Optimal. Leaf size=284 \[ \frac{2 b \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 (b g-a h) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}} \]
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Rubi [A] time = 1.03449, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ \frac{2 b \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 (b g-a h) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
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Rubi in Sympy [A] time = 116.765, size = 241, normalized size = 0.85 \[ \frac{2 b \sqrt{\frac{d \left (- e - f x\right )}{c f - d e}} \sqrt{g + h x} \sqrt{c f - d e} E\left (\operatorname{asin}{\left (\frac{\sqrt{f} \sqrt{c + d x}}{\sqrt{c f - d e}} \right )}\middle | \frac{h \left (c f - d e\right )}{f \left (c h - d g\right )}\right )}{d \sqrt{f} h \sqrt{\frac{d \left (- g - h x\right )}{c h - d g}} \sqrt{e + f x}} + \frac{2 \sqrt{\frac{d \left (- e - f x\right )}{c f - d e}} \sqrt{\frac{d \left (- g - h x\right )}{c h - d g}} \left (a h - b g\right ) \sqrt{c f - d e} F\left (\operatorname{asin}{\left (\frac{\sqrt{f} \sqrt{c + d x}}{\sqrt{c f - d e}} \right )}\middle | \frac{h \left (c f - d e\right )}{f \left (c h - d g\right )}\right )}{d \sqrt{f} h \sqrt{e + f x} \sqrt{g + h x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
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Mathematica [C] time = 3.03145, size = 319, normalized size = 1.12 \[ -\frac{2 \left (i d h (c+d x)^{3/2} (b e-a f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )-b d^2 (e+f x) (g+h x) \sqrt{\frac{d e}{f}-c}-i b h (c+d x)^{3/2} (d e-c f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )\right )}{d^2 f h \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \sqrt{\frac{d e}{f}-c}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
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Maple [B] time = 0.058, size = 559, normalized size = 2. \[ 2\,{\frac{\sqrt{dx+c}\sqrt{fx+e}\sqrt{hx+g}}{{d}^{2}fh \left ( dfh{x}^{3}+cfh{x}^{2}+deh{x}^{2}+dfg{x}^{2}+cehx+cfgx+degx+ceg \right ) } \left ({\it EllipticF} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) acdfh-{\it EllipticF} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) a{d}^{2}eh-{\it EllipticF} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcdfg+{\it EllipticF} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{d}^{2}eg-{\it EllipticE} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{c}^{2}fh+{\it EllipticE} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcdeh+{\it EllipticE} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) bcdfg-{\it EllipticE} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) b{d}^{2}eg \right ) \sqrt{-{\frac{d \left ( fx+e \right ) }{cf-de}}}\sqrt{-{\frac{d \left ( hx+g \right ) }{ch-dg}}}\sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b x + a}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")
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