Optimal. Leaf size=198 \[ \frac{\sqrt{2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}} \]
[Out]
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Rubi [A] time = 0.677084, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\sqrt{2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]/(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 82.6199, size = 185, normalized size = 0.93 \[ - \frac{\sqrt{2} \sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d + e x}}{\sqrt{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{c} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} \sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d + e x}}{\sqrt{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{c} \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.750661, size = 175, normalized size = 0.88 \[ \frac{\sqrt{2} \left (\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )-\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )\right )}{\sqrt{c} \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2),x]
[Out]
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Maple [B] time = 0.032, size = 545, normalized size = 2.8 \[{{e}^{2}\sqrt{2}b\arctan \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }}}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}-2\,{\frac{ce\sqrt{2}d}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }+{e\sqrt{2}\arctan \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}+{{e}^{2}\sqrt{2}b{\it Artanh} \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }}}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}}-2\,{\frac{ce\sqrt{2}d}{\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }-{e\sqrt{2}{\it Artanh} \left ({c\sqrt{2}\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)/(c*x^2+b*x+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.294236, size = 965, normalized size = 4.87 \[ -\frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e +{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\sqrt{2}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt{\frac{2 \, c d - b e +{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e +{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\sqrt{2}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt{\frac{2 \, c d - b e +{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e -{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\sqrt{2}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt{\frac{2 \, c d - b e -{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e\right ) - \frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e -{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\sqrt{2}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt{\frac{2 \, c d - b e -{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 21.9676, size = 155, normalized size = 0.78 \[ 2 e \operatorname{RootSum}{\left (t^{4} \left (256 a^{2} c^{3} e^{4} - 128 a b^{2} c^{2} e^{4} + 16 b^{4} c e^{4}\right ) + t^{2} \left (- 16 a b c e^{3} + 32 a c^{2} d e^{2} + 4 b^{3} e^{3} - 8 b^{2} c d e^{2}\right ) + a e^{2} - b d e + c d^{2}, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} e^{2} - 16 t^{3} b^{2} c e^{2} - 2 t b e + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(1/2)/(c*x**2+b*x+a),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(e*x + d)/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]