3.1066 \(\int \frac{1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

sage0*x