3.226 \(\int \frac {x^2}{a-b x^2} \, dx\)

Optimal. Leaf size=31 \[ \frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {x}{b} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {321, 208} \[ \frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {x}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a - b*x^2),x]

[Out]

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

Rule 208

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

Rule 321

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.00 \[ \frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {x}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a - b*x^2),x]

[Out]

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

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fricas [A]  time = 1.06, size = 80, normalized size = 2.58 \[ \left [\frac {\sqrt {\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {\frac {a}{b}} + a}{b x^{2} - a}\right ) - 2 \, x}{2 \, b}, -\frac {\sqrt {-\frac {a}{b}} \arctan \left (\frac {b x \sqrt {-\frac {a}{b}}}{a}\right ) + x}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.63, size = 29, normalized size = 0.94 \[ -\frac {a \arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{\sqrt {-a b} b} - \frac {x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*b) - x/b

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maple [A]  time = 0.00, size = 27, normalized size = 0.87 \[ \frac {a \arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}-\frac {x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.97, size = 42, normalized size = 1.35 \[ -\frac {a \log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{2 \, \sqrt {a b} b} - \frac {x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*log((b*x - sqrt(a*b))/(b*x + sqrt(a*b)))/(sqrt(a*b)*b) - x/b

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mupad [B]  time = 4.57, size = 23, normalized size = 0.74 \[ \frac {\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a - b*x^2),x)

[Out]

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

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sympy [A]  time = 0.15, size = 49, normalized size = 1.58 \[ - \frac {\sqrt {\frac {a}{b^{3}}} \log {\left (- b \sqrt {\frac {a}{b^{3}}} + x \right )}}{2} + \frac {\sqrt {\frac {a}{b^{3}}} \log {\left (b \sqrt {\frac {a}{b^{3}}} + x \right )}}{2} - \frac {x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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