Optimal. Leaf size=134 \[ \frac{\sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{3/4}}-\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{3/4}} \]
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Rubi [A] time = 0.315421, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{3/4}}-\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]/(a - c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 48.9836, size = 117, normalized size = 0.87 \[ - \frac{\sqrt{\sqrt{a} e - \sqrt{c} d} \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{3}{4}}} + \frac{\sqrt{\sqrt{a} e + \sqrt{c} d} \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{\sqrt{a} c^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)/(-c*x**2+a),x)
[Out]
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Mathematica [A] time = 0.155931, size = 127, normalized size = 0.95 \[ \frac{\sqrt{\sqrt{a} \sqrt{c} e+c d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )-\sqrt{c d-\sqrt{a} \sqrt{c} e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} c} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]/(a - c*x^2),x]
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Maple [B] time = 0.029, size = 203, normalized size = 1.5 \[{ced{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{e{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{ced\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{e\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)/(-c*x^2+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{e x + d}}{c x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(e*x + d)/(c*x^2 - a),x, algorithm="maxima")
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Fricas [A] time = 0.226173, size = 463, normalized size = 3.46 \[ \frac{1}{2} \, \sqrt{\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} + d}{a c}} \log \left (a c^{2} \sqrt{\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} + d}{a c}} \sqrt{\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) - \frac{1}{2} \, \sqrt{\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} + d}{a c}} \log \left (-a c^{2} \sqrt{\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} + d}{a c}} \sqrt{\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) - \frac{1}{2} \, \sqrt{-\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} - d}{a c}} \log \left (a c^{2} \sqrt{-\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} - d}{a c}} \sqrt{\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{-\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} - d}{a c}} \log \left (-a c^{2} \sqrt{-\frac{a c \sqrt{\frac{e^{2}}{a c^{3}}} - d}{a c}} \sqrt{\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(e*x + d)/(c*x^2 - a),x, algorithm="fricas")
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Sympy [A] time = 14.6218, size = 76, normalized size = 0.57 \[ - 2 e \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(1/2)/(-c*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 77.037, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(e*x + d)/(c*x^2 - a),x, algorithm="giac")
[Out]