3.1551 \(\int \frac{1}{\sqrt{\frac{-b+b c}{d}+b x} \sqrt{c+d x}} \, dx\)

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

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[-((b*(1 - c))/d) + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqr
t[b]*Sqrt[d])

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Rubi [A]  time = 0.0606125, 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 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{b x-\frac{b (1-c)}{d}}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[-((b*(1 - c))/d) + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqr
t[b]*Sqrt[d])

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Rubi in Sympy [A]  time = 7.98201, size = 44, normalized size = 0.85 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{b x + \frac{b \left (c - 1\right )}{d}}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b} \sqrt{d}} \]

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*atanh(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 = 0.0441413, size = 51, normalized size = 0.98 \[ \frac{2 \sqrt{c+d x-1} \log \left (\sqrt{c+d x-1}+\sqrt{c+d x}\right )}{d \sqrt{\frac{b (c+d x-1)}{d}}} \]

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

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Maple [B]  time = 0.021, size = 100, normalized size = 1.9 \[{1\sqrt{ \left ( bx+{\frac{b \left ( c-1 \right ) }{d}} \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{b \left ( c-1 \right ) }{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( b \left ( c-1 \right ) +bc \right ) x+{\frac{b \left ( c-1 \right ) c}{d}}} \right ){\frac{1}{\sqrt{bx+{\frac{b \left ( c-1 \right ) }{d}}}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}} \]

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*x+b*(c-1)/d)*(d*x+c))^(1/2)/(b*x+b*(c-1)/d)^(1/2)/(d*x+c)^(1/2)*ln((1/2*b*(c
-1)+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(b*(c-1)+b*c)*x+b*(c-1)/d*c)^(1/2))/(b*d
)^(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 = 0.223591, 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*sq
rt(-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 = 0.213063, size = 84, normalized size = 1.62 \[ -\frac{2 \, b{\rm ln}\left (-\sqrt{b d} \sqrt{b x + \frac{b c - b}{d}} + \sqrt{{\left (b x + \frac{b c - b}{d}\right )} b d + b^{2}}\right )}{\sqrt{b d}{\left | b \right |}} \]

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))