3.84 \(\int \frac{1}{x^3 (c+(a+b x)^2)} \, dx\)
Optimal. Leaf size=121 \[ \frac{b^2 \left (3 a^2-c\right ) \log (x)}{\left (a^2+c\right )^3}-\frac{b^2 \left (3 a^2-c\right ) \log \left ((a+b x)^2+c\right )}{2 \left (a^2+c\right )^3}-\frac{a b^2 \left (a^2-3 c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2+c\right )^3}+\frac{2 a b}{x \left (a^2+c\right )^2}-\frac{1}{2 x^2 \left (a^2+c\right )} \]
[Out]
-1/(2*(a^2 + c)*x^2) + (2*a*b)/((a^2 + c)^2*x) - (a*b^2*(a^2 - 3*c)*ArcTan[(a + b*x)/Sqrt[c]])/(Sqrt[c]*(a^2 +
c)^3) + (b^2*(3*a^2 - c)*Log[x])/(a^2 + c)^3 - (b^2*(3*a^2 - c)*Log[c + (a + b*x)^2])/(2*(a^2 + c)^3)
________________________________________________________________________________________
Rubi [A] time = 0.125346, antiderivative size = 121, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used =
{371, 710, 801, 635, 203, 260} \[ \frac{b^2 \left (3 a^2-c\right ) \log (x)}{\left (a^2+c\right )^3}-\frac{b^2 \left (3 a^2-c\right ) \log \left ((a+b x)^2+c\right )}{2 \left (a^2+c\right )^3}-\frac{a b^2 \left (a^2-3 c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2+c\right )^3}+\frac{2 a b}{x \left (a^2+c\right )^2}-\frac{1}{2 x^2 \left (a^2+c\right )} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^3*(c + (a + b*x)^2)),x]
[Out]
-1/(2*(a^2 + c)*x^2) + (2*a*b)/((a^2 + c)^2*x) - (a*b^2*(a^2 - 3*c)*ArcTan[(a + b*x)/Sqrt[c]])/(Sqrt[c]*(a^2 +
c)^3) + (b^2*(3*a^2 - c)*Log[x])/(a^2 + c)^3 - (b^2*(3*a^2 - c)*Log[c + (a + b*x)^2])/(2*(a^2 + c)^3)
Rule 371
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Rule 710
Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 +
a*e^2)), x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*(d - e*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]
Rule 801
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]
Rule 635
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && !NiceSqrtQ[-(a*c)]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 260
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (c+(a+b x)^2\right )} \, dx &=b^2 \operatorname{Subst}\left (\int \frac{1}{(-a+x)^3 \left (c+x^2\right )} \, dx,x,a+b x\right )\\ &=-\frac{1}{2 \left (a^2+c\right ) x^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{-a-x}{(-a+x)^2 \left (c+x^2\right )} \, dx,x,a+b x\right )}{a^2+c}\\ &=-\frac{1}{2 \left (a^2+c\right ) x^2}+\frac{b^2 \operatorname{Subst}\left (\int \left (-\frac{2 a}{\left (a^2+c\right ) (a-x)^2}+\frac{-3 a^2+c}{\left (a^2+c\right )^2 (a-x)}+\frac{-a \left (a^2-3 c\right )-\left (3 a^2-c\right ) x}{\left (a^2+c\right )^2 \left (c+x^2\right )}\right ) \, dx,x,a+b x\right )}{a^2+c}\\ &=-\frac{1}{2 \left (a^2+c\right ) x^2}+\frac{2 a b}{\left (a^2+c\right )^2 x}+\frac{b^2 \left (3 a^2-c\right ) \log (x)}{\left (a^2+c\right )^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{-a \left (a^2-3 c\right )-\left (3 a^2-c\right ) x}{c+x^2} \, dx,x,a+b x\right )}{\left (a^2+c\right )^3}\\ &=-\frac{1}{2 \left (a^2+c\right ) x^2}+\frac{2 a b}{\left (a^2+c\right )^2 x}+\frac{b^2 \left (3 a^2-c\right ) \log (x)}{\left (a^2+c\right )^3}-\frac{\left (a b^2 \left (a^2-3 c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+x^2} \, dx,x,a+b x\right )}{\left (a^2+c\right )^3}-\frac{\left (b^2 \left (3 a^2-c\right )\right ) \operatorname{Subst}\left (\int \frac{x}{c+x^2} \, dx,x,a+b x\right )}{\left (a^2+c\right )^3}\\ &=-\frac{1}{2 \left (a^2+c\right ) x^2}+\frac{2 a b}{\left (a^2+c\right )^2 x}-\frac{a b^2 \left (a^2-3 c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2+c\right )^3}+\frac{b^2 \left (3 a^2-c\right ) \log (x)}{\left (a^2+c\right )^3}-\frac{b^2 \left (3 a^2-c\right ) \log \left (c+(a+b x)^2\right )}{2 \left (a^2+c\right )^3}\\ \end{align*}
Mathematica [A] time = 0.128787, size = 106, normalized size = 0.88 \[ -\frac{b^2 \left (3 a^2-c\right ) \log \left (a^2+2 a b x+b^2 x^2+c\right )+2 b^2 \left (c-3 a^2\right ) \log (x)+\frac{2 a b^2 \left (a^2-3 c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c}}+\frac{\left (a^2+c\right ) \left (a^2-4 a b x+c\right )}{x^2}}{2 \left (a^2+c\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^3*(c + (a + b*x)^2)),x]
[Out]
-(((a^2 + c)*(a^2 + c - 4*a*b*x))/x^2 + (2*a*b^2*(a^2 - 3*c)*ArcTan[(a + b*x)/Sqrt[c]])/Sqrt[c] + 2*b^2*(-3*a^
2 + c)*Log[x] + b^2*(3*a^2 - c)*Log[a^2 + c + 2*a*b*x + b^2*x^2])/(2*(a^2 + c)^3)
________________________________________________________________________________________
Maple [A] time = 0.01, size = 198, normalized size = 1.6 \begin{align*} -{\frac{1}{ \left ( 2\,{a}^{2}+2\,c \right ){x}^{2}}}+3\,{\frac{{b}^{2}\ln \left ( x \right ){a}^{2}}{ \left ({a}^{2}+c \right ) ^{3}}}-{\frac{{b}^{2}\ln \left ( x \right ) c}{ \left ({a}^{2}+c \right ) ^{3}}}+2\,{\frac{ab}{ \left ({a}^{2}+c \right ) ^{2}x}}-{\frac{3\,{b}^{2}\ln \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2}+c \right ){a}^{2}}{2\, \left ({a}^{2}+c \right ) ^{3}}}+{\frac{{b}^{2}\ln \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2}+c \right ) c}{2\, \left ({a}^{2}+c \right ) ^{3}}}-{\frac{{a}^{3}{b}^{2}}{ \left ({a}^{2}+c \right ) ^{3}}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+3\,{\frac{{b}^{2}\sqrt{c}a}{ \left ({a}^{2}+c \right ) ^{3}}\arctan \left ( 1/2\,{\frac{2\,{b}^{2}x+2\,ab}{b\sqrt{c}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^3/(c+(b*x+a)^2),x)
[Out]
-1/2/(a^2+c)/x^2+3*b^2/(a^2+c)^3*ln(x)*a^2-b^2/(a^2+c)^3*ln(x)*c+2*a*b/(a^2+c)^2/x-3/2*b^2/(a^2+c)^3*ln(b^2*x^
2+2*a*b*x+a^2+c)*a^2+1/2*b^2/(a^2+c)^3*ln(b^2*x^2+2*a*b*x+a^2+c)*c-b^2/(a^2+c)^3/c^(1/2)*arctan(1/2*(2*b^2*x+2
*a*b)/b/c^(1/2))*a^3+3*b^2/(a^2+c)^3*c^(1/2)*arctan(1/2*(2*b^2*x+2*a*b)/b/c^(1/2))*a
________________________________________________________________________________________
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+(b*x+a)^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [A] time = 1.92599, size = 811, normalized size = 6.7 \begin{align*} \left [-\frac{a^{4} c -{\left (a^{3} b^{2} - 3 \, a b^{2} c\right )} \sqrt{-c} x^{2} \log \left (\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - 2 \,{\left (b x + a\right )} \sqrt{-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) + 2 \, a^{2} c^{2} +{\left (3 \, a^{2} b^{2} c - b^{2} c^{2}\right )} x^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \,{\left (3 \, a^{2} b^{2} c - b^{2} c^{2}\right )} x^{2} \log \left (x\right ) + c^{3} - 4 \,{\left (a^{3} b c + a b c^{2}\right )} x}{2 \,{\left (a^{6} c + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{3} + c^{4}\right )} x^{2}}, -\frac{a^{4} c + 2 \,{\left (a^{3} b^{2} - 3 \, a b^{2} c\right )} \sqrt{c} x^{2} \arctan \left (\frac{b x + a}{\sqrt{c}}\right ) + 2 \, a^{2} c^{2} +{\left (3 \, a^{2} b^{2} c - b^{2} c^{2}\right )} x^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \,{\left (3 \, a^{2} b^{2} c - b^{2} c^{2}\right )} x^{2} \log \left (x\right ) + c^{3} - 4 \,{\left (a^{3} b c + a b c^{2}\right )} x}{2 \,{\left (a^{6} c + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{3} + c^{4}\right )} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(c+(b*x+a)^2),x, algorithm="fricas")
[Out]
[-1/2*(a^4*c - (a^3*b^2 - 3*a*b^2*c)*sqrt(-c)*x^2*log((b^2*x^2 + 2*a*b*x + a^2 - 2*(b*x + a)*sqrt(-c) - c)/(b^
2*x^2 + 2*a*b*x + a^2 + c)) + 2*a^2*c^2 + (3*a^2*b^2*c - b^2*c^2)*x^2*log(b^2*x^2 + 2*a*b*x + a^2 + c) - 2*(3*
a^2*b^2*c - b^2*c^2)*x^2*log(x) + c^3 - 4*(a^3*b*c + a*b*c^2)*x)/((a^6*c + 3*a^4*c^2 + 3*a^2*c^3 + c^4)*x^2),
-1/2*(a^4*c + 2*(a^3*b^2 - 3*a*b^2*c)*sqrt(c)*x^2*arctan((b*x + a)/sqrt(c)) + 2*a^2*c^2 + (3*a^2*b^2*c - b^2*c
^2)*x^2*log(b^2*x^2 + 2*a*b*x + a^2 + c) - 2*(3*a^2*b^2*c - b^2*c^2)*x^2*log(x) + c^3 - 4*(a^3*b*c + a*b*c^2)*
x)/((a^6*c + 3*a^4*c^2 + 3*a^2*c^3 + c^4)*x^2)]
________________________________________________________________________________________
Sympy [B] time = 7.24595, size = 3284, normalized size = 27.14 \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+(b*x+a)**2),x)
[Out]
b**2*(3*a**2 - c)*log(x + (-4*a**16*b**4*c*(3*a**2 - c)**2/(a**2 + c)**6 + 24*a**14*b**4*c**2*(3*a**2 - c)**2/
(a**2 + c)**6 + 216*a**12*b**4*c**3*(3*a**2 - c)**2/(a**2 + c)**6 - 14*a**12*b**4*c*(3*a**2 - c)/(a**2 + c)**3
+ 568*a**10*b**4*c**4*(3*a**2 - c)**2/(a**2 + c)**6 - 44*a**10*b**4*c**2*(3*a**2 - c)/(a**2 + c)**3 + 720*a**
8*b**4*c**5*(3*a**2 - c)**2/(a**2 + c)**6 - 42*a**8*b**4*c**3*(3*a**2 - c)/(a**2 + c)**3 + 78*a**8*b**4*c + 45
6*a**6*b**4*c**6*(3*a**2 - c)**2/(a**2 + c)**6 - 8*a**6*b**4*c**4*(3*a**2 - c)/(a**2 + c)**3 - 464*a**6*b**4*c
**2 + 104*a**4*b**4*c**7*(3*a**2 - c)**2/(a**2 + c)**6 - 2*a**4*b**4*c**5*(3*a**2 - c)/(a**2 + c)**3 + 380*a**
4*b**4*c**3 - 24*a**2*b**4*c**8*(3*a**2 - c)**2/(a**2 + c)**6 - 12*a**2*b**4*c**6*(3*a**2 - c)/(a**2 + c)**3 -
96*a**2*b**4*c**4 - 12*b**4*c**9*(3*a**2 - c)**2/(a**2 + c)**6 - 6*b**4*c**7*(3*a**2 - c)/(a**2 + c)**3 + 6*b
**4*c**5)/(a**9*b**5 + 72*a**7*b**5*c - 270*a**5*b**5*c**2 + 144*a**3*b**5*c**3 - 27*a*b**5*c**4))/(a**2 + c)*
*3 + (-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2
+ c)**3))*log(x + (-4*a**16*c*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b*
*2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 + 24*a**14*c**2*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3
*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 - 14*a**12*b**2*c*(-a*b**2*sqrt(-c)*(a**2 - 3*c)
/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3)) + 216*a**12*c**3*(-a*b**2
*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2
- 44*a**10*b**2*c**2*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**
2 - c)/(2*(a**2 + c)**3)) + 568*a**10*c**4*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2
+ c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 + 78*a**8*b**4*c - 42*a**8*b**2*c**3*(-a*b**2*sqrt(-c)*(a**
2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3)) + 720*a**8*c**5*(
-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**
3))**2 - 464*a**6*b**4*c**2 - 8*a**6*b**2*c**4*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c
**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3)) + 456*a**6*c**6*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6
+ 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 + 380*a**4*b**4*c**3 - 2*a**4*b**2
*c**5*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2
+ c)**3)) + 104*a**4*c**7*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*
(3*a**2 - c)/(2*(a**2 + c)**3))**2 - 96*a**2*b**4*c**4 - 12*a**2*b**2*c**6*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c
*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3)) - 24*a**2*c**8*(-a*b**2*sqrt(-
c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 + 6*b**
4*c**5 - 6*b**2*c**7*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**
2 - c)/(2*(a**2 + c)**3)) - 12*c**9*(-a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)
) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2)/(a**9*b**5 + 72*a**7*b**5*c - 270*a**5*b**5*c**2 + 144*a**3*b**5*
c**3 - 27*a*b**5*c**4)) + (a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3
*a**2 - c)/(2*(a**2 + c)**3))*log(x + (-4*a**16*c*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2
*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 + 24*a**14*c**2*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a
**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 - 14*a**12*b**2*c*(a*b**2*sqrt
(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3)) + 216*a*
*12*c**3*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a*
*2 + c)**3))**2 - 44*a**10*b**2*c**2*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)
) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3)) + 568*a**10*c**4*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c
+ 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 + 78*a**8*b**4*c - 42*a**8*b**2*c**3*(a*b**2*
sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3)) + 72
0*a**8*c**5*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*
(a**2 + c)**3))**2 - 464*a**6*b**4*c**2 - 8*a**6*b**2*c**4*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c
+ 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3)) + 456*a**6*c**6*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(
2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 + 380*a**4*b**4*c**3 - 2
*a**4*b**2*c**5*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)
/(2*(a**2 + c)**3)) + 104*a**4*c**7*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3))
- b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 - 96*a**2*b**4*c**4 - 12*a**2*b**2*c**6*(a*b**2*sqrt(-c)*(a**2 - 3*
c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3)) - 24*a**2*c**8*(a*b**2*
sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2 +
6*b**4*c**5 - 6*b**2*c**7*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c**3)) - b**2*(
3*a**2 - c)/(2*(a**2 + c)**3)) - 12*c**9*(a*b**2*sqrt(-c)*(a**2 - 3*c)/(2*c*(a**6 + 3*a**4*c + 3*a**2*c**2 + c
**3)) - b**2*(3*a**2 - c)/(2*(a**2 + c)**3))**2)/(a**9*b**5 + 72*a**7*b**5*c - 270*a**5*b**5*c**2 + 144*a**3*b
**5*c**3 - 27*a*b**5*c**4)) + (-a**2 + 4*a*b*x - c)/(x**2*(2*a**4 + 4*a**2*c + 2*c**2))
________________________________________________________________________________________
Giac [A] time = 1.16012, size = 263, normalized size = 2.17 \begin{align*} -\frac{{\left (3 \, a^{2} b^{2} - b^{2} c\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \,{\left (a^{6} + 3 \, a^{4} c + 3 \, a^{2} c^{2} + c^{3}\right )}} + \frac{{\left (3 \, a^{2} b^{2} - b^{2} c\right )} \log \left ({\left | x \right |}\right )}{a^{6} + 3 \, a^{4} c + 3 \, a^{2} c^{2} + c^{3}} - \frac{{\left (a^{3} b^{3} - 3 \, a b^{3} c\right )} \arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{{\left (a^{6} + 3 \, a^{4} c + 3 \, a^{2} c^{2} + c^{3}\right )} b \sqrt{c}} - \frac{a^{4} + 2 \, a^{2} c + c^{2} - 4 \,{\left (a^{3} b + a b c\right )} x}{2 \,{\left (a^{2} + c\right )}^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(c+(b*x+a)^2),x, algorithm="giac")
[Out]
-1/2*(3*a^2*b^2 - b^2*c)*log(b^2*x^2 + 2*a*b*x + a^2 + c)/(a^6 + 3*a^4*c + 3*a^2*c^2 + c^3) + (3*a^2*b^2 - b^2
*c)*log(abs(x))/(a^6 + 3*a^4*c + 3*a^2*c^2 + c^3) - (a^3*b^3 - 3*a*b^3*c)*arctan((b*x + a)/sqrt(c))/((a^6 + 3*
a^4*c + 3*a^2*c^2 + c^3)*b*sqrt(c)) - 1/2*(a^4 + 2*a^2*c + c^2 - 4*(a^3*b + a*b*c)*x)/((a^2 + c)^3*x^2)