3.1555 \(\int \frac{1}{\sqrt{\frac{b-b c}{d}+b x} \sqrt{c-d x}} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{b (1-c)}{d}+b x}}{\sqrt{b} \sqrt{c-d x}}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

(2*ArcTan[(Sqrt[d]*Sqrt[(b*(1 - c))/d + b*x])/(Sqrt[b]*Sqrt[c - d*x])])/(Sqrt[b]
*Sqrt[d])

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Rubi [A]  time = 0.0615046, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{b (1-c)}{d}+b x}}{\sqrt{b} \sqrt{c-d x}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[(b - b*c)/d + b*x]*Sqrt[c - d*x]),x]

[Out]

(2*ArcTan[(Sqrt[d]*Sqrt[(b*(1 - c))/d + b*x])/(Sqrt[b]*Sqrt[c - d*x])])/(Sqrt[b]
*Sqrt[d])

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Rubi in Sympy [A]  time = 7.33741, size = 44, normalized size = 0.85 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{b x - \frac{b \left (c - 1\right )}{d}}}{\sqrt{b} \sqrt{c - d x}} \right )}}{\sqrt{b} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt(d)*sqrt(b*x - b*(c - 1)/d)/(sqrt(b)*sqrt(c - d*x)))/(sqrt(b)*sqrt(d)
)

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Mathematica [A]  time = 0.0524695, size = 45, normalized size = 0.87 \[ -\frac{2 \sqrt{-c+d x+1} \sin ^{-1}\left (\sqrt{c-d x}\right )}{d \sqrt{\frac{b (-c+d x+1)}{d}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[(b - b*c)/d + b*x]*Sqrt[c - d*x]),x]

[Out]

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

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Maple [B]  time = 0.046, size = 118, normalized size = 2.3 \[{1\sqrt{ \left ({\frac{b \left ( 1-c \right ) }{d}}+bx \right ) \left ( -dx+c \right ) }\arctan \left ({1\sqrt{bd} \left ( x-{\frac{-b \left ( 1-c \right ) +bc}{2\,bd}} \right ){\frac{1}{\sqrt{-d{x}^{2}b+ \left ( -b \left ( 1-c \right ) +bc \right ) x+{\frac{b \left ( 1-c \right ) c}{d}}}}}} \right ){\frac{1}{\sqrt{{\frac{b \left ( 1-c \right ) }{d}}+bx}}}{\frac{1}{\sqrt{-dx+c}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((-b*c+b)/d+b*x)^(1/2)/(-d*x+c)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22558, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (4 \,{\left (2 \, d^{2} x -{\left (2 \, c - 1\right )} d\right )} \sqrt{-d x + c} \sqrt{\frac{b d x - b c + b}{d}} +{\left (8 \, d^{2} x^{2} - 8 \,{\left (2 \, c - 1\right )} d x + 8 \, c^{2} - 8 \, c + 1\right )} \sqrt{-b d}\right )}{2 \, \sqrt{-b d}}, \frac{\arctan \left (\frac{\sqrt{b d}{\left (2 \, d x - 2 \, c + 1\right )}}{2 \, \sqrt{-d x + c} d \sqrt{\frac{b d x - b c + b}{d}}}\right )}{\sqrt{b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log(4*(2*d^2*x - (2*c - 1)*d)*sqrt(-d*x + c)*sqrt((b*d*x - b*c + b)/d) + (8
*d^2*x^2 - 8*(2*c - 1)*d*x + 8*c^2 - 8*c + 1)*sqrt(-b*d))/sqrt(-b*d), arctan(1/2
*sqrt(b*d)*(2*d*x - 2*c + 1)/(sqrt(-d*x + c)*d*sqrt((b*d*x - b*c + b)/d)))/sqrt(
b*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219282, size = 90, normalized size = 1.73 \[ -\frac{2 \, b{\rm ln}\left (-\sqrt{-b d} \sqrt{b x - \frac{b c - b}{d}} + \sqrt{-{\left (b x - \frac{b c - b}{d}\right )} b d + b^{2}}\right )}{\sqrt{-b d}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b*ln(-sqrt(-b*d)*sqrt(b*x - (b*c - b)/d) + sqrt(-(b*x - (b*c - b)/d)*b*d + b^
2))/(sqrt(-b*d)*abs(b))