3.2104 \(\int (d+e x)^{-5-2 p} (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\)

Optimal. Leaf size=288 \[ \frac{6 c^2 d^2 (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^3}+\frac{6 c^3 d^3 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^4}+\frac{(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}+\frac{3 c d (d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) (p+4) \left (c d^2-a e^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.164065, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {658, 650} \[ \frac{6 c^2 d^2 (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^3}+\frac{6 c^3 d^3 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^4}+\frac{(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}+\frac{3 c d (d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) (p+4) \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(-5 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

((d + e*x)^(-5 - 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)*(4 + p)) + (6*c^2*d^2*
(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)^3*(2 + p)*(3 + p)*(4 +
p)) + (6*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p))/((c*d^2 - a*e^2)^4*(1 + p)*(2 + p)*(3 + p)*(
4 + p)*(d + e*x)^(2*(1 + p))) + (3*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*(3
+ p)*(4 + 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)^{-5-2 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)^{-5-2 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 ) (4+p)}+\frac{(3 c d) \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}{\left (c d^2-a e^2\right ) (4+p)}\\ &=\frac{(d+e x)^{-5-2 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 ) (4+p)}+\frac{3 c d (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 )^2 (3+p) (4+p)}+\frac{\left (6 c^2 d^2\right ) \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 )^2 (3+p) (4+p)}\\ &=\frac{(d+e x)^{-5-2 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 ) (4+p)}+\frac{6 c^2 d^2 (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 )^3 (2+p) (3+p) (4+p)}+\frac{3 c d (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 )^2 (3+p) (4+p)}+\frac{\left (6 c^3 d^3\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 )^3 (2+p) (3+p) (4+p)}\\ &=\frac{(d+e x)^{-5-2 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 ) (4+p)}+\frac{6 c^2 d^2 (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 )^3 (2+p) (3+p) (4+p)}+\frac{6 c^3 d^3 (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 )^4 (1+p) (2+p) (3+p) (4+p)}+\frac{3 c d (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 )^2 (3+p) (4+p)}\\ \end{align*}

Mathematica [A]  time = 0.149112, size = 217, normalized size = 0.75 \[ \frac{(d+e x)^{-2 p-5} ((d+e x) (a e+c d x))^{p+1} \left (3 a^2 c d e^4 \left (p^2+3 p+2\right ) (d (p+4)+e x)-a^3 e^6 \left (p^3+6 p^2+11 p+6\right )-3 a c^2 d^2 e^2 (p+1) \left (d^2 \left (p^2+7 p+12\right )+2 d e (p+4) x+2 e^2 x^2\right )+c^3 d^3 \left (3 d^2 e \left (p^2+7 p+12\right ) x+d^3 \left (p^3+9 p^2+26 p+24\right )+6 d e^2 (p+4) x^2+6 e^3 x^3\right )\right )}{(p+1) (p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(-5 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

((d + e*x)^(-5 - 2*p)*((a*e + c*d*x)*(d + e*x))^(1 + p)*(-(a^3*e^6*(6 + 11*p + 6*p^2 + p^3)) + 3*a^2*c*d*e^4*(
2 + 3*p + p^2)*(d*(4 + p) + e*x) - 3*a*c^2*d^2*e^2*(1 + p)*(d^2*(12 + 7*p + p^2) + 2*d*e*(4 + p)*x + 2*e^2*x^2
) + c^3*d^3*(d^3*(24 + 26*p + 9*p^2 + p^3) + 3*d^2*e*(12 + 7*p + p^2)*x + 6*d*e^2*(4 + p)*x^2 + 6*e^3*x^3)))/(
(c*d^2 - a*e^2)^4*(1 + p)*(2 + p)*(3 + p)*(4 + p))

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Maple [B]  time = 0.048, size = 745, normalized size = 2.6 \begin{align*} -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{-4-2\,p} \left ({a}^{3}{e}^{6}{p}^{3}-3\,{a}^{2}c{d}^{2}{e}^{4}{p}^{3}-3\,{a}^{2}cd{e}^{5}{p}^{2}x+3\,a{c}^{2}{d}^{4}{e}^{2}{p}^{3}+6\,a{c}^{2}{d}^{3}{e}^{3}{p}^{2}x+6\,a{c}^{2}{d}^{2}{e}^{4}p{x}^{2}-{c}^{3}{d}^{6}{p}^{3}-3\,{c}^{3}{d}^{5}e{p}^{2}x-6\,{c}^{3}{d}^{4}{e}^{2}p{x}^{2}-6\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+6\,{a}^{3}{e}^{6}{p}^{2}-21\,{a}^{2}c{d}^{2}{e}^{4}{p}^{2}-9\,{a}^{2}cd{e}^{5}px+24\,a{c}^{2}{d}^{4}{e}^{2}{p}^{2}+30\,a{c}^{2}{d}^{3}{e}^{3}px+6\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-9\,{c}^{3}{d}^{6}{p}^{2}-21\,{c}^{3}{d}^{5}epx-24\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+11\,{a}^{3}{e}^{6}p-42\,{a}^{2}c{d}^{2}{e}^{4}p-6\,{a}^{2}cd{e}^{5}x+57\,a{c}^{2}{d}^{4}{e}^{2}p+24\,a{c}^{2}{d}^{3}{e}^{3}x-26\,{c}^{3}{d}^{6}p-36\,{c}^{3}{d}^{5}ex+6\,{a}^{3}{e}^{6}-24\,{a}^{2}c{d}^{2}{e}^{4}+36\,a{c}^{2}{d}^{4}{e}^{2}-24\,{c}^{3}{d}^{6} \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{p}}{{a}^{4}{e}^{8}{p}^{4}-4\,{a}^{3}c{d}^{2}{e}^{6}{p}^{4}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{p}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}{p}^{4}+{c}^{4}{d}^{8}{p}^{4}+10\,{a}^{4}{e}^{8}{p}^{3}-40\,{a}^{3}c{d}^{2}{e}^{6}{p}^{3}+60\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{p}^{3}-40\,a{c}^{3}{d}^{6}{e}^{2}{p}^{3}+10\,{c}^{4}{d}^{8}{p}^{3}+35\,{a}^{4}{e}^{8}{p}^{2}-140\,{a}^{3}c{d}^{2}{e}^{6}{p}^{2}+210\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{p}^{2}-140\,a{c}^{3}{d}^{6}{e}^{2}{p}^{2}+35\,{c}^{4}{d}^{8}{p}^{2}+50\,{a}^{4}{e}^{8}p-200\,{a}^{3}c{d}^{2}{e}^{6}p+300\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}p-200\,a{c}^{3}{d}^{6}{e}^{2}p+50\,{c}^{4}{d}^{8}p+24\,{a}^{4}{e}^{8}-96\,{a}^{3}c{d}^{2}{e}^{6}+144\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-96\,a{c}^{3}{d}^{6}{e}^{2}+24\,{c}^{4}{d}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(-5-2*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)^(-4-2*p)*(a^3*e^6*p^3-3*a^2*c*d^2*e^4*p^3-3*a^2*c*d*e^5*p^2*x+3*a*c^2*d^4*e^2*p^3+6*a*c^2
*d^3*e^3*p^2*x+6*a*c^2*d^2*e^4*p*x^2-c^3*d^6*p^3-3*c^3*d^5*e*p^2*x-6*c^3*d^4*e^2*p*x^2-6*c^3*d^3*e^3*x^3+6*a^3
*e^6*p^2-21*a^2*c*d^2*e^4*p^2-9*a^2*c*d*e^5*p*x+24*a*c^2*d^4*e^2*p^2+30*a*c^2*d^3*e^3*p*x+6*a*c^2*d^2*e^4*x^2-
9*c^3*d^6*p^2-21*c^3*d^5*e*p*x-24*c^3*d^4*e^2*x^2+11*a^3*e^6*p-42*a^2*c*d^2*e^4*p-6*a^2*c*d*e^5*x+57*a*c^2*d^4
*e^2*p+24*a*c^2*d^3*e^3*x-26*c^3*d^6*p-36*c^3*d^5*e*x+6*a^3*e^6-24*a^2*c*d^2*e^4+36*a*c^2*d^4*e^2-24*c^3*d^6)*
(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^p/(a^4*e^8*p^4-4*a^3*c*d^2*e^6*p^4+6*a^2*c^2*d^4*e^4*p^4-4*a*c^3*d^6*e^2*p^4
+c^4*d^8*p^4+10*a^4*e^8*p^3-40*a^3*c*d^2*e^6*p^3+60*a^2*c^2*d^4*e^4*p^3-40*a*c^3*d^6*e^2*p^3+10*c^4*d^8*p^3+35
*a^4*e^8*p^2-140*a^3*c*d^2*e^6*p^2+210*a^2*c^2*d^4*e^4*p^2-140*a*c^3*d^6*e^2*p^2+35*c^4*d^8*p^2+50*a^4*e^8*p-2
00*a^3*c*d^2*e^6*p+300*a^2*c^2*d^4*e^4*p-200*a*c^3*d^6*e^2*p+50*c^4*d^8*p+24*a^4*e^8-96*a^3*c*d^2*e^6+144*a^2*
c^2*d^4*e^4-96*a*c^3*d^6*e^2+24*c^4*d^8)

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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 - 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-5-2*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 - 5), x)

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Fricas [B]  time = 2.48054, size = 2107, normalized size = 7.32 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-5-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x, algorithm="fricas")

[Out]

(6*c^4*d^4*e^4*x^5 + 24*a*c^3*d^7*e - 36*a^2*c^2*d^5*e^3 + 24*a^3*c*d^3*e^5 - 6*a^4*d*e^7 + 6*(5*c^4*d^5*e^3 +
(c^4*d^5*e^3 - a*c^3*d^3*e^5)*p)*x^4 + (a*c^3*d^7*e - 3*a^2*c^2*d^5*e^3 + 3*a^3*c*d^3*e^5 - a^4*d*e^7)*p^3 +
3*(20*c^4*d^6*e^2 + (c^4*d^6*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*p^2 + (9*c^4*d^6*e^2 - 10*a*c^3*d^4*e^4
+ a^2*c^2*d^2*e^6)*p)*x^3 + 3*(3*a*c^3*d^7*e - 8*a^2*c^2*d^5*e^3 + 7*a^3*c*d^3*e^5 - 2*a^4*d*e^7)*p^2 + (60*c^
4*d^7*e + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*p^3 + 3*(4*c^4*d^7*e - 9*a*c^3*d^5*e
^3 + 6*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*p^2 + (47*c^4*d^7*e - 60*a*c^3*d^5*e^3 + 15*a^2*c^2*d^3*e^5 - 2*a^3*c*d*
e^7)*p)*x^2 + (26*a*c^3*d^7*e - 57*a^2*c^2*d^5*e^3 + 42*a^3*c*d^3*e^5 - 11*a^4*d*e^7)*p + (24*c^4*d^8 + 24*a*c
^3*d^6*e^2 - 36*a^2*c^2*d^4*e^4 + 24*a^3*c*d^2*e^6 - 6*a^4*e^8 + (c^4*d^8 - 2*a*c^3*d^6*e^2 + 2*a^3*c*d^2*e^6
- a^4*e^8)*p^3 + 3*(3*c^4*d^8 - 4*a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4 + 6*a^3*c*d^2*e^6 - 2*a^4*e^8)*p^2 + (26*c
^4*d^8 - 10*a*c^3*d^6*e^2 - 45*a^2*c^2*d^4*e^4 + 40*a^3*c*d^2*e^6 - 11*a^4*e^8)*p)*x)*(c*d*e*x^2 + a*d*e + (c*
d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 5)/(24*c^4*d^8 - 96*a*c^3*d^6*e^2 + 144*a^2*c^2*d^4*e^4 - 96*a^3*c*d^2*e^6
+ 24*a^4*e^8 + (c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*p^4 + 10*(c^4*d^8
- 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*p^3 + 35*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2
*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*p^2 + 50*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^
2*e^6 + a^4*e^8)*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)**(-5-2*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 - 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(-5-2*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 - 5), x)