### 3.2103 $$\int (d+e x)^{-4-2 p} (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx$$

Optimal. Leaf size=206 $\frac{2 c^2 d^2 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}+\frac{2 c d (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) \left (c d^2-a e^2\right )^2}+\frac{(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )}$

[Out]

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

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Rubi [A]  time = 0.0931066, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.051, Rules used = {658, 650} $\frac{2 c^2 d^2 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}+\frac{2 c d (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) \left (c d^2-a e^2\right )^2}+\frac{(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(-4 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

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

Rule 658

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]

Rule 650

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]

Rubi steps

\begin{align*} \int (d+e x)^{-4-2 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 (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 ) (3+p)}+\frac{(2 c d) \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^2-a e^2\right ) (3+p)}\\ &=\frac{2 c d (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 (2+p) (3+p)}+\frac{(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 )^{1+p}}{\left (c d^2-a e^2\right ) (3+p)}+\frac{\left (2 c^2 d^2\right ) \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 (2+p) (3+p)}\\ &=\frac{2 c d (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 (2+p) (3+p)}+\frac{2 c^2 d^2 (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 )^3 (1+p) (2+p) (3+p)}+\frac{(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 )^{1+p}}{\left (c d^2-a e^2\right ) (3+p)}\\ \end{align*}

Mathematica [A]  time = 0.0906075, size = 131, normalized size = 0.64 $\frac{(d+e x)^{-2 (p+2)} ((d+e x) (a e+c d x))^{p+1} \left (a^2 e^4 \left (p^2+3 p+2\right )-2 a c d e^2 (p+1) (d (p+3)+e x)+c^2 d^2 \left (d^2 \left (p^2+5 p+6\right )+2 d e (p+3) x+2 e^2 x^2\right )\right )}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(-4 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(1 + p)*(a^2*e^4*(2 + 3*p + p^2) - 2*a*c*d*e^2*(1 + p)*(d*(3 + p) + e*x) + c^2*d^2*
(d^2*(6 + 5*p + p^2) + 2*d*e*(3 + p)*x + 2*e^2*x^2)))/((c*d^2 - a*e^2)^3*(1 + p)*(2 + p)*(3 + p)*(d + e*x)^(2*
(2 + p)))

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Maple [A]  time = 0.044, size = 381, normalized size = 1.9 \begin{align*} -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{-3-2\,p} \left ({a}^{2}{e}^{4}{p}^{2}-2\,ac{d}^{2}{e}^{2}{p}^{2}-2\,acd{e}^{3}px+{c}^{2}{d}^{4}{p}^{2}+2\,{c}^{2}{d}^{3}epx+2\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+3\,{a}^{2}{e}^{4}p-8\,ac{d}^{2}{e}^{2}p-2\,acd{e}^{3}x+5\,{c}^{2}{d}^{4}p+6\,{c}^{2}{d}^{3}ex+2\,{a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}+6\,{c}^{2}{d}^{4} \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{p}}{{a}^{3}{e}^{6}{p}^{3}-3\,{a}^{2}c{d}^{2}{e}^{4}{p}^{3}+3\,a{c}^{2}{d}^{4}{e}^{2}{p}^{3}-{c}^{3}{d}^{6}{p}^{3}+6\,{a}^{3}{e}^{6}{p}^{2}-18\,{a}^{2}c{d}^{2}{e}^{4}{p}^{2}+18\,a{c}^{2}{d}^{4}{e}^{2}{p}^{2}-6\,{c}^{3}{d}^{6}{p}^{2}+11\,{a}^{3}{e}^{6}p-33\,{a}^{2}c{d}^{2}{e}^{4}p+33\,a{c}^{2}{d}^{4}{e}^{2}p-11\,{c}^{3}{d}^{6}p+6\,{a}^{3}{e}^{6}-18\,{a}^{2}c{d}^{2}{e}^{4}+18\,a{c}^{2}{d}^{4}{e}^{2}-6\,{c}^{3}{d}^{6}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(-4-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x)

[Out]

-(c*d*x+a*e)*(e*x+d)^(-3-2*p)*(a^2*e^4*p^2-2*a*c*d^2*e^2*p^2-2*a*c*d*e^3*p*x+c^2*d^4*p^2+2*c^2*d^3*e*p*x+2*c^2
*d^2*e^2*x^2+3*a^2*e^4*p-8*a*c*d^2*e^2*p-2*a*c*d*e^3*x+5*c^2*d^4*p+6*c^2*d^3*e*x+2*a^2*e^4-6*a*c*d^2*e^2+6*c^2
*d^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^p/(a^3*e^6*p^3-3*a^2*c*d^2*e^4*p^3+3*a*c^2*d^4*e^2*p^3-c^3*d^6*p^3+6*a
^3*e^6*p^2-18*a^2*c*d^2*e^4*p^2+18*a*c^2*d^4*e^2*p^2-6*c^3*d^6*p^2+11*a^3*e^6*p-33*a^2*c*d^2*e^4*p+33*a*c^2*d^
4*e^2*p-11*c^3*d^6*p+6*a^3*e^6-18*a^2*c*d^2*e^4+18*a*c^2*d^4*e^2-6*c^3*d^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 - 4}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-4-2*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 - 4), x)

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Fricas [B]  time = 2.40401, size = 1145, normalized size = 5.56 \begin{align*} \frac{{\left (2 \, c^{3} d^{3} e^{3} x^{4} + 6 \, a c^{2} d^{5} e - 6 \, a^{2} c d^{3} e^{3} + 2 \, a^{3} d e^{5} + 2 \,{\left (4 \, c^{3} d^{4} e^{2} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} p\right )} x^{3} +{\left (a c^{2} d^{5} e - 2 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} p^{2} +{\left (12 \, c^{3} d^{5} e +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p^{2} +{\left (7 \, c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p\right )} x^{2} +{\left (5 \, a c^{2} d^{5} e - 8 \, a^{2} c d^{3} e^{3} + 3 \, a^{3} d e^{5}\right )} p +{\left (6 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} +{\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} p^{2} +{\left (5 \, c^{3} d^{6} - a c^{2} d^{4} e^{2} - 7 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6}\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 - 4}}{6 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 6 \, a^{3} e^{6} +{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{3} + 6 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{2} + 11 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-4-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x, algorithm="fricas")

[Out]

(2*c^3*d^3*e^3*x^4 + 6*a*c^2*d^5*e - 6*a^2*c*d^3*e^3 + 2*a^3*d*e^5 + 2*(4*c^3*d^4*e^2 + (c^3*d^4*e^2 - a*c^2*d
^2*e^4)*p)*x^3 + (a*c^2*d^5*e - 2*a^2*c*d^3*e^3 + a^3*d*e^5)*p^2 + (12*c^3*d^5*e + (c^3*d^5*e - 2*a*c^2*d^3*e^
3 + a^2*c*d*e^5)*p^2 + (7*c^3*d^5*e - 8*a*c^2*d^3*e^3 + a^2*c*d*e^5)*p)*x^2 + (5*a*c^2*d^5*e - 8*a^2*c*d^3*e^3
+ 3*a^3*d*e^5)*p + (6*c^3*d^6 + 6*a*c^2*d^4*e^2 - 6*a^2*c*d^2*e^4 + 2*a^3*e^6 + (c^3*d^6 - a*c^2*d^4*e^2 - a^
2*c*d^2*e^4 + a^3*e^6)*p^2 + (5*c^3*d^6 - a*c^2*d^4*e^2 - 7*a^2*c*d^2*e^4 + 3*a^3*e^6)*p)*x)*(c*d*e*x^2 + a*d*
e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4)/(6*c^3*d^6 - 18*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 - 6*a^3*e^6 + (
c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*p^3 + 6*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 -
a^3*e^6)*p^2 + 11*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*p)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(-4-2*p)*(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 - 4}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-4-2*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 - 4), x)