Optimal. Leaf size=31 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^3 e} \]
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Rubi [A] time = 0.0696238, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^3 e} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 18.5959, size = 29, normalized size = 0.94 \[ \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**5/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.00861778, size = 23, normalized size = 0.74 \[ \frac{x (d+e x)}{c^2 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.002, size = 32, normalized size = 1. \[{ \left ( ex+d \right ) ^{5}x \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x)
[Out]
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Maxima [A] time = 0.699544, size = 266, normalized size = 8.58 \[ \frac{e^{3} x^{4}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} - \frac{4 \, c^{2} d^{5} e^{4}}{\left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} - \frac{6 \, d^{2} e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} + \frac{32 \, c d^{4} e^{3}}{3 \, \left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} - \frac{17 \, d^{4}}{3 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e} - \frac{8 \, d^{3} e^{2}}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}} + \frac{4 \, d^{5}}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210172, size = 51, normalized size = 1.65 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c^{3} e x + c^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.70877, size = 42, normalized size = 1.35 \[ \begin{cases} \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{3} e} & \text{for}\: e \neq 0 \\\frac{d^{5} x}{\left (c d^{2}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**5/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.274326, size = 126, normalized size = 4.06 \[ \frac{2 \, C_{0} d^{3} e^{\left (-3\right )} - \frac{3 \, d^{4} e^{\left (-1\right )}}{c} +{\left (6 \, C_{0} d^{2} e^{\left (-2\right )} - \frac{8 \, d^{3}}{c} +{\left (6 \, C_{0} d e^{\left (-1\right )} +{\left (2 \, C_{0} + \frac{x e^{3}}{c}\right )} x - \frac{6 \, d^{2} e}{c}\right )} x\right )} x}{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="giac")
[Out]