Optimal. Leaf size=616 \[ \frac{2 \sqrt{g+h x} (a C d-b B d+b c C) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right ),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right )}{\sqrt{c+d x} (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{2 b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 b d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{c+d x} (b c-a d) (b e-a f) (b g-a h)}-\frac{2 b \sqrt{a+b x} (b B-2 a C) \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{\sqrt{g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]
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Rubi [A] time = 1.17029, antiderivative size = 616, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 62, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {24, 1599, 1602, 12, 170, 419, 176, 424} \[ -\frac{2 b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{a+b x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 b d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{\sqrt{c+d x} (b c-a d) (b e-a f) (b g-a h)}+\frac{2 \sqrt{g+h x} (a C d-b B d+b c C) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{c+d x} (b c-a d) \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac{2 b \sqrt{a+b x} (b B-2 a C) \sqrt{d g-c h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{\sqrt{g+h x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}} \]
Antiderivative was successfully verified.
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Rule 24
Rule 1599
Rule 1602
Rule 12
Rule 170
Rule 419
Rule 176
Rule 424
Rubi steps
\begin{align*} \int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{\int \frac{b^2 (b B-a C)+b^3 C x}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{b^2}\\ &=-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt{a+b x}}+\frac{\int \frac{b^2 \left (b^2 C (b c e g-a (d e g+c f g+c e h))-a (b B-a C) (a d f h-b (d f g+d e h+c f h))\right )+b^3 (b B-2 a C) (a d f h+b (d f g+d e h+c f h)) x+2 b^4 (b B-2 a C) d f h x^2}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{b^2 (b c-a d) (b e-a f) (b g-a h)}\\ &=\frac{2 b (b B-2 a C) d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt{c+d x}}-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt{a+b x}}+\frac{\int \frac{2 b^3 d (b c C-b B d+a C d) f (b e-a f) h (b g-a h)}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{2 b^3 d (b c-a d) f (b e-a f) h (b g-a h)}+\frac{(b (b B-2 a C) (d e-c f) (d g-c h)) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{(b c-a d) (b e-a f) (b g-a h)}\\ &=\frac{2 b (b B-2 a C) d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt{c+d x}}-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt{a+b x}}+\frac{(b c C-b B d+a C d) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{b c-a d}-\frac{\left (2 b (b B-2 a C) (d g-c 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 )}{(b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}\\ &=\frac{2 b (b B-2 a C) d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt{c+d x}}-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt{a+b x}}-\frac{2 b (b B-2 a C) \sqrt{d g-c h} \sqrt{f g-e 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 )}{(b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{\left (2 (b c C-b B d+a C d) \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 )}{(b c-a d) (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{2 b (b B-2 a C) d \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt{c+d x}}-\frac{2 b^2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt{a+b x}}-\frac{2 b (b B-2 a C) \sqrt{d g-c h} \sqrt{f g-e 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 )}{(b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{2 (b c C-b B d+a C d) \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 c-a d) \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)}}}\\ \end{align*}
Mathematica [A] time = 11.9875, size = 340, normalized size = 0.55 \[ \frac{2 (e+f x)^{3/2} (g+h x)^{3/2} (b e-a f) \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \left ((b g-a h) (a C d-b B d+b c C) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}\right ),\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )+b (b B-2 a C) (d g-c h) E\left (\sin ^{-1}\left (\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )\right )}{(a+b x)^{5/2} \sqrt{c+d x} (b c-a d) (f g-e h)^3 \left (-\frac{(e+f x) (g+h x) (b e-a f) (b g-a h)}{(a+b x)^2 (f g-e h)^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.156, size = 9443, normalized size = 15.3 \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 b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\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 b x - C a + B b\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}{b^{2} d f h x^{5} + a^{2} c e g +{\left (b^{2} d f g +{\left (b^{2} d e +{\left (b^{2} c + 2 \, a b d\right )} f\right )} h\right )} x^{4} +{\left ({\left (b^{2} d e +{\left (b^{2} c + 2 \, a b d\right )} f\right )} g +{\left ({\left (b^{2} c + 2 \, a b d\right )} e +{\left (2 \, a b c + a^{2} d\right )} f\right )} h\right )} x^{3} +{\left ({\left ({\left (b^{2} c + 2 \, a b d\right )} e +{\left (2 \, a b c + a^{2} d\right )} f\right )} g +{\left (a^{2} c f +{\left (2 \, a b c + a^{2} d\right )} e\right )} h\right )} x^{2} +{\left (a^{2} c e h +{\left (a^{2} c f +{\left (2 \, a b c + a^{2} 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 b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\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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