3.2083 \(\int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.150892, antiderivative size = 152, normalized size of antiderivative = 1.6, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{2 c d (p+1)}-\frac{\left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{2 c d (p+1)} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 44.6467, size = 121, normalized size = 1.27 \[ - \frac{\left (\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}\right )^{- p} \left (a e + c d x\right )^{- p} \left (a e + c d x\right )^{p + 1} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - 1, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{e \left (a e + c d x\right )}{a e^{2} - c d^{2}}} \right )}}{c^{2} d^{2} \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [C]  time = 0.470049, size = 237, normalized size = 2.49 \[ \frac{d ((d+e x) (a e+c d x))^p \left (\frac{3 a e^3 x^2 F_1\left (2;-p,-p;3;-\frac{c d x}{a e},-\frac{e x}{d}\right )}{2 p x \left (c d^2 F_1\left (3;1-p,-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )+a e^2 F_1\left (3;-p,1-p;4;-\frac{c d x}{a e},-\frac{e x}{d}\right )\right )+6 a d e F_1\left (2;-p,-p;3;-\frac{c d x}{a e},-\frac{e x}{d}\right )}+\frac{(d+e x) \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} \, _2F_1\left (-p,p+1;p+2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{p+1}\right )}{e} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)*(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*((3*a*e^3*x^2*AppellF1[2, -p, -p, 3, -((c*d*x)/(a
*e)), -((e*x)/d)])/(6*a*d*e*AppellF1[2, -p, -p, 3, -((c*d*x)/(a*e)), -((e*x)/d)]
+ 2*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)])) + ((d + e*x)*Hypergeom
etric2F1[-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)))/e

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Maple [F]  time = 0.121, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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