3.12.5 \(\int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx\) [1105]

Optimal. Leaf size=62 \[ -\frac {3 \sqrt {1-a^2 x^2}}{2 a}-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3 \sin ^{-1}(a x)}{2 a} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {809, 679, 222} \begin {gather*} -\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac {3 \sqrt {1-a^2 x^2}}{2 a}+\frac {3 \text {ArcSin}(a x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + a*x)*Sqrt[1 - a^2*x^2])/(1 - a*x),x]

[Out]

(-3*Sqrt[1 - a^2*x^2])/(2*a) - (1 - a^2*x^2)^(3/2)/(2*a*(1 - a*x)) + (3*ArcSin[a*x])/(2*a)

Rule 222

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

Rule 679

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]

Rule 809

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

Rubi steps

\begin {align*} \int \frac {(1+a x) \sqrt {1-a^2 x^2}}{1-a x} \, dx &=-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3}{2} \int \frac {\sqrt {1-a^2 x^2}}{1-a x} \, dx\\ &=-\frac {3 \sqrt {1-a^2 x^2}}{2 a}-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3}{2} \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {3 \sqrt {1-a^2 x^2}}{2 a}-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}+\frac {3 \sin ^{-1}(a x)}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 69, normalized size = 1.11 \begin {gather*} \frac {(-4-a x) \sqrt {1-a^2 x^2}}{2 a}-\frac {3 \log \left (-\sqrt {-a^2} x+\sqrt {1-a^2 x^2}\right )}{2 \sqrt {-a^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + a*x)*Sqrt[1 - a^2*x^2])/(1 - a*x),x]

[Out]

((-4 - a*x)*Sqrt[1 - a^2*x^2])/(2*a) - (3*Log[-(Sqrt[-a^2]*x) + Sqrt[1 - a^2*x^2]])/(2*Sqrt[-a^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs. \(2(52)=104\).
time = 0.16, size = 120, normalized size = 1.94

method	result	size
risch	\(\frac {\left (a x +4\right ) \left (a^{2} x^{2}-1\right )}{2 a \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\)	\(60\)
default	\(-\frac {x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}-\frac {2 \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{a}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 42, normalized size = 0.68 \begin {gather*} -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} x + \frac {3 \, \arcsin \left (a x\right )}{2 \, a} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*x + 3/2*arcsin(a*x)/a - 2*sqrt(-a^2*x^2 + 1)/a

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Fricas [A]
time = 1.10, size = 48, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x + 4\right )} + 6 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(-a^2*x^2 + 1)*(a*x + 4) + 6*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a

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Sympy [A]
time = 2.99, size = 76, normalized size = 1.23 \begin {gather*} - \begin {cases} - \frac {- \sqrt {- a^{2} x^{2} + 1} + \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases} - \begin {cases} - \frac {- \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} - \sqrt {- a^{2} x^{2} + 1} + \frac {\operatorname {asin}{\left (a x \right )}}{2}}{a} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-(-sqrt(-a**2*x**2 + 1) + asin(a*x))/a, (a*x > -1) & (a*x < 1))) - Piecewise((-(-a*x*sqrt(-a**2*x*
*2 + 1)/2 - sqrt(-a**2*x**2 + 1) + asin(a*x)/2)/a, (a*x > -1) & (a*x < 1)))

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Giac [A]
time = 1.16, size = 34, normalized size = 0.55 \begin {gather*} -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (x + \frac {4}{a}\right )} + \frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-a^2*x^2 + 1)*(x + 4/a) + 3/2*arcsin(a*x)*sgn(a)/abs(a)

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Mupad [B]
time = 0.15, size = 55, normalized size = 0.89 \begin {gather*} \frac {\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2}+\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a}{\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2}\right )}{\sqrt {-a^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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