Optimal. Leaf size=83 \[ \frac{\sqrt{a+b x+c x^2}}{2 c d}-\frac{\sqrt{b^2-4 a c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{4 c^{3/2} d} \]
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Rubi [A] time = 0.166222, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{a+b x+c x^2}}{2 c d}-\frac{\sqrt{b^2-4 a c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{4 c^{3/2} d} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x),x]
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Rubi in Sympy [A] time = 35.8945, size = 71, normalized size = 0.86 \[ \frac{\sqrt{a + b x + c x^{2}}}{2 c d} - \frac{\sqrt{- 4 a c + b^{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{4 c^{\frac{3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d),x)
[Out]
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Mathematica [A] time = 0.225447, size = 137, normalized size = 1.65 \[ \frac{\left (b^2-4 a c\right ) \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )+2 \sqrt{c} \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \log (b+2 c x)}{4 c^{3/2} d \sqrt{4 a c-b^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x),x]
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Maple [B] time = 0.018, size = 244, normalized size = 2.9 \[{\frac{1}{4\,cd}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}}-{\frac{a}{cd}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}}+{\frac{{b}^{2}}{4\,{c}^{2}d}\ln \left ({1 \left ({\frac{4\,ac-{b}^{2}}{2\,c}}+{\frac{1}{2}\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}}} \right ) \left ( x+{\frac{b}{2\,c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d),x, algorithm="maxima")
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Fricas [A] time = 0.24789, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{-\frac{b^{2} - 4 \, a c}{c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a} c \sqrt{-\frac{b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, \sqrt{c x^{2} + b x + a}}{8 \, c d}, -\frac{\sqrt{\frac{b^{2} - 4 \, a c}{c}} \arctan \left (-\frac{b^{2} - 4 \, a c}{2 \, \sqrt{c x^{2} + b x + a} c \sqrt{\frac{b^{2} - 4 \, a c}{c}}}\right ) - 2 \, \sqrt{c x^{2} + b x + a}}{4 \, c d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{\sqrt{a + b x + c x^{2}}}{b + 2 c x}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d),x)
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GIAC/XCAS [A] time = 0.223106, size = 131, normalized size = 1.58 \[ -\frac{{\left (b^{2} - 4 \, a c\right )} \arctan \left (-\frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c + b \sqrt{c}}{\sqrt{b^{2} c - 4 \, a c^{2}}}\right )}{2 \, \sqrt{b^{2} c - 4 \, a c^{2}} c d} + \frac{\sqrt{c x^{2} + b x + a}}{2 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d),x, algorithm="giac")
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