3.2089 \(\int (d+e x)^{-1-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

Optimal. Leaf size=107 \[ -\frac{(d+e x)^{-2 p} \left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p,-p;1-p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.242658, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{(d+e x)^{-2 p} \left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p,-p;1-p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 49.143, size = 88, normalized size = 0.82 \[ - \frac{\left (\frac{e \left (a e + c d x\right )}{a e^{2} - c d^{2}}\right )^{- p} \left (d + e x\right )^{- 2 p} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ - p + 1 \end{matrix}\middle |{\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}} \right )}}{e p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0986911, size = 95, normalized size = 0.89 \[ -\frac{(d+e x)^{-2 p} \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (-p,-p;1-p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.209, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-1-2\,p} \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x+d)^(-1-2*p)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 1), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 1), 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)**(-1-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 1), x)