Optimal. Leaf size=204 \[ \frac{2 \sqrt{f} \sqrt{a+b x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|-\frac{b (d e-c f)}{(b c-a d) f}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) \sqrt{-\frac{d (a+b x)}{b c-a d}}}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d) (b e-a f)} \]
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Rubi [A] time = 0.591382, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 \sqrt{f} \sqrt{a+b x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|-\frac{b (d e-c f)}{(b c-a d) f}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) \sqrt{-\frac{d (a+b x)}{b c-a d}}}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
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Rubi in Sympy [A] time = 74.3414, size = 167, normalized size = 0.82 \[ - \frac{2 b \sqrt{c + d x} \sqrt{e + f x}}{\sqrt{a + b x} \left (a d - b c\right ) \left (a f - b e\right )} + \frac{2 \sqrt{f} \sqrt{\frac{d \left (- e - f x\right )}{c f - d e}} \sqrt{a + b x} \sqrt{c f - d e} E\left (\operatorname{asin}{\left (\frac{\sqrt{f} \sqrt{c + d x}}{\sqrt{c f - d e}} \right )}\middle | \frac{b \left (- c f + d e\right )}{f \left (a d - b c\right )}\right )}{\sqrt{\frac{d \left (a + b x\right )}{a d - b c}} \sqrt{e + f x} \left (a d - b c\right ) \left (a f - b e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
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Mathematica [C] time = 1.75697, size = 201, normalized size = 0.99 \[ \frac{2 b \sqrt{c+d x} \sqrt{e+f x} \left (-1-\frac{i \sqrt{\frac{d (a+b x)}{b (c+d x)}} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{d (a+b x)}{b c-a d}}\right )|\frac{b c f-a d f}{b d e-a d f}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{d (a+b x)}{b c-a d}}\right )|\frac{b c f-a d f}{b d e-a d f}\right )\right )}{\sqrt{\frac{b (e+f x)}{b e-a f}}}\right )}{\sqrt{a+b x} (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
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Maple [B] time = 0.049, size = 1011, normalized size = 5. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \sqrt{e + f x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="giac")
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