3.2077 \(\int \frac{(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=61 \[ -\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(1-m) \left (c d^2-a e^2\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0874469, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(1-m) \left (c d^2-a e^2\right )^2} \]

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)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.3772, size = 49, normalized size = 0.8 \[ - \frac{e \left (d + e x\right )^{m - 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m - 1 \\ m \end{matrix}\middle |{\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}} \right )}}{\left (- m + 1\right ) \left (a e^{2} - c d^{2}\right )^{2}} \]

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)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.208154, size = 95, normalized size = 1.56 \[ \frac{c d (d+e x)^m \left (\frac{c d (d+e x)}{e (a e+c d x)}\right )^{-m} \, _2F_1\left (2-m,3-m;4-m;\frac{a e^2-c d^2}{e (a e+c d x)}\right )}{e^2 (m-3) (a e+c d x)^3} \]

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)^2,x]

[Out]

(c*d*(d + e*x)^m*Hypergeometric2F1[2 - m, 3 - m, 4 - m, (-(c*d^2) + a*e^2)/(e*(a
*e + c*d*x))])/(e^2*(-3 + m)*(a*e + c*d*x)^3*((c*d*(d + e*x))/(e*(a*e + c*d*x)))
^m)

_______________________________________________________________________________________

Maple [F]  time = 0.211, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{2}}}\, dx \]

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)^2,x)

[Out]

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

_______________________________________________________________________________________

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

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)^2,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c^{2} d^{2} e^{2} x^{4} + a^{2} d^{2} e^{2} + 2 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{3} +{\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x}, x\right ) \]

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)^2,x, algorithm="fricas")

[Out]

integral((e*x + d)^m/(c^2*d^2*e^2*x^4 + a^2*d^2*e^2 + 2*(c^2*d^3*e + a*c*d*e^3)*
x^3 + (c^2*d^4 + 4*a*c*d^2*e^2 + a^2*e^4)*x^2 + 2*(a*c*d^3*e + a^2*d*e^3)*x), x)

_______________________________________________________________________________________

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)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

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

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)^2,x, algorithm="giac")

[Out]

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