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

Optimal. Leaf size=62 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{\log (x)}{a} \]

[Out]

(b*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + Log[x]/a - Log[a + b*x + c*x^2]/(2*a)

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Rubi [A]  time = 0.0476562, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {1354, 705, 29, 634, 618, 206, 628} \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + a/x^2 + b/x)*x^3),x]

[Out]

(b*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + Log[x]/a - Log[a + b*x + c*x^2]/(2*a)

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 705

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x]
/; FreeQ[{a, b, c, d, e}, 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]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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^3} \, dx &=\int \frac{1}{x \left (a+b x+c x^2\right )} \, dx\\ &=\frac{\int \frac{1}{x} \, dx}{a}+\frac{\int \frac{-b-c x}{a+b x+c x^2} \, dx}{a}\\ &=\frac{\log (x)}{a}-\frac{\int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 a}-\frac{b \int \frac{1}{a+b x+c x^2} \, dx}{2 a}\\ &=\frac{\log (x)}{a}-\frac{\log \left (a+b x+c x^2\right )}{2 a}+\frac{b \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a}\\ &=\frac{b \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c}}+\frac{\log (x)}{a}-\frac{\log \left (a+b x+c x^2\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0680658, size = 61, normalized size = 0.98 \[ -\frac{\frac{2 b \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\log (a+x (b+c x))-2 \log (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*b*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 2*Log[x] + Log[a + x*(b + c*x)])/(2*a)

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Maple [A]  time = 0.005, size = 62, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{2\,a}}-{\frac{b}{a}\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^3,x)

[Out]

ln(x)/a-1/2*ln(c*x^2+b*x+a)/a-1/a*b/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))

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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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.89301, size = 494, normalized size = 7.97 \begin{align*} \left [\frac{\sqrt{b^{2} - 4 \, a c} b \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 ) -{\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right ) + 2 \,{\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} b \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right ) + 2 \,{\left (b^{2} - 4 \, a c\right )} \log \left (x\right )}{2 \,{\left (a b^{2} - 4 \, a^{2} c\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(sqrt(b^2 - 4*a*c)*b*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)) - (b^2 - 4*a*c)*log(c*x^2 + b*x + a) + 2*(b^2 - 4*a*c)*log(x))/(a*b^2 - 4*a^2*c), 1/2*(2*sqrt(-b^2 + 4*
a*c)*b*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (b^2 - 4*a*c)*log(c*x^2 + b*x + a) + 2*(b^2 - 4
*a*c)*log(x))/(a*b^2 - 4*a^2*c)]

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Sympy [B]  time = 1.90599, size = 564, normalized size = 9.1 \begin{align*} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) \log{\left (x + \frac{24 a^{4} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) + 2 a^{2} b^{4} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) \log{\left (x + \frac{24 a^{4} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) + 2 a^{2} b^{4} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac{1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \frac{\log{\left (x \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))*log(x + (24*a**4*c**2*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*
a*c - b**2)) - 1/(2*a))**2 - 14*a**3*b**2*c*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 12*a*
*3*c**2*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a)) + 2*a**2*b**4*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*
a*c - b**2)) - 1/(2*a))**2 + 3*a**2*b**2*c*(-b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a)) - 12*a**2*c
**2 + 11*a*b**2*c - 2*b**4)/(9*a*b*c**2 - 2*b**3*c)) + (b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))*
log(x + (24*a**4*c**2*(b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 14*a**3*b**2*c*(b*sqrt(-4*a*
c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 - 12*a**3*c**2*(b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2
*a)) + 2*a**2*b**4*(b*sqrt(-4*a*c + b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a))**2 + 3*a**2*b**2*c*(b*sqrt(-4*a*c +
b**2)/(2*a*(4*a*c - b**2)) - 1/(2*a)) - 12*a**2*c**2 + 11*a*b**2*c - 2*b**4)/(9*a*b*c**2 - 2*b**3*c)) + log(x)
/a

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Giac [A]  time = 1.13397, size = 84, normalized size = 1.35 \begin{align*} -\frac{b \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} a} - \frac{\log \left (c x^{2} + b x + a\right )}{2 \, a} + \frac{\log \left ({\left | x \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a) - 1/2*log(c*x^2 + b*x + a)/a + log(abs(x))/a