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

Optimal. Leaf size=46 \[ \frac {\sqrt {a^2-b^2 x^2}}{b}+\frac {a \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b} \]

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Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 = {665, 217, 203} \begin {gather*} \frac {\sqrt {a^2-b^2 x^2}}{b}+\frac {a \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 - b^2*x^2]/(a + b*x),x]

[Out]

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

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])

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 665

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 + 2*p + 1)), x] - Dist[(2*c*d*p)/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*(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] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[
m + 2*p + 1, 0] && IntegerQ[2*p]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 43, normalized size = 0.93 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2}+a \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.20, size = 65, normalized size = 1.41 \begin {gather*} \frac {\sqrt {a^2-b^2 x^2}}{b}+\frac {a \sqrt {-b^2} \log \left (\sqrt {a^2-b^2 x^2}-\sqrt {-b^2} x\right )}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 52, normalized size = 1.13 \begin {gather*} -\frac {2 \, a \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) - \sqrt {-b^{2} x^{2} + a^{2}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 36, normalized size = 0.78 \begin {gather*} \frac {a \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (b)}{{\left | b \right |}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 77, normalized size = 1.67 \begin {gather*} \frac {a \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {2 \left (x +\frac {a}{b}\right ) a b -\left (x +\frac {a}{b}\right )^{2} b^{2}}}\right )}{\sqrt {b^{2}}}+\frac {\sqrt {2 \left (x +\frac {a}{b}\right ) a b -\left (x +\frac {a}{b}\right )^{2} b^{2}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.89, size = 31, normalized size = 0.67 \begin {gather*} \frac {a \arcsin \left (\frac {b x}{a}\right )}{b} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {a^2-b^2\,x^2}}{a+b\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{a + b x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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