Optimal. Leaf size=48 \[ -\frac{2 (d+e x)^{3/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0783846, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026 \[ -\frac{2 (d+e x)^{3/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 17.1524, size = 44, normalized size = 0.92 \[ - \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]
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)**(5/2),x)
[Out]
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Mathematica [A] time = 0.051914, size = 37, normalized size = 0.77 \[ -\frac{2 (d+e x)^{3/2}}{3 c d ((d+e x) (a e+c d x))^{3/2}} \]
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)^(5/2),x]
[Out]
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Maple [A] time = 0.003, size = 50, normalized size = 1. \[ -{\frac{2\,cdx+2\,ae}{3\,cd} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{-{\frac{5}{2}}}} \]
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)^(5/2),x)
[Out]
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Maxima [A] time = 0.829463, size = 38, normalized size = 0.79 \[ -\frac{2}{3 \,{\left (c^{2} d^{2} x + a c d e\right )} \sqrt{c d x + a e}} \]
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)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210471, size = 144, normalized size = 3. \[ -\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{3 \,{\left (c^{3} d^{3} e x^{3} + a^{2} c d^{2} e^{2} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} x^{2} +{\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x\right )}} \]
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)^(5/2),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)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.664421, size = 4, normalized size = 0.08 \[ \mathit{sage}_{0} x \]
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)^(5/2),x, algorithm="giac")
[Out]