Optimal. Leaf size=31 \[ -\frac{1}{5 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0673158, 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 \[ -\frac{1}{5 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]
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Rubi in Sympy [A] time = 18.3687, size = 31, normalized size = 1. \[ - \frac{1}{5 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0518667, size = 20, normalized size = 0.65 \[ -\frac{1}{5 e \left (c (d+e x)^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.007, size = 28, normalized size = 0.9 \[ -{\frac{1}{5\,e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x)
[Out]
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Maxima [A] time = 0.681326, size = 101, normalized size = 3.26 \[ -\frac{1}{5 \,{\left (c^{\frac{5}{2}} e^{6} x^{5} + 5 \, c^{\frac{5}{2}} d e^{5} x^{4} + 10 \, c^{\frac{5}{2}} d^{2} e^{4} x^{3} + 10 \, c^{\frac{5}{2}} d^{3} e^{3} x^{2} + 5 \, c^{\frac{5}{2}} d^{4} e^{2} x + c^{\frac{5}{2}} d^{5} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)*(e*x + d)),x, algorithm="maxima")
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Fricas [A] time = 0.228652, size = 150, normalized size = 4.84 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{5 \,{\left (c^{3} e^{7} x^{6} + 6 \, c^{3} d e^{6} x^{5} + 15 \, c^{3} d^{2} e^{5} x^{4} + 20 \, c^{3} d^{3} e^{4} x^{3} + 15 \, c^{3} d^{4} e^{3} x^{2} + 6 \, c^{3} d^{5} e^{2} x + c^{3} d^{6} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.46354, size = 42, normalized size = 1.35 \[ \begin{cases} - \frac{1}{5 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}} & \text{for}\: e \neq 0 \\\frac{x}{d \left (c d^{2}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)*(e*x + d)),x, algorithm="giac")
[Out]