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

Optimal. Leaf size=239 \[ \frac {3 b \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {3 b \log (x)}{a^4}-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 x \left (b^2-4 a c\right )^2}+\frac {20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{2 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c x}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

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Rubi [A]  time = 0.31, antiderivative size = 239, 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 {20 a^2 c^2+3 b c x \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{2 a^2 x \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {3 \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 x \left (b^2-4 a c\right )^2}+\frac {3 b \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {3 b \log (x)}{a^4}+\frac {-2 a c+b^2+b c x}{2 a x \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^2*(a + b*x + c*x^2)^3),x]

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(a^3*(b^2 - 4*a*c)^2*x) + (b^2 - 2*a*c + b*c*x)/(2*a*(b^2 - 4*a*c)*x*(a + b*x
+ c*x^2)^2) + (3*b^4 - 20*a*b^2*c + 20*a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x)/(2*a^2*(b^2 - 4*a*c)^2*x*(a + b*x + c
*x^2)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^4*(b^2
- 4*a*c)^(5/2)) - (3*b*Log[x])/a^4 + (3*b*Log[a + b*x + c*x^2])/(2*a^4)

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^2 \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 \left (a+b x+c x^2\right )^2}-\frac {\int \frac {-3 b^2+10 a c-4 b c x}{x^2 \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 \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}+\frac {\int \frac {6 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )+6 b c \left (b^2-6 a c\right ) x}{x^2 \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 \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {6 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a x^2}-\frac {6 b \left (-b^2+4 a c\right )^2}{a^2 x}+\frac {6 \left (b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3+b c \left (b^2-4 a c\right )^2 x\right )}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 \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 \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac {3 b \log (x)}{a^4}+\frac {3 \int \frac {b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3+b c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx}{a^4 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 \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 \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac {3 b \log (x)}{a^4}+\frac {(3 b) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^4}+\frac {\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 a^4 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 \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 \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {\left (3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 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^4 \left (b^2-4 a c\right )^2}\\ &=-\frac {3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{a^3 \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 \left (a+b x+c x^2\right )^2}+\frac {3 b^4-20 a b^2 c+20 a^2 c^2+3 b c \left (b^2-6 a c\right ) x}{2 a^2 \left (b^2-4 a c\right )^2 x \left (a+b x+c x^2\right )}-\frac {3 \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log \left (a+b x+c x^2\right )}{2 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 221, normalized size = 0.92 \begin {gather*} \frac {\frac {a^2 \left (-3 a b c-2 a c^2 x+b^3+b^2 c x\right )}{\left (4 a c-b^2\right ) (a+x (b+c x))^2}-\frac {a \left (46 a^2 b c^2+28 a^2 c^3 x-29 a b^3 c-26 a b^2 c^2 x+4 b^5+4 b^4 c x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {6 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+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}}+3 b \log (a+x (b+c x))-\frac {2 a}{x}-6 b \log (x)}{2 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a)/x + (a^2*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((-b^2 + 4*a*c)*(a + x*(b + c*x))^2) - (a*(4*b^5 - 29*
a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x - 26*a*b^2*c^2*x + 28*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (6*
(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2)
- 6*b*Log[x] + 3*b*Log[a + x*(b + c*x)])/(2*a^4)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.08, size = 2280, normalized size = 9.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*a^3*b^6 - 24*a^4*b^4*c + 96*a^5*b^2*c^2 - 128*a^6*c^3 + 6*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^
4 - 40*a^4*c^5)*x^4 + 3*(4*a*b^7*c - 45*a^2*b^5*c^2 + 162*a^3*b^3*c^3 - 184*a^4*b*c^4)*x^3 + 2*(3*a*b^8 - 30*a
^2*b^6*c + 79*a^3*b^4*c^2 + 22*a^4*b^2*c^3 - 200*a^5*c^4)*x^2 + 3*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 -
20*a^3*c^5)*x^5 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^4 + (b^8 - 8*a*b^6*c + 10*a^2*b^4
*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^3 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*x^2 + (a^2*
b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*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)) + (9*a^2*b^7 - 104*a^3*b^5*c + 394*a^4*b^3*c^2 - 488*a^5*b
*c^3)*x - 3*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b
^4*c^3 - 64*a^3*b^2*c^4)*x^4 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^3 + 2*(a
*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^2 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5
*b*c^3)*x)*log(c*x^2 + b*x + a) + 6*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 2*(b^8*c -
12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^4 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 1
28*a^4*b*c^4)*x^3 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^2 + (a^2*b^7 - 12*a^3*b^5*c +
48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*log(x))/((a^4*b^6*c^2 - 12*a^5*b^4*c^3 + 48*a^6*b^2*c^4 - 64*a^7*c^5)*x^5 +
2*(a^4*b^7*c - 12*a^5*b^5*c^2 + 48*a^6*b^3*c^3 - 64*a^7*b*c^4)*x^4 + (a^4*b^8 - 10*a^5*b^6*c + 24*a^6*b^4*c^2
+ 32*a^7*b^2*c^3 - 128*a^8*c^4)*x^3 + 2*(a^5*b^7 - 12*a^6*b^5*c + 48*a^7*b^3*c^2 - 64*a^8*b*c^3)*x^2 + (a^6*b
^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*x), -1/2*(2*a^3*b^6 - 24*a^4*b^4*c + 96*a^5*b^2*c^2 - 128*a^6
*c^3 + 6*(a*b^6*c^2 - 11*a^2*b^4*c^3 + 38*a^3*b^2*c^4 - 40*a^4*c^5)*x^4 + 3*(4*a*b^7*c - 45*a^2*b^5*c^2 + 162*
a^3*b^3*c^3 - 184*a^4*b*c^4)*x^3 + 2*(3*a*b^8 - 30*a^2*b^6*c + 79*a^3*b^4*c^2 + 22*a^4*b^2*c^3 - 200*a^5*c^4)*
x^2 + 6*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^5 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3
- 20*a^3*b*c^4)*x^4 + (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^3 + 2*(a*b^7 - 10*a^
2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*x^2 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c^3)*x)*sqrt(
-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (9*a^2*b^7 - 104*a^3*b^5*c + 394*a^4*b^3
*c^2 - 488*a^5*b*c^3)*x - 3*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 2*(b^8*c - 12*a*b^
6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^4 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32*a^3*b^3*c^3 - 128*a^4*b
*c^4)*x^3 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^2 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*
b^3*c^2 - 64*a^5*b*c^3)*x)*log(c*x^2 + b*x + a) + 6*((b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*
x^5 + 2*(b^8*c - 12*a*b^6*c^2 + 48*a^2*b^4*c^3 - 64*a^3*b^2*c^4)*x^4 + (b^9 - 10*a*b^7*c + 24*a^2*b^5*c^2 + 32
*a^3*b^3*c^3 - 128*a^4*b*c^4)*x^3 + 2*(a*b^8 - 12*a^2*b^6*c + 48*a^3*b^4*c^2 - 64*a^4*b^2*c^3)*x^2 + (a^2*b^7
- 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*log(x))/((a^4*b^6*c^2 - 12*a^5*b^4*c^3 + 48*a^6*b^2*c^4 - 6
4*a^7*c^5)*x^5 + 2*(a^4*b^7*c - 12*a^5*b^5*c^2 + 48*a^6*b^3*c^3 - 64*a^7*b*c^4)*x^4 + (a^4*b^8 - 10*a^5*b^6*c
+ 24*a^6*b^4*c^2 + 32*a^7*b^2*c^3 - 128*a^8*c^4)*x^3 + 2*(a^5*b^7 - 12*a^6*b^5*c + 48*a^7*b^3*c^2 - 64*a^8*b*c
^3)*x^2 + (a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*x)]

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giac [A]  time = 0.18, size = 309, normalized size = 1.29 \begin {gather*} \frac {3 \, {\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, b \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {2 \, a^{3} b^{4} - 16 \, a^{4} b^{2} c + 32 \, a^{5} c^{2} + 6 \, {\left (a b^{4} c^{2} - 7 \, a^{2} b^{2} c^{3} + 10 \, a^{3} c^{4}\right )} x^{4} + 3 \, {\left (4 \, a b^{5} c - 29 \, a^{2} b^{3} c^{2} + 46 \, a^{3} b c^{3}\right )} x^{3} + 2 \, {\left (3 \, a b^{6} - 18 \, a^{2} b^{4} c + 7 \, a^{3} b^{2} c^{2} + 50 \, a^{4} c^{3}\right )} x^{2} + {\left (9 \, a^{2} b^{5} - 68 \, a^{3} b^{3} c + 122 \, a^{4} b c^{2}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} a^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^4 - 8*a^5*b^
2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a*c)) + 3/2*b*log(c*x^2 + b*x + a)/a^4 - 3*b*log(abs(x))/a^4 - 1/2*(2*a^3*b^4
- 16*a^4*b^2*c + 32*a^5*c^2 + 6*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*x^4 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 +
46*a^3*b*c^3)*x^3 + 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 50*a^4*c^3)*x^2 + (9*a^2*b^5 - 68*a^3*b^3*c +
122*a^4*b*c^2)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*a^4*x)

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maple [B]  time = 0.08, size = 954, normalized size = 3.99 \begin {gather*} -\frac {14 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}+\frac {13 b^{2} 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^{2}}-\frac {2 b^{4} 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^{3}}-\frac {37 b \,c^{3} 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 {55 b^{3} 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^{2}}-\frac {4 b^{5} 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^{3}}-\frac {7 b^{2} 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}+\frac {12 b^{4} 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^{2}}-\frac {2 b^{6} 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 {18 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 )}+\frac {18 b^{3} c}{\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 {60 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}-\frac {5 b^{5}}{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 {90 b^{2} 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^{2}}-\frac {30 b^{4} 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^{3}}+\frac {3 b^{6} \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 {29 b \,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 )}+\frac {24 b \,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^{2}}-\frac {12 b^{3} c \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 {3 b^{5} \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 \ln \relax (x )}{a^{4}}-\frac {1}{a^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-14/a/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+13/a^2/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)
*x^3*b^2-2/a^3/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*b^4-37/a/(c*x^2+b*x+a)^2*b*c^3/(16*a^2*c^2-8
*a*b^2*c+b^4)*x^2+55/2/a^2/(c*x^2+b*x+a)^2*b^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-4/a^3/(c*x^2+b*x+a)^2*b^5*c/
(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-18/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*c^3-7/a/(c*x^2+b*x+a)^2/(16*a^2
*c^2-8*a*b^2*c+b^4)*x*b^2*c^2+12/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^4*c-2/a^3/(c*x^2+b*x+a)^2/
(16*a^2*c^2-8*a*b^2*c+b^4)*x*b^6-29/(c*x^2+b*x+a)^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2+18/a/(c*x^2+b*x+a)^2*b^3/
(16*a^2*c^2-8*a*b^2*c+b^4)*c-5/2/a^2/(c*x^2+b*x+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)+24/a^2/(16*a^2*c^2-8*a*b^2
*c+b^4)*c^2*ln(c*x^2+b*x+a)*b-12/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln(c*x^2+b*x+a)*b^3+3/2/a^4/(16*a^2*c^2-8*a*
b^2*c+b^4)*ln(c*x^2+b*x+a)*b^5-60/a/(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^3+90/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^2*c^2-30/
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^4*c+3/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^6-1/a^3/x-3/a^4*b*ln(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^2/(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 = 2.02, size = 1255, normalized size = 5.25 \begin {gather*} -\frac {\frac {1}{a}+\frac {x^2\,\left (50\,a^3\,c^3+7\,a^2\,b^2\,c^2-18\,a\,b^4\,c+3\,b^6\right )}{a^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (122\,a^2\,b\,c^2-68\,a\,b^3\,c+9\,b^5\right )}{2\,a^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,x^3\,\left (46\,a^2\,b\,c^3-29\,a\,b^3\,c^2+4\,b^5\,c\right )}{2\,a^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,c^2\,x^4\,\left (10\,a^2\,c^2-7\,a\,b^2\,c+b^4\right )}{a^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^3\,\left (b^2+2\,a\,c\right )+a^2\,x+c^2\,x^5+2\,a\,b\,x^2+2\,b\,c\,x^4}-\frac {3\,b\,\ln \relax (x)}{a^4}-\frac {3\,\ln \left (2\,a\,b^{11}+2\,b^{12}\,x+2\,a\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-39\,a^2\,b^9\,c-1696\,a^6\,b\,c^5+320\,a^6\,c^6\,x+2\,b^7\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+303\,a^3\,b^7\,c^2-1170\,a^4\,b^5\,c^3+2240\,a^5\,b^3\,c^4-10\,a^4\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-17\,a^2\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+321\,a^2\,b^8\,c^2\,x-1296\,a^3\,b^6\,c^3\,x+2660\,a^4\,b^4\,c^4\,x-2336\,a^5\,b^2\,c^5\,x-40\,a\,b^{10}\,c\,x+39\,a^3\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-20\,a\,b^5\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-58\,a^3\,b\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+63\,a^2\,b^3\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (b^{11}+b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-1024\,a^5\,b\,c^5+160\,a^2\,b^7\,c^2-640\,a^3\,b^5\,c^3+1280\,a^4\,b^3\,c^4-20\,a^3\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-20\,a\,b^9\,c+30\,a^2\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-10\,a\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^4\,{\left (4\,a\,c-b^2\right )}^5}-\frac {3\,\ln \left (2\,a\,b^{11}+2\,b^{12}\,x-2\,a\,b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-39\,a^2\,b^9\,c-1696\,a^6\,b\,c^5+320\,a^6\,c^6\,x-2\,b^7\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+303\,a^3\,b^7\,c^2-1170\,a^4\,b^5\,c^3+2240\,a^5\,b^3\,c^4+10\,a^4\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+17\,a^2\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+321\,a^2\,b^8\,c^2\,x-1296\,a^3\,b^6\,c^3\,x+2660\,a^4\,b^4\,c^4\,x-2336\,a^5\,b^2\,c^5\,x-40\,a\,b^{10}\,c\,x-39\,a^3\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+20\,a\,b^5\,c\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+58\,a^3\,b\,c^3\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-63\,a^2\,b^3\,c^2\,x\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )\,\left (b^{11}-b^6\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-1024\,a^5\,b\,c^5+160\,a^2\,b^7\,c^2-640\,a^3\,b^5\,c^3+1280\,a^4\,b^3\,c^4+20\,a^3\,c^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}-20\,a\,b^9\,c-30\,a^2\,b^2\,c^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}+10\,a\,b^4\,c\,\sqrt {-{\left (4\,a\,c-b^2\right )}^5}\right )}{2\,a^4\,{\left (4\,a\,c-b^2\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x + c*x^2)^3),x)

[Out]

- (1/a + (x^2*(3*b^6 + 50*a^3*c^3 + 7*a^2*b^2*c^2 - 18*a*b^4*c))/(a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x*(9*
b^5 + 122*a^2*b*c^2 - 68*a*b^3*c))/(2*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*x^3*(4*b^5*c - 29*a*b^3*c^2 + 4
6*a^2*b*c^3))/(2*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*c^2*x^4*(b^4 + 10*a^2*c^2 - 7*a*b^2*c))/(a^3*(b^4 +
16*a^2*c^2 - 8*a*b^2*c)))/(x^3*(2*a*c + b^2) + a^2*x + c^2*x^5 + 2*a*b*x^2 + 2*b*c*x^4) - (3*b*log(x))/a^4 - (
3*log(2*a*b^11 + 2*b^12*x + 2*a*b^6*(-(4*a*c - b^2)^5)^(1/2) - 39*a^2*b^9*c - 1696*a^6*b*c^5 + 320*a^6*c^6*x +
2*b^7*x*(-(4*a*c - b^2)^5)^(1/2) + 303*a^3*b^7*c^2 - 1170*a^4*b^5*c^3 + 2240*a^5*b^3*c^4 - 10*a^4*c^3*(-(4*a*
c - b^2)^5)^(1/2) - 17*a^2*b^4*c*(-(4*a*c - b^2)^5)^(1/2) + 321*a^2*b^8*c^2*x - 1296*a^3*b^6*c^3*x + 2660*a^4*
b^4*c^4*x - 2336*a^5*b^2*c^5*x - 40*a*b^10*c*x + 39*a^3*b^2*c^2*(-(4*a*c - b^2)^5)^(1/2) - 20*a*b^5*c*x*(-(4*a
*c - b^2)^5)^(1/2) - 58*a^3*b*c^3*x*(-(4*a*c - b^2)^5)^(1/2) + 63*a^2*b^3*c^2*x*(-(4*a*c - b^2)^5)^(1/2))*(b^1
1 + b^6*(-(4*a*c - b^2)^5)^(1/2) - 1024*a^5*b*c^5 + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 20*
a^3*c^3*(-(4*a*c - b^2)^5)^(1/2) - 20*a*b^9*c + 30*a^2*b^2*c^2*(-(4*a*c - b^2)^5)^(1/2) - 10*a*b^4*c*(-(4*a*c
- b^2)^5)^(1/2)))/(2*a^4*(4*a*c - b^2)^5) - (3*log(2*a*b^11 + 2*b^12*x - 2*a*b^6*(-(4*a*c - b^2)^5)^(1/2) - 39
*a^2*b^9*c - 1696*a^6*b*c^5 + 320*a^6*c^6*x - 2*b^7*x*(-(4*a*c - b^2)^5)^(1/2) + 303*a^3*b^7*c^2 - 1170*a^4*b^
5*c^3 + 2240*a^5*b^3*c^4 + 10*a^4*c^3*(-(4*a*c - b^2)^5)^(1/2) + 17*a^2*b^4*c*(-(4*a*c - b^2)^5)^(1/2) + 321*a
^2*b^8*c^2*x - 1296*a^3*b^6*c^3*x + 2660*a^4*b^4*c^4*x - 2336*a^5*b^2*c^5*x - 40*a*b^10*c*x - 39*a^3*b^2*c^2*(
-(4*a*c - b^2)^5)^(1/2) + 20*a*b^5*c*x*(-(4*a*c - b^2)^5)^(1/2) + 58*a^3*b*c^3*x*(-(4*a*c - b^2)^5)^(1/2) - 63
*a^2*b^3*c^2*x*(-(4*a*c - b^2)^5)^(1/2))*(b^11 - b^6*(-(4*a*c - b^2)^5)^(1/2) - 1024*a^5*b*c^5 + 160*a^2*b^7*c
^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 + 20*a^3*c^3*(-(4*a*c - b^2)^5)^(1/2) - 20*a*b^9*c - 30*a^2*b^2*c^2*(-
(4*a*c - b^2)^5)^(1/2) + 10*a*b^4*c*(-(4*a*c - b^2)^5)^(1/2)))/(2*a^4*(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**2/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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