Optimal. Leaf size=94 \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x}}{c d (a e+c d x)} \]
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Rubi [A] time = 0.155438, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x}}{c d (a e+c d x)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)/(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 = 40.4508, size = 78, normalized size = 0.83 \[ - \frac{\sqrt{d + e x}}{c d \left (a e + c d x\right )} + \frac{e \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{3}{2}} d^{\frac{3}{2}} \sqrt{a e^{2} - c d^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)/(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 = 0.158628, size = 94, normalized size = 1. \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x}}{c d (a e+c d x)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)/(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 = 0.017, size = 84, normalized size = 0.9 \[ -{\frac{e}{cd \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+{\frac{e}{cd}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)/(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((e*x + d)^(5/2)/(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 = 0.224263, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (c d e x + a e^{2}\right )} \log \left (\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{c d x + a e}\right ) - 2 \, \sqrt{c^{2} d^{3} - a c d e^{2}} \sqrt{e x + d}}{2 \, \sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c^{2} d^{2} x + a c d e\right )}}, -\frac{{\left (c d e x + a e^{2}\right )} \arctan \left (-\frac{c d^{2} - a e^{2}}{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}\right ) + \sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}{\sqrt{-c^{2} d^{3} + a c d e^{2}}{\left (c^{2} d^{2} x + a c d e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(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)**(5/2)/(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((e*x + d)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")
[Out]