3.1038 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=34 \[ -\frac{c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0662464, antiderivative size = 34, 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 \[ -\frac{c^3}{3 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

-c^3/(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 = 17.9012, size = 32, normalized size = 0.94 \[ - \frac{c^{3}}{3 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2)/(e*x+d)**7,x)

[Out]

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

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Mathematica [A]  time = 0.017201, size = 27, normalized size = 0.79 \[ -\frac{\left (c (d+e x)^2\right )^{3/2}}{3 e (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

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

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Maple [A]  time = 0.005, size = 35, normalized size = 1. \[ -{\frac{1}{3\, \left ( ex+d \right ) ^{6}e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^7,x)

[Out]

-1/3/(e*x+d)^6/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)^7,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.226261, size = 93, normalized size = 2.74 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c}{3 \,{\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} \]

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)^7,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{7}}\, dx \]

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)**7,x)

[Out]

Integral((c*(d + e*x)**2)**(3/2)/(d + e*x)**7, x)

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GIAC/XCAS [A]  time = 0.416803, size = 1, normalized size = 0.03 \[ \mathit{Done} \]

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)^7,x, algorithm="giac")

[Out]

Done