\(\int (d+e x)^{-2 p} (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [2099]

Optimal result

Integrand size = 37, antiderivative size = 113 \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(a e+c d x) (d+e x)^{-2 p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{c d (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {693, 691, 72, 71} \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-2 p} (a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \operatorname {Hypergeometric2F1}\left (p,p+1,p+2,-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{c d (p+1)} \]

[In]

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

[Out]

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

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 691

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^m*((a + b*x + c*x^2
)^FracPart[p]/((1 + e*(x/d))^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])), Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)
*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Int
egerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (IntegerQ[3*p] || IntegerQ[4*p]))

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^IntPart[m]*((d + e*
x)^FracPart[m]/(1 + e*(x/d))^FracPart[m]), Int[(1 + e*(x/d))^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c,
d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] || GtQ
[d, 0])

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-1-2 p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p ((a e+c d x) (d+e x))^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d (1+p)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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