3.1037 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=41 \[ -\frac{c^2}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0683986, antiderivative size = 41, 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 \[ -\frac{c^2}{2 e (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 17.9547, size = 36, normalized size = 0.88 \[ - \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2)/(e*x+d)**6,x)

[Out]

-(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(3/2)/(2*e*(d + e*x)**5)

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Mathematica [A]  time = 0.0180413, size = 27, normalized size = 0.66 \[ -\frac{\left (c (d+e x)^2\right )^{3/2}}{2 e (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

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

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Maple [A]  time = 0.004, size = 35, normalized size = 0.9 \[ -{\frac{1}{2\, \left ( ex+d \right ) ^{5}e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^6,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.243703, size = 78, normalized size = 1.9 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c}{2 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)/(e*x + d)^6,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2)/(e*x+d)**6,x)

[Out]

Integral((c*(d + e*x)**2)**(3/2)/(d + e*x)**6, x)

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GIAC/XCAS [A]  time = 0.367985, size = 1, normalized size = 0.02 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)/(e*x + d)^6,x, algorithm="giac")

[Out]

Done