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

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

[Out]

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

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Rubi [A]  time = 0.0202435, 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} \[ \frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{3/2}}-\frac{x (b c-a d)}{2 c d \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/(c + d*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0443304, size = 63, normalized size = 1. \[ \frac{(a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{3/2}}-\frac{x (b c-a d)}{2 c d \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 68, normalized size = 1.1 \begin{align*}{\frac{ \left ( ad-bc \right ) x}{2\,cd \left ( d{x}^{2}+c \right ) }}+{\frac{a}{2\,c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{b}{2\,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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62434, size = 381, normalized size = 6.05 \begin{align*} \left [-\frac{{\left (b c^{2} + a c d +{\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) + 2 \,{\left (b c^{2} d - a c d^{2}\right )} x}{4 \,{\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}}, \frac{{\left (b c^{2} + a c d +{\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) -{\left (b c^{2} d - a c d^{2}\right )} x}{2 \,{\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.516933, size = 112, normalized size = 1.78 \begin{align*} \frac{x \left (a d - b c\right )}{2 c^{2} d + 2 c d^{2} x^{2}} - \frac{\sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d + b c\right ) \log{\left (- c^{2} d \sqrt{- \frac{1}{c^{3} d^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{c^{3} d^{3}}} \left (a d + b c\right ) \log{\left (c^{2} d \sqrt{- \frac{1}{c^{3} d^{3}}} + x \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.26224, size = 77, normalized size = 1.22 \begin{align*} \frac{{\left (b c + a d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} c d} - \frac{b c x - a d x}{2 \,{\left (d x^{2} + c\right )} c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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