3.3.26 \(\int \frac {x^2}{a-b x^2} \, dx\) [226]

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

[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 = {327, 214} \begin {gather*} \frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {x}{b} \end {gather*}

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 214

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

Rule 327

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 \begin {gather*} -\frac {x}{b}+\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \end {gather*}

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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Maple [A]
time = 0.03, size = 27, normalized size = 0.87

method	result	size
default	\(-\frac {x}{b}+\frac {a \arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{b \sqrt {a b}}\)	\(27\)
risch	\(-\frac {x}{b}-\frac {\sqrt {a b}\, \ln \left (\sqrt {a b}\, x -a \right )}{2 b^{2}}+\frac {\sqrt {a b}\, \ln \left (-\sqrt {a b}\, x -a \right )}{2 b^{2}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 42, normalized size = 1.35 \begin {gather*} -\frac {a \log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{2 \, \sqrt {a b} b} - \frac {x}{b} \end {gather*}

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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Fricas [A]
time = 0.90, size = 80, normalized size = 2.58 \begin {gather*} \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 ] \end {gather*}

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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Sympy [A]
time = 0.05, size = 49, normalized size = 1.58 \begin {gather*} - \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} \end {gather*}

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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Giac [A]
time = 1.50, size = 29, normalized size = 0.94 \begin {gather*} -\frac {a \arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{\sqrt {-a b} b} - \frac {x}{b} \end {gather*}

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

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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