3.2500 \(\int \frac{1}{\sqrt{a+b x} (e+f x) \sqrt{2 b e-a f+b f x}} \, dx\)

Optimal. Leaf size=59 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x} \sqrt{-a f+2 b e+b f x}}{b e-a f}\right )}{\sqrt{f} (b e-a f)} \]

[Out]

ArcTan[(Sqrt[f]*Sqrt[a + b*x]*Sqrt[2*b*e - a*f + b*f*x])/(b*e - a*f)]/(Sqrt[f]*(
b*e - a*f))

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Rubi [A]  time = 0.178832, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x} \sqrt{-a f+2 b e+b f x}}{b e-a f}\right )}{\sqrt{f} (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*x]*(e + f*x)*Sqrt[2*b*e - a*f + b*f*x]),x]

[Out]

ArcTan[(Sqrt[f]*Sqrt[a + b*x]*Sqrt[2*b*e - a*f + b*f*x])/(b*e - a*f)]/(Sqrt[f]*(
b*e - a*f))

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Rubi in Sympy [A]  time = 16.4549, size = 49, normalized size = 0.83 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{f} \sqrt{a + b x} \sqrt{- a f + 2 b e + b f x}}{a f - b e} \right )}}{\sqrt{f} \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(f)*sqrt(a + b*x)*sqrt(-a*f + 2*b*e + b*f*x)/(a*f - b*e))/(sqrt(f)*(a*f
- b*e))

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Mathematica [C]  time = 0.164553, size = 81, normalized size = 1.37 \[ \frac{i \log \left (\frac{2 f \sqrt{a+b x} \sqrt{-a f+2 b e+b f x}}{e+f x}-\frac{2 i \sqrt{f} (a f-b e)}{e+f x}\right )}{\sqrt{f} (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b*x]*(e + f*x)*Sqrt[2*b*e - a*f + b*f*x]),x]

[Out]

(I*Log[((-2*I)*Sqrt[f]*(-(b*e) + a*f))/(e + f*x) + (2*f*Sqrt[a + b*x]*Sqrt[2*b*e
- a*f + b*f*x])/(e + f*x)])/(Sqrt[f]*(b*e - a*f))

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Maple [B]  time = 0.073, size = 154, normalized size = 2.6 \[ -{\frac{1}{f}\ln \left ( -2\,{\frac{1}{fx+e} \left ({a}^{2}{f}^{2}-2\,abef+{b}^{2}{e}^{2}-\sqrt{-{\frac{ \left ( af-be \right ) ^{2}}{f}}}\sqrt{{b}^{2}f{x}^{2}+2\,{b}^{2}ex-{a}^{2}f+2\,abe}f \right ) } \right ) \sqrt{bfx-af+2\,be}\sqrt{bx+a}{\frac{1}{\sqrt{-{\frac{ \left ( af-be \right ) ^{2}}{f}}}}}{\frac{1}{\sqrt{{b}^{2}f{x}^{2}+2\,{b}^{2}ex-{a}^{2}f+2\,abe}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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*f*x + 2*b*e - a*f)*sqrt(b*x + a)*(f*x + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268886, size = 1, normalized size = 0.02 \[ \left [-\frac{\log \left (\frac{2 \,{\left (b e f - a f^{2}\right )} \sqrt{b f x + 2 \, b e - a f} \sqrt{b x + a} -{\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x - b^{2} e^{2} + 4 \, a b e f - 2 \, a^{2} f^{2}\right )} \sqrt{-f}}{f^{2} x^{2} + 2 \, e f x + e^{2}}\right )}{2 \,{\left (b e - a f\right )} \sqrt{-f}}, \frac{\arctan \left (-\frac{b e - a f}{\sqrt{b f x + 2 \, b e - a f} \sqrt{b x + a} \sqrt{f}}\right )}{{\left (b e - a f\right )} \sqrt{f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*f*x + 2*b*e - a*f)*sqrt(b*x + a)*(f*x + e)),x, algorithm="fricas")

[Out]

[-1/2*log((2*(b*e*f - a*f^2)*sqrt(b*f*x + 2*b*e - a*f)*sqrt(b*x + a) - (b^2*f^2*
x^2 + 2*b^2*e*f*x - b^2*e^2 + 4*a*b*e*f - 2*a^2*f^2)*sqrt(-f))/(f^2*x^2 + 2*e*f*
x + e^2))/((b*e - a*f)*sqrt(-f)), arctan(-(b*e - a*f)/(sqrt(b*f*x + 2*b*e - a*f)
*sqrt(b*x + a)*sqrt(f)))/((b*e - a*f)*sqrt(f))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + b x} \left (e + f x\right ) \sqrt{- a f + 2 b e + b f x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(a + b*x)*(e + f*x)*sqrt(-a*f + 2*b*e + b*f*x)), x)

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GIAC/XCAS [A]  time = 0.226551, size = 130, normalized size = 2.2 \[ -\frac{2 \, f^{\frac{3}{2}} \arctan \left (\frac{{\left (\sqrt{b f x - a f + 2 \, b e} \sqrt{f} - \sqrt{2 \, a f^{2} - 2 \, b f e +{\left (b f x - a f + 2 \, b e\right )} f}\right )}^{2}}{2 \,{\left (a f^{2} - b f e\right )}}\right )}{{\left (a f^{2} - b f e\right )}{\left | f \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*f*x + 2*b*e - a*f)*sqrt(b*x + a)*(f*x + e)),x, algorithm="giac")

[Out]

-2*f^(3/2)*arctan(1/2*(sqrt(b*f*x - a*f + 2*b*e)*sqrt(f) - sqrt(2*a*f^2 - 2*b*f*
e + (b*f*x - a*f + 2*b*e)*f))^2/(a*f^2 - b*f*e))/((a*f^2 - b*f*e)*abs(f))