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

Optimal. Leaf size=118 \[ -\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{x \left (b^2-a c\right )}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c} \]

[Out]

((b^2 - a*c)*x)/c^3 - (b*x^2)/(2*c^2) + x^3/(3*c) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^
2 - 4*a*c]])/(c^4*Sqrt[b^2 - 4*a*c]) - (b*(b^2 - 2*a*c)*Log[a + b*x + c*x^2])/(2*c^4)

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Rubi [A]  time = 0.104779, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1354, 701, 634, 618, 206, 628} \[ -\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{b \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{x \left (b^2-a c\right )}{c^3}-\frac{b x^2}{2 c^2}+\frac{x^3}{3 c} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(c + a/x^2 + b/x),x]

[Out]

((b^2 - a*c)*x)/c^3 - (b*x^2)/(2*c^2) + x^3/(3*c) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*ArcTanh[(b + 2*c*x)/Sqrt[b^
2 - 4*a*c]])/(c^4*Sqrt[b^2 - 4*a*c]) - (b*(b^2 - 2*a*c)*Log[a + b*x + c*x^2])/(2*c^4)

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 701

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)
^m, 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] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

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

Mathematica [A]  time = 0.084465, size = 112, normalized size = 0.95 \[ \frac{\frac{6 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+c x \left (-6 a c+6 b^2-3 b c x+2 c^2 x^2\right )-3 \left (b^3-2 a b c\right ) \log (a+x (b+c x))}{6 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(c + a/x^2 + b/x),x]

[Out]

(c*x*(6*b^2 - 6*a*c - 3*b*c*x + 2*c^2*x^2) + (6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4
*a*c]])/Sqrt[-b^2 + 4*a*c] - 3*(b^3 - 2*a*b*c)*Log[a + x*(b + c*x)])/(6*c^4)

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Maple [A]  time = 0.004, size = 190, normalized size = 1.6 \begin{align*}{\frac{{x}^{3}}{3\,c}}-{\frac{b{x}^{2}}{2\,{c}^{2}}}-{\frac{ax}{{c}^{2}}}+{\frac{{b}^{2}x}{{c}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ab}{{c}^{3}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}}{2\,{c}^{4}}}+2\,{\frac{{a}^{2}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{a{b}^{2}}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}}{{c}^{4}}\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(x^2/(c+a/x^2+b/x),x)

[Out]

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

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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(x^2/(c+a/x^2+b/x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.82093, size = 829, normalized size = 7.03 \begin{align*} \left [\frac{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} - 3 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + 3 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt{b^{2} - 4 \, a c} \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 ) + 6 \,{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} x - 3 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{6 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} - 3 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} - 6 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \,{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} x - 3 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} \log \left (c x^{2} + b x + a\right )}{6 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(2*(b^2*c^3 - 4*a*c^4)*x^3 - 3*(b^3*c^2 - 4*a*b*c^3)*x^2 + 3*(b^4 - 4*a*b^2*c + 2*a^2*c^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)) + 6*(b^4*c - 5*
a*b^2*c^2 + 4*a^2*c^3)*x - 3*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*log(c*x^2 + b*x + a))/(b^2*c^4 - 4*a*c^5), 1/6*(2
*(b^2*c^3 - 4*a*c^4)*x^3 - 3*(b^3*c^2 - 4*a*b*c^3)*x^2 - 6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*sqrt(-b^2 + 4*a*c)*ar
ctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 6*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*x - 3*(b^5 - 6*a*b^3
*c + 8*a^2*b*c^2)*log(c*x^2 + b*x + a))/(b^2*c^4 - 4*a*c^5)]

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Sympy [B]  time = 0.963526, size = 496, normalized size = 4.2 \begin{align*} - \frac{b x^{2}}{2 c^{2}} + \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 3 a^{2} b c + a b^{3} + 4 a c^{4} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) - b^{2} c^{3} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right )}{2 a^{2} c^{2} - 4 a b^{2} c + b^{4}} \right )} + \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 3 a^{2} b c + a b^{3} + 4 a c^{4} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right ) - b^{2} c^{3} \left (\frac{b \left (2 a c - b^{2}\right )}{2 c^{4}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right )}{2 c^{4} \left (4 a c - b^{2}\right )}\right )}{2 a^{2} c^{2} - 4 a b^{2} c + b^{4}} \right )} + \frac{x^{3}}{3 c} - \frac{x \left (a c - b^{2}\right )}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*x**2/(2*c**2) + (b*(2*a*c - b**2)/(2*c**4) - sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*c**4*
(4*a*c - b**2)))*log(x + (-3*a**2*b*c + a*b**3 + 4*a*c**4*(b*(2*a*c - b**2)/(2*c**4) - sqrt(-4*a*c + b**2)*(2*
a**2*c**2 - 4*a*b**2*c + b**4)/(2*c**4*(4*a*c - b**2))) - b**2*c**3*(b*(2*a*c - b**2)/(2*c**4) - sqrt(-4*a*c +
b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*c**4*(4*a*c - b**2))))/(2*a**2*c**2 - 4*a*b**2*c + b**4)) + (b*(2*
a*c - b**2)/(2*c**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c + b**4)/(2*c**4*(4*a*c - b**2)))*log(x +
(-3*a**2*b*c + a*b**3 + 4*a*c**4*(b*(2*a*c - b**2)/(2*c**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a*b**2*c +
b**4)/(2*c**4*(4*a*c - b**2))) - b**2*c**3*(b*(2*a*c - b**2)/(2*c**4) + sqrt(-4*a*c + b**2)*(2*a**2*c**2 - 4*a
*b**2*c + b**4)/(2*c**4*(4*a*c - b**2))))/(2*a**2*c**2 - 4*a*b**2*c + b**4)) + x**3/(3*c) - x*(a*c - b**2)/c**
3

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Giac [A]  time = 1.12624, size = 153, normalized size = 1.3 \begin{align*} \frac{2 \, c^{2} x^{3} - 3 \, b c x^{2} + 6 \, b^{2} x - 6 \, a c x}{6 \, c^{3}} - \frac{{\left (b^{3} - 2 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*c^2*x^3 - 3*b*c*x^2 + 6*b^2*x - 6*a*c*x)/c^3 - 1/2*(b^3 - 2*a*b*c)*log(c*x^2 + b*x + a)/c^4 + (b^4 - 4*
a*b^2*c + 2*a^2*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4)