3.662 \(\int \frac{\sqrt{d+e x}}{(f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=140 \[ \frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)}+\frac{c d \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{\sqrt{g} (c d f-a e g)^{3/2}} \]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/((c*d*f - a*e*g)*Sqrt[d + e*x]*(f +
g*x)) + (c*d*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[
c*d*f - a*e*g]*Sqrt[d + e*x])])/(Sqrt[g]*(c*d*f - a*e*g)^(3/2))

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Rubi [A]  time = 0.640196, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} (f+g x) (c d f-a e g)}+\frac{c d \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{\sqrt{g} (c d f-a e g)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/((c*d*f - a*e*g)*Sqrt[d + e*x]*(f +
g*x)) + (c*d*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[
c*d*f - a*e*g]*Sqrt[d + e*x])])/(Sqrt[g]*(c*d*f - a*e*g)^(3/2))

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Rubi in Sympy [A]  time = 56.4043, size = 131, normalized size = 0.94 \[ \frac{c d \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{\sqrt{g} \left (a e g - c d f\right )^{\frac{3}{2}}} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \left (f + g x\right ) \left (a e g - c d f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

c*d*atanh(sqrt(g)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(sqrt(d + e*x)*
sqrt(a*e*g - c*d*f)))/(sqrt(g)*(a*e*g - c*d*f)**(3/2)) - sqrt(a*d*e + c*d*e*x**2
+ x*(a*e**2 + c*d**2))/(sqrt(d + e*x)*(f + g*x)*(a*e*g - c*d*f))

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Mathematica [A]  time = 0.264266, size = 138, normalized size = 0.99 \[ -\frac{\sqrt{d+e x} \left (\sqrt{g} (a e+c d x) \sqrt{a e g-c d f}-c d (f+g x) \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )\right )}{\sqrt{g} (f+g x) \sqrt{(d+e x) (a e+c d x)} (a e g-c d f)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

-((Sqrt[d + e*x]*(Sqrt[g]*Sqrt[-(c*d*f) + a*e*g]*(a*e + c*d*x) - c*d*Sqrt[a*e +
c*d*x]*(f + g*x)*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d*f) + a*e*g]]))/(
Sqrt[g]*(-(c*d*f) + a*e*g)^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)))

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Maple [A]  time = 0.029, size = 168, normalized size = 1.2 \[{\frac{1}{ \left ( aeg-cdf \right ) \left ( gx+f \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ({\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) xcdg+{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) cdf-\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d
*f)*g)^(1/2))*x*c*d*g+arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c*d*f
-(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2))/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/(a*e
*g-c*d*f)/(g*x+f)/((a*e*g-c*d*f)*g)^(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(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30394, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (c d e g x^{2} + c d^{2} f +{\left (c d e f + c d^{2} g\right )} x\right )} \log \left (\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d f g - a e g^{2}\right )} \sqrt{e x + d} -{\left (c d e g x^{2} - c d^{2} f + 2 \, a d e g -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x\right )} \sqrt{-c d f g + a e g^{2}}}{e g x^{2} + d f +{\left (e f + d g\right )} x}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d f g + a e g^{2}} \sqrt{e x + d}}{2 \,{\left (c d^{2} f^{2} - a d e f g +{\left (c d e f g - a e^{2} g^{2}\right )} x^{2} +{\left (c d e f^{2} - a d e g^{2} +{\left (c d^{2} - a e^{2}\right )} f g\right )} x\right )} \sqrt{-c d f g + a e g^{2}}}, -\frac{{\left (c d e g x^{2} + c d^{2} f +{\left (c d e f + c d^{2} g\right )} x\right )} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d f g - a e g^{2}} \sqrt{e x + d}}{c d e g x^{2} + a d e g +{\left (c d^{2} + a e^{2}\right )} g x}\right ) - \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d f g - a e g^{2}} \sqrt{e x + d}}{{\left (c d^{2} f^{2} - a d e f g +{\left (c d e f g - a e^{2} g^{2}\right )} x^{2} +{\left (c d e f^{2} - a d e g^{2} +{\left (c d^{2} - a e^{2}\right )} f g\right )} x\right )} \sqrt{c d f g - a e g^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(1/2)/(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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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(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^2),x, algorithm="giac")

[Out]

Timed out