3.186 \(\int \frac{1}{x^2 (b x^2+c x^4)} \, dx\)

Optimal. Leaf size=43 \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{c}{b^2 x}-\frac{1}{3 b x^3} \]

[Out]

-1/(3*b*x^3) + c/(b^2*x) + (c^(3/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(5/2)

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Rubi [A]  time = 0.0226364, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1584, 325, 205} \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{c}{b^2 x}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(b*x^2 + c*x^4)),x]

[Out]

-1/(3*b*x^3) + c/(b^2*x) + (c^(3/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(5/2)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (b x^2+c x^4\right )} \, dx &=\int \frac{1}{x^4 \left (b+c x^2\right )} \, dx\\ &=-\frac{1}{3 b x^3}-\frac{c \int \frac{1}{x^2 \left (b+c x^2\right )} \, dx}{b}\\ &=-\frac{1}{3 b x^3}+\frac{c}{b^2 x}+\frac{c^2 \int \frac{1}{b+c x^2} \, dx}{b^2}\\ &=-\frac{1}{3 b x^3}+\frac{c}{b^2 x}+\frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0200254, size = 43, normalized size = 1. \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{c}{b^2 x}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(b*x^2 + c*x^4)),x]

[Out]

-1/(3*b*x^3) + c/(b^2*x) + (c^(3/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(5/2)

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Maple [A]  time = 0.05, size = 39, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,b{x}^{3}}}+{\frac{c}{{b}^{2}x}}+{\frac{{c}^{2}}{{b}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(c*x^4+b*x^2),x)

[Out]

-1/3/b/x^3+c/b^2/x+c^2/b^2/(b*c)^(1/2)*arctan(x*c/(b*c)^(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/x^2/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.39671, size = 234, normalized size = 5.44 \begin{align*} \left [\frac{3 \, c x^{3} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} + 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) + 6 \, c x^{2} - 2 \, b}{6 \, b^{2} x^{3}}, \frac{3 \, c x^{3} \sqrt{\frac{c}{b}} \arctan \left (x \sqrt{\frac{c}{b}}\right ) + 3 \, c x^{2} - b}{3 \, b^{2} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

[1/6*(3*c*x^3*sqrt(-c/b)*log((c*x^2 + 2*b*x*sqrt(-c/b) - b)/(c*x^2 + b)) + 6*c*x^2 - 2*b)/(b^2*x^3), 1/3*(3*c*
x^3*sqrt(c/b)*arctan(x*sqrt(c/b)) + 3*c*x^2 - b)/(b^2*x^3)]

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Sympy [B]  time = 0.402925, size = 87, normalized size = 2.02 \begin{align*} - \frac{\sqrt{- \frac{c^{3}}{b^{5}}} \log{\left (- \frac{b^{3} \sqrt{- \frac{c^{3}}{b^{5}}}}{c^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{c^{3}}{b^{5}}} \log{\left (\frac{b^{3} \sqrt{- \frac{c^{3}}{b^{5}}}}{c^{2}} + x \right )}}{2} + \frac{- b + 3 c x^{2}}{3 b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(c*x**4+b*x**2),x)

[Out]

-sqrt(-c**3/b**5)*log(-b**3*sqrt(-c**3/b**5)/c**2 + x)/2 + sqrt(-c**3/b**5)*log(b**3*sqrt(-c**3/b**5)/c**2 + x
)/2 + (-b + 3*c*x**2)/(3*b**2*x**3)

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Giac [A]  time = 1.27426, size = 54, normalized size = 1.26 \begin{align*} \frac{c^{2} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} b^{2}} + \frac{3 \, c x^{2} - b}{3 \, b^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

c^2*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b^2) + 1/3*(3*c*x^2 - b)/(b^2*x^3)