\(\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx\) [782]

Optimal result

Integrand size = 24, antiderivative size = 54 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx=-\frac {2 \sqrt {a^2-b^2 x^2}}{b (a+b x)}-\frac {\arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {677, 223, 209} \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx=-\frac {\arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b}-\frac {2 \sqrt {a^2-b^2 x^2}}{b (a+b x)} \]

[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 209

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

Rule 223

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 677

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]

Rubi steps \begin{align*} \text {integral}& = -\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)}-\text {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] (verified)

Time = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx=\frac {2 \left (-\frac {\sqrt {a^2-b^2 x^2}}{a+b x}+\arctan \left (\frac {b x}{\sqrt {a^2}-\sqrt {a^2-b^2 x^2}}\right )\right )}{b} \]

[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] - Sqrt[a^2 - b^2*x^2])]))/b

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(50)=100\).

Time = 2.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.41

method	result	size
default	\(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{a b \left (x +\frac {a}{b}\right )^{2}}-\frac {b \left (\sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}+\frac {a b \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}\right )}{\sqrt {b^{2}}}\right )}{a}}{b^{2}}\)	\(130\)

[In]

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

[Out]

1/b^2*(-1/a/b/(x+a/b)^2*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(3/2)-b/a*((-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(1/2)+a*b/(b^
2)^(1/2)*arctan((b^2)^(1/2)*x/(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx=-\frac {2 \, {\left (b x - {\left (b x + a\right )} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) + a + \sqrt {-b^{2} x^{2} + a^{2}}\right )}}{b^{2} x + a b} \]

[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)

Sympy [F]

\[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx=\int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{2}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx=-\frac {\arcsin \left (\frac {b x}{a}\right )}{b} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{b^{2} x + a b} \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx=\text {Exception raised: NotImplementedError} \]

[In]

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

[Out]

Exception raised: NotImplementedError >> unable to parse Giac output: abs(sageVARb)*(-(2*atan(i)-2*i)/sageVARb
^2*sign((sageVARb*sageVARx+sageVARa)^-1)*sign(sageVARb)-2*sageVARa*(sqrt(2*sageVARa*sageVARb*(sageVARb*sageVAR
x+sageVARa)^-1/sageVA

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^2} \, dx=\int \frac {\sqrt {a^2-b^2\,x^2}}{{\left (a+b\,x\right )}^2} \,d x \]

[In]

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

[Out]

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