Optimal. Leaf size=134 \[ \frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}} \]
[Out]
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Rubi [A] time = 0.326858, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt{d} \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 42.8128, size = 114, normalized size = 0.85 \[ \frac{2 \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \sqrt{e + f x} \sqrt{a d - b c} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{a d - b c}} \right )}\middle | \frac{f \left (a d - b c\right )}{d \left (a f - b e\right )}\right )}{b \sqrt{d} \sqrt{\frac{b \left (- e - f x\right )}{a f - b e}} \sqrt{c + d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x+e)**(1/2)/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 1.29506, size = 154, normalized size = 1.15 \[ \frac{2 \sqrt{c+d x} \left (\frac{(a f-b e) \sqrt{\frac{b (e+f x)}{f (a+b x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{a-\frac{b e}{f}}}{\sqrt{a+b x}}\right )|\frac{b c f-a d f}{b d e-a d f}\right )}{b \sqrt{a-\frac{b e}{f}} \sqrt{\frac{b (c+d x)}{d (a+b x)}}}+\frac{e+f x}{\sqrt{a+b x}}\right )}{d \sqrt{e+f x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]),x]
[Out]
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Maple [A] time = 0., size = 209, normalized size = 1.6 \[ -2\,{\frac{ \left ({a}^{2}df-abcf-bead+{b}^{2}ce \right ) \sqrt{dx+c}\sqrt{bx+a}\sqrt{fx+e}}{{b}^{2}d \left ( bdf{x}^{3}+adf{x}^{2}+bcf{x}^{2}+bde{x}^{2}+acfx+adex+bcex+ace \right ) }{\it EllipticE} \left ( \sqrt{{\frac{d \left ( bx+a \right ) }{ad-bc}}},\sqrt{{\frac{ \left ( ad-bc \right ) f}{d \left ( af-be \right ) }}} \right ) \sqrt{-{\frac{ \left ( dx+c \right ) b}{ad-bc}}}\sqrt{-{\frac{ \left ( fx+e \right ) b}{af-be}}}\sqrt{{\frac{d \left ( bx+a \right ) }{ad-bc}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x+e)^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{f x + e}}{\sqrt{b x + a} \sqrt{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(f*x + e)/(sqrt(b*x + a)*sqrt(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{f x + e}}{\sqrt{b x + a} \sqrt{d x + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(f*x + e)/(sqrt(b*x + a)*sqrt(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e + f x}}{\sqrt{a + b x} \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x+e)**(1/2)/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{f x + e}}{\sqrt{b x + a} \sqrt{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(f*x + e)/(sqrt(b*x + a)*sqrt(d*x + c)),x, algorithm="giac")
[Out]