\(\int (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [2095]

Optimal result

Integrand size = 27, antiderivative size = 113 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=-\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-1-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {638} \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=-\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(p+1) \left (c d^2-a e^2\right )} \]

[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)*Hypergeomet
ric2F1[-p, 1 + p, 2 + p, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/((c*d^2 - a*e^2)*(1 + p)))

Rule 638

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(a + b*x + c*
x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/
(2*q)], x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[4*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-1-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (1+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} ((a e+c d x) (d+e x))^p \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d (1+p)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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