3.1169 \(\int \frac{1}{\sqrt{a+b x} \sqrt{-a d+b d x}} \, dx\)

Optimal. Leaf size=39 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b d x-a d}}\right )}{b \sqrt{d}} \]

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(a*d) + b*d*x]])/(b*Sqrt[d])

_______________________________________________________________________________________

Rubi [A]  time = 0.048064, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b d x-a d}}\right )}{b \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(a*d) + b*d*x]])/(b*Sqrt[d])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.33575, size = 34, normalized size = 0.87 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{- a d + b d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atanh(sqrt(-a*d + b*d*x)/(sqrt(d)*sqrt(a + b*x)))/(b*sqrt(d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0631371, size = 43, normalized size = 1.1 \[ \frac{\log \left (\sqrt{d} \sqrt{a+b x} \sqrt{-d (a-b x)}+b d x\right )}{b \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 76, normalized size = 2. \[{1\sqrt{ \left ( bx+a \right ) \left ( bdx-ad \right ) }\ln \left ({{b}^{2}dx{\frac{1}{\sqrt{{b}^{2}d}}}}+\sqrt{{b}^{2}d{x}^{2}-{a}^{2}d} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{bdx-ad}}}{\frac{1}{\sqrt{{b}^{2}d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b*x+a)*(b*d*x-a*d))^(1/2)/(b*x+a)^(1/2)/(b*d*x-a*d)^(1/2)*ln(b^2*d*x/(b^2*d)^(
1/2)+(b^2*d*x^2-a^2*d)^(1/2))/(b^2*d)^(1/2)

_______________________________________________________________________________________

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*d*x - a*d)*sqrt(b*x + a)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.215054, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (2 \, \sqrt{b d x - a d} \sqrt{b x + a} b x +{\left (2 \, b^{2} x^{2} - a^{2}\right )} \sqrt{d}\right )}{2 \, b \sqrt{d}}, \frac{\arctan \left (\frac{b \sqrt{-d} x}{\sqrt{b d x - a d} \sqrt{b x + a}}\right )}{b \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log(2*sqrt(b*d*x - a*d)*sqrt(b*x + a)*b*x + (2*b^2*x^2 - a^2)*sqrt(d))/(b*s
qrt(d)), arctan(b*sqrt(-d)*x/(sqrt(b*d*x - a*d)*sqrt(b*x + a)))/(b*sqrt(-d))]

_______________________________________________________________________________________

Sympy [A]  time = 10.76, size = 88, normalized size = 2.26 \[ \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{d}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{a^{2} e^{2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), a**2/(b*
*2*x**2))/(4*pi**(3/2)*b*sqrt(d)) - I*meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ())
, ((-1/4, 1/4), (-1/2, 0, 0, 0)), a**2*exp_polar(2*I*pi)/(b**2*x**2))/(4*pi**(3/
2)*b*sqrt(d))

_______________________________________________________________________________________

GIAC/XCAS [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/(sqrt(b*d*x - a*d)*sqrt(b*x + a)),x, algorithm="giac")

[Out]

Timed out