Optimal. Leaf size=444 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 e} \]
[Out]
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Rubi [A] time = 1.03496, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 e} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 166.304, size = 418, normalized size = 0.94 \[ \frac{2 \sqrt{d + e x} \sqrt{a + b x + c x^{2}}}{3 e} + \frac{4 \sqrt{2} \sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{- 4 a c + b^{2}} \left (a e^{2} - b d e + c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{3 c e^{2} \sqrt{d + e x} \sqrt{a + b x + c x^{2}}} + \frac{\sqrt{2} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{d + e x} \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{3 c e^{2} \sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**(1/2),x)
[Out]
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Mathematica [C] time = 12.5552, size = 591, normalized size = 1.33 \[ \frac{2 \left (\frac{i (d+e x) \sqrt{1-\frac{2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt{\frac{2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}+1} \left (\left (-2 c d \sqrt{e^2 \left (b^2-4 a c\right )}+b e \sqrt{e^2 \left (b^2-4 a c\right )}+4 a c e^2-b^2 e^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )-(b e-2 c d) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt{2} c \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}+e^2 \sqrt{d+e x} (a+x (b+c x))+\frac{e^2 (a+x (b+c x)) (b e-2 c d)}{c \sqrt{d+e x}}\right )}{3 e^3 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x],x]
[Out]
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Maple [B] time = 0.048, size = 1854, normalized size = 4.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(1/2)/(e*x+d)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}}{\sqrt{e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}}{\sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x + c x^{2}}}{\sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/sqrt(e*x + d),x, algorithm="giac")
[Out]