3.2517 \(\int \frac{1}{\sqrt{a+b x} (e+f x) \sqrt{2 b e-a f+b f x}} \, dx\)

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 = 0.0465872, 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, Rules used = {92, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x} \sqrt{-a f+2 b e+b f x}}{b e-a f}\right )}{\sqrt{f} (b e-a f)} \]

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))

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+b x} (e+f x) \sqrt{2 b e-a f+b f x}} \, dx &=(b f) \operatorname{Subst}\left (\int \frac{1}{b f (b e-a f)^2+b f^2 x^2} \, dx,x,\sqrt{a+b x} \sqrt{2 b e-a f+b f x}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x} \sqrt{2 b e-a f+b f x}}{b e-a f}\right )}{\sqrt{f} (b e-a f)}\\ \end{align*}

Mathematica [A]  time = 0.052782, size = 82, normalized size = 1.39 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{f (b e-a f)}}{\sqrt{b e-a f} \sqrt{-a f+2 b e+b f x}}\right )}{\sqrt{b e-a f} \sqrt{f (b e-a f)}} \]

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]

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

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Maple [B]  time = 0.054, size = 154, normalized size = 2.6 \begin{align*} -{\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\,aeb}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\,aeb}}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.11885, size = 455, normalized size = 7.71 \begin{align*} \left [\frac{\sqrt{-f} \log \left (-\frac{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} + 2 \, \sqrt{b f x + 2 \, b e - a f}{\left (b e - a f\right )} \sqrt{b x + a} \sqrt{-f}}{f^{2} x^{2} + 2 \, e f x + e^{2}}\right )}{2 \,{\left (b e f - a f^{2}\right )}}, \frac{\sqrt{f} \arctan \left (-\frac{\sqrt{b f x + 2 \, b e - a f}{\left (b e - a f\right )} \sqrt{b x + a} \sqrt{f}}{b^{2} f^{2} x^{2} + 2 \, b^{2} e f x + 2 \, a b e f - a^{2} f^{2}}\right )}{b e f - a f^{2}}\right ] \end{align*}

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, algorithm="fricas")

[Out]

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

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

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 [A]  time = 1.49232, size = 130, normalized size = 2.2 \begin{align*} -\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 |}} \end{align*}

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, 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))