Optimal. Leaf size=184 \[ \frac{2 \sqrt{f} \sqrt{c+d x} \sqrt{a f-b e} \sqrt{\frac{b (e+f x)}{b e-a f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x}}{\sqrt{a f-b e}}\right )|\frac{d (b e-a f)}{(b c-a d) f}\right )}{b \sqrt{e+f x} (b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d)} \]
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Rubi [A] time = 0.53936, antiderivative size = 184, 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 \[ \frac{2 \sqrt{f} \sqrt{c+d x} \sqrt{a f-b e} \sqrt{\frac{b (e+f x)}{b e-a f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x}}{\sqrt{a f-b e}}\right )|\frac{d (b e-a f)}{(b c-a d) f}\right )}{b \sqrt{e+f x} (b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[e + f*x]/((a + b*x)^(3/2)*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 59.0382, size = 150, normalized size = 0.82 \[ \frac{2 \sqrt{c + d x} \sqrt{e + f x}}{\sqrt{a + b x} \left (a d - b c\right )} - \frac{2 \sqrt{\frac{f \left (a + b x\right )}{a f - b e}} \sqrt{c + d x} \sqrt{- a f + b e} E\left (\operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{e + f x}}{\sqrt{- a f + b e}} \right )}\middle | \frac{d \left (a f - b e\right )}{b \left (c f - d e\right )}\right )}{\sqrt{b} \sqrt{\frac{f \left (c + d x\right )}{c f - d e}} \sqrt{a + b x} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x+e)**(1/2)/(b*x+a)**(3/2)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 1.29133, size = 126, normalized size = 0.68 \[ -\frac{2 \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{d (a+b x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{a-\frac{b c}{d}}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )}{b \sqrt{c+d x} \sqrt{a-\frac{b c}{d}} \sqrt{\frac{b (e+f x)}{f (a+b x)}}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[e + f*x]/((a + b*x)^(3/2)*Sqrt[c + d*x]),x]
[Out]
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Maple [B] time = 0.082, size = 1022, normalized size = 5.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x+e)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(f*x + e)/((b*x + a)^(3/2)*sqrt(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(f*x + e)/((b*x + a)^(3/2)*sqrt(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e + f x}}{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x+e)**(1/2)/(b*x+a)**(3/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(f*x + e)/((b*x + a)^(3/2)*sqrt(d*x + c)),x, algorithm="giac")
[Out]