Optimal. Leaf size=128 \[ -\frac{3 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}-\frac{1}{\sqrt{d+e x} \left (c d^2-a e^2\right ) (a e+c d x)}+\frac{3 \sqrt{c} \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]
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Rubi [A] time = 0.216524, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{3 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}-\frac{1}{\sqrt{d+e x} \left (c d^2-a e^2\right ) (a e+c d x)}+\frac{3 \sqrt{c} \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 50.8267, size = 112, normalized size = 0.88 \[ - \frac{3 \sqrt{c} \sqrt{d} e \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} - \frac{3 e}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{\sqrt{d + e x} \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.269767, size = 118, normalized size = 0.92 \[ \frac{3 \sqrt{c} \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}-\frac{2 a e^2+c d (d+3 e x)}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2 (a e+c d x)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
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Maple [A] time = 0.026, size = 129, normalized size = 1. \[ -2\,{\frac{e}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ex+d}}}-{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-3\,{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
[Out]
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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(e*x + d)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227678, size = 1, normalized size = 0.01 \[ \left [-\frac{6 \, c d e x + 2 \, c d^{2} + 4 \, a e^{2} - 3 \,{\left (c d e x + a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} + 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right )}{2 \,{\left (a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} +{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x\right )} \sqrt{e x + d}}, -\frac{3 \, c d e x + c d^{2} + 2 \, a e^{2} - 3 \,{\left (c d e x + a e^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{\sqrt{e x + d} c d}\right )}{{\left (a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} +{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x\right )} \sqrt{e x + d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,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*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")
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