Optimal. Leaf size=146 \[ \frac{3 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{5/2}} \]
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Rubi [A] time = 0.249161, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ \frac{3 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{5/2}} \]
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)^3,x]
[Out]
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Rubi in Sympy [A] time = 53.8161, size = 128, normalized size = 0.88 \[ \frac{3 e \sqrt{d + e x}}{4 \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{2}} + \frac{\sqrt{d + e x}}{2 \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )} + \frac{3 e^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 \sqrt{c} \sqrt{d} \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} \]
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)**3,x)
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Mathematica [A] time = 0.350775, size = 125, normalized size = 0.86 \[ \frac{1}{4} \left (\frac{\sqrt{d+e x} \left (5 a e^2+c d (3 e x-2 d)\right )}{\left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{5/2}}\right ) \]
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)^3,x]
[Out]
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Maple [A] time = 0.014, size = 144, normalized size = 1. \[{\frac{{e}^{2}}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ) \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]
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)^3,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)^3,x, algorithm="maxima")
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Fricas [A] time = 0.230514, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c^{2} d^{3} - a c d e^{2}}{\left (3 \, c d e x - 2 \, c d^{2} + 5 \, a e^{2}\right )} \sqrt{e x + d} + 3 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{c d x + a e}\right )}{8 \,{\left (a^{2} c^{2} d^{4} e^{2} - 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6} +{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x\right )} \sqrt{c^{2} d^{3} - a c d e^{2}}}, \frac{\sqrt{-c^{2} d^{3} + a c d e^{2}}{\left (3 \, c d e x - 2 \, c d^{2} + 5 \, a e^{2}\right )} \sqrt{e x + d} - 3 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \arctan \left (-\frac{c d^{2} - a e^{2}}{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}\right )}{4 \,{\left (a^{2} c^{2} d^{4} e^{2} - 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6} +{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x\right )} \sqrt{-c^{2} d^{3} + a c d e^{2}}}\right ] \]
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)^3,x, algorithm="fricas")
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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)**3,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)^3,x, algorithm="giac")
[Out]