Optimal. Leaf size=38 \[ -\frac{1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0661968, antiderivative size = 38, 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 \[ -\frac{1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 17.8803, size = 36, normalized size = 0.95 \[ - \frac{1}{4 e \left (d + e x\right ) \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**2/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.039874, size = 26, normalized size = 0.68 \[ -\frac{c (d+e x)}{4 e \left (c (d+e x)^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.006, size = 35, normalized size = 0.9 \[ -{\frac{1}{4\,e \left ( ex+d \right ) } \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.683406, size = 82, normalized size = 2.16 \[ -\frac{1}{4 \,{\left (c^{\frac{3}{2}} e^{5} x^{4} + 4 \, c^{\frac{3}{2}} d e^{4} x^{3} + 6 \, c^{\frac{3}{2}} d^{2} e^{3} x^{2} + 4 \, c^{\frac{3}{2}} d^{3} e^{2} x + c^{\frac{3}{2}} d^{4} e\right )}} \]
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)^(3/2)*(e*x + d)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.2151, size = 131, normalized size = 3.45 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (c^{2} e^{6} x^{5} + 5 \, c^{2} d e^{5} x^{4} + 10 \, c^{2} d^{2} e^{4} x^{3} + 10 \, c^{2} d^{3} e^{3} x^{2} + 5 \, c^{2} d^{4} e^{2} x + c^{2} d^{5} e\right )}} \]
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)^(3/2)*(e*x + d)^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**2/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.664846, size = 4, normalized size = 0.11 \[ \mathit{sage}_{0} x \]
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)^(3/2)*(e*x + d)^2),x, algorithm="giac")
[Out]