Optimal. Leaf size=757 \[ \frac{\sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \left (a^2 C f h+a b C (e h+f g)-b^2 (C e g-2 A f h)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b^2 f h \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{C (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (a d f h+b (c f h+d e h+d f g)) \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b^2 d f h^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}+\frac{C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{b f h \sqrt{c+d x}}-\frac{C \sqrt{a+b x} \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{b d f h \sqrt{g+h x} \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]
[Out]
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Rubi [A] time = 4.04754, antiderivative size = 757, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.204 \[ \frac{\sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \left (a^2 C f h+a b C (e h+f g)-b^2 (C e g-2 A f h)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b^2 f h \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{C (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (a d f h+b (c f h+d e h+d f g)) \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b^2 d f h^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}+\frac{C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{b f h \sqrt{c+d x}}-\frac{C \sqrt{a+b x} \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{b d f h \sqrt{g+h x} \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]
Warning: Unable to verify antiderivative.
[In] Int[(A + C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((C*x**2+A)/(b*x+a)**(1/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
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Mathematica [B] time = 16.5267, size = 6207, normalized size = 8.2 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(A + C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
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Maple [B] time = 0.12, size = 15875, normalized size = 21. \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*x^2+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{C x^{2} + A}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),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((C*x^2 + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),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((C*x**2+A)/(b*x+a)**(1/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{C x^{2} + A}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")
[Out]