Optimal. Leaf size=309 \[ \frac{2 C \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right )}{d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 (b B-2 a C) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)} \]
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Rubi [A] time = 0.82074, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 60, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {24, 1607, 169, 538, 537, 12, 121, 120} \[ \frac{2 C \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 (b B-2 a C) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 24
Rule 1607
Rule 169
Rule 538
Rule 537
Rule 12
Rule 121
Rule 120
Rubi steps
\begin{align*} \int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^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) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{b^2}\\ &=\frac{\int \frac{b^2 C}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{b^2}+(b B-2 a C) \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\\ &=C \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx-(2 (b B-2 a C)) \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 )\\ &=\frac{\left (C \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}{\sqrt{e+f x}}-\frac{\left (2 (b B-2 a C) \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 )}{\sqrt{e+f x}}\\ &=\frac{\left (C \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}{\sqrt{e+f x} \sqrt{g+h x}}-\frac{\left (2 (b B-2 a C) \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 )}{\sqrt{e+f x} \sqrt{g+h x}}\\ &=\frac{2 C \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 )}{d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 (b B-2 a C) \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}} \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 c-a d) \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}\\ \end{align*}
Mathematica [C] time = 1.75386, size = 248, normalized size = 0.8 \[ \frac{2 i \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \left (d (2 a C-b B) \Pi \left (\frac{(b c-a d) f}{b (c f-d e)};i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )-(a C d-b B d+b c C) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right ),\frac{d f g-c f h}{d e h-c f h}\right )\right )}{f \sqrt{g+h x} (a d-b c) \sqrt{\frac{d e}{f}-c} \sqrt{\frac{d (e+f x)}{f (c+d x)}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 663, normalized size = 2.2 \begin{align*} 2\,{\frac{\sqrt{hx+g}\sqrt{fx+e}\sqrt{dx+c}}{ \left ( ad-bc \right ) df \left ( dfh{x}^{3}+cfh{x}^{2}+deh{x}^{2}+dfg{x}^{2}+cehx+cfgx+degx+ceg \right ) }\sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}}\sqrt{-{\frac{ \left ( hx+g \right ) d}{ch-dg}}}\sqrt{-{\frac{ \left ( fx+e \right ) d}{cf-de}}} \left ( B{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcdf-B{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{d}^{2}e+C{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) acdf-C{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) a{d}^{2}e-C{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{c}^{2}f+C{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcde-2\,C{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) acdf+2\,C{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) a{d}^{2}e \right ) } \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 )}^{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{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\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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