3.2082 \(\int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

Optimal. Leaf size=95 \[ -\frac{(d+e x)^3 (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (1,2 (p+2);p+4;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(p+3) \left (c d^2-a e^2\right )} \]

[Out]

-(((a*e + c*d*x)*(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p*Hypergeom
etric2F1[1, 2*(2 + p), 4 + p, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/((c*d^2 - a*e^2)
*(3 + p)))

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Rubi [A]  time = 0.188336, antiderivative size = 124, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{\left (c d^2-a e^2\right )^2 (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p-2,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c^3 d^3 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

((c*d^2 - a*e^2)^2*(a*e + c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p*Hyper
geometric2F1[-2 - p, 1 + p, 2 + p, -((e*(a*e + c*d*x))/(c*d^2 - a*e^2))])/(c^3*d
^3*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)

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Rubi in Sympy [A]  time = 67.9219, size = 121, normalized size = 1.27 \[ \frac{\left (\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}\right )^{- p} \left (a e + c d x\right )^{- p} \left (a e + c d x\right )^{p + 1} \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - 2, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{e \left (a e + c d x\right )}{a e^{2} - c d^{2}}} \right )}}{c^{3} d^{3} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(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)*(a*e + c*d*x)**(-p)*(a*e + c*d*x)**(p +
1)*(a*e**2 - c*d**2)**2*(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))/(c**3*d**3*(p + 1))

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Mathematica [C]  time = 0.939775, size = 385, normalized size = 4.05 \[ \frac{d ((d+e x) (a e+c d x))^p \left (\frac{4 a e^4 x^3 F_1\left (3;-p,-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )}{c d^2 p x F_1\left (4;1-p,-p;5;-\frac{c d x}{a e},-\frac{e x}{d}\right )+a e^2 p x F_1\left (4;-p,1-p;5;-\frac{c d x}{a e},-\frac{e x}{d}\right )+4 a d e F_1\left (3;-p,-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )}+\frac{9 a d e^3 x^2 F_1\left (2;-p,-p;3;-\frac{c d x}{a e},-\frac{e x}{d}\right )}{p x \left (c d^2 F_1\left (3;1-p,-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )+a e^2 F_1\left (3;-p,1-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )\right )+3 a d e F_1\left (2;-p,-p;3;-\frac{c d x}{a e},-\frac{e x}{d}\right )}+\frac{3 d (d+e x) \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{p+1}\right )}{3 e} \]

Warning: Unable to verify antiderivative.

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

[Out]

(d*((a*e + c*d*x)*(d + e*x))^p*((9*a*d*e^3*x^2*AppellF1[2, -p, -p, 3, -((c*d*x)/
(a*e)), -((e*x)/d)])/(3*a*d*e*AppellF1[2, -p, -p, 3, -((c*d*x)/(a*e)), -((e*x)/d
)] + p*x*(c*d^2*AppellF1[3, 1 - p, -p, 4, -((c*d*x)/(a*e)), -((e*x)/d)] + a*e^2*
AppellF1[3, -p, 1 - p, 4, -((c*d*x)/(a*e)), -((e*x)/d)])) + (4*a*e^4*x^3*AppellF
1[3, -p, -p, 4, -((c*d*x)/(a*e)), -((e*x)/d)])/(4*a*d*e*AppellF1[3, -p, -p, 4, -
((c*d*x)/(a*e)), -((e*x)/d)] + c*d^2*p*x*AppellF1[4, 1 - p, -p, 5, -((c*d*x)/(a*
e)), -((e*x)/d)] + a*e^2*p*x*AppellF1[4, -p, 1 - p, 5, -((c*d*x)/(a*e)), -((e*x)
/d)]) + (3*d*(d + e*x)*Hypergeometric2F1[-p, 1 + p, 2 + p, (c*d*(d + e*x))/(c*d^
2 - a*e^2)])/((1 + p)*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^p)))/(3*e)

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Maple [F]  time = 0.138, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2} \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x+d)^2*(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. \[ \int{\left (e x + d\right )}^{2}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^2*(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. \[{\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^2*x^2 + 2*d*e*x + d^2)*(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((e*x+d)**2*(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. \[ \int{\left (e x + d\right )}^{2}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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