3.1465 \(\int \frac{A+B x}{\sqrt{d+e x} (1-x^2)} \, dx\)

Optimal. Leaf size=66 \[ \frac{(A+B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d+e}}\right )}{\sqrt{d+e}}-\frac{(A-B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d-e}}\right )}{\sqrt{d-e}} \]

[Out]

-(((A - B)*ArcTanh[Sqrt[d + e*x]/Sqrt[d - e]])/Sqrt[d - e]) + ((A + B)*ArcTanh[Sqrt[d + e*x]/Sqrt[d + e]])/Sqr
t[d + e]

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Rubi [A]  time = 0.102499, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {827, 1166, 206} \[ \frac{(A+B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d+e}}\right )}{\sqrt{d+e}}-\frac{(A-B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d-e}}\right )}{\sqrt{d-e}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[d + e*x]*(1 - x^2)),x]

[Out]

-(((A - B)*ArcTanh[Sqrt[d + e*x]/Sqrt[d - e]])/Sqrt[d - e]) + ((A + B)*ArcTanh[Sqrt[d + e*x]/Sqrt[d + e]])/Sqr
t[d + e]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{A+B x}{\sqrt{d+e x} \left (1-x^2\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{-B d+A e+B x^2}{-d^2+e^2+2 d x^2-x^4} \, dx,x,\sqrt{d+e x}\right )\\ &=(-A+B) \operatorname{Subst}\left (\int \frac{1}{d-e-x^2} \, dx,x,\sqrt{d+e x}\right )+(A+B) \operatorname{Subst}\left (\int \frac{1}{d+e-x^2} \, dx,x,\sqrt{d+e x}\right )\\ &=-\frac{(A-B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d-e}}\right )}{\sqrt{d-e}}+\frac{(A+B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d+e}}\right )}{\sqrt{d+e}}\\ \end{align*}

Mathematica [A]  time = 0.0781069, size = 66, normalized size = 1. \[ \frac{(A+B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d+e}}\right )}{\sqrt{d+e}}-\frac{(A-B) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d-e}}\right )}{\sqrt{d-e}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[d + e*x]*(1 - x^2)),x]

[Out]

-(((A - B)*ArcTanh[Sqrt[d + e*x]/Sqrt[d - e]])/Sqrt[d - e]) + ((A + B)*ArcTanh[Sqrt[d + e*x]/Sqrt[d + e]])/Sqr
t[d + e]

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Maple [A]  time = 0.026, size = 95, normalized size = 1.4 \begin{align*}{A{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d+e}}}} \right ){\frac{1}{\sqrt{d+e}}}}+{B{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d+e}}}} \right ){\frac{1}{\sqrt{d+e}}}}+{A\arctan \left ({\sqrt{ex+d}{\frac{1}{\sqrt{-d+e}}}} \right ){\frac{1}{\sqrt{-d+e}}}}-{B\arctan \left ({\sqrt{ex+d}{\frac{1}{\sqrt{-d+e}}}} \right ){\frac{1}{\sqrt{-d+e}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(-x^2+1)/(e*x+d)^(1/2),x)

[Out]

1/(d+e)^(1/2)*arctanh((e*x+d)^(1/2)/(d+e)^(1/2))*A+1/(d+e)^(1/2)*arctanh((e*x+d)^(1/2)/(d+e)^(1/2))*B+1/(-d+e)
^(1/2)*arctan((e*x+d)^(1/2)/(-d+e)^(1/2))*A-1/(-d+e)^(1/2)*arctan((e*x+d)^(1/2)/(-d+e)^(1/2))*B

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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*x+A)/(-x^2+1)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.07172, size = 1077, normalized size = 16.32 \begin{align*} \left [-\frac{{\left ({\left (A - B\right )} d +{\left (A - B\right )} e\right )} \sqrt{d - e} \log \left (\frac{e x + 2 \, \sqrt{e x + d} \sqrt{d - e} + 2 \, d - e}{x + 1}\right ) -{\left ({\left (A + B\right )} d -{\left (A + B\right )} e\right )} \sqrt{d + e} \log \left (\frac{e x + 2 \, \sqrt{e x + d} \sqrt{d + e} + 2 \, d + e}{x - 1}\right )}{2 \,{\left (d^{2} - e^{2}\right )}}, -\frac{2 \,{\left ({\left (A - B\right )} d +{\left (A - B\right )} e\right )} \sqrt{-d + e} \arctan \left (-\frac{\sqrt{e x + d} \sqrt{-d + e}}{d - e}\right ) -{\left ({\left (A + B\right )} d -{\left (A + B\right )} e\right )} \sqrt{d + e} \log \left (\frac{e x + 2 \, \sqrt{e x + d} \sqrt{d + e} + 2 \, d + e}{x - 1}\right )}{2 \,{\left (d^{2} - e^{2}\right )}}, -\frac{2 \,{\left ({\left (A + B\right )} d -{\left (A + B\right )} e\right )} \sqrt{-d - e} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d - e}}{d + e}\right ) +{\left ({\left (A - B\right )} d +{\left (A - B\right )} e\right )} \sqrt{d - e} \log \left (\frac{e x + 2 \, \sqrt{e x + d} \sqrt{d - e} + 2 \, d - e}{x + 1}\right )}{2 \,{\left (d^{2} - e^{2}\right )}}, -\frac{{\left ({\left (A - B\right )} d +{\left (A - B\right )} e\right )} \sqrt{-d + e} \arctan \left (-\frac{\sqrt{e x + d} \sqrt{-d + e}}{d - e}\right ) +{\left ({\left (A + B\right )} d -{\left (A + B\right )} e\right )} \sqrt{-d - e} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d - e}}{d + e}\right )}{d^{2} - e^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(-x^2+1)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[-1/2*(((A - B)*d + (A - B)*e)*sqrt(d - e)*log((e*x + 2*sqrt(e*x + d)*sqrt(d - e) + 2*d - e)/(x + 1)) - ((A +
B)*d - (A + B)*e)*sqrt(d + e)*log((e*x + 2*sqrt(e*x + d)*sqrt(d + e) + 2*d + e)/(x - 1)))/(d^2 - e^2), -1/2*(2
*((A - B)*d + (A - B)*e)*sqrt(-d + e)*arctan(-sqrt(e*x + d)*sqrt(-d + e)/(d - e)) - ((A + B)*d - (A + B)*e)*sq
rt(d + e)*log((e*x + 2*sqrt(e*x + d)*sqrt(d + e) + 2*d + e)/(x - 1)))/(d^2 - e^2), -1/2*(2*((A + B)*d - (A + B
)*e)*sqrt(-d - e)*arctan(sqrt(e*x + d)*sqrt(-d - e)/(d + e)) + ((A - B)*d + (A - B)*e)*sqrt(d - e)*log((e*x +
2*sqrt(e*x + d)*sqrt(d - e) + 2*d - e)/(x + 1)))/(d^2 - e^2), -(((A - B)*d + (A - B)*e)*sqrt(-d + e)*arctan(-s
qrt(e*x + d)*sqrt(-d + e)/(d - e)) + ((A + B)*d - (A + B)*e)*sqrt(-d - e)*arctan(sqrt(e*x + d)*sqrt(-d - e)/(d
+ e)))/(d^2 - e^2)]

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Sympy [A]  time = 29.5768, size = 78, normalized size = 1.18 \begin{align*} \frac{\left (- A - B\right ) \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d + e}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{1}{d + e}} \left (d + e\right )} + \frac{\left (A - B\right ) \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d - e}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{1}{d - e}} \left (d - e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(-x**2+1)/(e*x+d)**(1/2),x)

[Out]

(-A - B)*atan(1/(sqrt(-1/(d + e))*sqrt(d + e*x)))/(sqrt(-1/(d + e))*(d + e)) + (A - B)*atan(1/(sqrt(-1/(d - e)
)*sqrt(d + e*x)))/(sqrt(-1/(d - e))*(d - e))

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Giac [A]  time = 1.27226, size = 92, normalized size = 1.39 \begin{align*} \frac{{\left (A - B\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d + e}}\right )}{\sqrt{-d + e}} - \frac{{\left (A + B\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d - e}}\right )}{\sqrt{-d - e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(-x^2+1)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

(A - B)*arctan(sqrt(x*e + d)/sqrt(-d + e))/sqrt(-d + e) - (A + B)*arctan(sqrt(x*e + d)/sqrt(-d - e))/sqrt(-d -
e)