Optimal. Leaf size=1395 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 5.74231, antiderivative size = 1376, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 10, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1601, 1600, 1602, 1598, 170, 419, 165, 537, 176, 424} \[ \frac{C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)^{3/2}}{3 d f h}-\frac{\sqrt{c h-d g} \left ((a d f h+b (d f g+d e h+c f h)) \left (24 A d^2 f^2 h^2 b^2+15 C (d f g+d e h+c f h)^2 b^2-16 C d f h (d e g+c f g+c e h) b^2-22 a C d f h (d f g+d e h+c f h) b+3 a^2 C d^2 f^2 h^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (2 C (d f g+d e h+c f h) a^2-b (12 A d f h-5 C (d e g+c f g+c e h)) a+3 b^2 c C e g\right )\right )\right ) \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 ) (a+b x)}{24 b^2 d^3 \sqrt{b c-a d} f^3 h^4 \sqrt{c+d x} \sqrt{e+f x}}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} \left (24 A d^2 f^2 h^2 b^2+15 C (d f g+d e h+c f h)^2 b^2-16 C d f h (d e g+c f g+c e h) b^2-22 a C d f h (d f g+d e h+c f h) b+3 a^2 C d^2 f^2 h^2\right ) \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 ) \sqrt{a+b x}}{24 b d^3 f^3 h^3 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \sqrt{a+b x}}{12 d^2 f^2 h^2}+\frac{\left (24 A b f h d^2+\frac{3 a^2 C f h d^2}{b}-16 b C (d e g+c f g+c e h) d-22 a C (d f g+d e h+c f h) d+\frac{15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt{e+f x} \sqrt{g+h x} \sqrt{a+b x}}{24 d^2 f^2 h^2 \sqrt{c+d x}}+\frac{(b e-a f) \sqrt{b g-a h} \left (-\left (24 A d^2 f^2 h^2+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2+6 a C d f h (c f h+2 d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\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 )}{24 b^2 d^2 f^3 h^3 \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)}}} \]
Antiderivative was successfully verified.
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Rule 1601
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} \left (A+C x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{C (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}+\frac{\int \frac{\sqrt{a+b x} \left (-3 b c C e g+6 a A d f h-a C (d e g+c f g+c e h)+2 (3 A b d f h-2 b C (d e g+c f g+c e h)-a C (d f g+d e h+c f h)) x+C (3 a d f h-5 b (d f g+d e h+c f h)) x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{6 d f h}\\ &=\frac{C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{12 d^2 f^2 h^2}+\frac{C (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}+\frac{\int \frac{-4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))-C (b c e g+a (d e g+c f g+c e h)) (3 a d f h-5 b (d f g+d e h+c f h))-2 \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right ) x+\left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right ) 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{\left (24 A b d^2 f h+\frac{3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac{15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{24 d^2 f^2 h^2 \sqrt{c+d x}}+\frac{C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{12 d^2 f^2 h^2}+\frac{C (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}+\frac{\int \frac{-(b d e g+a c f h) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )-2 b d f h (4 a d f h (3 b c C e g-6 a A d f h+a C (d e g+c f g+c e h))+C (b c e g+a (d e g+c f g+c e h)) (3 a d f h-5 b (d f g+d e h+c f h)))-\left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) 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{\left ((d e-c f) (d g-c h) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )\right ) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{48 b d^3 f^3 h^3}\\ &=\frac{\left (24 A b d^2 f h+\frac{3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac{15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{24 d^2 f^2 h^2 \sqrt{c+d x}}+\frac{C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{12 d^2 f^2 h^2}+\frac{C (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}+\frac{\left ((b e-a f) (b g-a h) \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{48 b^2 d^2 f^3 h^3}-\frac{\left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{48 b^2 d^3 f^3 h^3}-\frac{\left ((d g-c h) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\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 )}{24 b d^3 f^3 h^3 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}\\ &=\frac{\left (24 A b d^2 f h+\frac{3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac{15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{24 d^2 f^2 h^2 \sqrt{c+d x}}+\frac{C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{12 d^2 f^2 h^2}+\frac{C (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\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 )}{24 b d^3 f^3 h^3 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{\left (\left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) (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 )}{24 b^2 d^3 f^3 h^3 \sqrt{c+d x} \sqrt{e+f x}}+\frac{\left ((b e-a f) (b g-a h) \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\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 )}{24 b^2 d^2 f^3 h^3 (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 (24 A b d^2 f h+\frac{3 a^2 C d^2 f h}{b}-16 b C d (d e g+c f g+c e h)-22 a C d (d f g+d e h+c f h)+\frac{15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{24 d^2 f^2 h^2 \sqrt{c+d x}}+\frac{C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{12 d^2 f^2 h^2}+\frac{C (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\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 )}{24 b d^3 f^3 h^3 \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} \left (3 a^2 C d^2 f^2 h^2+6 a b C d f h (c f h+2 d (f g+e h))-b^2 \left (24 A d^2 f^2 h^2+C \left (5 c^2 f^2 h^2+4 c d f h (f g+e h)+d^2 \left (15 f^2 g^2+14 e f g h+15 e^2 h^2\right )\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 )}{24 b^2 d^2 f^3 h^3 \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} \left ((a d f h+b (d f g+d e h+c f h)) \left (24 A b^2 d^2 f^2 h^2+3 a^2 C d^2 f^2 h^2-16 b^2 C d f h (d e g+c f g+c e h)-22 a b C d f h (d f g+d e h+c f h)+15 b^2 C (d f g+d e h+c f h)^2\right )+4 b d f h \left (C (b (d e g+c f g+c e h)+a (d f g+d e h+c f h)) (3 a d f h-5 b (d f g+d e h+c f h))+2 d f h \left (3 b^2 c C e g+2 a^2 C (d f g+d e h+c f h)-a b (12 A d f h-5 C (d e g+c f g+c e h))\right )\right )\right ) (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 )}{24 b^2 d^3 \sqrt{b c-a d} f^3 h^4 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}
Mathematica [B] time = 25.9062, size = 38310, normalized size = 27.46 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.24, size = 89498, normalized size = 64.2 \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 (C x^{2} + 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 (C x^{2} + 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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