Optimal. Leaf size=36 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e} \]
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Rubi [A] time = 0.0222116, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e} \]
Antiderivative was successfully verified.
[In] Int[(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 = 2.56578, size = 36, normalized size = 1. \[ \frac{\left (2 d + 2 e x\right ) \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}}{12 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.00355501, size = 25, normalized size = 0.69 \[ \frac{(d+e x) \left (c (d+e x)^2\right )^{5/2}}{6 e} \]
Antiderivative was successfully verified.
[In] Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.005, size = 84, normalized size = 2.3 \[{\frac{x \left ({e}^{5}{x}^{5}+6\,d{e}^{4}{x}^{4}+15\,{d}^{2}{e}^{3}{x}^{3}+20\,{d}^{3}{e}^{2}{x}^{2}+15\,{d}^{4}ex+6\,{d}^{5} \right ) }{6\, \left ( ex+d \right ) ^{5}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),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((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 = 0.209603, size = 139, normalized size = 3.86 \[ \frac{{\left (c^{2} e^{5} x^{6} + 6 \, c^{2} d e^{4} x^{5} + 15 \, c^{2} d^{2} e^{3} x^{4} + 20 \, c^{2} d^{3} e^{2} x^{3} + 15 \, c^{2} d^{4} e x^{2} + 6 \, c^{2} d^{5} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{6 \,{\left (e x + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((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 = 0.221259, size = 120, normalized size = 3.33 \[ \frac{1}{6} \,{\left (c^{2} d^{5} e^{\left (-1\right )} +{\left (5 \, c^{2} d^{4} +{\left (10 \, c^{2} d^{3} e +{\left (10 \, c^{2} d^{2} e^{2} +{\left (c^{2} x e^{4} + 5 \, c^{2} d e^{3}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="giac")
[Out]