3.1082 \(\int \frac{(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=24 \[ -\frac{(d+e x)^{m-5}}{c^3 e (5-m)} \]

[Out]

-((d + e*x)^(-5 + m)/(c^3*e*(5 - m)))

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Rubi [A]  time = 0.0294007, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{(d+e x)^{m-5}}{c^3 e (5-m)} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)^(-5 + m)/(c^3*e*(5 - m)))

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Rubi in Sympy [A]  time = 20.5232, size = 17, normalized size = 0.71 \[ - \frac{\left (d + e x\right )^{m - 5}}{c^{3} e \left (- m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)**(m - 5)/(c**3*e*(-m + 5))

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Mathematica [A]  time = 0.0213384, size = 21, normalized size = 0.88 \[ \frac{(d+e x)^{m-5}}{c^3 e (m-5)} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x)^(-5 + m)/(c^3*e*(-5 + m))

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Maple [A]  time = 0.007, size = 40, normalized size = 1.7 \[{\frac{ \left ( ex+d \right ) ^{-1+m}}{ \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) ^{2}{c}^{3}e \left ( -5+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.692892, size = 134, normalized size = 5.58 \[ \frac{{\left (e x + d\right )}^{m}}{c^{3} e^{6}{\left (m - 5\right )} x^{5} + 5 \, c^{3} d e^{5}{\left (m - 5\right )} x^{4} + 10 \, c^{3} d^{2} e^{4}{\left (m - 5\right )} x^{3} + 10 \, c^{3} d^{3} e^{3}{\left (m - 5\right )} x^{2} + 5 \, c^{3} d^{4} e^{2}{\left (m - 5\right )} x + c^{3} d^{5} e{\left (m - 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x + d)^m/(c^3*e^6*(m - 5)*x^5 + 5*c^3*d*e^5*(m - 5)*x^4 + 10*c^3*d^2*e^4*(m -
5)*x^3 + 10*c^3*d^3*e^3*(m - 5)*x^2 + 5*c^3*d^4*e^2*(m - 5)*x + c^3*d^5*e*(m -
5))

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Fricas [A]  time = 0.243474, size = 211, normalized size = 8.79 \[ \frac{{\left (e x + d\right )}^{m}}{c^{3} d^{5} e m - 5 \, c^{3} d^{5} e +{\left (c^{3} e^{6} m - 5 \, c^{3} e^{6}\right )} x^{5} + 5 \,{\left (c^{3} d e^{5} m - 5 \, c^{3} d e^{5}\right )} x^{4} + 10 \,{\left (c^{3} d^{2} e^{4} m - 5 \, c^{3} d^{2} e^{4}\right )} x^{3} + 10 \,{\left (c^{3} d^{3} e^{3} m - 5 \, c^{3} d^{3} e^{3}\right )} x^{2} + 5 \,{\left (c^{3} d^{4} e^{2} m - 5 \, c^{3} d^{4} e^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x + d)^m/(c^3*d^5*e*m - 5*c^3*d^5*e + (c^3*e^6*m - 5*c^3*e^6)*x^5 + 5*(c^3*d*
e^5*m - 5*c^3*d*e^5)*x^4 + 10*(c^3*d^2*e^4*m - 5*c^3*d^2*e^4)*x^3 + 10*(c^3*d^3*
e^3*m - 5*c^3*d^3*e^3)*x^2 + 5*(c^3*d^4*e^2*m - 5*c^3*d^4*e^2)*x)

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Sympy [A]  time = 12.048, size = 201, normalized size = 8.38 \[ \begin{cases} \frac{x}{c^{3} d} & \text{for}\: e = 0 \wedge m = 5 \\\frac{d^{m} x}{c^{3} d^{6}} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c^{3} e} & \text{for}\: m = 5 \\\frac{\left (d + e x\right )^{m}}{c^{3} d^{5} e m - 5 c^{3} d^{5} e + 5 c^{3} d^{4} e^{2} m x - 25 c^{3} d^{4} e^{2} x + 10 c^{3} d^{3} e^{3} m x^{2} - 50 c^{3} d^{3} e^{3} x^{2} + 10 c^{3} d^{2} e^{4} m x^{3} - 50 c^{3} d^{2} e^{4} x^{3} + 5 c^{3} d e^{5} m x^{4} - 25 c^{3} d e^{5} x^{4} + c^{3} e^{6} m x^{5} - 5 c^{3} e^{6} x^{5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((x/(c**3*d), Eq(e, 0) & Eq(m, 5)), (d**m*x/(c**3*d**6), Eq(e, 0)), (lo
g(d/e + x)/(c**3*e), Eq(m, 5)), ((d + e*x)**m/(c**3*d**5*e*m - 5*c**3*d**5*e + 5
*c**3*d**4*e**2*m*x - 25*c**3*d**4*e**2*x + 10*c**3*d**3*e**3*m*x**2 - 50*c**3*d
**3*e**3*x**2 + 10*c**3*d**2*e**4*m*x**3 - 50*c**3*d**2*e**4*x**3 + 5*c**3*d*e**
5*m*x**4 - 25*c**3*d*e**5*x**4 + c**3*e**6*m*x**5 - 5*c**3*e**6*x**5), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^m/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^3, x)