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

Optimal. Leaf size=40 \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0198071, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {388, 205} \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/(c + d*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0225555, size = 40, normalized size = 1. \[ \frac{b x}{d}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/(c + d*x^2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 45, normalized size = 1.1 \begin{align*}{\frac{bx}{d}}+{a\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{bc}{d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)/(d*x^2+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77492, size = 223, normalized size = 5.58 \begin{align*} \left [\frac{2 \, b c d x +{\left (b c - a d\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right )}{2 \, c d^{2}}, \frac{b c d x -{\left (b c - a d\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{c d^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.387133, size = 82, normalized size = 2.05 \begin{align*} \frac{b x}{d} - \frac{\sqrt{- \frac{1}{c d^{3}}} \left (a d - b c\right ) \log{\left (- c d \sqrt{- \frac{1}{c d^{3}}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c d^{3}}} \left (a d - b c\right ) \log{\left (c d \sqrt{- \frac{1}{c d^{3}}} + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06938, size = 46, normalized size = 1.15 \begin{align*} \frac{b x}{d} - \frac{{\left (b c - a d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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