3.2086 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx\)

Optimal. Leaf size=92 \[ \frac{(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (1,2 p;p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(1-p) (d+e x) \left (c d^2-a e^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.190735, antiderivative size = 120, normalized size of antiderivative = 1.3, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{c d (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (2-p,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{(p+1) \left (c d^2-a e^2\right )^2} \]

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,x]

[Out]

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

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Rubi in Sympy [A]  time = 31.5857, size = 116, normalized size = 1.26 \[ \frac{c d \left (\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}\right )^{- p} \left (a e + c d x\right )^{- p} \left (a e + c d x\right )^{p + 1} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p + 2, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{e \left (a e + c d x\right )}{a e^{2} - c d^{2}}} \right )}}{\left (p + 1\right ) \left (a e^{2} - c d^{2}\right )^{2}} \]

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,x)

[Out]

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

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Mathematica [A]  time = 0.125294, size = 90, normalized size = 0.98 \[ \frac{\left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (p-1,-p;p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e (p-1) (d+e x)} \]

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,x]

[Out]

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

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Maple [F]  time = 0.156, size = 0, normalized size = 0. \[ \int{\frac{ \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \]

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,x)

[Out]

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

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

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,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, x)

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

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,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

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,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, x)