3.22.4 \(\int (d+e x)^{-5-2 p} (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [2104]

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

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 288, normalized size of antiderivative = 1.00, 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 = {672, 664} \begin {gather*} \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 {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 {(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} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.13, size = 217, normalized size = 0.75 \begin {gather*} \frac {(d+e x)^{-5-2 p} ((a e+c d x) (d+e x))^{1+p} \left (-a^3 e^6 \left (6+11 p+6 p^2+p^3\right )+3 a^2 c d e^4 \left (2+3 p+p^2\right ) (d (4+p)+e x)-3 a c^2 d^2 e^2 (1+p) \left (d^2 \left (12+7 p+p^2\right )+2 d e (4+p) x+2 e^2 x^2\right )+c^3 d^3 \left (d^3 \left (24+26 p+9 p^2+p^3\right )+3 d^2 e \left (12+7 p+p^2\right ) x+6 d e^2 (4+p) x^2+6 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^4 (1+p) (2+p) (3+p) (4+p)} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(744\) vs. \(2(292)=584\).
time = 0.90, size = 745, normalized size = 2.59

method	result	size
gosper	\(-\frac {\left (c d x +a e \right ) \left (e x +d \right )^{-4-2 p} \left (a^{3} e^{6} p^{3}-3 a^{2} c \,d^{2} e^{4} p^{3}-3 a^{2} c d \,e^{5} p^{2} x +3 a \,c^{2} d^{4} e^{2} p^{3}+6 a \,c^{2} d^{3} e^{3} p^{2} x +6 a \,c^{2} d^{2} e^{4} p \,x^{2}-c^{3} d^{6} p^{3}-3 c^{3} d^{5} e \,p^{2} x -6 c^{3} d^{4} e^{2} p \,x^{2}-6 c^{3} d^{3} e^{3} x^{3}+6 a^{3} e^{6} p^{2}-21 a^{2} c \,d^{2} e^{4} p^{2}-9 a^{2} c d \,e^{5} p x +24 a \,c^{2} d^{4} e^{2} p^{2}+30 a \,c^{2} d^{3} e^{3} p x +6 a \,c^{2} d^{2} e^{4} x^{2}-9 c^{3} d^{6} p^{2}-21 c^{3} d^{5} e p x -24 c^{3} d^{4} e^{2} x^{2}+11 a^{3} e^{6} p -42 a^{2} c \,d^{2} e^{4} p -6 a^{2} c d \,e^{5} x +57 a \,c^{2} d^{4} e^{2} p +24 a \,c^{2} d^{3} e^{3} x -26 c^{3} d^{6} p -36 c^{3} d^{5} e x +6 e^{6} a^{3}-24 e^{4} d^{2} a^{2} c +36 d^{4} e^{2} c^{2} a -24 d^{6} c^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p}}{a^{4} e^{8} p^{4}-4 a^{3} c \,d^{2} e^{6} p^{4}+6 a^{2} c^{2} d^{4} e^{4} p^{4}-4 a \,c^{3} d^{6} e^{2} p^{4}+c^{4} d^{8} p^{4}+10 a^{4} e^{8} p^{3}-40 a^{3} c \,d^{2} e^{6} p^{3}+60 a^{2} c^{2} d^{4} e^{4} p^{3}-40 a \,c^{3} d^{6} e^{2} p^{3}+10 c^{4} d^{8} p^{3}+35 a^{4} e^{8} p^{2}-140 a^{3} c \,d^{2} e^{6} p^{2}+210 a^{2} c^{2} d^{4} e^{4} p^{2}-140 a \,c^{3} d^{6} e^{2} p^{2}+35 c^{4} d^{8} p^{2}+50 a^{4} e^{8} p -200 a^{3} c \,d^{2} e^{6} p +300 a^{2} c^{2} d^{4} e^{4} p -200 a \,c^{3} d^{6} e^{2} p +50 c^{4} d^{8} p +24 a^{4} e^{8}-96 a^{3} c \,d^{2} e^{6}+144 a^{2} c^{2} d^{4} e^{4}-96 a \,c^{3} d^{6} e^{2}+24 c^{4} d^{8}}\)	\(745\)

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,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1017 vs. \(2 (296) = 592\).
time = 4.85, size = 1017, normalized size = 3.53 \begin {gather*} -\frac {{\left ({\left (a^{4} p^{3} + 6 \, a^{4} p^{2} + 11 \, a^{4} p + 6 \, a^{4}\right )} x e^{8} - {\left (c^{4} d^{8} p^{3} + 9 \, c^{4} d^{8} p^{2} + 26 \, c^{4} d^{8} p + 24 \, c^{4} d^{8}\right )} x + {\left (a^{4} d p^{3} + 6 \, a^{4} d p^{2} + 11 \, a^{4} d p + 6 \, a^{4} d + {\left (a^{3} c d p^{3} + 3 \, a^{3} c d p^{2} + 2 \, a^{3} c d p\right )} x^{2}\right )} e^{7} - {\left (3 \, {\left (a^{2} c^{2} d^{2} p^{2} + a^{2} c^{2} d^{2} p\right )} x^{3} + 2 \, {\left (a^{3} c d^{2} p^{3} + 9 \, a^{3} c d^{2} p^{2} + 20 \, a^{3} c d^{2} p + 12 \, a^{3} c d^{2}\right )} x\right )} e^{6} + 3 \, {\left (2 \, a c^{3} d^{3} p x^{4} - a^{3} c d^{3} p^{3} - 7 \, a^{3} c d^{3} p^{2} - 14 \, a^{3} c d^{3} p - 8 \, a^{3} c d^{3} - {\left (a^{2} c^{2} d^{3} p^{3} + 6 \, a^{2} c^{2} d^{3} p^{2} + 5 \, a^{2} c^{2} d^{3} p\right )} x^{2}\right )} e^{5} - 3 \, {\left (2 \, c^{4} d^{4} x^{5} - 2 \, {\left (a c^{3} d^{4} p^{2} + 5 \, a c^{3} d^{4} p\right )} x^{3} - 3 \, {\left (a^{2} c^{2} d^{4} p^{2} + 5 \, a^{2} c^{2} d^{4} p + 4 \, a^{2} c^{2} d^{4}\right )} x\right )} e^{4} + 3 \, {\left (a^{2} c^{2} d^{5} p^{3} + 8 \, a^{2} c^{2} d^{5} p^{2} + 19 \, a^{2} c^{2} d^{5} p + 12 \, a^{2} c^{2} d^{5} - 2 \, {\left (c^{4} d^{5} p + 5 \, c^{4} d^{5}\right )} x^{4} + {\left (a c^{3} d^{5} p^{3} + 9 \, a c^{3} d^{5} p^{2} + 20 \, a c^{3} d^{5} p\right )} x^{2}\right )} e^{3} - {\left (3 \, {\left (c^{4} d^{6} p^{2} + 9 \, c^{4} d^{6} p + 20 \, c^{4} d^{6}\right )} x^{3} - 2 \, {\left (a c^{3} d^{6} p^{3} + 6 \, a c^{3} d^{6} p^{2} + 5 \, a c^{3} d^{6} p - 12 \, a c^{3} d^{6}\right )} x\right )} e^{2} - {\left (a c^{3} d^{7} p^{3} + 9 \, a c^{3} d^{7} p^{2} + 26 \, a c^{3} d^{7} p + 24 \, a c^{3} d^{7} + {\left (c^{4} d^{7} p^{3} + 12 \, c^{4} d^{7} p^{2} + 47 \, c^{4} d^{7} p + 60 \, c^{4} d^{7}\right )} x^{2}\right )} e\right )} {\left (c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e\right )}^{p} {\left (x e + d\right )}^{-2 \, p - 5}}{c^{4} d^{8} p^{4} + 10 \, c^{4} d^{8} p^{3} + 35 \, c^{4} d^{8} p^{2} + 50 \, c^{4} d^{8} p + 24 \, c^{4} d^{8} + {\left (a^{4} p^{4} + 10 \, a^{4} p^{3} + 35 \, a^{4} p^{2} + 50 \, a^{4} p + 24 \, a^{4}\right )} e^{8} - 4 \, {\left (a^{3} c d^{2} p^{4} + 10 \, a^{3} c d^{2} p^{3} + 35 \, a^{3} c d^{2} p^{2} + 50 \, a^{3} c d^{2} p + 24 \, a^{3} c d^{2}\right )} e^{6} + 6 \, {\left (a^{2} c^{2} d^{4} p^{4} + 10 \, a^{2} c^{2} d^{4} p^{3} + 35 \, a^{2} c^{2} d^{4} p^{2} + 50 \, a^{2} c^{2} d^{4} p + 24 \, a^{2} c^{2} d^{4}\right )} e^{4} - 4 \, {\left (a c^{3} d^{6} p^{4} + 10 \, a c^{3} d^{6} p^{3} + 35 \, a c^{3} d^{6} p^{2} + 50 \, a c^{3} d^{6} p + 24 \, a c^{3} d^{6}\right )} e^{2}} \end {gather*}

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]

-((a^4*p^3 + 6*a^4*p^2 + 11*a^4*p + 6*a^4)*x*e^8 - (c^4*d^8*p^3 + 9*c^4*d^8*p^2 + 26*c^4*d^8*p + 24*c^4*d^8)*x
+ (a^4*d*p^3 + 6*a^4*d*p^2 + 11*a^4*d*p + 6*a^4*d + (a^3*c*d*p^3 + 3*a^3*c*d*p^2 + 2*a^3*c*d*p)*x^2)*e^7 - (3
*(a^2*c^2*d^2*p^2 + a^2*c^2*d^2*p)*x^3 + 2*(a^3*c*d^2*p^3 + 9*a^3*c*d^2*p^2 + 20*a^3*c*d^2*p + 12*a^3*c*d^2)*x
)*e^6 + 3*(2*a*c^3*d^3*p*x^4 - a^3*c*d^3*p^3 - 7*a^3*c*d^3*p^2 - 14*a^3*c*d^3*p - 8*a^3*c*d^3 - (a^2*c^2*d^3*p
^3 + 6*a^2*c^2*d^3*p^2 + 5*a^2*c^2*d^3*p)*x^2)*e^5 - 3*(2*c^4*d^4*x^5 - 2*(a*c^3*d^4*p^2 + 5*a*c^3*d^4*p)*x^3
- 3*(a^2*c^2*d^4*p^2 + 5*a^2*c^2*d^4*p + 4*a^2*c^2*d^4)*x)*e^4 + 3*(a^2*c^2*d^5*p^3 + 8*a^2*c^2*d^5*p^2 + 19*a
^2*c^2*d^5*p + 12*a^2*c^2*d^5 - 2*(c^4*d^5*p + 5*c^4*d^5)*x^4 + (a*c^3*d^5*p^3 + 9*a*c^3*d^5*p^2 + 20*a*c^3*d^
5*p)*x^2)*e^3 - (3*(c^4*d^6*p^2 + 9*c^4*d^6*p + 20*c^4*d^6)*x^3 - 2*(a*c^3*d^6*p^3 + 6*a*c^3*d^6*p^2 + 5*a*c^3
*d^6*p - 12*a*c^3*d^6)*x)*e^2 - (a*c^3*d^7*p^3 + 9*a*c^3*d^7*p^2 + 26*a*c^3*d^7*p + 24*a*c^3*d^7 + (c^4*d^7*p^
3 + 12*c^4*d^7*p^2 + 47*c^4*d^7*p + 60*c^4*d^7)*x^2)*e)*(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)^p*(x*e + d)^(-
2*p - 5)/(c^4*d^8*p^4 + 10*c^4*d^8*p^3 + 35*c^4*d^8*p^2 + 50*c^4*d^8*p + 24*c^4*d^8 + (a^4*p^4 + 10*a^4*p^3 +
35*a^4*p^2 + 50*a^4*p + 24*a^4)*e^8 - 4*(a^3*c*d^2*p^4 + 10*a^3*c*d^2*p^3 + 35*a^3*c*d^2*p^2 + 50*a^3*c*d^2*p
+ 24*a^3*c*d^2)*e^6 + 6*(a^2*c^2*d^4*p^4 + 10*a^2*c^2*d^4*p^3 + 35*a^2*c^2*d^4*p^2 + 50*a^2*c^2*d^4*p + 24*a^2
*c^2*d^4)*e^4 - 4*(a*c^3*d^6*p^4 + 10*a*c^3*d^6*p^3 + 35*a*c^3*d^6*p^2 + 50*a*c^3*d^6*p + 24*a*c^3*d^6)*e^2)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 4371 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [B]
time = 2.06, size = 1036, normalized size = 3.60 \begin {gather*} {\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p\,\left (\frac {6\,c^4\,d^4\,e^4\,x^5}{{\left (a\,e^2-c\,d^2\right )}^4\,{\left (d+e\,x\right )}^{2\,p+5}\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}-\frac {x\,\left (a^4\,e^8\,p^3+6\,a^4\,e^8\,p^2+11\,a^4\,e^8\,p+6\,a^4\,e^8-2\,a^3\,c\,d^2\,e^6\,p^3-18\,a^3\,c\,d^2\,e^6\,p^2-40\,a^3\,c\,d^2\,e^6\,p-24\,a^3\,c\,d^2\,e^6+9\,a^2\,c^2\,d^4\,e^4\,p^2+45\,a^2\,c^2\,d^4\,e^4\,p+36\,a^2\,c^2\,d^4\,e^4+2\,a\,c^3\,d^6\,e^2\,p^3+12\,a\,c^3\,d^6\,e^2\,p^2+10\,a\,c^3\,d^6\,e^2\,p-24\,a\,c^3\,d^6\,e^2-c^4\,d^8\,p^3-9\,c^4\,d^8\,p^2-26\,c^4\,d^8\,p-24\,c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^4\,{\left (d+e\,x\right )}^{2\,p+5}\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}-\frac {a\,d\,e\,\left (a^3\,e^6\,p^3+6\,a^3\,e^6\,p^2+11\,a^3\,e^6\,p+6\,a^3\,e^6-3\,a^2\,c\,d^2\,e^4\,p^3-21\,a^2\,c\,d^2\,e^4\,p^2-42\,a^2\,c\,d^2\,e^4\,p-24\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2\,p^3+24\,a\,c^2\,d^4\,e^2\,p^2+57\,a\,c^2\,d^4\,e^2\,p+36\,a\,c^2\,d^4\,e^2-c^3\,d^6\,p^3-9\,c^3\,d^6\,p^2-26\,c^3\,d^6\,p-24\,c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^4\,{\left (d+e\,x\right )}^{2\,p+5}\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}+\frac {6\,c^3\,d^3\,e^3\,x^4\,\left (5\,c\,d^2-a\,e^2\,p+c\,d^2\,p\right )}{{\left (a\,e^2-c\,d^2\right )}^4\,{\left (d+e\,x\right )}^{2\,p+5}\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}+\frac {3\,c^2\,d^2\,e^2\,x^3\,\left (a^2\,e^4\,p^2+a^2\,e^4\,p-2\,a\,c\,d^2\,e^2\,p^2-10\,a\,c\,d^2\,e^2\,p+c^2\,d^4\,p^2+9\,c^2\,d^4\,p+20\,c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^4\,{\left (d+e\,x\right )}^{2\,p+5}\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}+\frac {c\,d\,e\,x^2\,\left (-a^3\,e^6\,p^3-3\,a^3\,e^6\,p^2-2\,a^3\,e^6\,p+3\,a^2\,c\,d^2\,e^4\,p^3+18\,a^2\,c\,d^2\,e^4\,p^2+15\,a^2\,c\,d^2\,e^4\,p-3\,a\,c^2\,d^4\,e^2\,p^3-27\,a\,c^2\,d^4\,e^2\,p^2-60\,a\,c^2\,d^4\,e^2\,p+c^3\,d^6\,p^3+12\,c^3\,d^6\,p^2+47\,c^3\,d^6\,p+60\,c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^4\,{\left (d+e\,x\right )}^{2\,p+5}\,\left (p^4+10\,p^3+35\,p^2+50\,p+24\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p/(d + e*x)^(2*p + 5),x)

[Out]

(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p*((6*c^4*d^4*e^4*x^5)/((a*e^2 - c*d^2)^4*(d + e*x)^(2*p + 5)*(50*p +
35*p^2 + 10*p^3 + p^4 + 24)) - (x*(6*a^4*e^8 - 24*c^4*d^8 + 11*a^4*e^8*p - 26*c^4*d^8*p + 6*a^4*e^8*p^2 + a^4*
e^8*p^3 - 9*c^4*d^8*p^2 - c^4*d^8*p^3 - 24*a*c^3*d^6*e^2 - 24*a^3*c*d^2*e^6 + 36*a^2*c^2*d^4*e^4 + 9*a^2*c^2*d
^4*e^4*p^2 + 10*a*c^3*d^6*e^2*p - 40*a^3*c*d^2*e^6*p + 45*a^2*c^2*d^4*e^4*p + 12*a*c^3*d^6*e^2*p^2 - 18*a^3*c*
d^2*e^6*p^2 + 2*a*c^3*d^6*e^2*p^3 - 2*a^3*c*d^2*e^6*p^3))/((a*e^2 - c*d^2)^4*(d + e*x)^(2*p + 5)*(50*p + 35*p^
2 + 10*p^3 + p^4 + 24)) - (a*d*e*(6*a^3*e^6 - 24*c^3*d^6 + 11*a^3*e^6*p - 26*c^3*d^6*p + 6*a^3*e^6*p^2 + a^3*e
^6*p^3 - 9*c^3*d^6*p^2 - c^3*d^6*p^3 + 36*a*c^2*d^4*e^2 - 24*a^2*c*d^2*e^4 + 57*a*c^2*d^4*e^2*p - 42*a^2*c*d^2
*e^4*p + 24*a*c^2*d^4*e^2*p^2 - 21*a^2*c*d^2*e^4*p^2 + 3*a*c^2*d^4*e^2*p^3 - 3*a^2*c*d^2*e^4*p^3))/((a*e^2 - c
*d^2)^4*(d + e*x)^(2*p + 5)*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) + (6*c^3*d^3*e^3*x^4*(5*c*d^2 - a*e^2*p + c*d
^2*p))/((a*e^2 - c*d^2)^4*(d + e*x)^(2*p + 5)*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) + (3*c^2*d^2*e^2*x^3*(20*c^
2*d^4 + a^2*e^4*p + 9*c^2*d^4*p + a^2*e^4*p^2 + c^2*d^4*p^2 - 10*a*c*d^2*e^2*p - 2*a*c*d^2*e^2*p^2))/((a*e^2 -
c*d^2)^4*(d + e*x)^(2*p + 5)*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)) + (c*d*e*x^2*(60*c^3*d^6 - 2*a^3*e^6*p + 47
*c^3*d^6*p - 3*a^3*e^6*p^2 - a^3*e^6*p^3 + 12*c^3*d^6*p^2 + c^3*d^6*p^3 - 60*a*c^2*d^4*e^2*p + 15*a^2*c*d^2*e^
4*p - 27*a*c^2*d^4*e^2*p^2 + 18*a^2*c*d^2*e^4*p^2 - 3*a*c^2*d^4*e^2*p^3 + 3*a^2*c*d^2*e^4*p^3))/((a*e^2 - c*d^
2)^4*(d + e*x)^(2*p + 5)*(50*p + 35*p^2 + 10*p^3 + p^4 + 24)))

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