\(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [2094]

Optimal result

Integrand size = 33, antiderivative size = 95 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=-\frac {(a e+c d x) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,3+2 p,3+p,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (2+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {654, 638} \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{2 c d (p+1)}-\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{2 c d (p+1)} \]

[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))

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]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {\left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-1-p} ((a e+c d x) (d+e x))^{1+p} \operatorname {Hypergeometric2F1}\left (-1-p,1+p,2+p,\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d (1+p)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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