3.420 \(\int \frac{1}{(c+\frac{a}{x^2}+\frac{b}{x}) x^5} \, dx\)
Optimal. Leaf size=104 \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}+\frac{b}{a^2 x}-\frac{1}{2 a x^2} \]
[Out]
-1/(2*a*x^2) + b/(a^2*x) + (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^3*Sqrt[b^2 - 4*a*c]) +
((b^2 - a*c)*Log[x])/a^3 - ((b^2 - a*c)*Log[a + b*x + c*x^2])/(2*a^3)
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Rubi [A] time = 0.152542, antiderivative size = 104, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used =
{1354, 709, 800, 634, 618, 206, 628} \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}+\frac{b}{a^2 x}-\frac{1}{2 a x^2} \]
Antiderivative was successfully verified.
[In]
Int[1/((c + a/x^2 + b/x)*x^5),x]
[Out]
-1/(2*a*x^2) + b/(a^2*x) + (b*(b^2 - 3*a*c)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^3*Sqrt[b^2 - 4*a*c]) +
((b^2 - a*c)*Log[x])/a^3 - ((b^2 - a*c)*Log[a + b*x + c*x^2])/(2*a^3)
Rule 1354
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + 2*n*p)*(c + b/x^n +
a/x^(2*n))^p, x] /; FreeQ[{a, b, c, m, n}, x] && EqQ[n2, 2*n] && ILtQ[p, 0] && NegQ[n]
Rule 709
Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d - b*e - c
*e*x, x])/(a + b*x + c*x^2), 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[m, -1]
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}{\left (c+\frac{a}{x^2}+\frac{b}{x}\right ) x^5} \, dx &=\int \frac{1}{x^3 \left (a+b x+c x^2\right )} \, dx\\ &=-\frac{1}{2 a x^2}+\frac{\int \frac{-b-c x}{x^2 \left (a+b x+c x^2\right )} \, dx}{a}\\ &=-\frac{1}{2 a x^2}+\frac{\int \left (-\frac{b}{a x^2}+\frac{b^2-a c}{a^2 x}+\frac{-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx}{a}\\ &=-\frac{1}{2 a x^2}+\frac{b}{a^2 x}+\frac{\left (b^2-a c\right ) \log (x)}{a^3}+\frac{\int \frac{-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x}{a+b x+c x^2} \, dx}{a^3}\\ &=-\frac{1}{2 a x^2}+\frac{b}{a^2 x}+\frac{\left (b^2-a c\right ) \log (x)}{a^3}-\frac{\left (b \left (b^2-3 a c\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 a^3}-\frac{\left (b^2-a c\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 a^3}\\ &=-\frac{1}{2 a x^2}+\frac{b}{a^2 x}+\frac{\left (b^2-a c\right ) \log (x)}{a^3}-\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac{\left (b \left (b^2-3 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^3}\\ &=-\frac{1}{2 a x^2}+\frac{b}{a^2 x}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a^3 \sqrt{b^2-4 a c}}+\frac{\left (b^2-a c\right ) \log (x)}{a^3}-\frac{\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.136071, size = 102, normalized size = 0.98 \[ \frac{-\frac{a^2}{x^2}+2 \log (x) \left (b^2-a c\right )+\left (a c-b^2\right ) \log (a+x (b+c x))-\frac{2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{2 a b}{x}}{2 a^3} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((c + a/x^2 + b/x)*x^5),x]
[Out]
(-(a^2/x^2) + (2*a*b)/x - (2*b*(b^2 - 3*a*c)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 2*(b
^2 - a*c)*Log[x] + (-b^2 + a*c)*Log[a + x*(b + c*x)])/(2*a^3)
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Maple [A] time = 0.007, size = 150, normalized size = 1.4 \begin{align*} -{\frac{1}{2\,a{x}^{2}}}-{\frac{\ln \left ( x \right ) c}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}}}+{\frac{b}{x{a}^{2}}}+{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) }{2\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}}{2\,{a}^{3}}}+3\,{\frac{bc}{{a}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}}{{a}^{3}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(c+a/x^2+b/x)/x^5,x)
[Out]
-1/2/a/x^2-1/a^2*ln(x)*c+1/a^3*ln(x)*b^2+b/a^2/x+1/2/a^2*c*ln(c*x^2+b*x+a)-1/2/a^3*ln(c*x^2+b*x+a)*b^2+3/a^2/(
4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c-1/a^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^
(1/2))*b^3
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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/(c+a/x^2+b/x)/x^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.11755, size = 798, normalized size = 7.67 \begin{align*} \left [-\frac{{\left (b^{3} - 3 \, a b c\right )} \sqrt{b^{2} - 4 \, a c} x^{2} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + a^{2} b^{2} - 4 \, a^{3} c +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}, \frac{2 \,{\left (b^{3} - 3 \, a b c\right )} \sqrt{-b^{2} + 4 \, a c} x^{2} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - a^{2} b^{2} + 4 \, a^{3} c -{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (c x^{2} + b x + a\right ) + 2 \,{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (x\right ) + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+a/x^2+b/x)/x^5,x, algorithm="fricas")
[Out]
[-1/2*((b^3 - 3*a*b*c)*sqrt(b^2 - 4*a*c)*x^2*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)) + a^2*b^2 - 4*a^3*c + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*x^2*log(c*x^2 + b*x + a) - 2*(b^
4 - 5*a*b^2*c + 4*a^2*c^2)*x^2*log(x) - 2*(a*b^3 - 4*a^2*b*c)*x)/((a^3*b^2 - 4*a^4*c)*x^2), 1/2*(2*(b^3 - 3*a*
b*c)*sqrt(-b^2 + 4*a*c)*x^2*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - a^2*b^2 + 4*a^3*c - (b^4 -
5*a*b^2*c + 4*a^2*c^2)*x^2*log(c*x^2 + b*x + a) + 2*(b^4 - 5*a*b^2*c + 4*a^2*c^2)*x^2*log(x) + 2*(a*b^3 - 4*a
^2*b*c)*x)/((a^3*b^2 - 4*a^4*c)*x^2)]
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Sympy [B] time = 5.23101, size = 1525, normalized size = 14.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+a/x**2+b/x)/x**5,x)
[Out]
(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))*log(x + (24*a**9*c**3*
(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 - 42*a**8*b**2*c**2
*(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 + 17*a**7*b**4*c*(
-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 + 12*a**7*c**4*(-b*s
qrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) - 2*a**6*b**6*(-b*sqrt(-4*a
*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 - 15*a**6*b**2*c**3*(-b*sqrt(-4*
a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) + 7*a**5*b**4*c**2*(-b*sqrt(-4*a*c
+ b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) - 12*a**5*c**5 - a**4*b**6*c*(-b*sqrt
(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) + 63*a**4*b**2*c**4 - 103*a**3
*b**4*c**3 + 70*a**2*b**6*c**2 - 20*a*b**8*c + 2*b**10)/(27*a**4*b*c**5 - 63*a**3*b**3*c**4 + 54*a**2*b**5*c**
3 - 18*a*b**7*c**2 + 2*b**9*c)) + (b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)
/(2*a**3))*log(x + (24*a**9*c**3*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/
(2*a**3))**2 - 42*a**8*b**2*c**2*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/
(2*a**3))**2 + 17*a**7*b**4*c*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*
a**3))**2 + 12*a**7*c**4*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)
) - 2*a**6*b**6*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3))**2 - 15
*a**6*b**2*c**3*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) + 7*a**
5*b**4*c**2*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) - 12*a**5*c
**5 - a**4*b**6*c*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(2*a**3*(4*a*c - b**2)) + (a*c - b**2)/(2*a**3)) + 63*
a**4*b**2*c**4 - 103*a**3*b**4*c**3 + 70*a**2*b**6*c**2 - 20*a*b**8*c + 2*b**10)/(27*a**4*b*c**5 - 63*a**3*b**
3*c**4 + 54*a**2*b**5*c**3 - 18*a*b**7*c**2 + 2*b**9*c)) + (-a + 2*b*x)/(2*a**2*x**2) - (a*c - b**2)*log(x + (
-12*a**5*c**5 + 63*a**4*b**2*c**4 - 12*a**4*c**4*(a*c - b**2) - 103*a**3*b**4*c**3 + 15*a**3*b**2*c**3*(a*c -
b**2) + 24*a**3*c**3*(a*c - b**2)**2 + 70*a**2*b**6*c**2 - 7*a**2*b**4*c**2*(a*c - b**2) - 42*a**2*b**2*c**2*(
a*c - b**2)**2 - 20*a*b**8*c + a*b**6*c*(a*c - b**2) + 17*a*b**4*c*(a*c - b**2)**2 + 2*b**10 - 2*b**6*(a*c - b
**2)**2)/(27*a**4*b*c**5 - 63*a**3*b**3*c**4 + 54*a**2*b**5*c**3 - 18*a*b**7*c**2 + 2*b**9*c))/a**3
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Giac [A] time = 1.12457, size = 142, normalized size = 1.37 \begin{align*} -\frac{{\left (b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac{{\left (b^{2} - a c\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a^{3}} + \frac{2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c+a/x^2+b/x)/x^5,x, algorithm="giac")
[Out]
-1/2*(b^2 - a*c)*log(c*x^2 + b*x + a)/a^3 + (b^2 - a*c)*log(abs(x))/a^3 - (b^3 - 3*a*b*c)*arctan((2*c*x + b)/s
qrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^3) + 1/2*(2*a*b*x - a^2)/(a^3*x^2)