3.650 \(\int \frac{1}{x^3 (a+c x^4)} \, dx\)

Optimal. Leaf size=40 \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a x^2} \]

[Out]

-1/(2*a*x^2) - (Sqrt[c]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/2))

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Rubi [A]  time = 0.0173497, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 325, 205} \[ -\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + c*x^4)),x]

[Out]

-1/(2*a*x^2) - (Sqrt[c]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*a^(3/2))

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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

Mathematica [A]  time = 0.0312544, size = 79, normalized size = 1.98 \[ \frac{\sqrt{c} x^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt{c} x^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-\sqrt{a}}{2 a^{3/2} x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + c*x^4)),x]

[Out]

(-Sqrt[a] + Sqrt[c]*x^2*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[c]*x^2*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a
^(1/4)])/(2*a^(3/2)*x^2)

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \begin{align*} -{\frac{c}{2\,a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{1}{2\,a{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(c*x^4+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.708, size = 201, normalized size = 5.02 \begin{align*} \left [\frac{x^{2} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} - 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) - 2}{4 \, a x^{2}}, \frac{x^{2} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) - 1}{2 \, a x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.509701, size = 71, normalized size = 1.78 \begin{align*} \frac{\sqrt{- \frac{c}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{c}{a^{3}}}}{c} + x^{2} \right )}}{4} - \frac{\sqrt{- \frac{c}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{c}{a^{3}}}}{c} + x^{2} \right )}}{4} - \frac{1}{2 a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(c*x**4+a),x)

[Out]

sqrt(-c/a**3)*log(-a**2*sqrt(-c/a**3)/c + x**2)/4 - sqrt(-c/a**3)*log(a**2*sqrt(-c/a**3)/c + x**2)/4 - 1/(2*a*
x**2)

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Giac [A]  time = 1.62223, size = 42, normalized size = 1.05 \begin{align*} -\frac{c \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a} - \frac{1}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4+a),x, algorithm="giac")

[Out]

-1/2*c*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*a) - 1/2/(a*x^2)