Optimal. Leaf size=465 \[ -\frac{2 \left (\frac{a^2 C}{b^2}+A\right ) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)}-\frac{2 C (a h+b g) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 C \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}} \]
[Out]
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Rubi [A] time = 3.02211, antiderivative size = 465, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \left (\frac{a^2 C}{b^2}+A\right ) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)}-\frac{2 C (a h+b g) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt{f} h \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 C \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt{f} h \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}} \]
Antiderivative was successfully verified.
[In] Int[(A + C*x^2)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((C*x**2+A)/(b*x+a)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
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Mathematica [C] time = 16.3669, size = 13075, normalized size = 28.12 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(A + C*x^2)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
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Maple [B] time = 0.049, size = 1368, normalized size = 2.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*x^2+A)/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{C x^{2} + A}{{\left (b x + a\right )} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + A)/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + A)/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + C x^{2}}{\left (a + b x\right ) \sqrt{c + d x} \sqrt{e + f x} \sqrt{g + h x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x**2+A)/(b*x+a)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{C x^{2} + A}{{\left (b x + a\right )} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + A)/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")
[Out]