3.270 \(\int \frac{a+\frac{b}{x^2}}{c+\frac{d}{x^2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0200836, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {374, 388, 205} \[ \frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{c^{3/2} \sqrt{d}}+\frac{a x}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/(c + d/x^2),x]

[Out]

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

Rule 374

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

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0256489, size = 40, normalized size = 1.03 \[ \frac{a x}{c}-\frac{(a d-b c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{c^{3/2} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/(c + d/x^2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 45, normalized size = 1.2 \begin{align*}{\frac{ax}{c}}-{\frac{ad}{c}\arctan \left ({cx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{b\arctan \left ({cx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)/(c+d/x^2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.23323, size = 223, normalized size = 5.72 \begin{align*} \left [\frac{2 \, a c d x +{\left (b c - a d\right )} \sqrt{-c d} \log \left (\frac{c x^{2} + 2 \, \sqrt{-c d} x - d}{c x^{2} + d}\right )}{2 \, c^{2} d}, \frac{a c d x +{\left (b c - a d\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{d}\right )}{c^{2} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.40391, size = 82, normalized size = 2.1 \begin{align*} \frac{a x}{c} + \frac{\sqrt{- \frac{1}{c^{3} d}} \left (a d - b c\right ) \log{\left (- c d \sqrt{- \frac{1}{c^{3} d}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{3} d}} \left (a d - b c\right ) \log{\left (c d \sqrt{- \frac{1}{c^{3} d}} + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13505, size = 45, normalized size = 1.15 \begin{align*} \frac{a x}{c} + \frac{{\left (b c - a d\right )} \arctan \left (\frac{c x}{\sqrt{c d}}\right )}{\sqrt{c d} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/c + (b*c - a*d)*arctan(c*x/sqrt(c*d))/(sqrt(c*d)*c)