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

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

[Out]

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

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Rubi [A]  time = 0.014759, antiderivative size = 39, 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} \[ \frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}}+\frac{d x}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(a + b*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0255159, size = 40, normalized size = 1.03 \[ \frac{d x}{b}-\frac{(a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)/(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 45, normalized size = 1.2 \begin{align*}{\frac{dx}{b}}-{\frac{ad}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{c\arctan \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((d*x^2+c)/(b*x^2+a),x)

[Out]

d*x/b-1/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*a*d+c/(a*b)^(1/2)*arctan(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((d*x^2+c)/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50152, size = 223, normalized size = 5.72 \begin{align*} \left [\frac{2 \, a b d x + \sqrt{-a b}{\left (b c - a d\right )} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{2 \, a b^{2}}, \frac{a b d x + \sqrt{a b}{\left (b c - a d\right )} \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((d*x^2+c)/(b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.40774, size = 82, normalized size = 2.1 \begin{align*} \frac{\sqrt{- \frac{1}{a b^{3}}} \left (a d - b c\right ) \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a b^{3}}} \left (a d - b c\right ) \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x \right )}}{2} + \frac{d x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13574, size = 45, normalized size = 1.15 \begin{align*} \frac{d x}{b} + \frac{{\left (b c - a d\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((d*x^2+c)/(b*x^2+a),x, algorithm="giac")

[Out]

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