3.781 \(\int (d+e x)^m \left (c d^2 e g-e \left (c d^2+a e^2\right ) g-c d e^2 g x\right )^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx\)

Optimal. Leaf size=78 \[ -\frac{(d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \log (a e+c d x) \left (-a e^3 g-c d e^2 g x\right )^m}{c d e^2 g} \]

[Out]

-(((d + e*x)^m*(-(a*e^3*g) - c*d*e^2*g*x)^m*Log[a*e + c*d*x])/(c*d*e^2*g*(a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^m))

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Rubi [A]  time = 0.326491, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 73, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.041 \[ -\frac{(d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \log (a e+c d x) \left (-a e^3 g-c d e^2 g x\right )^m}{c d e^2 g} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^m*(c*d^2*e*g - e*(c*d^2 + a*e^2)*g - c*d*e^2*g*x)^(-1 + m))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]

[Out]

-(((d + e*x)^m*(-(a*e^3*g) - c*d*e^2*g*x)^m*Log[a*e + c*d*x])/(c*d*e^2*g*(a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^m))

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Rubi in Sympy [A]  time = 174.401, size = 73, normalized size = 0.94 \[ - \frac{\left (d + e x\right )^{m} \left (- a e^{3} g - c d e^{2} g x\right )^{m} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{- m} \log{\left (a e + c d x \right )}}{c d e^{2} g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(c*d**2*e*g-e*(a*e**2+c*d**2)*g-c*d*e**2*g*x)**(-1+m)/((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**m),x)

[Out]

-(d + e*x)**m*(-a*e**3*g - c*d*e**2*g*x)**m*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*
d**2))**(-m)*log(a*e + c*d*x)/(c*d*e**2*g)

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Mathematica [A]  time = 0.0552863, size = 64, normalized size = 0.82 \[ -\frac{(d+e x)^m ((d+e x) (a e+c d x))^{-m} \log (a e+c d x) \left (-e^2 g (a e+c d x)\right )^m}{c d e^2 g} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^m*(c*d^2*e*g - e*(c*d^2 + a*e^2)*g - c*d*e^2*g*x)^(-1 + m))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]

[Out]

-(((-(e^2*g*(a*e + c*d*x)))^m*(d + e*x)^m*Log[a*e + c*d*x])/(c*d*e^2*g*((a*e + c
*d*x)*(d + e*x))^m))

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Maple [F]  time = 0.311, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{d}^{2}eg-e \left ( a{e}^{2}+c{d}^{2} \right ) g-cd{e}^{2}gx \right ) ^{-1+m}}{ \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{m}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*d^2*e*g-e*(a*e^2+c*d^2)*g-c*d*e^2*g*x)^(-1+m)/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x)

[Out]

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

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Maxima [A]  time = 0.720535, size = 43, normalized size = 0.55 \[ -\frac{e^{2 \, m - 2} \left (-g\right )^{m} \log \left (c d x + a e\right )}{c d g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*d*e^2*g*x + c*d^2*e*g - (c*d^2 + a*e^2)*e*g)^(m - 1)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.274066, size = 47, normalized size = 0.6 \[ -\frac{\log \left (c d x + a e\right )}{c d e^{2} g \left (-\frac{1}{e^{2} g}\right )^{m}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*d*e^2*g*x + c*d^2*e*g - (c*d^2 + a*e^2)*e*g)^(m - 1)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,x, algorithm="fricas")

[Out]

-log(c*d*x + a*e)/(c*d*e^2*g*(-1/(e^2*g))^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*(c*d**2*e*g-e*(a*e**2+c*d**2)*g-c*d*e**2*g*x)**(-1+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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-c d e^{2} g x + c d^{2} e g -{\left (c d^{2} + a e^{2}\right )} e g\right )}^{m - 1}{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*d*e^2*g*x + c*d^2*e*g - (c*d^2 + a*e^2)*e*g)^(m - 1)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,x, algorithm="giac")

[Out]

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