3.116 \(\int \frac{A+B x^2}{a-b x^2} \, dx\)

Optimal. Leaf size=39 \[ \frac{(a B+A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{B x}{b} \]

[Out]

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

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Rubi [A]  time = 0.018227, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {388, 208} \[ \frac{(a B+A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{B x}{b} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(a - b*x^2),x]

[Out]

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

Rule 388

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0214197, size = 39, normalized size = 1. \[ \frac{(a B+A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}-\frac{B x}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(a - b*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 37, normalized size = 1. \begin{align*} -{\frac{Bx}{b}}-{\frac{-Ab-Ba}{b}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.20187, size = 225, normalized size = 5.77 \begin{align*} \left [-\frac{2 \, B a b x -{\left (B a + A b\right )} \sqrt{a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{a b} x + a}{b x^{2} - a}\right )}{2 \, a b^{2}}, -\frac{B a b x +{\left (B a + A b\right )} \sqrt{-a b} \arctan \left (\frac{\sqrt{-a b} x}{a}\right )}{a b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.408331, size = 75, normalized size = 1.92 \begin{align*} - \frac{B x}{b} - \frac{\sqrt{\frac{1}{a b^{3}}} \left (A b + B a\right ) \log{\left (- a b \sqrt{\frac{1}{a b^{3}}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{a b^{3}}} \left (A b + B a\right ) \log{\left (a b \sqrt{\frac{1}{a b^{3}}} + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17008, size = 49, normalized size = 1.26 \begin{align*} -\frac{B x}{b} - \frac{{\left (B a + A b\right )} \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{\sqrt{-a b} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*x/b - (B*a + A*b)*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*b)