3.1847 \(\int \frac {x^2}{a+\frac {b}{x^2}} \, dx\)

Optimal. Leaf size=42 \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{5/2}}-\frac {b x}{a^2}+\frac {x^3}{3 a} \]

[Out]

-b*x/a^2+1/3*x^3/a+b^(3/2)*arctan(x*a^(1/2)/b^(1/2))/a^(5/2)

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Rubi [A]  time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 302, 205} \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{5/2}}-\frac {b x}{a^2}+\frac {x^3}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x^2),x]

[Out]

-((b*x)/a^2) + x^3/(3*a) + (b^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(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 263

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 42, normalized size = 1.00 \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{5/2}}-\frac {b x}{a^2}+\frac {x^3}{3 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x^2),x]

[Out]

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

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fricas [A]  time = 0.68, size = 99, normalized size = 2.36 \[ \left [\frac {2 \, a x^{3} + 3 \, b \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) - 6 \, b x}{6 \, a^{2}}, \frac {a x^{3} + 3 \, b \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) - 3 \, b x}{3 \, a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 40, normalized size = 0.95 \[ \frac {b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {a^{2} x^{3} - 3 \, a b x}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 38, normalized size = 0.90 \[ \frac {x^{3}}{3 a}+\frac {b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {b x}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b/x^2),x)

[Out]

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

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maxima [A]  time = 1.85, size = 37, normalized size = 0.88 \[ \frac {b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {a x^{3} - 3 \, b x}{3 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 32, normalized size = 0.76 \[ \frac {x^3}{3\,a}+\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{a^{5/2}}-\frac {b\,x}{a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/x^2),x)

[Out]

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

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sympy [B]  time = 0.18, size = 80, normalized size = 1.90 \[ - \frac {\sqrt {- \frac {b^{3}}{a^{5}}} \log {\left (- \frac {a^{2} \sqrt {- \frac {b^{3}}{a^{5}}}}{b} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{3}}{a^{5}}} \log {\left (\frac {a^{2} \sqrt {- \frac {b^{3}}{a^{5}}}}{b} + x \right )}}{2} + \frac {x^{3}}{3 a} - \frac {b x}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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