^3*b^6*c + 887*a^4*b^4*c^2 - 1300*a^5*b^2*c^3 + 288*a^6*c^4)*x^2 - 6*((
2*b^7*c^2 - 21*a*b^5*c^3 + 70*a^2*b^3*c^4 - 70*a^3*b*c^5)*x^6 + 2*(2*b^8*c - 21*a*b^6*c^2 + 70*a^2*b^4*c^3 - 7
0*a^3*b^2*c^4)*x^5 + (2*b^9 - 17*a*b^7*c + 28*a^2*b^5*c^2 + 70*a^3*b^3*c^3 - 140*a^4*b*c^4)*x^4 + 2*(2*a*b^8 -
21*a^2*b^6*c + 70*a^3*b^4*c^2 - 70*a^4*b^2*c^3)*x^3 + (2*a^2*b^7 - 21*a^3*b^5*c + 70*a^4*b^3*c^2 - 70*a^5*b*c
^3)*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 4*(a^3*b^7 - 12*a^4*b^5*c
+ 48*a^5*b^3*c^2 - 64*a^6*b*c^3)*x + 3*((2*b^8*c^2 - 25*a*b^6*c^3 + 108*a^2*b^4*c^4 - 176*a^3*b^2*c^5 + 64*a^4
*c^6)*x^6 + 2*(2*b^9*c - 25*a*b^7*c^2 + 108*a^2*b^5*c^3 - 176*a^3*b^3*c^4 + 64*a^4*b*c^5)*x^5 + (2*b^10 - 21*a
*b^8*c + 58*a^2*b^6*c^2 + 40*a^3*b^4*c^3 - 288*a^4*b^2*c^4 + 128*a^5*c^5)*x^4 + 2*(2*a*b^9 - 25*a^2*b^7*c + 10
8*a^3*b^5*c^2 - 176*a^4*b^3*c^3 + 64*a^5*b*c^4)*x^3 + (2*a^2*b^8 - 25*a^3*b^6*c + 108*a^4*b^4*c^2 - 176*a^5*b^
2*c^3 + 64*a^6*c^4)*x^2)*log(c*x^2 + b*x + a) - 6*((2*b^8*c^2 - 25*a*b^6*c^3 + 108*a^2*b^4*c^4 - 176*a^3*b^2*c
^5 + 64*a^4*c^6)*x^6 + 2*(2*b^9*c - 25*a*b^7*c^2 + 108*a^2*b^5*c^3 - 176*a^3*b^3*c^4 + 64*a^4*b*c^5)*x^5 + (2*
b^10 - 21*a*b^8*c + 58*a^2*b^6*c^2 + 40*a^3*b^4*c^3 - 288*a^4*b^2*c^4 + 128*a^5*c^5)*x^4 + 2*(2*a*b^9 - 25*a^2
*b^7*c + 108*a^3*b^5*c^2 - 176*a^4*b^3*c^3 + 64*a^5*b*c^4)*x^3 + (2*a^2*b^8 - 25*a^3*b^6*c + 108*a^4*b^4*c^2 -
176*a^5*b^2*c^3 + 64*a^6*c^4)*x^2)*log(x))/((a^5*b^6*c^2 - 12*a^6*b^4*c^3 + 48*a^7*b^2*c^4 - 64*a^8*c^5)*x^6
+ 2*(a^5*b^7*c - 12*a^6*b^5*c^2 + 48*a^7*b^3*c^3 - 64*a^8*b*c^4)*x^5 + (a^5*b^8 - 10*a^6*b^6*c + 24*a^7*b^4*c^
2 + 32*a^8*b^2*c^3 - 128*a^9*c^4)*x^4 + 2*(a^6*b^7 - 12*a^7*b^5*c + 48*a^8*b^3*c^2 - 64*a^9*b*c^3)*x^3 + (a^7*
b^6 - 12*a^8*b^4*c + 48*a^9*b^2*c^2 - 64*a^10*c^3)*x^2)]

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Sympy [B]  time = 52.2431, size = 7465, normalized size = 24.4 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(c*x**2+b*x+a)**3,x)

[Out]

(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c*
*5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/
(2*a**5))*log(x + (98304*a**19*c**9*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b*
*4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*
b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 429056*a**18*b**2*c**8*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a
**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*
b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 645888*a**17*b**4*c**
7*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*
c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2
)/(2*a**5))**2 - 508032*a**16*b**6*c**6*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*
a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 2
0*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 241376*a**15*b**8*c**5*(-3*b*sqrt(-(4*a*c - b**2)**5)*(
70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a
**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 147456*a**15*c**1
0*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*
c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2
)/(2*a**5)) - 73436*a**14*b**10*c**4*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b
**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a
*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 542016*a**14*b**2*c**9*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*
a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3
*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 14479*a**13*b**12*c**3*
(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c*
*5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/
(2*a**5))**2 + 923760*a**13*b**4*c**8*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*
b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*
a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 1797*a**12*b**14*c**2*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**
3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b*
*4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 849948*a**12*b**6*c**7*
(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c*
*5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/
(2*a**5)) + 128*a**11*b**16*c*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c -
2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c
- b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 464829*a**11*b**8*c**6*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c*
*3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c
**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 442368*a**11*c**11 - 4*a**10*b
**18*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a*
*5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b
**2)/(2*a**5))**2 - 159318*a**10*b**10*c**5*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 +
21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2
+ 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 4889664*a**10*b**2*c**10 + 34731*a**9*b**12*c**4*(-3*b
*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 -
1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a*
*5)) - 18774576*a**9*b**4*c**9 - 4695*a**8*b**14*c**3*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b
**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2
*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 35177868*a**8*b**6*c**8 + 360*a**7*b**16*c**
2*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*
c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2
)/(2*a**5)) - 37219329*a**7*b**8*c**7 - 12*a**6*b**18*c*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2
*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a*
*2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 24372684*a**6*b**10*c**6 - 10403442*a**5*b
**12*c**5 + 2958642*a**4*b**14*c**4 - 557838*a**3*b**16*c**3 + 67140*a**2*b**18*c**2 - 4680*a*b**20*c + 144*b*
*22)/(1451520*a**10*b*c**11 - 8300250*a**9*b**3*c**10 + 19711566*a**8*b**5*c**9 - 24401871*a**7*b**7*c**8 + 17
859492*a**6*b**9*c**7 - 8284248*a**5*b**11*c**6 + 2513700*a**4*b**13*c**5 - 499338*a**3*b**15*c**4 + 62748*a**
2*b**17*c**3 - 4536*a*b**19*c**2 + 144*b**21*c)) + (3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*
c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**
6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))*log(x + (98304*a**19*c**9*(3*b*sqrt(-(4*a*c - b**2
)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4
+ 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 429056*a**
18*b**2*c**8*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(
1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*
c - 2*b**2)/(2*a**5))**2 + 645888*a**17*b**4*c**7*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c
**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6
*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 508032*a**16*b**6*c**6*(3*b*sqrt(-(4*a*c - b**
2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4
+ 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 241376*a*
*15*b**8*c**5*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*
(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a
*c - 2*b**2)/(2*a**5))**2 + 147456*a**15*c**10*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2
+ 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c*
*2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 73436*a**14*b**10*c**4*(3*b*sqrt(-(4*a*c - b**2)**5)
*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640
*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 542016*a**14*b*
*2*c**9*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*
a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2
*b**2)/(2*a**5)) + 14479*a**13*b**12*c**3*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21
*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 +
20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 923760*a**13*b**4*c**8*(3*b*sqrt(-(4*a*c - b**2)**5)*(
70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a
**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 1797*a**12*b**14*c**
2*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c
**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)
/(2*a**5))**2 - 849948*a**12*b**6*c**7*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*
b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*
a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 128*a**11*b**16*c*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**
3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c*
*3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 + 464829*a**11*b**8*c**6*(3*b*
sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1
280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**
5)) - 442368*a**11*c**11 - 4*a**10*b**18*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*
a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 2
0*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5))**2 - 159318*a**10*b**10*c**5*(3*b*sqrt(-(4*a*c - b**2)**5)*(
70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a
**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 4889664*a**10*b**2*c
**10 + 34731*a**9*b**12*c**4*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2
*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c -
b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 18774576*a**9*b**4*c**9 - 4695*a**8*b**14*c**3*(3*b*sqrt(-(4*a*c - b**
2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4
+ 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 35177868*a**
8*b**6*c**8 + 360*a**7*b**16*c**2*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*
c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**
8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) - 37219329*a**7*b**8*c**7 - 12*a**6*b**18*c*(3*b*sqrt(-(4*a*c - b**
2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*a**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4
+ 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 3*(a*c - 2*b**2)/(2*a**5)) + 24372684*a**
6*b**10*c**6 - 10403442*a**5*b**12*c**5 + 2958642*a**4*b**14*c**4 - 557838*a**3*b**16*c**3 + 67140*a**2*b**18*
c**2 - 4680*a*b**20*c + 144*b**22)/(1451520*a**10*b*c**11 - 8300250*a**9*b**3*c**10 + 19711566*a**8*b**5*c**9
- 24401871*a**7*b**7*c**8 + 17859492*a**6*b**9*c**7 - 8284248*a**5*b**11*c**6 + 2513700*a**4*b**13*c**5 - 4993
38*a**3*b**15*c**4 + 62748*a**2*b**17*c**3 - 4536*a*b**19*c**2 + 144*b**21*c)) + (-16*a**5*c**2 + 8*a**4*b**2*
c - a**3*b**4 + x**5*(162*a**2*b*c**4 - 90*a*b**3*c**3 + 12*b**5*c**2) + x**4*(-48*a**3*c**4 + 363*a**2*b**2*c
**3 - 186*a*b**4*c**2 + 24*b**6*c) + x**3*(206*a**3*b*c**3 + 64*a**2*b**3*c**2 - 78*a*b**5*c + 12*b**7) + x**2
*(-72*a**4*c**3 + 307*a**3*b**2*c**2 - 145*a**2*b**4*c + 18*a*b**6) + x*(64*a**4*b*c**2 - 32*a**3*b**3*c + 4*a
**2*b**5))/(x**6*(32*a**6*c**4 - 16*a**5*b**2*c**3 + 2*a**4*b**4*c**2) + x**5*(64*a**6*b*c**3 - 32*a**5*b**3*c
**2 + 4*a**4*b**5*c) + x**4*(64*a**7*c**3 - 12*a**5*b**4*c + 2*a**4*b**6) + x**3*(64*a**7*b*c**2 - 32*a**6*b**
3*c + 4*a**5*b**5) + x**2*(32*a**8*c**2 - 16*a**7*b**2*c + 2*a**6*b**4)) - 3*(a*c - 2*b**2)*log(x + (-442368*a
**11*c**11 + 4889664*a**10*b**2*c**10 - 442368*a**10*c**10*(a*c - 2*b**2) - 18774576*a**9*b**4*c**9 + 1626048*
a**9*b**2*c**9*(a*c - 2*b**2) + 884736*a**9*c**9*(a*c - 2*b**2)**2 + 35177868*a**8*b**6*c**8 - 2771280*a**8*b*
*4*c**8*(a*c - 2*b**2) - 3861504*a**8*b**2*c**8*(a*c - 2*b**2)**2 - 37219329*a**7*b**8*c**7 + 2549844*a**7*b**
6*c**7*(a*c - 2*b**2) + 5812992*a**7*b**4*c**7*(a*c - 2*b**2)**2 + 24372684*a**6*b**10*c**6 - 1394487*a**6*b**
8*c**6*(a*c - 2*b**2) - 4572288*a**6*b**6*c**6*(a*c - 2*b**2)**2 - 10403442*a**5*b**12*c**5 + 477954*a**5*b**1
0*c**5*(a*c - 2*b**2) + 2172384*a**5*b**8*c**5*(a*c - 2*b**2)**2 + 2958642*a**4*b**14*c**4 - 104193*a**4*b**12
*c**4*(a*c - 2*b**2) - 660924*a**4*b**10*c**4*(a*c - 2*b**2)**2 - 557838*a**3*b**16*c**3 + 14085*a**3*b**14*c*
*3*(a*c - 2*b**2) + 130311*a**3*b**12*c**3*(a*c - 2*b**2)**2 + 67140*a**2*b**18*c**2 - 1080*a**2*b**16*c**2*(a
*c - 2*b**2) - 16173*a**2*b**14*c**2*(a*c - 2*b**2)**2 - 4680*a*b**20*c + 36*a*b**18*c*(a*c - 2*b**2) + 1152*a
*b**16*c*(a*c - 2*b**2)**2 + 144*b**22 - 36*b**18*(a*c - 2*b**2)**2)/(1451520*a**10*b*c**11 - 8300250*a**9*b**
3*c**10 + 19711566*a**8*b**5*c**9 - 24401871*a**7*b**7*c**8 + 17859492*a**6*b**9*c**7 - 8284248*a**5*b**11*c**
6 + 2513700*a**4*b**13*c**5 - 499338*a**3*b**15*c**4 + 62748*a**2*b**17*c**3 - 4536*a*b**19*c**2 + 144*b**21*c
))/a**5

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Giac [A]  time = 1.12907, size = 554, normalized size = 1.81 \begin{align*} -\frac{3 \,{\left (2 \, b^{7} - 21 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 70 \, a^{3} b c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{4} - 8 \, a^{6} b^{2} c + 16 \, a^{7} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, b^{5} c^{2} x^{5} - 90 \, a b^{3} c^{3} x^{5} + 162 \, a^{2} b c^{4} x^{5} + 24 \, b^{6} c x^{4} - 186 \, a b^{4} c^{2} x^{4} + 363 \, a^{2} b^{2} c^{3} x^{4} - 48 \, a^{3} c^{4} x^{4} + 12 \, b^{7} x^{3} - 78 \, a b^{5} c x^{3} + 64 \, a^{2} b^{3} c^{2} x^{3} + 206 \, a^{3} b c^{3} x^{3} + 18 \, a b^{6} x^{2} - 145 \, a^{2} b^{4} c x^{2} + 307 \, a^{3} b^{2} c^{2} x^{2} - 72 \, a^{4} c^{3} x^{2} + 4 \, a^{2} b^{5} x - 32 \, a^{3} b^{3} c x + 64 \, a^{4} b c^{2} x - a^{3} b^{4} + 8 \, a^{4} b^{2} c - 16 \, a^{5} c^{2}}{2 \,{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )}{\left (c x^{3} + b x^{2} + a x\right )}^{2}} - \frac{3 \,{\left (2 \, b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{5}} + \frac{3 \,{\left (2 \, b^{2} - a c\right )} \log \left ({\left | x \right |}\right )}{a^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-3*(2*b^7 - 21*a*b^5*c + 70*a^2*b^3*c^2 - 70*a^3*b*c^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^5*b^4 - 8*a
^6*b^2*c + 16*a^7*c^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(12*b^5*c^2*x^5 - 90*a*b^3*c^3*x^5 + 162*a^2*b*c^4*x^5 + 24*b
^6*c*x^4 - 186*a*b^4*c^2*x^4 + 363*a^2*b^2*c^3*x^4 - 48*a^3*c^4*x^4 + 12*b^7*x^3 - 78*a*b^5*c*x^3 + 64*a^2*b^3
*c^2*x^3 + 206*a^3*b*c^3*x^3 + 18*a*b^6*x^2 - 145*a^2*b^4*c*x^2 + 307*a^3*b^2*c^2*x^2 - 72*a^4*c^3*x^2 + 4*a^2
*b^5*x - 32*a^3*b^3*c*x + 64*a^4*b*c^2*x - a^3*b^4 + 8*a^4*b^2*c - 16*a^5*c^2)/((a^4*b^4 - 8*a^5*b^2*c + 16*a^
6*c^2)*(c*x^3 + b*x^2 + a*x)^2) - 3/2*(2*b^2 - a*c)*log(c*x^2 + b*x + a)/a^5 + 3*(2*b^2 - a*c)*log(abs(x))/a^5