3.782 \(\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}-\frac{\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0140286, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {663, 217, 203} \[ -\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}-\frac{\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^2,x]

[Out]

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

Rule 663

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1
, 0] && IntegerQ[2*p]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0551268, size = 51, normalized size = 0.94 \[ -\frac{\frac{2 \sqrt{a^2-b^2 x^2}}{a+b x}+\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^2,x]

[Out]

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

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Maple [B]  time = 0.049, size = 126, normalized size = 2.3 \begin{align*} -{\frac{1}{a{b}^{3}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-2}}-{\frac{1}{ab}\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab}}-{\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\, \left ( x+{\frac{a}{b}} \right ) ab}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b^3/a/(x+1/b*a)^2*(-(x+1/b*a)^2*b^2+2*(x+1/b*a)*a*b)^(3/2)-1/b/a*(-(x+1/b*a)^2*b^2+2*(x+1/b*a)*a*b)^(1/2)-1
/(b^2)^(1/2)*arctan((b^2)^(1/2)*x/(-(x+1/b*a)^2*b^2+2*(x+1/b*a)*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((-b^2*x^2+a^2)^(1/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.93435, size = 142, normalized size = 2.63 \begin{align*} -\frac{2 \,{\left (b x -{\left (b x + a\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right ) + a + \sqrt{-b^{2} x^{2} + a^{2}}\right )}}{b^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(-a + b*x)*(a + b*x))/(a + b*x)**2, x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError