Optimal. Leaf size=167 \[ -\frac{2 \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )} \]
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Rubi [A] time = 1.41636, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{2 \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]
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Rubi in Sympy [A] time = 18.5421, size = 201, normalized size = 1.2 \[ \frac{2 \sqrt{\frac{g \left (- \sqrt{c} x - \sqrt{- a}\right )}{\sqrt{c} f - g \sqrt{- a}}} \sqrt{\frac{g \left (- \sqrt{c} x + \sqrt{- a}\right )}{\sqrt{c} f + g \sqrt{- a}}} \Pi \left (- \frac{e \left (\sqrt{c} f + g \sqrt{- a}\right )}{\sqrt{c} \left (d g - e f\right )}; \operatorname{asin}{\left (\sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{f + g x} \right )}\middle | \frac{\sqrt{c} f + g \sqrt{- a}}{\sqrt{c} f - g \sqrt{- a}}\right )}{\sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{a + c x^{2}} \left (d g - e f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)
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Mathematica [C] time = 0.903257, size = 311, normalized size = 1.86 \[ -\frac{2 i (f+g x) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )-\Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{\sqrt{a+c x^2} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]
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Maple [A] time = 0.052, size = 235, normalized size = 1.4 \[ 2\,{\frac{ \left ( cf-g\sqrt{-ac} \right ) \sqrt{c{x}^{2}+a}\sqrt{gx+f}}{c \left ( dg-ef \right ) \left ( cg{x}^{3}+cf{x}^{2}+agx+fa \right ) }{\it EllipticPi} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},{\frac{ \left ( g\sqrt{-ac}-cf \right ) e}{c \left ( dg-ef \right ) }},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) \sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}+cf}}}\sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )} \sqrt{g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f)),x, algorithm="maxima")
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Fricas [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(c*x^2 + a)*(e*x + d)*sqrt(g*x + f)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + c x^{2}} \left (d + e x\right ) \sqrt{f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )} \sqrt{g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + a)*(e*x + d)*sqrt(g*x + f)),x, algorithm="giac")
[Out]