3.2084 \(\int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0599325, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ -\frac{\left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(p+1) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.12039, size = 95, normalized size = 0.84 \[ - \frac{\left (\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}\right )^{- p - 1} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{e \left (a e + c d x\right )}{a e^{2} - c d^{2}}} \right )}}{\left (p + 1\right ) \left (a e^{2} - c d^{2}\right )} \]

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,x)

[Out]

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

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Mathematica [A]  time = 0.127009, size = 90, normalized size = 0.8 \[ \frac{(d+e x) \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (-p,p+1;p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-p, 1 + 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 = 0.101, size = 0, normalized size = 0. \[ \int \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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,x, algorithm="maxima")

[Out]

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

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

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,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

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,x, algorithm="giac")

[Out]

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