Optimal. Leaf size=629 \[ \frac{e \log \left (-\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right )}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \log \left (\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right )}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac{e \tanh ^{-1}\left (\frac{-2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \tanh ^{-1}\left (\frac{2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}} \]
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Rubi [A] time = 2.56405, antiderivative size = 629, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{e \log \left (-\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right )}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \log \left (\sqrt{d+e x} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}+\sqrt{c d^2-e (-a e+i b d)}+\sqrt{c} (d+e x)\right )}{2 \sqrt{c} \sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}+\frac{e \tanh ^{-1}\left (\frac{-2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}-\frac{e \tanh ^{-1}\left (\frac{2 \sqrt{c} \sqrt{d+e x}+\sqrt{2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}{\sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}}\right )}{\sqrt{c} \sqrt{-2 \sqrt{c} \sqrt{c d^2-e (-a e+i b d)}-i b e+2 c d}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]/(a + I*b*x + c*x^2),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)/(a+I*b*x+c*x**2),x)
[Out]
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Mathematica [A] time = 0.729125, size = 197, normalized size = 0.31 \[ \frac{\sqrt{2} \left (\sqrt{2 c d-e \left (\sqrt{-4 a c-b^2}+i b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{-4 a c-b^2}+i b\right )}}\right )-\sqrt{2 c d+e \left (\sqrt{-4 a c-b^2}-i b\right )} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{-4 a c-b^2}-i b e+2 c d}}\right )\right )}{\sqrt{c} \sqrt{-4 a c-b^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]/(a + I*b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.147, size = 610, normalized size = 1. \[{\frac{e}{2}\ln \left ( - \left ( ex+d \right ) \sqrt{c}+\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}\sqrt{ex+d}-\sqrt{c{d}^{2}-ibde+a{e}^{2}} \right ){\frac{1}{\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}}}{\frac{1}{\sqrt{c}}}}-{e\arctan \left ({1 \left ( -2\,\sqrt{c}\sqrt{ex+d}+\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd} \right ){\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{c{d}^{2}-ibde+a{e}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{c{d}^{2}-ibde+a{e}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}}-{\frac{e}{2}\ln \left ( \left ( ex+d \right ) \sqrt{c}+\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}\sqrt{ex+d}+\sqrt{c{d}^{2}-ibde+a{e}^{2}} \right ){\frac{1}{\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd}}}{\frac{1}{\sqrt{c}}}}+{e\arctan \left ({1 \left ( 2\,\sqrt{c}\sqrt{ex+d}+\sqrt{2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }-ibe+2\,cd} \right ){\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{c{d}^{2}-ibde+a{e}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}} \right ){\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{4\,\sqrt{c}\sqrt{c{d}^{2}-ibde+a{e}^{2}}-2\,\sqrt{-c \left ( ibde-a{e}^{2}-c{d}^{2} \right ) }+ibe-2\,cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)/(a+I*b*x+c*x^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{c x^{2} + i \, b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/(c*x^2 + I*b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247233, size = 990, normalized size = 1.57 \[ -\frac{1}{2} \, \sqrt{-\frac{4 \, c d - 2 i \, b e + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (\frac{{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt{-\frac{4 \, c d - 2 i \, b e + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e}{2 \, e}\right ) + \frac{1}{2} \, \sqrt{-\frac{4 \, c d - 2 i \, b e + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (-\frac{{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt{-\frac{4 \, c d - 2 i \, b e + 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} - 2 \, \sqrt{e x + d} e}{2 \, e}\right ) + \frac{1}{2} \, \sqrt{-\frac{4 \, c d - 2 i \, b e - 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (\frac{{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt{-\frac{4 \, c d - 2 i \, b e - 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} + 2 \, \sqrt{e x + d} e}{2 \, e}\right ) - \frac{1}{2} \, \sqrt{-\frac{4 \, c d - 2 i \, b e - 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} \log \left (-\frac{{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}} \sqrt{-\frac{4 \, c d - 2 i \, b e - 2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} \sqrt{-\frac{e^{2}}{b^{2} c^{2} + 4 \, a c^{3}}}}{b^{2} c + 4 \, a c^{2}}} - 2 \, \sqrt{e x + d} e}{2 \, e}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/(c*x^2 + I*b*x + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{a + i b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(1/2)/(a+I*b*x+c*x**2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/(c*x^2 + I*b*x + a),x, algorithm="giac")
[Out]