Optimal. Leaf size=291 \[ -\frac{\sqrt{2} \left (2 c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\sqrt{2} \left (b e \left (\sqrt{b^2-4 a c}+b\right )-2 c \left (d \sqrt{b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+4 \sqrt{d+e x} \]
[Out]
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Rubi [A] time = 1.94434, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\sqrt{2} \left (2 c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\sqrt{2} \left (b e \left (\sqrt{b^2-4 a c}+b\right )-2 c \left (d \sqrt{b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+4 \sqrt{d+e x} \]
Antiderivative was successfully verified.
[In] Int[((b + 2*c*x)*Sqrt[d + e*x])/(a + 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((2*c*x+b)*(e*x+d)**(1/2)/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.401044, size = 288, normalized size = 0.99 \[ \frac{\sqrt{2} \left (c \left (4 a e-2 d \sqrt{b^2-4 a c}\right )+b e \left (\sqrt{b^2-4 a c}-b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{\sqrt{2} \left (b e \left (\sqrt{b^2-4 a c}+b\right )-2 c \left (d \sqrt{b^2-4 a c}+2 a e\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+4 \sqrt{d+e x} \]
Antiderivative was successfully verified.
[In] Integrate[((b + 2*c*x)*Sqrt[d + e*x])/(a + b*x + c*x^2),x]
[Out]
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Maple [B] time = 0.045, size = 724, normalized size = 2.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(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. \[ \int \frac{{\left (2 \, c x + b\right )} \sqrt{e x + d}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)*sqrt(e*x + d)/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286441, size = 497, normalized size = 1.71 \[ -\frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} \log \left (\sqrt{2} \sqrt{\frac{2 \, c d - b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} + 2 \, \sqrt{e x + d}\right ) + \frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} \log \left (-\sqrt{2} \sqrt{\frac{2 \, c d - b e + c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} + 2 \, \sqrt{e x + d}\right ) - \frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} \log \left (\sqrt{2} \sqrt{\frac{2 \, c d - b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} + 2 \, \sqrt{e x + d}\right ) + \frac{1}{2} \, \sqrt{2} \sqrt{\frac{2 \, c d - b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} \log \left (-\sqrt{2} \sqrt{\frac{2 \, c d - b e - c \sqrt{\frac{{\left (b^{2} - 4 \, a c\right )} e^{2}}{c^{2}}}}{c}} + 2 \, \sqrt{e x + d}\right ) + 4 \, \sqrt{e x + d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)*sqrt(e*x + d)/(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 142.52, size = 1431, normalized size = 4.92 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(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((2*c*x + b)*sqrt(e*x + d)/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]