Optimal. Leaf size=757 \[ \frac{\sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \left (a^2 C f h+a b C (e h+f g)-b^2 (C e g-2 A f h)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right ),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right )}{b^2 f h \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{C (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (a d f h+b (c f h+d e h+d f g)) \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b^2 d f h^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}+\frac{C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{b f h \sqrt{c+d x}}-\frac{C \sqrt{a+b x} \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{b d f h \sqrt{g+h x} \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]
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Rubi [A] time = 1.02598, antiderivative size = 757, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1603, 1598, 170, 419, 165, 537, 176, 424} \[ \frac{\sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \left (a^2 C f h+a b C (e h+f g)-b^2 (C e g-2 A f h)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{b^2 f h \sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{C (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} (a d f h+b (c f h+d e h+d f g)) \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{b^2 d f h^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}}+\frac{C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{b f h \sqrt{c+d x}}-\frac{C \sqrt{a+b x} \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{b d f h \sqrt{g+h x} \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]
Antiderivative was successfully verified.
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Rule 1603
Rule 1598
Rule 170
Rule 419
Rule 165
Rule 537
Rule 176
Rule 424
Rubi steps
\begin{align*} \int \frac{A+C x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{b f h \sqrt{c+d x}}+\frac{\int \frac{2 A b d f h-C (b d e g+a c f h)-C (a d f h+b (d f g+d e h+c f h)) x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 b d f h}+\frac{(C (d e-c f) (d g-c h)) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 b d f h}\\ &=\frac{C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{b f h \sqrt{c+d x}}+\frac{\left (a^2 C f h+a b C (f g+e h)-b^2 (C e g-2 A f h)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 b^2 f h}-\frac{(C (a d f h+b (d f g+d e h+c f h))) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 b^2 d f h}-\frac{\left (C (d g-c h) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{(-b c+a d) x^2}{b e-a f}}}{\sqrt{1-\frac{(d g-c h) x^2}{f g-e h}}} \, dx,x,\frac{\sqrt{e+f x}}{\sqrt{c+d x}}\right )}{b d f h \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}\\ &=\frac{C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{b f h \sqrt{c+d x}}-\frac{C \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} 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{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{\left (C (a d f h+b (d f g+d e h+c f h)) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (h-b x^2\right ) \sqrt{1+\frac{(b c-a d) x^2}{d g-c h}} \sqrt{1+\frac{(b e-a f) x^2}{f g-e h}}} \, dx,x,\frac{\sqrt{g+h x}}{\sqrt{a+b x}}\right )}{b^2 d f h \sqrt{c+d x} \sqrt{e+f x}}+\frac{\left (\left (a^2 C f h+a b C (f g+e h)-b^2 (C e g-2 A f h)\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{(b c-a d) x^2}{d e-c f}} \sqrt{1-\frac{(b g-a h) x^2}{f g-e h}}} \, dx,x,\frac{\sqrt{e+f x}}{\sqrt{a+b x}}\right )}{b^2 f h (f g-e h) \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}\\ &=\frac{C \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{b f h \sqrt{c+d x}}-\frac{C \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} 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{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{\left (a^2 C f h+a b C (f g+e h)-b^2 (C e g-2 A f h)\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} 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{b g-a h} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac{C \sqrt{-d g+c h} (a d f h+b (d f g+d e h+c f h)) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \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{-d g+c h} \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 \sqrt{b c-a d} f h^2 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}
Mathematica [B] time = 15.1693, size = 6207, normalized size = 8.2 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.096, size = 15875, normalized size = 21. \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C x^{2} + A}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C x^{2} + A}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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