Optimal. Leaf size=678 \[ -\frac{\sqrt{f} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right )}{b \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}+\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (3 a^2 A b d f h+a^3 (-B) d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}+\frac{\sqrt{f} \sqrt{g+h x} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}} \]
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Rubi [A] time = 1.53538, antiderivative size = 678, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1599, 1607, 169, 538, 537, 158, 114, 113, 121, 120} \[ \frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (3 a^2 A b d f h+a^3 (-B) d f h+a b^2 (B (c e h+c f g+d e g)-2 A (c f h+d e h+d f g))-b^3 (2 B c e g-A (c e h+c f g+d e g))\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (A b-a B)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}-\frac{\sqrt{f} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}+\frac{\sqrt{f} \sqrt{g+h x} (A b-a B) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}} \]
Antiderivative was successfully verified.
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Rule 1599
Rule 1607
Rule 169
Rule 538
Rule 537
Rule 158
Rule 114
Rule 113
Rule 121
Rule 120
Rubi steps
\begin{align*} \int \frac{A+B x}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=-\frac{b (A b-a B) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{\int \frac{-2 a^2 A d f h+b^2 (2 B c e g-A (d e g+c f g+c e h))-a b (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))+2 a (A b-a B) d f h x+b (A b-a B) d f h x^2}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac{b (A b-a B) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{\int \frac{a A d f h-\frac{a^2 B d f h}{b}+(A b d f h-a B d f h) x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}-\frac{\left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 b (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac{b (A b-a B) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}-\frac{((A b-a B) d f) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 b (b c-a d) (b e-a f)}+\frac{((A b-a B) d f) \int \frac{\sqrt{g+h x}}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h)}+\frac{\left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \operatorname{Subst}\left (\int \frac{1}{\left (b c-a d-b x^2\right ) \sqrt{e-\frac{c f}{d}+\frac{f x^2}{d}} \sqrt{g-\frac{c h}{d}+\frac{h x^2}{d}}} \, dx,x,\sqrt{c+d x}\right )}{b (b c-a d) (b e-a f) (b g-a h)}\\ &=-\frac{b (A b-a B) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}-\frac{\left ((A b-a B) d f \sqrt{\frac{d (e+f x)}{d e-c f}}\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}} \sqrt{g+h x}} \, dx}{2 b (b c-a d) (b e-a f) \sqrt{e+f x}}+\frac{\left (\left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \sqrt{\frac{d (e+f x)}{d e-c f}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (b c-a d-b x^2\right ) \sqrt{1+\frac{f x^2}{d \left (e-\frac{c f}{d}\right )}} \sqrt{g-\frac{c h}{d}+\frac{h x^2}{d}}} \, dx,x,\sqrt{c+d x}\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x}}+\frac{\left ((A b-a B) d f \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x}\right ) \int \frac{\sqrt{\frac{d g}{d g-c h}+\frac{d h x}{d g-c h}}}{\sqrt{c+d x} \sqrt{\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}}} \, dx}{2 (b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}\\ &=-\frac{b (A b-a B) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{(A b-a B) \sqrt{f} \sqrt{-d e+c f} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{\left ((A b-a B) d f \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}}\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}} \sqrt{\frac{d g}{d g-c h}+\frac{d h x}{d g-c h}}} \, dx}{2 b (b c-a d) (b e-a f) \sqrt{e+f x} \sqrt{g+h x}}+\frac{\left (\left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (b c-a d-b x^2\right ) \sqrt{1+\frac{f x^2}{d \left (e-\frac{c f}{d}\right )}} \sqrt{1+\frac{h x^2}{d \left (g-\frac{c h}{d}\right )}}} \, dx,x,\sqrt{c+d x}\right )}{b (b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{g+h x}}\\ &=-\frac{b (A b-a B) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) (a+b x)}+\frac{(A b-a B) \sqrt{f} \sqrt{-d e+c f} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{(A b-a B) \sqrt{f} \sqrt{-d e+c f} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d) (b e-a f) \sqrt{e+f x} \sqrt{g+h x}}+\frac{\sqrt{-d e+c f} \left (3 a^2 A b d f h-a^3 B d f h-b^3 (2 B c e g-A (d e g+c f g+c e h))+a b^2 (B (d e g+c f g+c e h)-2 A (d f g+d e h+c f h))\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{-d e+c f}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b (b c-a d)^2 \sqrt{f} (b e-a f) (b g-a h) \sqrt{e+f x} \sqrt{g+h x}}\\ \end{align*}
Mathematica [C] time = 16.3642, size = 14516, normalized size = 21.41 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.098, size = 13380, normalized size = 19.7 \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{B x + A}{{\left (b x + a\right )}^{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{B x + A}{{\left (b x + a\right )}^{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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