\(\int (d+e x)^{-3-2 p} (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [2102]

Optimal result

Integrand size = 39, antiderivative size = 128 \[ \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (2+p)}+\frac {c d (d+e x)^{-2 (1+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^2 (1+p) (2+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {672, 664} \[ \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) \left (c d^2-a e^2\right )}+\frac {c d (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) \left (c d^2-a e^2\right )^2} \]

[In]

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

[Out]

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

Rule 664

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

Rule 672

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

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (2+p)}+\frac {(c d) \int (d+e x)^{-2-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx}{\left (c d^2-a e^2\right ) (2+p)} \\ & = \frac {(d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (2+p)}+\frac {c d (d+e x)^{-2 (1+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^2 (1+p) (2+p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.59 \[ \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-3-2 p} ((a e+c d x) (d+e x))^{1+p} \left (-a e^2 (1+p)+c d (d (2+p)+e x)\right )}{\left (c d^2-a e^2\right )^2 (1+p) (2+p)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.98 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.33

method	result	size
gosper	\(-\frac {\left (e x +d \right )^{-2-2 p} \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} \left (a \,e^{2} p -c \,d^{2} p -x c d e +e^{2} a -2 c \,d^{2}\right )}{a^{2} e^{4} p^{2}-2 a c \,d^{2} e^{2} p^{2}+c^{2} d^{4} p^{2}+3 a^{2} e^{4} p -6 a c \,d^{2} e^{2} p +3 c^{2} d^{4} p +2 a^{2} e^{4}-4 a c \,d^{2} e^{2}+2 c^{2} d^{4}}\)	\(170\)
parallelrisch	\(\frac {x^{3} \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} c^{3} d^{3} e^{3}-x^{2} \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} a \,c^{2} d^{2} e^{4} p +x^{2} \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} c^{3} d^{4} e^{2} p +3 x^{2} \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} c^{3} d^{4} e^{2}-x \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} a^{2} c d \,e^{5} p +x \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} c^{3} d^{5} e p -x \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} a^{2} c d \,e^{5}+2 x \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} a \,c^{2} d^{3} e^{3}+2 x \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} c^{3} d^{5} e -\left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} a^{2} c \,d^{2} e^{4} p +\left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} a \,c^{2} d^{4} e^{2} p -\left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} a^{2} c \,d^{2} e^{4}+2 \left (e x +d \right )^{-3-2 p} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} a \,c^{2} d^{4} e^{2}}{c d e \left (a^{2} e^{4} p^{2}-2 a c \,d^{2} e^{2} p^{2}+c^{2} d^{4} p^{2}+3 a^{2} e^{4} p -6 a c \,d^{2} e^{2} p +3 c^{2} d^{4} p +2 a^{2} e^{4}-4 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right )}\)	\(746\)

[In]

int((e*x+d)^(-3-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x,method=_RETURNVERBOSE)

[Out]

-(e*x+d)^(-2-2*p)*(c*d*x+a*e)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^p*(a*e^2*p-c*d^2*p-c*d*e*x+a*e^2-2*c*d^2)/(a^2
*e^4*p^2-2*a*c*d^2*e^2*p^2+c^2*d^4*p^2+3*a^2*e^4*p-6*a*c*d^2*e^2*p+3*c^2*d^4*p+2*a^2*e^4-4*a*c*d^2*e^2+2*c^2*d
^4)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.99 \[ \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {{\left (c^{2} d^{2} e^{2} x^{3} + 2 \, a c d^{3} e - a^{2} d e^{3} + {\left (3 \, c^{2} d^{3} e + {\left (c^{2} d^{3} e - a c d e^{3}\right )} p\right )} x^{2} + {\left (a c d^{3} e - a^{2} d e^{3}\right )} p + {\left (2 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4} + {\left (c^{2} d^{4} - a^{2} e^{4}\right )} p\right )} x\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}}{2 \, c^{2} d^{4} - 4 \, a c d^{2} e^{2} + 2 \, a^{2} e^{4} + {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} p^{2} + 3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} p} \]

[In]

integrate((e*x+d)^(-3-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x, algorithm="fricas")

[Out]

(c^2*d^2*e^2*x^3 + 2*a*c*d^3*e - a^2*d*e^3 + (3*c^2*d^3*e + (c^2*d^3*e - a*c*d*e^3)*p)*x^2 + (a*c*d^3*e - a^2*
d*e^3)*p + (2*c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4 + (c^2*d^4 - a^2*e^4)*p)*x)*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^
2)*x)^p*(e*x + d)^(-2*p - 3)/(2*c^2*d^4 - 4*a*c*d^2*e^2 + 2*a^2*e^4 + (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*p^2
+ 3*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*p)

Sympy [F]

\[ \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{p} \left (d + e x\right )^{- 2 p - 3}\, dx \]

[In]

integrate((e*x+d)**(-3-2*p)*(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)**(-2*p - 3), x)

Maxima [F]

\[ \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int { {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3} \,d x } \]

[In]

integrate((e*x+d)^(-3-2*p)*(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*(e*x + d)^(-2*p - 3), x)

Giac [F]

\[ \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int { {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3} \,d x } \]

[In]

integrate((e*x+d)^(-3-2*p)*(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*(e*x + d)^(-2*p - 3), x)

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.29 \[ \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx={\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p\,\left (\frac {x\,\left (2\,c^2\,d^4-a^2\,e^4-a^2\,e^4\,p+c^2\,d^4\,p+2\,a\,c\,d^2\,e^2\right )}{{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}+\frac {c^2\,d^2\,e^2\,x^3}{{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}-\frac {a\,d\,e\,\left (a\,e^2-2\,c\,d^2+a\,e^2\,p-c\,d^2\,p\right )}{{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}+\frac {c\,d\,e\,x^2\,\left (3\,c\,d^2-a\,e^2\,p+c\,d^2\,p\right )}{{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{2\,p+3}\,\left (p^2+3\,p+2\right )}\right ) \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p/(d + e*x)^(2*p + 3),x)

[Out]

(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p*((x*(2*c^2*d^4 - a^2*e^4 - a^2*e^4*p + c^2*d^4*p + 2*a*c*d^2*e^2))/(
(a*e^2 - c*d^2)^2*(d + e*x)^(2*p + 3)*(3*p + p^2 + 2)) + (c^2*d^2*e^2*x^3)/((a*e^2 - c*d^2)^2*(d + e*x)^(2*p +
3)*(3*p + p^2 + 2)) - (a*d*e*(a*e^2 - 2*c*d^2 + a*e^2*p - c*d^2*p))/((a*e^2 - c*d^2)^2*(d + e*x)^(2*p + 3)*(3
*p + p^2 + 2)) + (c*d*e*x^2*(3*c*d^2 - a*e^2*p + c*d^2*p))/((a*e^2 - c*d^2)^2*(d + e*x)^(2*p + 3)*(3*p + p^2 +
2)))