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

Optimal. Leaf size=99 \[ \frac{(d+e x)^m (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \, _2F_1\left (1,1-m;2-m;-\frac{g (a e+c d x)}{c d f-a e g}\right )}{(1-m) (c d f-a e g)} \]

[Out]

((a*e + c*d*x)*(d + e*x)^m*Hypergeometric2F1[1, 1 - m, 2 - m, -((g*(a*e + c*d*x)
)/(c*d*f - a*e*g))])/((c*d*f - a*e*g)*(1 - m)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^m)

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Rubi [A]  time = 0.167592, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{(d+e x)^m (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \, _2F_1\left (1,1-m;2-m;-\frac{g (a e+c d x)}{c d f-a e g}\right )}{(1-m) (c d f-a e g)} \]

Antiderivative was successfully verified.

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

[Out]

((a*e + c*d*x)*(d + e*x)^m*Hypergeometric2F1[1, 1 - m, 2 - m, -((g*(a*e + c*d*x)
)/(c*d*f - a*e*g))])/((c*d*f - a*e*g)*(1 - m)*(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 = 35.3771, size = 95, normalized size = 0.96 \[ - \frac{\left (d + e x\right )^{m} \left (a e + c d x\right )^{m} \left (a e + c d x\right )^{- m + 1} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} 1, - m + 1 \\ - m + 2 \end{matrix}\middle |{\frac{g \left (a e + c d x\right )}{a e g - c d f}} \right )}}{\left (- m + 1\right ) \left (a e g - c d f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0953098, size = 87, normalized size = 0.88 \[ -\frac{(d+e x)^m ((d+e x) (a e+c d x))^{-m} \left (\frac{a e g+c d g x}{c d f+c d g x}\right )^m \, _2F_1\left (m,m;m+1;\frac{c d f-a e g}{c d f+c d g x}\right )}{g m} \]

Antiderivative was successfully verified.

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

[Out]

-(((d + e*x)^m*((a*e*g + c*d*g*x)/(c*d*f + c*d*g*x))^m*Hypergeometric2F1[m, m, 1
+ m, (c*d*f - a*e*g)/(c*d*f + c*d*g*x)])/(g*m*((a*e + c*d*x)*(d + e*x))^m))

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Maple [F]  time = 0.134, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( gx+f \right ) \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{m}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^m/((g*x + f)*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m), x)

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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/(g*x+f)/((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. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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