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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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]

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 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]

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 = 0.02, size = 43, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(b*x^2 + c*x^4)),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (b x^2+c x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^2*(b*x^2 + c*x^4)),x]

[Out]

IntegrateAlgebraic[1/(x^2*(b*x^2 + c*x^4)), x]

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 106, normalized size = 2.47 \begin {gather*} \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 {gather*}

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)]

________________________________________________________________________________________

giac [A]  time = 0.17, size = 40, normalized size = 0.93 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 39, normalized size = 0.91 \begin {gather*} \frac {c^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{2}}+\frac {c}{b^{2} x}-\frac {1}{3 b \,x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 2.96, size = 40, normalized size = 0.93 \begin {gather*} \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 {gather*}

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]

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

________________________________________________________________________________________

mupad [B]  time = 4.14, size = 37, normalized size = 0.86 \begin {gather*} \frac {c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{b^{5/2}}-\frac {\frac {1}{3\,b}-\frac {c\,x^2}{b^2}}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(b*x^2 + c*x^4)),x)

[Out]

(c^(3/2)*atan((c^(1/2)*x)/b^(1/2)))/b^(5/2) - (1/(3*b) - (c*x^2)/b^2)/x^3

________________________________________________________________________________________

sympy [B]  time = 0.25, size = 87, normalized size = 2.02 \begin {gather*} - \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 {gather*}

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)

________________________________________________________________________________________