3.1077 \(\int \frac{1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{c}{7 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \]

[Out]

-c/(7*e*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(7/2))

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Rubi [A]  time = 0.0658176, 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 \[ -\frac{c}{7 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]

[Out]

-c/(7*e*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(7/2))

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Rubi in Sympy [A]  time = 17.5971, size = 31, normalized size = 0.97 \[ - \frac{c}{7 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

-c/(7*e*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(7/2))

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Mathematica [A]  time = 0.0676057, size = 21, normalized size = 0.66 \[ -\frac{c}{7 e \left (c (d+e x)^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]

[Out]

-c/(7*e*(c*(d + e*x)^2)^(7/2))

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Maple [A]  time = 0.008, size = 35, normalized size = 1.1 \[ -{\frac{1}{7\, \left ( ex+d \right ) ^{2}e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x)

[Out]

-1/7/(e*x+d)^2/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)

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Maxima [A]  time = 0.686743, size = 139, normalized size = 4.34 \[ -\frac{1}{7 \,{\left (c^{\frac{5}{2}} e^{8} x^{7} + 7 \, c^{\frac{5}{2}} d e^{7} x^{6} + 21 \, c^{\frac{5}{2}} d^{2} e^{6} x^{5} + 35 \, c^{\frac{5}{2}} d^{3} e^{5} x^{4} + 35 \, c^{\frac{5}{2}} d^{4} e^{4} x^{3} + 21 \, c^{\frac{5}{2}} d^{5} e^{3} x^{2} + 7 \, c^{\frac{5}{2}} d^{6} e^{2} x + c^{\frac{5}{2}} d^{7} e\right )}} \]

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)^(5/2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

-1/7/(c^(5/2)*e^8*x^7 + 7*c^(5/2)*d*e^7*x^6 + 21*c^(5/2)*d^2*e^6*x^5 + 35*c^(5/2
)*d^3*e^5*x^4 + 35*c^(5/2)*d^4*e^4*x^3 + 21*c^(5/2)*d^5*e^3*x^2 + 7*c^(5/2)*d^6*
e^2*x + c^(5/2)*d^7*e)

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Fricas [A]  time = 0.232412, size = 188, normalized size = 5.88 \[ -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{7 \,{\left (c^{3} e^{9} x^{8} + 8 \, c^{3} d e^{8} x^{7} + 28 \, c^{3} d^{2} e^{7} x^{6} + 56 \, c^{3} d^{3} e^{6} x^{5} + 70 \, c^{3} d^{4} e^{5} x^{4} + 56 \, c^{3} d^{5} e^{4} x^{3} + 28 \, c^{3} d^{6} e^{3} x^{2} + 8 \, c^{3} d^{7} e^{2} x + c^{3} d^{8} e\right )}} \]

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)^(5/2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

-1/7*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^3*e^9*x^8 + 8*c^3*d*e^8*x^7 + 28*c^3
*d^2*e^7*x^6 + 56*c^3*d^3*e^6*x^5 + 70*c^3*d^4*e^5*x^4 + 56*c^3*d^5*e^4*x^3 + 28
*c^3*d^6*e^3*x^2 + 8*c^3*d^7*e^2*x + c^3*d^8*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c \left (d + e x\right )^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

Integral(1/((c*(d + e*x)**2)**(5/2)*(d + e*x)**3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

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)^(5/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, 1]