3.74 \(\int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{1-f^2 x^2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{2 \sqrt{\frac{f (c+d x)}{c f+d}} \Pi \left (\frac{2 b}{b+a f};\sin ^{-1}\left (\frac{\sqrt{1-f x}}{\sqrt{2}}\right )|\frac{2 d}{d+c f}\right )}{(a f+b) \sqrt{c+d x}} \]

[Out]

(-2*Sqrt[(f*(c + d*x))/(d + c*f)]*EllipticPi[(2*b)/(b + a*f), ArcSin[Sqrt[1 - f*x]/Sqrt[2]], (2*d)/(d + c*f)])
/((b + a*f)*Sqrt[c + d*x])

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Rubi [A]  time = 0.175884, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {932, 168, 538, 537} \[ -\frac{2 \sqrt{\frac{f (c+d x)}{c f+d}} \Pi \left (\frac{2 b}{b+a f};\sin ^{-1}\left (\frac{\sqrt{1-f x}}{\sqrt{2}}\right )|\frac{2 d}{d+c f}\right )}{(a f+b) \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[(f*(c + d*x))/(d + c*f)]*EllipticPi[(2*b)/(b + a*f), ArcSin[Sqrt[1 - f*x]/Sqrt[2]], (2*d)/(d + c*f)])
/((b + a*f)*Sqrt[c + d*x])

Rule 932

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[1/Sqrt[a], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, c,
d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{1-f^2 x^2}} \, dx &=\int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{1-f x} \sqrt{1+f x}} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (b+a f-b x^2\right ) \sqrt{c+\frac{d}{f}-\frac{d x^2}{f}}} \, dx,x,\sqrt{1-f x}\right )\right )\\ &=-\frac{\left (2 \sqrt{\frac{f (c+d x)}{d+c f}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (b+a f-b x^2\right ) \sqrt{1-\frac{d x^2}{\left (c+\frac{d}{f}\right ) f}}} \, dx,x,\sqrt{1-f x}\right )}{\sqrt{c+d x}}\\ &=-\frac{2 \sqrt{\frac{f (c+d x)}{d+c f}} \Pi \left (\frac{2 b}{b+a f};\sin ^{-1}\left (\frac{\sqrt{1-f x}}{\sqrt{2}}\right )|\frac{2 d}{d+c f}\right )}{(b+a f) \sqrt{c+d x}}\\ \end{align*}

Mathematica [C]  time = 0.102766, size = 203, normalized size = 2.74 \[ \frac{2 i (c+d x) \sqrt{\frac{d (f x-1)}{f (c+d x)}} \sqrt{\frac{d f x+d}{c f+d f x}} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{c f+d}{f}}}{\sqrt{c+d x}}\right ),\frac{c f-d}{c f+d}\right )-\Pi \left (\frac{b c f-a d f}{b d+b c f};i \sinh ^{-1}\left (\frac{\sqrt{-\frac{d+c f}{f}}}{\sqrt{c+d x}}\right )|\frac{c f-d}{d+c f}\right )\right )}{\sqrt{1-f^2 x^2} \sqrt{-\frac{c f+d}{f}} (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*(c + d*x)*Sqrt[(d*(-1 + f*x))/(f*(c + d*x))]*Sqrt[(d + d*f*x)/(c*f + d*f*x)]*(EllipticF[I*ArcSinh[Sqrt[
-((d + c*f)/f)]/Sqrt[c + d*x]], (-d + c*f)/(d + c*f)] - EllipticPi[(b*c*f - a*d*f)/(b*d + b*c*f), I*ArcSinh[Sq
rt[-((d + c*f)/f)]/Sqrt[c + d*x]], (-d + c*f)/(d + c*f)]))/((-(b*c) + a*d)*Sqrt[-((d + c*f)/f)]*Sqrt[1 - f^2*x
^2])

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Maple [B]  time = 0.025, size = 181, normalized size = 2.5 \begin{align*} -2\,{\frac{ \left ( cf-d \right ) \sqrt{-{f}^{2}{x}^{2}+1}\sqrt{dx+c}}{ \left ( ad-bc \right ) f \left ( d{f}^{2}{x}^{3}+c{f}^{2}{x}^{2}-dx-c \right ) }{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-d}}},-{\frac{ \left ( cf-d \right ) b}{ \left ( ad-bc \right ) f}},\sqrt{{\frac{cf-d}{cf+d}}} \right ) \sqrt{-{\frac{ \left ( fx+1 \right ) d}{cf-d}}}\sqrt{-{\frac{ \left ( fx-1 \right ) d}{cf+d}}}\sqrt{{\frac{ \left ( dx+c \right ) f}{cf-d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(c*f-d)*EllipticPi(((d*x+c)*f/(c*f-d))^(1/2),-(c*f-d)*b/f/(a*d-b*c),((c*f-d)/(c*f+d))^(1/2))*(-(f*x+1)*d/(c
*f-d))^(1/2)*(-(f*x-1)*d/(c*f+d))^(1/2)*((d*x+c)*f/(c*f-d))^(1/2)*(-f^2*x^2+1)^(1/2)*(d*x+c)^(1/2)/f/(a*d-b*c)
/(d*f^2*x^3+c*f^2*x^2-d*x-c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-f^{2} x^{2} + 1}{\left (b x + a\right )} \sqrt{d x + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-f^2*x^2 + 1)*(b*x + a)*sqrt(d*x + c)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (f x - 1\right ) \left (f x + 1\right )} \left (a + b x\right ) \sqrt{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-(f*x - 1)*(f*x + 1))*(a + b*x)*sqrt(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-f^{2} x^{2} + 1}{\left (b x + a\right )} \sqrt{d x + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-f^2*x^2 + 1)*(b*x + a)*sqrt(d*x + c)), x)