Optimal. Leaf size=968 \[ -\frac{\sqrt{d g-c h} \sqrt{f g-e h} \sqrt{a+b x} \sqrt{\frac{(d e-c f) (g+h x)}{(d g-c h) (e+f x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{f g-e h} \sqrt{c+d x}}{\sqrt{d g-c h} \sqrt{e+f x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right ) b}{d f h \sqrt{-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt{g+h x}}+\frac{(d e-c f) (b f g+b e h-2 a f h) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right ),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right ) b}{d f^2 h \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{\sqrt{b g-a h} (a d f h-b (d f g+d e h-c f h)) \sqrt{\frac{(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt{\frac{(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \Pi \left (\frac{f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{g+h x}}{\sqrt{b g-a h} \sqrt{e+f x}}\right )|\frac{(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right ) b}{d f^2 \sqrt{b e-a f} h^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \sqrt{g+h x} b}{d h \sqrt{e+f x}}-\frac{2 \sqrt{b c-a d} \sqrt{c h-d g} (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d h \sqrt{c+d x} \sqrt{e+f x}} \]
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Rubi [A] time = 0.882066, antiderivative size = 968, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216, Rules used = {166, 173, 176, 424, 170, 419, 165, 537} \[ -\frac{\sqrt{d g-c h} \sqrt{f g-e h} \sqrt{a+b x} \sqrt{\frac{(d e-c f) (g+h x)}{(d g-c h) (e+f x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{f g-e h} \sqrt{c+d x}}{\sqrt{d g-c h} \sqrt{e+f x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right ) b}{d f h \sqrt{-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt{g+h x}}+\frac{(d e-c f) (b f g+b e h-2 a f h) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right ) b}{d f^2 h \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{\sqrt{b g-a h} (a d f h-b (d f g+d e h-c f h)) \sqrt{\frac{(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt{\frac{(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \Pi \left (\frac{f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{g+h x}}{\sqrt{b g-a h} \sqrt{e+f x}}\right )|\frac{(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right ) b}{d f^2 \sqrt{b e-a f} h^2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \sqrt{g+h x} b}{d h \sqrt{e+f x}}-\frac{2 \sqrt{b c-a d} \sqrt{c h-d g} (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d h \sqrt{c+d x} \sqrt{e+f x}} \]
Antiderivative was successfully verified.
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Rule 166
Rule 173
Rule 176
Rule 424
Rule 170
Rule 419
Rule 165
Rule 537
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{b \int \frac{\sqrt{a+b x} \sqrt{c+d x}}{\sqrt{e+f x} \sqrt{g+h x}} \, dx}{d}-\frac{(b c-a d) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{d}\\ &=\frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{g+h x}}{d h \sqrt{e+f x}}-\frac{(b (d e-c f) (f g-e h)) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x} (e+f x)^{3/2} \sqrt{g+h x}} \, dx}{2 d f h}+\frac{(b (d e-c f) (b f g+b e h-2 a f h)) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 d f^2 h}+\frac{(b (a d f h-b (d f g+d e h-c f h))) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{g+h x}} \, dx}{2 d f^2 h}-\frac{\left (2 (b c-a d) (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 )}{d \sqrt{c+d x} \sqrt{e+f x}}\\ &=\frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{g+h x}}{d h \sqrt{e+f x}}-\frac{2 \sqrt{b c-a d} \sqrt{-d g+c 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 )}{d h \sqrt{c+d x} \sqrt{e+f x}}+\frac{\left (b (a d f h-b (d f g+d e h-c f h)) \sqrt{\frac{(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt{\frac{(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (h-f x^2\right ) \sqrt{1+\frac{(-b e+a f) x^2}{b g-a h}} \sqrt{1+\frac{(-d e+c f) x^2}{d g-c h}}} \, dx,x,\frac{\sqrt{g+h x}}{\sqrt{e+f x}}\right )}{d f^2 h \sqrt{a+b x} \sqrt{c+d x}}+\frac{\left (b (d e-c f) (b f g+b e h-2 a f h) \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 )}{d f^2 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{\left (b (d e-c f) (f g-e h) \sqrt{a+b x} \sqrt{-\frac{(-d e+c f) (g+h x)}{(d g-c h) (e+f x)}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{(-b e+a f) x^2}{b c-a d}}}{\sqrt{1-\frac{(f g-e h) x^2}{d g-c h}}} \, dx,x,\frac{\sqrt{c+d x}}{\sqrt{e+f x}}\right )}{d f (-d e+c f) h \sqrt{\frac{(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt{g+h x}}\\ &=\frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{g+h x}}{d h \sqrt{e+f x}}-\frac{b \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{a+b x} \sqrt{\frac{(d e-c f) (g+h x)}{(d g-c h) (e+f x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{f g-e h} \sqrt{c+d x}}{\sqrt{d g-c h} \sqrt{e+f x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{d f h \sqrt{-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}} \sqrt{g+h x}}+\frac{b (d e-c f) (b f g+b e h-2 a f h) \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 )}{d f^2 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{b \sqrt{b g-a h} (a d f h-b (d f g+d e h-c f h)) \sqrt{\frac{(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt{\frac{(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \Pi \left (\frac{f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{g+h x}}{\sqrt{b g-a h} \sqrt{e+f x}}\right )|\frac{(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{d f^2 \sqrt{b e-a f} h^2 \sqrt{a+b x} \sqrt{c+d x}}-\frac{2 \sqrt{b c-a d} \sqrt{-d g+c 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 )}{d h \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}
Mathematica [B] time = 14.2736, size = 6638, normalized size = 6.86 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.087, size = 16526, normalized size = 17.1 \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{{\left (b x + a\right )}^{\frac{3}{2}}}{\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{{\left (b x + a\right )}^{\frac{3}{2}}}{\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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