Optimal. Leaf size=188 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]
[Out]
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Rubi [A] time = 0.274175, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]/Sqrt[a + b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 44.2403, size = 177, normalized size = 0.94 \[ \frac{\sqrt{2} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{d + e x} \sqrt{- 4 a c + b^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{c \sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)/(c*x**2+b*x+a)**(1/2),x)
[Out]
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Mathematica [C] time = 1.43931, size = 365, normalized size = 1.94 \[ \frac{i \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right ) \sqrt{\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}} \sqrt{1-\frac{2 c (d+e x)}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}} \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d}} \sqrt{d+e x}\right )|\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d}} \sqrt{d+e x}\right )|\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )\right )}{\sqrt{2} c e \sqrt{a+x (b+c x)} \sqrt{\frac{c}{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]/Sqrt[a + b*x + c*x^2],x]
[Out]
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Maple [B] time = 0.047, size = 747, normalized size = 4. \[{\frac{\sqrt{2}}{2\,e \left ( ce{x}^{3}+be{x}^{2}+cd{x}^{2}+aex+bdx+ad \right ){c}^{2}}\sqrt{ex+d}\sqrt{c{x}^{2}+bx+a} \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \sqrt{-{ \left ( ex+d \right ) c \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}}\sqrt{{e \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( 2\,cd-be+e\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}}\sqrt{{e \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}} \left ({\it EllipticF} \left ( \sqrt{2}\sqrt{-{ \left ( ex+d \right ) c \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}},\sqrt{-{1 \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( 2\,cd-be+e\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) eb-2\,d{\it EllipticF} \left ( \sqrt{2}\sqrt{-{\frac{ \left ( ex+d \right ) c}{e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd}}},\sqrt{-{\frac{e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd}{2\,cd-be+e\sqrt{-4\,ac+{b}^{2}}}}} \right ) c-{\it EllipticF} \left ( \sqrt{2}\sqrt{-{ \left ( ex+d \right ) c \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}},\sqrt{-{1 \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( 2\,cd-be+e\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) e\sqrt{-4\,ac+{b}^{2}}-{\it EllipticE} \left ( \sqrt{2}\sqrt{-{ \left ( ex+d \right ) c \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}},\sqrt{-{1 \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( 2\,cd-be+e\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) be+2\,{\it EllipticE} \left ( \sqrt{2}\sqrt{-{\frac{ \left ( ex+d \right ) c}{e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd}}},\sqrt{-{\frac{e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd}{2\,cd-be+e\sqrt{-4\,ac+{b}^{2}}}}} \right ) cd+{\it EllipticE} \left ( \sqrt{2}\sqrt{-{ \left ( ex+d \right ) c \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}},\sqrt{-{1 \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( 2\,cd-be+e\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) \sqrt{-4\,ac+{b}^{2}}e \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{\sqrt{a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(1/2)/(c*x**2+b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/sqrt(c*x^2 + b*x + a),x, algorithm="giac")
[Out]