Optimal. Leaf size=1070 \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (C a^2+A b^2\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 \sqrt{d g-c h} \sqrt{f g-e h} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{2 \left (-\left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^2+3 b \left (C c^2+A d^2\right ) (f g+e h) a-b^2 \left ((3 C e g-A f h) c^2+A d (f g+e h) c+2 A d^2 e g\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right ),-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{4 b \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{4 d \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}} \]
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Rubi [A] time = 3.45407, antiderivative size = 1070, 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 = {1605, 1599, 1602, 12, 170, 419, 176, 424} \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (C a^2+A b^2\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 \sqrt{d g-c h} \sqrt{f g-e h} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{2 \left (-\left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^2+3 b \left (C c^2+A d^2\right ) (f g+e h) a-b^2 \left ((3 C e g-A f h) c^2+A d (f g+e h) c+2 A d^2 e g\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{4 b \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{4 d \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 1605
Rule 1599
Rule 1602
Rule 12
Rule 170
Rule 419
Rule 176
Rule 424
Rubi steps
\begin{align*} \int \frac{A+C x^2}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=-\frac{2 \left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{\int \frac{-2 A b^2 (d e g+c f g+c e h)-3 a b (c C e g-A d f g-A d e h-A c f h)-a^2 (3 A d f h-C (d e g+c f g+c e h))+\left (2 a^2 C (d f g+d e h+c f h)+b^2 (3 c C e g-A d f g-A d e h-A c f h)+3 a b (A d f h-C (d e g+c f g+c e h))\right ) x}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 b \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}+\frac{\int \frac{b (b c e g-a (d e g+c f g+c e h)) \left (2 a^2 C (d f g+d e h+c f h)+b^2 (3 c C e g-A d f g-A d e h-A c f h)+3 a b (A d f h-C (d e g+c f g+c e h))\right )+a (a d f h-b (d f g+d e h+c f h)) \left (2 A b^2 (d e g+c f g+c e h)+3 a b (c C e g-A d f g-A d e h-A c f h)+a^2 (3 A d f h-C (d e g+c f g+c e h))\right )-2 (a d f h+b (d f g+d e h+c f h)) \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) x-4 b d f h \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2}\\ &=-\frac{4 d \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 \left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 b \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}+\frac{\int -\frac{2 b d f (b e-a f) h (b g-a h) \left (3 a b \left (c^2 C+A d^2\right ) (f g+e h)-b^2 \left (2 A d^2 e g+A c d (f g+e h)+c^2 (3 C e g-A f h)\right )-a^2 \left (3 A d^2 f h-C \left (d^2 e g-c d f g-c d e h-2 c^2 f h\right )\right )\right )}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{6 b d (b c-a d)^2 f (b e-a f)^2 h (b g-a h)^2}-\frac{\left (2 (d e-c f) (d g-c h) \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right )\right ) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2}\\ &=-\frac{4 d \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 \left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 b \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{\left (3 a b \left (c^2 C+A d^2\right ) (f g+e h)-b^2 \left (2 A d^2 e g+A c d (f g+e h)+c^2 (3 C e g-A f h)\right )-a^2 \left (3 A d^2 f h-C \left (d^2 e g-2 c^2 f h-c d (f g+e h)\right )\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 (b c-a d)^2 (b e-a f) (b g-a h)}+\frac{\left (4 (d g-c h) \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \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 )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}\\ &=-\frac{4 d \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 \left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 b \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}+\frac{4 \sqrt{d g-c h} \sqrt{f g-e h} \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \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 )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{\left (2 \left (3 a b \left (c^2 C+A d^2\right ) (f g+e h)-b^2 \left (2 A d^2 e g+A c d (f g+e h)+c^2 (3 C e g-A f h)\right )-a^2 \left (3 A d^2 f h-C \left (d^2 e g-2 c^2 f h-c d (f g+e h)\right )\right )\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 )}{3 (b c-a d)^2 (b e-a f) (b g-a 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{4 d \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 \left (A b^2+a^2 C\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 b \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}+\frac{4 \sqrt{d g-c h} \sqrt{f g-e h} \left (A b^3 (d e g+c f g+c e h)+a^3 C (d f g+d e h+c f h)+a^2 b (3 A d f h-2 C (d e g+c f g+c e h))-a b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h))\right ) \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 )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{2 \left (3 a b \left (c^2 C+A d^2\right ) (f g+e h)-b^2 \left (2 A d^2 e g+A c d (f g+e h)+c^2 (3 C e g-A f h)\right )-a^2 \left (3 A d^2 f h-C \left (d^2 e g-c d f g-c d e h-2 c^2 f h\right )\right )\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 )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \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)}}}\\ \end{align*}
Mathematica [B] time = 26.7135, size = 11160, normalized size = 10.43 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 1.214, size = 72702, normalized size = 68. \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}{{\left (b x + a\right )}^{\frac{5}{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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C x^{2} + A\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}{b^{3} d f h x^{6} + a^{3} c e g +{\left (b^{3} d f g +{\left (b^{3} d e +{\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} h\right )} x^{5} +{\left ({\left (b^{3} d e +{\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} g +{\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \,{\left (a b^{2} c + a^{2} b d\right )} f\right )} h\right )} x^{4} +{\left ({\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \,{\left (a b^{2} c + a^{2} b d\right )} f\right )} g +{\left (3 \,{\left (a b^{2} c + a^{2} b d\right )} e +{\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} h\right )} x^{3} +{\left ({\left (3 \,{\left (a b^{2} c + a^{2} b d\right )} e +{\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} g +{\left (a^{3} c f +{\left (3 \, a^{2} b c + a^{3} d\right )} e\right )} h\right )} x^{2} +{\left (a^{3} c e h +{\left (a^{3} c f +{\left (3 \, a^{2} b c + a^{3} d\right )} e\right )} g\right )} x}, x\right ) \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}{{\left (b x + a\right )}^{\frac{5}{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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