Optimal. Leaf size=387 \[ \frac{2 g^2 \sqrt{a+c x^2}}{\sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)}+\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{a+c x^2} \left (a g^2+c f^2\right ) (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 e \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) (e f-d g)} \]
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Rubi [A] time = 2.03397, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ \frac{2 g^2 \sqrt{a+c x^2}}{\sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)}+\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{a+c x^2} \left (a g^2+c f^2\right ) (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{2 e \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) (e f-d g)} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*(f + g*x)^(3/2)*Sqrt[a + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 115.104, size = 394, normalized size = 1.02 \[ - \frac{2 \sqrt{c} g \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{f + g x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{\sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{a + c x^{2}} \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )} - \frac{2 e \sqrt{\frac{g \left (- \sqrt{c} x - \sqrt{- a}\right )}{\sqrt{c} f - g \sqrt{- a}}} \sqrt{\frac{g \left (- \sqrt{c} x + \sqrt{- a}\right )}{\sqrt{c} f + g \sqrt{- a}}} \Pi \left (- \frac{e \left (\sqrt{c} f + g \sqrt{- a}\right )}{\sqrt{c} \left (d g - e f\right )}; \operatorname{asin}{\left (\sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{f + g x} \right )}\middle | \frac{\sqrt{c} f + g \sqrt{- a}}{\sqrt{c} f - g \sqrt{- a}}\right )}{\sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{a + c x^{2}} \left (d g - e f\right )^{2}} - \frac{2 g^{2} \sqrt{a + c x^{2}}}{\sqrt{f + g x} \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(g*x+f)**(3/2)/(c*x**2+a)**(1/2),x)
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Mathematica [C] time = 6.35817, size = 468, normalized size = 1.21 \[ \frac{2 i (f+g x) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (\left (\sqrt{c} (d g-2 e f)+i \sqrt{a} e g\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )+\sqrt{c} (e f-d g) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )+e \left (\sqrt{c} f-i \sqrt{a} g\right ) \Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{\sqrt{a+c x^2} \left (\sqrt{c} f-i \sqrt{a} g\right ) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*(f + g*x)^(3/2)*Sqrt[a + c*x^2]),x]
[Out]
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Maple [B] time = 0.093, size = 2011, normalized size = 5.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(g*x+f)^(3/2)/(c*x^2+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + a)*(e*x + d)*(g*x + f)^(3/2)),x, algorithm="maxima")
[Out]
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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)*(g*x + f)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + c x^{2}} \left (d + e x\right ) \left (f + g x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(g*x+f)**(3/2)/(c*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + a)*(e*x + d)*(g*x + f)^(3/2)),x, algorithm="giac")
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