Optimal. Leaf size=650 \[ -\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} (2 e f-3 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g-c d (2 e f-3 d g)\right ) \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{\sqrt{a+c x^2} \sqrt{f+g x}}{e (d+e x)}+\frac{3 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{3 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]
[Out]
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Rubi [A] time = 4.10413, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{\sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} (2 e f-3 d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (a e^2 g-c d (2 e f-3 d g)\right ) \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e^3 \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{\sqrt{a+c x^2} \sqrt{f+g x}}{e (d+e x)}+\frac{3 \sqrt{-a} \sqrt{c} f \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{f+g x}}-\frac{3 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x)^2,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + c x^{2}} \sqrt{f + g x}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)**(1/2)*(c*x**2+a)**(1/2)/(e*x+d)**2,x)
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Mathematica [C] time = 13.5394, size = 7969, normalized size = 12.26 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x)^2,x]
[Out]
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Maple [B] time = 0.064, size = 6044, normalized size = 9.3 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(e*x+d)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + a} \sqrt{g x + f}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*sqrt(g*x + f)/(e*x + d)^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(sqrt(c*x^2 + a)*sqrt(g*x + f)/(e*x + d)^2,x, algorithm="fricas")
[Out]
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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((g*x+f)**(1/2)*(c*x**2+a)**(1/2)/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*sqrt(g*x + f)/(e*x + d)^2,x, algorithm="giac")
[Out]