3.1071 \(\int \frac{(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{1}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

-(1/(c^2*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]))

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Rubi [A]  time = 0.0700318, antiderivative size = 32, 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 \[ -\frac{1}{c^2 e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

-(1/(c^2*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]))

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Rubi in Sympy [A]  time = 18.4883, size = 32, normalized size = 1. \[ - \frac{1}{c^{2} e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0293885, size = 21, normalized size = 0.66 \[ -\frac{1}{c^2 e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

-(1/(c^2*e*Sqrt[c*(d + e*x)^2]))

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Maple [A]  time = 0.004, size = 35, normalized size = 1.1 \[ -{\frac{ \left ( ex+d \right ) ^{4}}{e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x)

[Out]

-(e*x+d)^4/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)

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Maxima [A]  time = 0.690829, size = 215, normalized size = 6.72 \[ -\frac{c^{2} d^{3} e^{4}}{\left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} - \frac{e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} + \frac{8 \, c d^{2} e^{3}}{3 \, \left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} - \frac{5 \, d^{2}}{3 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e} - \frac{2 \, d e^{2}}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}} + \frac{d^{3}}{\left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="maxima")

[Out]

-c^2*d^3*e^4/((c*e^2)^(9/2)*(x + d/e)^4) - e*x^2/((c*e^2*x^2 + 2*c*d*e*x + c*d^2
)^(3/2)*c) + 8/3*c*d^2*e^3/((c*e^2)^(7/2)*(x + d/e)^3) - 5/3*d^2/((c*e^2*x^2 + 2
*c*d*e*x + c*d^2)^(3/2)*c*e) - 2*d*e^2/((c*e^2)^(5/2)*(x + d/e)^2) + d^3/((c*e^2
)^(5/2)*(x + d/e)^4)

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Fricas [A]  time = 0.221293, size = 74, normalized size = 2.31 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{3} e^{3} x^{2} + 2 \, c^{3} d e^{2} x + c^{3} d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.27603, size = 70, normalized size = 2.19 \[ \begin{cases} - \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{3} d^{2} e + 2 c^{3} d e^{2} x + c^{3} e^{3} x^{2}} & \text{for}\: e \neq 0 \\\frac{d^{3} x}{\left (c d^{2}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

Piecewise((-sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(c**3*d**2*e + 2*c**3*d*e**2*
x + c**3*e**3*x**2), Ne(e, 0)), (d**3*x/(c*d**2)**(5/2), True))

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GIAC/XCAS [A]  time = 0.267671, size = 107, normalized size = 3.34 \[ \frac{2 \, C_{0} d^{3} e^{\left (-3\right )} +{\left (6 \, C_{0} d^{2} e^{\left (-2\right )} +{\left (6 \, C_{0} d e^{\left (-1\right )} + 2 \, C_{0} x - \frac{e}{c}\right )} x - \frac{2 \, d}{c}\right )} x - \frac{d^{2} e^{\left (-1\right )}}{c}}{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="giac")

[Out]

(2*C_0*d^3*e^(-3) + (6*C_0*d^2*e^(-2) + (6*C_0*d*e^(-1) + 2*C_0*x - e/c)*x - 2*d
/c)*x - d^2*e^(-1)/c)/(c*x^2*e^2 + 2*c*d*x*e + c*d^2)^(3/2)