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

Optimal. Leaf size=35 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt{b} d} \]

[Out]

ArcTan[(Sqrt[b]*(c + d*x))/(-a)^(1/4)]/((-a)^(1/4)*Sqrt[b]*d)

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Rubi [A]  time = 0.0353452, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {247, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-a] + b*(c + d*x)^2)^(-1),x]

[Out]

ArcTan[(Sqrt[b]*(c + d*x))/(-a)^(1/4)]/((-a)^(1/4)*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}{\sqrt{-a}+b (c+d x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}+b x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt{b} d}\\ \end{align*}

Mathematica [A]  time = 0.0136602, size = 35, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[-a] + b*(c + d*x)^2)^(-1),x]

[Out]

ArcTan[(Sqrt[b]*(c + d*x))/(-a)^(1/4)]/((-a)^(1/4)*Sqrt[b]*d)

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Maple [A]  time = 0.006, size = 42, normalized size = 1.2 \begin{align*}{\frac{1}{d}\arctan \left ({\frac{2\,b{d}^{2}x+2\,bcd}{2\,d}{\frac{1}{\sqrt{\sqrt{-a}b}}}} \right ){\frac{1}{\sqrt{\sqrt{-a}b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.65837, size = 89, normalized size = 2.54 \begin{align*} \frac{\log \left (\frac{b d^{2} x + b c d - \sqrt{-\sqrt{-a} b} d}{b d^{2} x + b c d + \sqrt{-\sqrt{-a} b} d}\right )}{2 \, \sqrt{-\sqrt{-a} b} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.11544, size = 576, normalized size = 16.46 \begin{align*} \left [\frac{\sqrt{\frac{\sqrt{-a}}{a b}} \log \left (\frac{b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - 2 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt{-a} + 2 \,{\left (a b d x + a b c +{\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \sqrt{-a}\right )} \sqrt{\frac{\sqrt{-a}}{a b}} - a}{b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} + a}\right )}{2 \, d}, \frac{\sqrt{-\frac{\sqrt{-a}}{a b}} \arctan \left ({\left (b d x + b c\right )} \sqrt{-\frac{\sqrt{-a}}{a b}}\right )}{d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(sqrt(-a)/(a*b))*log((b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 - 2
*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sqrt(-a) + 2*(a*b*d*x + a*b*c + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*
x + b^2*c^3)*sqrt(-a))*sqrt(sqrt(-a)/(a*b)) - a)/(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^
3*d*x + b^2*c^4 + a))/d, sqrt(-sqrt(-a)/(a*b))*arctan((b*d*x + b*c)*sqrt(-sqrt(-a)/(a*b)))/d]

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Sympy [B]  time = 0.198407, size = 92, normalized size = 2.63 \begin{align*} \frac{- \frac{\sqrt{- \frac{1}{b \sqrt{- a}}} \log{\left (x + \frac{c - \sqrt{- a} \sqrt{- \frac{1}{b \sqrt{- a}}}}{d} \right )}}{2} + \frac{\sqrt{- \frac{1}{b \sqrt{- a}}} \log{\left (x + \frac{c + \sqrt{- a} \sqrt{- \frac{1}{b \sqrt{- a}}}}{d} \right )}}{2}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14624, size = 41, normalized size = 1.17 \begin{align*} \frac{\arctan \left (\frac{b d x + b c}{\left (-a\right )^{\frac{1}{4}} \sqrt{b}}\right )}{\left (-a\right )^{\frac{1}{4}} \sqrt{b} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan((b*d*x + b*c)/((-a)^(1/4)*sqrt(b)))/((-a)^(1/4)*sqrt(b)*d)