3.3.33 \(\int \frac {x^2}{(a-b x^2)^2} \, dx\) [233]

Optimal. Leaf size=46 \[ \frac {x}{2 b \left (a-b x^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 46, 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 = {294, 214} \begin {gather*} \frac {x}{2 b \left (a-b x^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a - b*x^2)^2,x]

[Out]

x/(2*b*(a - b*x^2)) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(2*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 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 47, normalized size = 1.02 \begin {gather*} -\frac {x}{2 b \left (-a+b x^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a - b*x^2)^2,x]

[Out]

-1/2*x/(b*(-a + b*x^2)) - ArcTanh[(Sqrt[b]*x)/Sqrt[a]]/(2*Sqrt[a]*b^(3/2))

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Maple [A]
time = 0.04, size = 37, normalized size = 0.80

method	result	size
default	\(\frac {x}{2 b \left (-b \,x^{2}+a \right )}-\frac {\arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}\)	\(37\)
risch	\(\frac {x}{2 b \left (-b \,x^{2}+a \right )}+\frac {\ln \left (b x -\sqrt {a b}\right )}{4 \sqrt {a b}\, b}-\frac {\ln \left (-b x -\sqrt {a b}\right )}{4 \sqrt {a b}\, b}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 52, normalized size = 1.13 \begin {gather*} -\frac {x}{2 \, {\left (b^{2} x^{2} - a b\right )}} + \frac {\log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{4 \, \sqrt {a b} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x/(b^2*x^2 - a*b) + 1/4*log((b*x - sqrt(a*b))/(b*x + sqrt(a*b)))/(sqrt(a*b)*b)

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Fricas [A]
time = 0.94, size = 127, normalized size = 2.76 \begin {gather*} \left [-\frac {2 \, a b x - {\left (b x^{2} - a\right )} \sqrt {a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {a b} x + a}{b x^{2} - a}\right )}{4 \, {\left (a b^{3} x^{2} - a^{2} b^{2}\right )}}, -\frac {a b x - {\left (b x^{2} - a\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b} x}{a}\right )}{2 \, {\left (a b^{3} x^{2} - a^{2} b^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*a*b*x - (b*x^2 - a)*sqrt(a*b)*log((b*x^2 - 2*sqrt(a*b)*x + a)/(b*x^2 - a)))/(a*b^3*x^2 - a^2*b^2), -1
/2*(a*b*x - (b*x^2 - a)*sqrt(-a*b)*arctan(sqrt(-a*b)*x/a))/(a*b^3*x^2 - a^2*b^2)]

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Sympy [A]
time = 0.09, size = 71, normalized size = 1.54 \begin {gather*} - \frac {x}{- 2 a b + 2 b^{2} x^{2}} + \frac {\sqrt {\frac {1}{a b^{3}}} \log {\left (- a b \sqrt {\frac {1}{a b^{3}}} + x \right )}}{4} - \frac {\sqrt {\frac {1}{a b^{3}}} \log {\left (a b \sqrt {\frac {1}{a b^{3}}} + x \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.52, size = 39, normalized size = 0.85 \begin {gather*} \frac {\arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{2 \, \sqrt {-a b} b} - \frac {x}{2 \, {\left (b x^{2} - a\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.68, size = 34, normalized size = 0.74 \begin {gather*} \frac {x}{2\,b\,\left (a-b\,x^2\right )}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,b^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a - b*x^2)^2,x)

[Out]

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

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