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

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

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Rubi [A]  time = 0.46, antiderivative size = 306, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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]

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 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 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 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])

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*}

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Mathematica [A]  time = 0.52, size = 269, normalized size = 0.88 \begin {gather*} \frac {\frac {a^2 \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x+b^4+b^3 c x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac {a^2}{x^2}-\frac {6 b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 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 \left (-32 a^3 c^3+97 a^2 b^2 c^2+66 a^2 b c^3 x-47 a b^4 c-42 a b^3 c^2 x+6 b^6+6 b^5 c x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \left (a+b x+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 2.07, size = 2669, normalized size = 8.72

result too large to display

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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giac [A]  time = 0.17, size = 410, normalized size = 1.34 \begin {gather*} -\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 {gather*}

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

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maple [B]  time = 0.07, size = 1110, normalized size = 3.63 \begin {gather*} \frac {33 b \,c^{4} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {21 b^{3} c^{3} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}+\frac {3 b^{5} c^{2} x^{3}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{4}}-\frac {16 c^{4} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {163 b^{2} c^{3} x^{2}}{2 \left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {89 b^{4} c^{2} x^{2}}{2 \left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}+\frac {6 b^{6} c \,x^{2}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{4}}+\frac {23 b \,c^{3} x}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {24 b^{3} c^{2} x}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {20 b^{5} c x}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}+\frac {3 b^{7} x}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{4}}+\frac {115 b^{2} c^{2}}{2 \left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {55 b^{4} c}{2 \left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}+\frac {210 b \,c^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {7 b^{6}}{2 \left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}-\frac {210 b^{3} c^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{3}}+\frac {63 b^{5} c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{4}}-\frac {6 b^{7} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}\, a^{5}}-\frac {20 c^{3}}{\left (c \,x^{2}+b x +a \right )^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {24 c^{3} \ln \left (c \,x^{2}+b x +a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {60 b^{2} c^{2} \ln \left (c \,x^{2}+b x +a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{3}}+\frac {51 b^{4} c \ln \left (c \,x^{2}+b x +a \right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{4}}-\frac {3 b^{6} \ln \left (c \,x^{2}+b x +a \right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{5}}-\frac {3 c \ln \relax (x )}{a^{4}}+\frac {6 b^{2} \ln \relax (x )}{a^{5}}+\frac {3 b}{a^{4} x}-\frac {1}{2 a^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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^7-16/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+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*b^2*ln(x)+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+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+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+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^3-3/a^4*c*ln(x)-1/2/a^3/x^2-20/(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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 2.31, size = 1404, normalized size = 4.59

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x + c*x^2)^3),x)

[Out]

((2*b*x)/a^2 - 1/(2*a) + (x^2*(18*b^6 - 72*a^3*c^3 + 307*a^2*b^2*c^2 - 145*a*b^4*c))/(2*a^3*(b^4 + 16*a^2*c^2
- 8*a*b^2*c)) + (3*x^4*(8*b^6*c - 16*a^3*c^4 - 62*a*b^4*c^2 + 121*a^2*b^2*c^3))/(2*a^4*(b^4 + 16*a^2*c^2 - 8*a
*b^2*c)) + (b*x^3*(6*b^6 + 103*a^3*c^3 + 32*a^2*b^2*c^2 - 39*a*b^4*c))/(a^4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) +
(3*b*c^2*x^5*(2*b^4 + 27*a^2*c^2 - 15*a*b^2*c))/(a^4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2
*x^2 + c^2*x^6 + 2*a*b*x^3 + 2*b*c*x^5) - (3*log(x)*(a*c - 2*b^2))/a^5 + (3*log(4*a*b^12 + 4*b^13*x + 1536*a^7
*c^6 - 4*a*b^7*(-(4*a*c - b^2)^5)^(1/2) - 80*a^2*b^10*c - 4*b^8*x*(-(4*a*c - b^2)^5)^(1/2) + 645*a^3*b^8*c^2 -
2643*a^4*b^6*c^3 + 5640*a^5*b^4*c^4 - 5552*a^6*b^2*c^5 + 36*a^2*b^5*c*(-(4*a*c - b^2)^5)^(1/2) + 59*a^4*b*c^3
*(-(4*a*c - b^2)^5)^(1/2) + 682*a^2*b^9*c^2*x - 2913*a^3*b^7*c^3*x + 6606*a^4*b^5*c^4*x - 7232*a^5*b^3*c^5*x -
48*a^4*c^4*x*(-(4*a*c - b^2)^5)^(1/2) - 82*a*b^11*c*x - 95*a^3*b^3*c^2*(-(4*a*c - b^2)^5)^(1/2) + 2656*a^6*b*
c^6*x + 42*a*b^6*c*x*(-(4*a*c - b^2)^5)^(1/2) - 146*a^2*b^4*c^2*x*(-(4*a*c - b^2)^5)^(1/2) + 179*a^3*b^2*c^3*x
*(-(4*a*c - b^2)^5)^(1/2))*(2*b^12 + 1024*a^6*c^6 - 2*b^7*(-(4*a*c - b^2)^5)^(1/2) + 340*a^2*b^8*c^2 - 1440*a^
3*b^6*c^3 + 3200*a^4*b^4*c^4 - 3328*a^5*b^2*c^5 - 41*a*b^10*c + 70*a^3*b*c^3*(-(4*a*c - b^2)^5)^(1/2) - 70*a^2
*b^3*c^2*(-(4*a*c - b^2)^5)^(1/2) + 21*a*b^5*c*(-(4*a*c - b^2)^5)^(1/2)))/(2*a^5*(4*a*c - b^2)^5) + (3*log(4*a
*b^12 + 4*b^13*x + 1536*a^7*c^6 + 4*a*b^7*(-(4*a*c - b^2)^5)^(1/2) - 80*a^2*b^10*c + 4*b^8*x*(-(4*a*c - b^2)^5
)^(1/2) + 645*a^3*b^8*c^2 - 2643*a^4*b^6*c^3 + 5640*a^5*b^4*c^4 - 5552*a^6*b^2*c^5 - 36*a^2*b^5*c*(-(4*a*c - b
^2)^5)^(1/2) - 59*a^4*b*c^3*(-(4*a*c - b^2)^5)^(1/2) + 682*a^2*b^9*c^2*x - 2913*a^3*b^7*c^3*x + 6606*a^4*b^5*c
^4*x - 7232*a^5*b^3*c^5*x + 48*a^4*c^4*x*(-(4*a*c - b^2)^5)^(1/2) - 82*a*b^11*c*x + 95*a^3*b^3*c^2*(-(4*a*c -
b^2)^5)^(1/2) + 2656*a^6*b*c^6*x - 42*a*b^6*c*x*(-(4*a*c - b^2)^5)^(1/2) + 146*a^2*b^4*c^2*x*(-(4*a*c - b^2)^5
)^(1/2) - 179*a^3*b^2*c^3*x*(-(4*a*c - b^2)^5)^(1/2))*(2*b^12 + 1024*a^6*c^6 + 2*b^7*(-(4*a*c - b^2)^5)^(1/2)
+ 340*a^2*b^8*c^2 - 1440*a^3*b^6*c^3 + 3200*a^4*b^4*c^4 - 3328*a^5*b^2*c^5 - 41*a*b^10*c - 70*a^3*b*c^3*(-(4*a
*c - b^2)^5)^(1/2) + 70*a^2*b^3*c^2*(-(4*a*c - b^2)^5)^(1/2) - 21*a*b^5*c*(-(4*a*c - b^2)^5)^(1/2)))/(2*a^5*(4
*a*c - b^2)^5)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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