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

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

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Rubi [A]  time = 0.16, 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 = {658, 650} \begin {gather*} \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} \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 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]

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]

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 = 0.14, size = 217, normalized size = 0.75 \begin {gather*} \frac {(d+e x)^{-2 p-5} ((d+e x) (a e+c d x))^{p+1} \left (-a^3 e^6 \left (p^3+6 p^2+11 p+6\right )+3 a^2 c d e^4 \left (p^2+3 p+2\right ) (d (p+4)+e x)-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 (d^3 \left (p^3+9 p^2+26 p+24\right )+3 d^2 e \left (p^2+7 p+12\right ) x+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} \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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IntegrateAlgebraic [F]  time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.46, size = 1051, normalized size = 3.65 \begin {gather*} \frac {{\left (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 \, {\left (5 \, c^{4} d^{5} e^{3} + {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} p\right )} x^{4} + {\left (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}\right )} p^{3} + 3 \, {\left (20 \, c^{4} d^{6} e^{2} + {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} p^{2} + {\left (9 \, c^{4} d^{6} e^{2} - 10 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} p\right )} x^{3} + 3 \, {\left (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}\right )} p^{2} + {\left (60 \, c^{4} d^{7} e + {\left (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}\right )} p^{3} + 3 \, {\left (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}\right )} p^{2} + {\left (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}\right )} p\right )} x^{2} + {\left (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}\right )} p + {\left (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} + {\left (c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} + 2 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\right )} p^{3} + 3 \, {\left (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}\right )} p^{2} + {\left (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}\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 - 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} + {\left (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}\right )} p^{4} + 10 \, {\left (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}\right )} p^{3} + 35 \, {\left (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}\right )} p^{2} + 50 \, {\left (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}\right )} p} \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]

(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {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*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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maple [B]  time = 0.06, size = 745, normalized size = 2.59 \begin {gather*} -\frac {\left (c d x +a e \right ) \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 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 (e x +d \right )^{-2 p -4} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*d*x+a*e)*(e*x+d)^(-2*p-4)*(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*} \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 {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*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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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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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