Optimal. Leaf size=132 \[ \frac{c d \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{3/2} \sqrt{c d f-a e g}}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt{d+e x} (f+g x)} \]
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Rubi [A] time = 0.546814, antiderivative size = 132, 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 \[ \frac{c d \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{3/2} \sqrt{c d f-a e g}}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt{d+e x} (f+g x)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 56.2263, size = 124, normalized size = 0.94 \[ - \frac{c d \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e g - c d f}} \right )}}{g^{\frac{3}{2}} \sqrt{a e g - c d f}} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{g \sqrt{d + e x} \left (f + g x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**2/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.22079, size = 111, normalized size = 0.84 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (-\frac{c d \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{a e g-c d f}}\right )}{\sqrt{a e+c d x} \sqrt{a e g-c d f}}-\frac{\sqrt{g}}{f+g x}\right )}{g^{3/2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(Sqrt[d + e*x]*(f + g*x)^2),x]
[Out]
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Maple [A] time = 0.032, size = 161, normalized size = 1.2 \[{\frac{1}{g \left ( gx+f \right ) } \left ( -{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) xcdg-{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) cdf-\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g} \right ) \sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^2/(e*x+d)^(1/2),x)
[Out]
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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(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.28509, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (c d e g x^{2} + c d^{2} f +{\left (c d e f + c d^{2} g\right )} x\right )} \log \left (-\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d f g - a e g^{2}\right )} \sqrt{e x + d} +{\left (c d e g x^{2} - c d^{2} f + 2 \, a d e g -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x\right )} \sqrt{-c d f g + a e g^{2}}}{e g x^{2} + d f +{\left (e f + d g\right )} x}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d f g + a e g^{2}} \sqrt{e x + d}}{2 \,{\left (e g^{2} x^{2} + d f g +{\left (e f g + d g^{2}\right )} x\right )} \sqrt{-c d f g + a e g^{2}}}, -\frac{{\left (c d e g x^{2} + c d^{2} f +{\left (c d e f + c d^{2} g\right )} x\right )} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d f g - a e g^{2}} \sqrt{e x + d}}{c d e g x^{2} + a d e g +{\left (c d^{2} + a e^{2}\right )} g x}\right ) + \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d f g - a e g^{2}} \sqrt{e x + d}}{{\left (e g^{2} x^{2} + d f g +{\left (e f g + d g^{2}\right )} x\right )} \sqrt{c d f g - a e g^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}{\sqrt{d + e x} \left (f + g x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**2/(e*x+d)**(1/2),x)
[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(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^2),x, algorithm="giac")
[Out]