3.2088 \(\int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

Optimal. Leaf size=113 \[ \frac{(d+e x)^{-2 p} (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^p \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (p,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c d (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.227392, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ \frac{(d+e x)^{-2 p} (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^p \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (p,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c d (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 60.9442, size = 110, normalized size = 0.97 \[ \frac{\left (\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}\right )^{p} \left (d + e x\right )^{- 2 p} \left (a e + c d x\right )^{- p} \left (a e + c d x\right )^{p + 1} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{e \left (a e + c d x\right )}{a e^{2} - c d^{2}}} \right )}}{c d \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.109851, size = 101, normalized size = 0.89 \[ -\frac{(d+e x)^{1-2 p} \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (1-p,-p;2-p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e (p-1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.213, size = 0, normalized size = 0. \[ \int{\frac{ \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2\,p}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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),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), x)

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

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

[Out]

Timed out

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

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),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), x)