3.2094 \(\int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx\)

Optimal. Leaf size=54 \[ \frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (1-m)} \]

[Out]

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

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Rubi [A]  time = 0.054998, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027 \[ \frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (1-m)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.3712, size = 42, normalized size = 0.78 \[ \frac{\left (d + e x\right )^{m - 1} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{- m + 1}}{c d \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0437657, size = 42, normalized size = 0.78 \[ -\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m}}{c d (m-1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 57, normalized size = 1.1 \[ -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{m}}{cd \left ( -1+m \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{m}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.820336, size = 45, normalized size = 0.83 \[ -\frac{{\left (c d x + a e\right )}{\left (c d x + a e\right )}^{-m}}{c d{\left (m - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229122, size = 77, normalized size = 1.43 \[ -\frac{{\left (c d x + a e\right )}{\left (e x + d\right )}^{m}}{{\left (c d m - c d\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216379, size = 68, normalized size = 1.26 \[ -\frac{c d x e^{\left (-m{\rm ln}\left (c d x + a e\right )\right )} + a e^{\left (-m{\rm ln}\left (c d x + a e\right ) + 1\right )}}{c d m - c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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