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

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]

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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fricas [A]  time = 0.72, size = 109, normalized size = 3.52 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {b d^{2} x^{2} + 2 \, b c d x + b c^{2} - 2 \, \sqrt {-a b} {\left (d x + c\right )} - a}{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}\right )}{2 \, a b d}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} {\left (d x + c\right )}}{a}\right )}{a b d}\right ] \]

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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giac [A]  time = 0.45, size = 24, normalized size = 0.77 \[ \frac {\arctan \left (\frac {b d x + b c}{\sqrt {a b}}\right )}{\sqrt {a b} d} \]

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)

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maple [A]  time = 0.01, size = 34, normalized size = 1.10 \[ \frac {\arctan \left (\frac {2 b \,d^{2} x +2 b d c}{2 \sqrt {a b}\, d}\right )}{\sqrt {a b}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.56, size = 30, normalized size = 0.97 \[ \frac {\arctan \left (\frac {b d^{2} x + b c d}{\sqrt {a b} d}\right )}{\sqrt {a b} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 27, normalized size = 0.87 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,c+\sqrt {b}\,d\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*(c + d*x)^2),x)

[Out]

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

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sympy [B]  time = 0.21, size = 61, normalized size = 1.97 \[ \frac {- \frac {\sqrt {- \frac {1}{a b}} \log {\left (x + \frac {- a \sqrt {- \frac {1}{a b}} + c}{d} \right )}}{2} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (x + \frac {a \sqrt {- \frac {1}{a b}} + c}{d} \right )}}{2}}{d} \]

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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