Optimal. Leaf size=39 \[ -\frac{c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0675535, antiderivative size = 39, 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 \[ -\frac{c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^4*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 17.9618, size = 37, normalized size = 0.95 \[ - \frac{1}{4 e \left (d + e x\right )^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0206706, size = 27, normalized size = 0.69 \[ -\frac{1}{4 e (d+e x)^3 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^4*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]),x]
[Out]
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Maple [A] time = 0.005, size = 35, normalized size = 0.9 \[ -{\frac{1}{4\, \left ( ex+d \right ) ^{3}e}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.68527, size = 82, normalized size = 2.1 \[ -\frac{1}{4 \,{\left (\sqrt{c} e^{5} x^{4} + 4 \, \sqrt{c} d e^{4} x^{3} + 6 \, \sqrt{c} d^{2} e^{3} x^{2} + 4 \, \sqrt{c} d^{3} e^{2} x + \sqrt{c} d^{4} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216903, size = 115, normalized size = 2.95 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (c e^{6} x^{5} + 5 \, c d e^{5} x^{4} + 10 \, c d^{2} e^{4} x^{3} + 10 \, c d^{3} e^{3} x^{2} + 5 \, c d^{4} e^{2} x + c d^{5} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c \left (d + e x\right )^{2}} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^4),x, algorithm="giac")
[Out]