Optimal. Leaf size=981 \[ \frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} B}{2 d f h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h))) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{(b e-a f) \sqrt{b g-a h} (3 a B d f h+b (4 A d f h-B (c f h+3 d (f g+e h)))) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right ),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{4 b d f^2 h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{\sqrt{c h-d g} (4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))-(a d f h+b (d f g+d e h+c f h)) (5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h)))) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b d^2 \sqrt{b c-a d} f^2 h^3 \sqrt{c+d x} \sqrt{e+f x}}+\frac{(5 a B d f h+b (4 A d f h-3 B (d f g+d e h+c f h))) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{4 d f^2 h^2 \sqrt{c+d x}} \]
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Rubi [A] time = 3.412, antiderivative size = 976, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 10, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {1597, 1600, 1602, 1598, 170, 419, 165, 537, 176, 424} \[ \frac{b \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} B}{2 d f h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{4 d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{(b e-a f) \sqrt{b g-a h} (4 A b d f h+3 a B d f h-b B (c f h+3 d (f g+e h))) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{4 b d f^2 h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}-\frac{\sqrt{c h-d g} ((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)))) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b d^2 \sqrt{b c-a d} f^2 h^3 \sqrt{c+d x} \sqrt{e+f x}}+\frac{(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{4 d f^2 h^2 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 1597
Rule 1600
Rule 1602
Rule 1598
Rule 170
Rule 419
Rule 165
Rule 537
Rule 176
Rule 424
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{\int \frac{\sqrt{a+b x} \left (6 a A d f h+6 (A b+a B) d f h x+6 b B d f h x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{6 d f h}\\ &=\frac{b B \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{2 d f h}+\frac{\int \frac{6 d f h \left (4 a^2 A d f h-b B (b c e g+a (d e g+c f g+c e h))\right )+12 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (d f g+d e h+c f h))) x+6 b d f h (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{24 d^2 f^2 h^2}\\ &=\frac{(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{4 d f^2 h^2 \sqrt{c+d x}}+\frac{b B \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{2 d f h}+\frac{\int \frac{-6 b d f h \left ((b d e g+a c f h) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-2 d f h \left (4 a^2 A d f h-b B (b c e g+a (d e g+c f g+c e h))\right )\right )-6 b d f h ((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (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}{48 b d^3 f^3 h^3}+\frac{((d e-c f) (d g-c h) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{8 d^2 f^2 h^2}\\ &=\frac{(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{4 d f^2 h^2 \sqrt{c+d x}}+\frac{b B \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{2 d f h}-\frac{((b e-a f) (b g-a h) (4 A b d f h+3 a B d f h-b B (c f h+3 d (f g+e h)))) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{8 b d f^2 h^2}-\frac{((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (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}{8 b d^2 f^2 h^2}-\frac{\left ((d g-c h) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f 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 )}{4 d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}\\ &=\frac{(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{4 d f^2 h^2 \sqrt{c+d x}}+\frac{b B \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{2 d f h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f 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 )}{4 d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{\left (((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (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 )}{4 b d^2 f^2 h^2 \sqrt{c+d x} \sqrt{e+f x}}-\frac{\left ((b e-a f) (b g-a h) (4 A b d f h+3 a B d f h-b B (c f h+3 d (f g+e 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 )}{4 b d f^2 h^2 (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{(4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{4 d f^2 h^2 \sqrt{c+d x}}+\frac{b B \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{2 d f h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f 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 )}{4 d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{(b e-a f) \sqrt{b g-a h} (4 A b d f h+3 a B d f h-b B (c f h+3 d (f g+e 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 )}{4 b d f^2 h^2 \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{-d g+c h} ((a d f h+b (d f g+d e h+c f h)) (4 A b d f h+5 a B d f h-3 b B (d f g+d e h+c f h))-4 d f h (2 a (2 A b+a B) d f h-b B (b (d e g+c f g+c e h)+a (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 )}{4 b d^2 \sqrt{b c-a d} f^2 h^3 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}
Mathematica [B] time = 19.0552, size = 21555, normalized size = 21.97 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.152, size = 54632, normalized size = 55.7 \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 )}{\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 )}{\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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