3.1081 \(\int \frac{(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=24 \[ -\frac{(d+e x)^{m-3}}{c^2 e (3-m)} \]

[Out]

-((d + e*x)^(-3 + m)/(c^2*e*(3 - m)))

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Rubi [A]  time = 0.0299091, 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 \[ -\frac{(d+e x)^{m-3}}{c^2 e (3-m)} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)^(-3 + m)/(c^2*e*(3 - m)))

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Rubi in Sympy [A]  time = 20.6195, size = 17, normalized size = 0.71 \[ - \frac{\left (d + e x\right )^{m - 3}}{c^{2} e \left (- m + 3\right )} \]

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)**2,x)

[Out]

-(d + e*x)**(m - 3)/(c**2*e*(-m + 3))

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Mathematica [A]  time = 0.0216884, size = 21, normalized size = 0.88 \[ \frac{(d+e x)^{m-3}}{c^2 e (m-3)} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x)^(-3 + m)/(c^2*e*(-3 + m))

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Maple [A]  time = 0.005, size = 40, normalized size = 1.7 \[{\frac{ \left ( ex+d \right ) ^{-1+m}}{ \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ){c}^{2}e \left ( -3+m \right ) }} \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 0.697206, size = 88, normalized size = 3.67 \[ \frac{{\left (e x + d\right )}^{m}}{c^{2} e^{4}{\left (m - 3\right )} x^{3} + 3 \, c^{2} d e^{3}{\left (m - 3\right )} x^{2} + 3 \, c^{2} d^{2} e^{2}{\left (m - 3\right )} x + c^{2} d^{3} e{\left (m - 3\right )}} \]

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.23173, size = 135, normalized size = 5.62 \[ \frac{{\left (e x + d\right )}^{m}}{c^{2} d^{3} e m - 3 \, c^{2} d^{3} e +{\left (c^{2} e^{4} m - 3 \, c^{2} e^{4}\right )} x^{3} + 3 \,{\left (c^{2} d e^{3} m - 3 \, c^{2} d e^{3}\right )} x^{2} + 3 \,{\left (c^{2} d^{2} e^{2} m - 3 \, c^{2} d^{2} e^{2}\right )} x} \]

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.80283, size = 136, normalized size = 5.67 \[ \begin{cases} \frac{x}{c^{2} d} & \text{for}\: e = 0 \wedge m = 3 \\\frac{d^{m} x}{c^{2} d^{4}} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c^{2} e} & \text{for}\: m = 3 \\\frac{\left (d + e x\right )^{m}}{c^{2} d^{3} e m - 3 c^{2} d^{3} e + 3 c^{2} d^{2} e^{2} m x - 9 c^{2} d^{2} e^{2} x + 3 c^{2} d e^{3} m x^{2} - 9 c^{2} d e^{3} x^{2} + c^{2} e^{4} m x^{3} - 3 c^{2} e^{4} x^{3}} & \text{otherwise} \end{cases} \]

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)**2,x)

[Out]

Piecewise((x/(c**2*d), Eq(e, 0) & Eq(m, 3)), (d**m*x/(c**2*d**4), Eq(e, 0)), (lo
g(d/e + x)/(c**2*e), Eq(m, 3)), ((d + e*x)**m/(c**2*d**3*e*m - 3*c**2*d**3*e + 3
*c**2*d**2*e**2*m*x - 9*c**2*d**2*e**2*x + 3*c**2*d*e**3*m*x**2 - 9*c**2*d*e**3*
x**2 + c**2*e**4*m*x**3 - 3*c**2*e**4*x**3), True))

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

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)^2,x, algorithm="giac")

[Out]

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