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

Optimal. Leaf size=63 \[ \frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}+\frac{x (b c-a d)}{2 a b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0222988, antiderivative size = 63, 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 = {385, 205} \[ \frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}+\frac{x (b c-a d)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(a + b*x^2)^2,x]

[Out]

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 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}{\left (a+b x^2\right )^2} \, dx &=\frac{(b c-a d) x}{2 a b \left (a+b x^2\right )}+\frac{(b c+a d) \int \frac{1}{a+b x^2} \, dx}{2 a b}\\ &=\frac{(b c-a d) x}{2 a b \left (a+b x^2\right )}+\frac{(b c+a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0437442, size = 63, normalized size = 1. \[ \frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{3/2}}-\frac{x (a d-b c)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)/(a + b*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0., size = 68, normalized size = 1.1 \begin{align*} -{\frac{ \left ( ad-bc \right ) x}{2\,ab \left ( b{x}^{2}+a \right ) }}+{\frac{d}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{c}{2\,a}\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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54424, size = 381, normalized size = 6.05 \begin{align*} \left [-\frac{{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) - 2 \,{\left (a b^{2} c - a^{2} b d\right )} x}{4 \,{\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, \frac{{\left (a b c + a^{2} d +{\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (a b^{2} c - a^{2} b d\right )} x}{2 \,{\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.53018, size = 112, normalized size = 1.78 \begin{align*} - \frac{x \left (a d - b c\right )}{2 a^{2} b + 2 a b^{2} x^{2}} - \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d + b c\right ) \log{\left (- a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \left (a d + b c\right ) \log{\left (a^{2} b \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19489, size = 77, normalized size = 1.22 \begin{align*} \frac{{\left (b c + a d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b} + \frac{b c x - a d x}{2 \,{\left (b x^{2} + a\right )} a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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