3.1065 \(\int \frac{1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=31 \[ -\frac{1}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

-1/(3*e*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))

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Rubi [A]  time = 0.0671008, 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 \[ -\frac{1}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)),x]

[Out]

-1/(3*e*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))

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Rubi in Sympy [A]  time = 18.5233, size = 31, normalized size = 1. \[ - \frac{1}{3 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

-1/(3*e*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(3/2))

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Mathematica [A]  time = 0.0437196, size = 20, normalized size = 0.65 \[ -\frac{1}{3 e \left (c (d+e x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)),x]

[Out]

-1/(3*e*(c*(d + e*x)^2)^(3/2))

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Maple [A]  time = 0.003, size = 28, normalized size = 0.9 \[ -{\frac{1}{3\,e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x)

[Out]

-1/3/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)

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Maxima [A]  time = 0.679801, size = 63, normalized size = 2.03 \[ -\frac{1}{3 \,{\left (c^{\frac{3}{2}} e^{4} x^{3} + 3 \, c^{\frac{3}{2}} d e^{3} x^{2} + 3 \, c^{\frac{3}{2}} d^{2} e^{2} x + c^{\frac{3}{2}} d^{3} e\right )}} \]

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)),x, algorithm="maxima")

[Out]

-1/3/(c^(3/2)*e^4*x^3 + 3*c^(3/2)*d*e^3*x^2 + 3*c^(3/2)*d^2*e^2*x + c^(3/2)*d^3*
e)

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Fricas [A]  time = 0.213647, size = 112, normalized size = 3.61 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \,{\left (c^{2} e^{5} x^{4} + 4 \, c^{2} d e^{4} x^{3} + 6 \, c^{2} d^{2} e^{3} x^{2} + 4 \, c^{2} d^{3} e^{2} x + c^{2} d^{4} e\right )}} \]

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)),x, algorithm="fricas")

[Out]

-1/3*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^2*e^5*x^4 + 4*c^2*d*e^4*x^3 + 6*c^2*
d^2*e^3*x^2 + 4*c^2*d^3*e^2*x + c^2*d^4*e)

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Sympy [A]  time = 5.05291, size = 42, normalized size = 1.35 \[ \begin{cases} - \frac{1}{3 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} & \text{for}\: e \neq 0 \\\frac{x}{d \left (c d^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

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)**(3/2),x)

[Out]

Piecewise((-1/(3*e*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(3/2)), Ne(e, 0)), (x/(d*
(c*d**2)**(3/2)), True))

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GIAC/XCAS [A]  time = 0.625464, size = 4, normalized size = 0.13 \[ \mathit{sage}_{0} x \]

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)),x, algorithm="giac")

[Out]

sage0*x