3.85 \(\int \frac{1}{a+b (c+d x)^2} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d} \]

[Out]

ArcTan[(Sqrt[b]*(c + d*x))/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)

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Rubi [A]  time = 0.0235124, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {247, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c + d*x)^2)^(-1),x]

[Out]

ArcTan[(Sqrt[b]*(c + d*x))/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

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

Mathematica [A]  time = 0.0090773, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c + d*x)^2)^(-1),x]

[Out]

ArcTan[(Sqrt[b]*(c + d*x))/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)

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Maple [A]  time = 0.006, size = 34, normalized size = 1.1 \begin{align*}{\frac{1}{d}\arctan \left ({\frac{2\,b{d}^{2}x+2\,bcd}{2\,d}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67146, size = 243, normalized size = 7.84 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{b d^{2} x^{2} + 2 \, b c d x + b c^{2} - 2 \, \sqrt{-a b}{\left (d x + c\right )} - a}{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}\right )}{2 \, a b d}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}{\left (d x + c\right )}}{a}\right )}{a b d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.201686, size = 61, normalized size = 1.97 \begin{align*} \frac{- \frac{\sqrt{- \frac{1}{a b}} \log{\left (x + \frac{- a \sqrt{- \frac{1}{a b}} + c}{d} \right )}}{2} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (x + \frac{a \sqrt{- \frac{1}{a b}} + c}{d} \right )}}{2}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12971, size = 32, normalized size = 1.03 \begin{align*} \frac{\arctan \left (\frac{b d x + b c}{\sqrt{a b}}\right )}{\sqrt{a b} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*(d*x+c)^2),x, algorithm="giac")

[Out]

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