3.1059 \(\int \frac{1}{(d+e x)^4 \sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

-c/(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 = 0.0675535, 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]

-c/(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 = 17.9618, 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]

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

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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]

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

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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]

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

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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]

-1/4/(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)

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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]

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

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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]

Integral(1/(sqrt(c*(d + e*x)**2)*(d + e*x)**4), x)

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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]

Exception raised: NotImplementedError