3.1080 \(\int \frac{(d+e x)^m}{c d^2+2 c d e x+c e^2 x^2} \, dx\)

Optimal. Leaf size=24 \[ -\frac{(d+e x)^{m-1}}{c e (1-m)} \]

[Out]

-((d + e*x)^(-1 + m)/(c*e*(1 - m)))

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Rubi [A]  time = 0.0324684, 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 \[ -\frac{(d+e x)^{m-1}}{c e (1-m)} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)^(-1 + m)/(c*e*(1 - m)))

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Rubi in Sympy [A]  time = 20.2682, size = 15, normalized size = 0.62 \[ - \frac{\left (d + e x\right )^{m - 1}}{c e \left (- m + 1\right )} \]

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),x)

[Out]

-(d + e*x)**(m - 1)/(c*e*(-m + 1))

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Mathematica [A]  time = 0.0207131, size = 21, normalized size = 0.88 \[ \frac{(d+e x)^{m-1}}{c e (m-1)} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x)^(-1 + m)/(c*e*(-1 + m))

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Maple [A]  time = 0.005, size = 22, normalized size = 0.9 \[{\frac{ \left ( ex+d \right ) ^{-1+m}}{ce \left ( -1+m \right ) }} \]

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),x)

[Out]

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

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Maxima [A]  time = 0.694647, size = 36, normalized size = 1.5 \[ \frac{{\left (e x + d\right )}^{m}}{c e^{2}{\left (m - 1\right )} x + c d e{\left (m - 1\right )}} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.236705, size = 49, normalized size = 2.04 \[ \frac{{\left (e x + d\right )}^{m}}{c d e m - c d e +{\left (c e^{2} m - c e^{2}\right )} x} \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.08711, size = 63, normalized size = 2.62 \[ \begin{cases} \mathrm{NaN} & \text{for}\: d = 0 \wedge e = 0 \wedge m = 1 \\0^{m} \tilde{\infty } x & \text{for}\: d = - e x \\\frac{d^{m} x}{c d^{2}} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c e} & \text{for}\: m = 1 \\\frac{\left (d + e x\right )^{m}}{c d e m - c d e + c e^{2} m x - c e^{2} x} & \text{otherwise} \end{cases} \]

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),x)

[Out]

Piecewise((nan, Eq(d, 0) & Eq(e, 0) & Eq(m, 1)), (0**m*zoo*x, Eq(d, -e*x)), (d**
m*x/(c*d**2), Eq(e, 0)), (log(d/e + x)/(c*e), Eq(m, 1)), ((d + e*x)**m/(c*d*e*m
- c*d*e + c*e**2*m*x - c*e**2*x), True))

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

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),x, algorithm="giac")

[Out]

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