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

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 = 0.648813, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \sqrt{\frac{f (c+d x)}{c f+d}} \Pi \left (\frac{2 b}{b+a f};\sin ^{-1}\left (\frac{\sqrt{1-f x}}{\sqrt{2}}\right )|\frac{2 d}{d+c f}\right )}{(a f+b) \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)*Sqrt[c + d*x]*Sqrt[1 - f*x]*Sqrt[1 + f*x]),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 in Sympy [A]  time = 17.292, size = 102, normalized size = 1.38 \[ - \frac{2 \sqrt{\frac{f \left (c + d x\right )}{c f + d}} \sqrt{\frac{f x}{2} + \frac{1}{2}} \Pi \left (\frac{b \left (c f + d\right )}{d \left (a f + b\right )}; \operatorname{asin}{\left (\sqrt{\frac{d}{c f + d}} \sqrt{- f x + 1} \right )}\middle | \frac{c f + d}{2 d}\right )}{\sqrt{\frac{d}{c f + d}} \sqrt{c + d x} \left (a f + b\right ) \sqrt{f x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)/(d*x+c)**(1/2)/(-f*x+1)**(1/2)/(f*x+1)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.855339, size = 203, normalized size = 2.74 \[ \frac{2 i (c+d x) \sqrt{\frac{d (f x-1)}{f (c+d x)}} \sqrt{\frac{d f x+d}{c f+d f x}} \left (F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{d+c f}{f}}}{\sqrt{c+d x}}\right )|\frac{c f-d}{d+c f}\right )-\Pi \left (\frac{b c f-a d f}{b d+b c f};i \sinh ^{-1}\left (\frac{\sqrt{-\frac{d+c f}{f}}}{\sqrt{c+d x}}\right )|\frac{c f-d}{d+c f}\right )\right )}{\sqrt{1-f^2 x^2} \sqrt{-\frac{c f+d}{f}} (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)*Sqrt[c + d*x]*Sqrt[1 - f*x]*Sqrt[1 + f*x]),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[Sqrt[-((d + c*f)/f)]/S
qrt[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 = 0.132, size = 184, normalized size = 2.5 \[ -2\,{\frac{ \left ( cf-d \right ) \sqrt{fx+1}\sqrt{-fx+1}\sqrt{dx+c}}{f \left ( ad-bc \right ) \left ( d{f}^{2}{x}^{3}+c{f}^{2}{x}^{2}-dx-c \right ) }{\it EllipticPi} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-d}}},-{\frac{b \left ( cf-d \right ) }{f \left ( ad-bc \right ) }},\sqrt{{\frac{cf-d}{cf+d}}} \right ) \sqrt{-{\frac{ \left ( fx+1 \right ) d}{cf-d}}}\sqrt{-{\frac{ \left ( fx-1 \right ) d}{cf+d}}}\sqrt{{\frac{f \left ( dx+c \right ) }{cf-d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(d*x+c)^(1/2)/(-f*x+1)^(1/2)/(f*x+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*x+1)^(1/2)*(-f*x+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. \[ \int \frac{1}{{\left (b x + a\right )} \sqrt{d x + c} \sqrt{f x + 1} \sqrt{-f x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + 1)*sqrt(-f*x + 1)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + 1)*sqrt(-f*x + 1)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )} \sqrt{d x + c} \sqrt{f x + 1} \sqrt{-f x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + 1)*sqrt(-f*x + 1)),x, algorithm="giac")

[Out]

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