3.18.32 \(\int (d+e x)^m (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1732]

Optimal. Leaf size=142 \[ \frac {(b d-a e)^4 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {4 b (b d-a e)^3 (d+e x)^{2+m}}{e^5 (2+m)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3+m}}{e^5 (3+m)}-\frac {4 b^3 (b d-a e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {b^4 (d+e x)^{5+m}}{e^5 (5+m)} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \begin {gather*} -\frac {4 b^3 (b d-a e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{m+3}}{e^5 (m+3)}+\frac {(b d-a e)^4 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {4 b (b d-a e)^3 (d+e x)^{m+2}}{e^5 (m+2)}+\frac {b^4 (d+e x)^{m+5}}{e^5 (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^m \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (d+e x)^m}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{1+m}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{2+m}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{3+m}}{e^4}+\frac {b^4 (d+e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac {(b d-a e)^4 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {4 b (b d-a e)^3 (d+e x)^{2+m}}{e^5 (2+m)}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3+m}}{e^5 (3+m)}-\frac {4 b^3 (b d-a e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {b^4 (d+e x)^{5+m}}{e^5 (5+m)}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 121, normalized size = 0.85 \begin {gather*} \frac {(d+e x)^{1+m} \left (\frac {(b d-a e)^4}{1+m}-\frac {4 b (b d-a e)^3 (d+e x)}{2+m}+\frac {6 b^2 (b d-a e)^2 (d+e x)^2}{3+m}-\frac {4 b^3 (b d-a e) (d+e x)^3}{4+m}+\frac {b^4 (d+e x)^4}{5+m}\right )}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(734\) vs. \(2(142)=284\).
time = 0.76, size = 735, normalized size = 5.18 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

b^4/(5+m)*x^5*exp(m*ln(e*x+d))+d*(a^4*e^4*m^4+14*a^4*e^4*m^3-4*a^3*b*d*e^3*m^3+71*a^4*e^4*m^2-48*a^3*b*d*e^3*m
^2+12*a^2*b^2*d^2*e^2*m^2+154*a^4*e^4*m-188*a^3*b*d*e^3*m+108*a^2*b^2*d^2*e^2*m-24*a*b^3*d^3*e*m+120*a^4*e^4-2
40*a^3*b*d*e^3+240*a^2*b^2*d^2*e^2-120*a*b^3*d^3*e+24*b^4*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)*exp(m
*ln(e*x+d))+(a^4*e^4*m^4+4*a^3*b*d*e^3*m^4+14*a^4*e^4*m^3+48*a^3*b*d*e^3*m^3-12*a^2*b^2*d^2*e^2*m^3+71*a^4*e^4
*m^2+188*a^3*b*d*e^3*m^2-108*a^2*b^2*d^2*e^2*m^2+24*a*b^3*d^3*e*m^2+154*a^4*e^4*m+240*a^3*b*d*e^3*m-240*a^2*b^
2*d^2*e^2*m+120*a*b^3*d^3*e*m-24*b^4*d^4*m+120*a^4*e^4)/e^4/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)*x*exp(m*ln(e
*x+d))+b^3*(4*a*e*m+b*d*m+20*a*e)/e/(m^2+9*m+20)*x^4*exp(m*ln(e*x+d))+2*(3*a^2*e^2*m^2+2*a*b*d*e*m^2+27*a^2*e^
2*m+10*a*b*d*e*m-2*b^2*d^2*m+60*a^2*e^2)*b^2/e^2/(m^3+12*m^2+47*m+60)*x^3*exp(m*ln(e*x+d))+2*(2*a^3*e^3*m^3+3*
a^2*b*d*e^2*m^3+24*a^3*e^3*m^2+27*a^2*b*d*e^2*m^2-6*a*b^2*d^2*e*m^2+94*a^3*e^3*m+60*a^2*b*d*e^2*m-30*a*b^2*d^2
*e*m+6*b^3*d^3*m+120*a^3*e^3)*b/e^3/(m^4+14*m^3+71*m^2+154*m+120)*x^2*exp(m*ln(e*x+d))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (146) = 292\).
time = 0.30, size = 394, normalized size = 2.77 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} a^{4} e^{\left (-1\right )}}{m + 1} + \frac {4 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} a^{3} b e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {6 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} a^{2} b^{2} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {4 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} e^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d x^{3} e^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} x^{2} e^{2} + 6 \, d^{3} m x e - 6 \, d^{4}\right )} a b^{3} e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} e^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d x^{4} e^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} x^{3} e^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} x^{2} e^{2} - 24 \, d^{4} m x e + 24 \, d^{5}\right )} b^{4} e^{\left (m \log \left (x e + d\right ) - 5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

(x*e + d)^(m + 1)*a^4*e^(-1)/(m + 1) + 4*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*a^3*b*e^(m*log(x*e + d) - 2)/(m^2 +
3*m + 2) + 6*((m^2 + 3*m + 2)*x^3*e^3 + (m^2 + m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)*a^2*b^2*e^(m*log(x*e + d)
- 3)/(m^3 + 6*m^2 + 11*m + 6) + 4*((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 +
m)*d^2*x^2*e^2 + 6*d^3*m*x*e - 6*d^4)*a*b^3*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + ((m^
4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*
x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x*e + 24*d^5)*b^4*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3
+ 225*m^2 + 274*m + 120)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 771 vs. \(2 (146) = 292\).
time = 2.94, size = 771, normalized size = 5.43 \begin {gather*} \frac {{\left (24 \, b^{4} d^{5} + {\left ({\left (b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}\right )} x^{5} + 4 \, {\left (a b^{3} m^{4} + 11 \, a b^{3} m^{3} + 41 \, a b^{3} m^{2} + 61 \, a b^{3} m + 30 \, a b^{3}\right )} x^{4} + 6 \, {\left (a^{2} b^{2} m^{4} + 12 \, a^{2} b^{2} m^{3} + 49 \, a^{2} b^{2} m^{2} + 78 \, a^{2} b^{2} m + 40 \, a^{2} b^{2}\right )} x^{3} + 4 \, {\left (a^{3} b m^{4} + 13 \, a^{3} b m^{3} + 59 \, a^{3} b m^{2} + 107 \, a^{3} b m + 60 \, a^{3} b\right )} x^{2} + {\left (a^{4} m^{4} + 14 \, a^{4} m^{3} + 71 \, a^{4} m^{2} + 154 \, a^{4} m + 120 \, a^{4}\right )} x\right )} e^{5} + {\left (a^{4} d m^{4} + 14 \, a^{4} d m^{3} + 71 \, a^{4} d m^{2} + 154 \, a^{4} d m + 120 \, a^{4} d + {\left (b^{4} d m^{4} + 6 \, b^{4} d m^{3} + 11 \, b^{4} d m^{2} + 6 \, b^{4} d m\right )} x^{4} + 4 \, {\left (a b^{3} d m^{4} + 8 \, a b^{3} d m^{3} + 17 \, a b^{3} d m^{2} + 10 \, a b^{3} d m\right )} x^{3} + 6 \, {\left (a^{2} b^{2} d m^{4} + 10 \, a^{2} b^{2} d m^{3} + 29 \, a^{2} b^{2} d m^{2} + 20 \, a^{2} b^{2} d m\right )} x^{2} + 4 \, {\left (a^{3} b d m^{4} + 12 \, a^{3} b d m^{3} + 47 \, a^{3} b d m^{2} + 60 \, a^{3} b d m\right )} x\right )} e^{4} - 4 \, {\left (a^{3} b d^{2} m^{3} + 12 \, a^{3} b d^{2} m^{2} + 47 \, a^{3} b d^{2} m + 60 \, a^{3} b d^{2} + {\left (b^{4} d^{2} m^{3} + 3 \, b^{4} d^{2} m^{2} + 2 \, b^{4} d^{2} m\right )} x^{3} + 3 \, {\left (a b^{3} d^{2} m^{3} + 6 \, a b^{3} d^{2} m^{2} + 5 \, a b^{3} d^{2} m\right )} x^{2} + 3 \, {\left (a^{2} b^{2} d^{2} m^{3} + 9 \, a^{2} b^{2} d^{2} m^{2} + 20 \, a^{2} b^{2} d^{2} m\right )} x\right )} e^{3} + 12 \, {\left (a^{2} b^{2} d^{3} m^{2} + 9 \, a^{2} b^{2} d^{3} m + 20 \, a^{2} b^{2} d^{3} + {\left (b^{4} d^{3} m^{2} + b^{4} d^{3} m\right )} x^{2} + 2 \, {\left (a b^{3} d^{3} m^{2} + 5 \, a b^{3} d^{3} m\right )} x\right )} e^{2} - 24 \, {\left (b^{4} d^{4} m x + a b^{3} d^{4} m + 5 \, a b^{3} d^{4}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

(24*b^4*d^5 + ((b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m + 24*b^4)*x^5 + 4*(a*b^3*m^4 + 11*a*b^3*m^3 + 41*
a*b^3*m^2 + 61*a*b^3*m + 30*a*b^3)*x^4 + 6*(a^2*b^2*m^4 + 12*a^2*b^2*m^3 + 49*a^2*b^2*m^2 + 78*a^2*b^2*m + 40*
a^2*b^2)*x^3 + 4*(a^3*b*m^4 + 13*a^3*b*m^3 + 59*a^3*b*m^2 + 107*a^3*b*m + 60*a^3*b)*x^2 + (a^4*m^4 + 14*a^4*m^
3 + 71*a^4*m^2 + 154*a^4*m + 120*a^4)*x)*e^5 + (a^4*d*m^4 + 14*a^4*d*m^3 + 71*a^4*d*m^2 + 154*a^4*d*m + 120*a^
4*d + (b^4*d*m^4 + 6*b^4*d*m^3 + 11*b^4*d*m^2 + 6*b^4*d*m)*x^4 + 4*(a*b^3*d*m^4 + 8*a*b^3*d*m^3 + 17*a*b^3*d*m
^2 + 10*a*b^3*d*m)*x^3 + 6*(a^2*b^2*d*m^4 + 10*a^2*b^2*d*m^3 + 29*a^2*b^2*d*m^2 + 20*a^2*b^2*d*m)*x^2 + 4*(a^3
*b*d*m^4 + 12*a^3*b*d*m^3 + 47*a^3*b*d*m^2 + 60*a^3*b*d*m)*x)*e^4 - 4*(a^3*b*d^2*m^3 + 12*a^3*b*d^2*m^2 + 47*a
^3*b*d^2*m + 60*a^3*b*d^2 + (b^4*d^2*m^3 + 3*b^4*d^2*m^2 + 2*b^4*d^2*m)*x^3 + 3*(a*b^3*d^2*m^3 + 6*a*b^3*d^2*m
^2 + 5*a*b^3*d^2*m)*x^2 + 3*(a^2*b^2*d^2*m^3 + 9*a^2*b^2*d^2*m^2 + 20*a^2*b^2*d^2*m)*x)*e^3 + 12*(a^2*b^2*d^3*
m^2 + 9*a^2*b^2*d^3*m + 20*a^2*b^2*d^3 + (b^4*d^3*m^2 + b^4*d^3*m)*x^2 + 2*(a*b^3*d^3*m^2 + 5*a*b^3*d^3*m)*x)*
e^2 - 24*(b^4*d^4*m*x + a*b^3*d^4*m + 5*a*b^3*d^4)*e)*(x*e + d)^m*e^(-5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 27
4*m + 120)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 8719 vs. \(2 (124) = 248\).
time = 1.99, size = 8719, normalized size = 61.40 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Piecewise((d**m*(a**4*x + 2*a**3*b*x**2 + 2*a**2*b**2*x**3 + a*b**3*x**4 + b**4*x**5/5), Eq(e, 0)), (-3*a**4*e
**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 4*a**3*b*d*e**3/(12*
d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 16*a**3*b*e**4*x/(12*d**4*e*
*5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*a**2*b**2*d**2*e**2/(12*d**4*e**5
+ 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 24*a**2*b**2*d*e**3*x/(12*d**4*e**5 +
48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 36*a**2*b**2*e**4*x**2/(12*d**4*e**5 +
48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 12*a*b**3*d**3*e/(12*d**4*e**5 + 48*d**3
*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 48*a*b**3*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e
**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 72*a*b**3*d*e**3*x**2/(12*d**4*e**5 + 48*d**3*e**
6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 48*a*b**3*e**4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x
+ 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*b**4*d**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*
x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 25*b**4*d**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2
*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*b**4*d**3*e*x*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 7
2*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 88*b**4*d**3*e*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*
e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 72*b**4*d**2*e**2*x**2*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*
x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 108*b**4*d**2*e**2*x**2/(12*d**4*e**5 + 48*d**3*e**6*
x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*b**4*d*e**3*x**3*log(d/e + x)/(12*d**4*e**5 + 48*d
**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*b**4*d*e**3*x**3/(12*d**4*e**5 + 48*d**3*
e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*b**4*e**4*x**4*log(d/e + x)/(12*d**4*e**5 + 4
8*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4), Eq(m, -5)), (-a**4*e**4/(3*d**3*e**5 + 9*d
**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 2*a**3*b*d*e**3/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e
**8*x**3) - 6*a**3*b*e**4*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 6*a**2*b**2*d**2*e**
2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 18*a**2*b**2*d*e**3*x/(3*d**3*e**5 + 9*d**2*e*
*6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 18*a**2*b**2*e**4*x**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*
e**8*x**3) + 12*a*b**3*d**3*e*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 22*a*
b**3*d**3*e/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 36*a*b**3*d**2*e**2*x*log(d/e + x)/(
3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 54*a*b**3*d**2*e**2*x/(3*d**3*e**5 + 9*d**2*e**6*
x + 9*d*e**7*x**2 + 3*e**8*x**3) + 36*a*b**3*d*e**3*x**2*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*
x**2 + 3*e**8*x**3) + 36*a*b**3*d*e**3*x**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 12*a
*b**3*e**4*x**3*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 12*b**4*d**4*log(d/
e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 22*b**4*d**4/(3*d**3*e**5 + 9*d**2*e**6*x
+ 9*d*e**7*x**2 + 3*e**8*x**3) - 36*b**4*d**3*e*x*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 +
3*e**8*x**3) - 54*b**4*d**3*e*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*b**4*d**2*e*
*2*x**2*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*b**4*d**2*e**2*x**2/(3*d
**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 12*b**4*d*e**3*x**3*log(d/e + x)/(3*d**3*e**5 + 9*d*
*2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 3*b**4*e**4*x**4/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e
**8*x**3), Eq(m, -4)), (-a**4*e**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*a**3*b*d*e**3/(2*d**2*e**5 + 4
*d*e**6*x + 2*e**7*x**2) - 8*a**3*b*e**4*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*a**2*b**2*d**2*e**2*l
og(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 18*a**2*b**2*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e*
*7*x**2) + 24*a**2*b**2*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*a**2*b**2*d*e**3*x
/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*a**2*b**2*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*
e**7*x**2) - 24*a*b**3*d**3*e*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 36*a*b**3*d**3*e/(2*d**2
*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 48*a*b**3*d**2*e**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2
) - 48*a*b**3*d**2*e**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*a*b**3*d*e**3*x**2*log(d/e + x)/(2*d**
2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 8*a*b**3*e**4*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*b**4*d**
4*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1529 vs. \(2 (146) = 292\).
time = 0.95, size = 1529, normalized size = 10.77 \begin {gather*} \frac {{\left (x e + d\right )}^{m} b^{4} m^{4} x^{5} e^{5} + {\left (x e + d\right )}^{m} b^{4} d m^{4} x^{4} e^{4} + 4 \, {\left (x e + d\right )}^{m} a b^{3} m^{4} x^{4} e^{5} + 10 \, {\left (x e + d\right )}^{m} b^{4} m^{3} x^{5} e^{5} + 4 \, {\left (x e + d\right )}^{m} a b^{3} d m^{4} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} b^{4} d m^{3} x^{4} e^{4} - 4 \, {\left (x e + d\right )}^{m} b^{4} d^{2} m^{3} x^{3} e^{3} + 6 \, {\left (x e + d\right )}^{m} a^{2} b^{2} m^{4} x^{3} e^{5} + 44 \, {\left (x e + d\right )}^{m} a b^{3} m^{3} x^{4} e^{5} + 35 \, {\left (x e + d\right )}^{m} b^{4} m^{2} x^{5} e^{5} + 6 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d m^{4} x^{2} e^{4} + 32 \, {\left (x e + d\right )}^{m} a b^{3} d m^{3} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} b^{4} d m^{2} x^{4} e^{4} - 12 \, {\left (x e + d\right )}^{m} a b^{3} d^{2} m^{3} x^{2} e^{3} - 12 \, {\left (x e + d\right )}^{m} b^{4} d^{2} m^{2} x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} b^{4} d^{3} m^{2} x^{2} e^{2} + 4 \, {\left (x e + d\right )}^{m} a^{3} b m^{4} x^{2} e^{5} + 72 \, {\left (x e + d\right )}^{m} a^{2} b^{2} m^{3} x^{3} e^{5} + 164 \, {\left (x e + d\right )}^{m} a b^{3} m^{2} x^{4} e^{5} + 50 \, {\left (x e + d\right )}^{m} b^{4} m x^{5} e^{5} + 4 \, {\left (x e + d\right )}^{m} a^{3} b d m^{4} x e^{4} + 60 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d m^{3} x^{2} e^{4} + 68 \, {\left (x e + d\right )}^{m} a b^{3} d m^{2} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} b^{4} d m x^{4} e^{4} - 12 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{2} m^{3} x e^{3} - 72 \, {\left (x e + d\right )}^{m} a b^{3} d^{2} m^{2} x^{2} e^{3} - 8 \, {\left (x e + d\right )}^{m} b^{4} d^{2} m x^{3} e^{3} + 24 \, {\left (x e + d\right )}^{m} a b^{3} d^{3} m^{2} x e^{2} + 12 \, {\left (x e + d\right )}^{m} b^{4} d^{3} m x^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} b^{4} d^{4} m x e + {\left (x e + d\right )}^{m} a^{4} m^{4} x e^{5} + 52 \, {\left (x e + d\right )}^{m} a^{3} b m^{3} x^{2} e^{5} + 294 \, {\left (x e + d\right )}^{m} a^{2} b^{2} m^{2} x^{3} e^{5} + 244 \, {\left (x e + d\right )}^{m} a b^{3} m x^{4} e^{5} + 24 \, {\left (x e + d\right )}^{m} b^{4} x^{5} e^{5} + {\left (x e + d\right )}^{m} a^{4} d m^{4} e^{4} + 48 \, {\left (x e + d\right )}^{m} a^{3} b d m^{3} x e^{4} + 174 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d m^{2} x^{2} e^{4} + 40 \, {\left (x e + d\right )}^{m} a b^{3} d m x^{3} e^{4} - 4 \, {\left (x e + d\right )}^{m} a^{3} b d^{2} m^{3} e^{3} - 108 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{2} m^{2} x e^{3} - 60 \, {\left (x e + d\right )}^{m} a b^{3} d^{2} m x^{2} e^{3} + 12 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{3} m^{2} e^{2} + 120 \, {\left (x e + d\right )}^{m} a b^{3} d^{3} m x e^{2} - 24 \, {\left (x e + d\right )}^{m} a b^{3} d^{4} m e + 24 \, {\left (x e + d\right )}^{m} b^{4} d^{5} + 14 \, {\left (x e + d\right )}^{m} a^{4} m^{3} x e^{5} + 236 \, {\left (x e + d\right )}^{m} a^{3} b m^{2} x^{2} e^{5} + 468 \, {\left (x e + d\right )}^{m} a^{2} b^{2} m x^{3} e^{5} + 120 \, {\left (x e + d\right )}^{m} a b^{3} x^{4} e^{5} + 14 \, {\left (x e + d\right )}^{m} a^{4} d m^{3} e^{4} + 188 \, {\left (x e + d\right )}^{m} a^{3} b d m^{2} x e^{4} + 120 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d m x^{2} e^{4} - 48 \, {\left (x e + d\right )}^{m} a^{3} b d^{2} m^{2} e^{3} - 240 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{2} m x e^{3} + 108 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{3} m e^{2} - 120 \, {\left (x e + d\right )}^{m} a b^{3} d^{4} e + 71 \, {\left (x e + d\right )}^{m} a^{4} m^{2} x e^{5} + 428 \, {\left (x e + d\right )}^{m} a^{3} b m x^{2} e^{5} + 240 \, {\left (x e + d\right )}^{m} a^{2} b^{2} x^{3} e^{5} + 71 \, {\left (x e + d\right )}^{m} a^{4} d m^{2} e^{4} + 240 \, {\left (x e + d\right )}^{m} a^{3} b d m x e^{4} - 188 \, {\left (x e + d\right )}^{m} a^{3} b d^{2} m e^{3} + 240 \, {\left (x e + d\right )}^{m} a^{2} b^{2} d^{3} e^{2} + 154 \, {\left (x e + d\right )}^{m} a^{4} m x e^{5} + 240 \, {\left (x e + d\right )}^{m} a^{3} b x^{2} e^{5} + 154 \, {\left (x e + d\right )}^{m} a^{4} d m e^{4} - 240 \, {\left (x e + d\right )}^{m} a^{3} b d^{2} e^{3} + 120 \, {\left (x e + d\right )}^{m} a^{4} x e^{5} + 120 \, {\left (x e + d\right )}^{m} a^{4} d e^{4}}{m^{5} e^{5} + 15 \, m^{4} e^{5} + 85 \, m^{3} e^{5} + 225 \, m^{2} e^{5} + 274 \, m e^{5} + 120 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*b^4*m^4*x^5*e^5 + (x*e + d)^m*b^4*d*m^4*x^4*e^4 + 4*(x*e + d)^m*a*b^3*m^4*x^4*e^5 + 10*(x*e + d)^
m*b^4*m^3*x^5*e^5 + 4*(x*e + d)^m*a*b^3*d*m^4*x^3*e^4 + 6*(x*e + d)^m*b^4*d*m^3*x^4*e^4 - 4*(x*e + d)^m*b^4*d^
2*m^3*x^3*e^3 + 6*(x*e + d)^m*a^2*b^2*m^4*x^3*e^5 + 44*(x*e + d)^m*a*b^3*m^3*x^4*e^5 + 35*(x*e + d)^m*b^4*m^2*
x^5*e^5 + 6*(x*e + d)^m*a^2*b^2*d*m^4*x^2*e^4 + 32*(x*e + d)^m*a*b^3*d*m^3*x^3*e^4 + 11*(x*e + d)^m*b^4*d*m^2*
x^4*e^4 - 12*(x*e + d)^m*a*b^3*d^2*m^3*x^2*e^3 - 12*(x*e + d)^m*b^4*d^2*m^2*x^3*e^3 + 12*(x*e + d)^m*b^4*d^3*m
^2*x^2*e^2 + 4*(x*e + d)^m*a^3*b*m^4*x^2*e^5 + 72*(x*e + d)^m*a^2*b^2*m^3*x^3*e^5 + 164*(x*e + d)^m*a*b^3*m^2*
x^4*e^5 + 50*(x*e + d)^m*b^4*m*x^5*e^5 + 4*(x*e + d)^m*a^3*b*d*m^4*x*e^4 + 60*(x*e + d)^m*a^2*b^2*d*m^3*x^2*e^
4 + 68*(x*e + d)^m*a*b^3*d*m^2*x^3*e^4 + 6*(x*e + d)^m*b^4*d*m*x^4*e^4 - 12*(x*e + d)^m*a^2*b^2*d^2*m^3*x*e^3
- 72*(x*e + d)^m*a*b^3*d^2*m^2*x^2*e^3 - 8*(x*e + d)^m*b^4*d^2*m*x^3*e^3 + 24*(x*e + d)^m*a*b^3*d^3*m^2*x*e^2
+ 12*(x*e + d)^m*b^4*d^3*m*x^2*e^2 - 24*(x*e + d)^m*b^4*d^4*m*x*e + (x*e + d)^m*a^4*m^4*x*e^5 + 52*(x*e + d)^m
*a^3*b*m^3*x^2*e^5 + 294*(x*e + d)^m*a^2*b^2*m^2*x^3*e^5 + 244*(x*e + d)^m*a*b^3*m*x^4*e^5 + 24*(x*e + d)^m*b^
4*x^5*e^5 + (x*e + d)^m*a^4*d*m^4*e^4 + 48*(x*e + d)^m*a^3*b*d*m^3*x*e^4 + 174*(x*e + d)^m*a^2*b^2*d*m^2*x^2*e
^4 + 40*(x*e + d)^m*a*b^3*d*m*x^3*e^4 - 4*(x*e + d)^m*a^3*b*d^2*m^3*e^3 - 108*(x*e + d)^m*a^2*b^2*d^2*m^2*x*e^
3 - 60*(x*e + d)^m*a*b^3*d^2*m*x^2*e^3 + 12*(x*e + d)^m*a^2*b^2*d^3*m^2*e^2 + 120*(x*e + d)^m*a*b^3*d^3*m*x*e^
2 - 24*(x*e + d)^m*a*b^3*d^4*m*e + 24*(x*e + d)^m*b^4*d^5 + 14*(x*e + d)^m*a^4*m^3*x*e^5 + 236*(x*e + d)^m*a^3
*b*m^2*x^2*e^5 + 468*(x*e + d)^m*a^2*b^2*m*x^3*e^5 + 120*(x*e + d)^m*a*b^3*x^4*e^5 + 14*(x*e + d)^m*a^4*d*m^3*
e^4 + 188*(x*e + d)^m*a^3*b*d*m^2*x*e^4 + 120*(x*e + d)^m*a^2*b^2*d*m*x^2*e^4 - 48*(x*e + d)^m*a^3*b*d^2*m^2*e
^3 - 240*(x*e + d)^m*a^2*b^2*d^2*m*x*e^3 + 108*(x*e + d)^m*a^2*b^2*d^3*m*e^2 - 120*(x*e + d)^m*a*b^3*d^4*e + 7
1*(x*e + d)^m*a^4*m^2*x*e^5 + 428*(x*e + d)^m*a^3*b*m*x^2*e^5 + 240*(x*e + d)^m*a^2*b^2*x^3*e^5 + 71*(x*e + d)
^m*a^4*d*m^2*e^4 + 240*(x*e + d)^m*a^3*b*d*m*x*e^4 - 188*(x*e + d)^m*a^3*b*d^2*m*e^3 + 240*(x*e + d)^m*a^2*b^2
*d^3*e^2 + 154*(x*e + d)^m*a^4*m*x*e^5 + 240*(x*e + d)^m*a^3*b*x^2*e^5 + 154*(x*e + d)^m*a^4*d*m*e^4 - 240*(x*
e + d)^m*a^3*b*d^2*e^3 + 120*(x*e + d)^m*a^4*x*e^5 + 120*(x*e + d)^m*a^4*d*e^4)/(m^5*e^5 + 15*m^4*e^5 + 85*m^3
*e^5 + 225*m^2*e^5 + 274*m*e^5 + 120*e^5)

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Mupad [B]
time = 1.04, size = 831, normalized size = 5.85 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (a^4\,d\,e^4\,m^4+14\,a^4\,d\,e^4\,m^3+71\,a^4\,d\,e^4\,m^2+154\,a^4\,d\,e^4\,m+120\,a^4\,d\,e^4-4\,a^3\,b\,d^2\,e^3\,m^3-48\,a^3\,b\,d^2\,e^3\,m^2-188\,a^3\,b\,d^2\,e^3\,m-240\,a^3\,b\,d^2\,e^3+12\,a^2\,b^2\,d^3\,e^2\,m^2+108\,a^2\,b^2\,d^3\,e^2\,m+240\,a^2\,b^2\,d^3\,e^2-24\,a\,b^3\,d^4\,e\,m-120\,a\,b^3\,d^4\,e+24\,b^4\,d^5\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (a^4\,e^5\,m^4+14\,a^4\,e^5\,m^3+71\,a^4\,e^5\,m^2+154\,a^4\,e^5\,m+120\,a^4\,e^5+4\,a^3\,b\,d\,e^4\,m^4+48\,a^3\,b\,d\,e^4\,m^3+188\,a^3\,b\,d\,e^4\,m^2+240\,a^3\,b\,d\,e^4\,m-12\,a^2\,b^2\,d^2\,e^3\,m^3-108\,a^2\,b^2\,d^2\,e^3\,m^2-240\,a^2\,b^2\,d^2\,e^3\,m+24\,a\,b^3\,d^3\,e^2\,m^2+120\,a\,b^3\,d^3\,e^2\,m-24\,b^4\,d^4\,e\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {b^4\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {2\,b^2\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (3\,a^2\,e^2\,m^2+27\,a^2\,e^2\,m+60\,a^2\,e^2+2\,a\,b\,d\,e\,m^2+10\,a\,b\,d\,e\,m-2\,b^2\,d^2\,m\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {b^3\,x^4\,{\left (d+e\,x\right )}^m\,\left (20\,a\,e+4\,a\,e\,m+b\,d\,m\right )\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {2\,b\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (2\,a^3\,e^3\,m^3+24\,a^3\,e^3\,m^2+94\,a^3\,e^3\,m+120\,a^3\,e^3+3\,a^2\,b\,d\,e^2\,m^3+27\,a^2\,b\,d\,e^2\,m^2+60\,a^2\,b\,d\,e^2\,m-6\,a\,b^2\,d^2\,e\,m^2-30\,a\,b^2\,d^2\,e\,m+6\,b^3\,d^3\,m\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^m*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

((d + e*x)^m*(24*b^4*d^5 + 120*a^4*d*e^4 - 240*a^3*b*d^2*e^3 + 71*a^4*d*e^4*m^2 + 14*a^4*d*e^4*m^3 + a^4*d*e^4
*m^4 + 240*a^2*b^2*d^3*e^2 - 120*a*b^3*d^4*e + 154*a^4*d*e^4*m - 24*a*b^3*d^4*e*m + 12*a^2*b^2*d^3*e^2*m^2 - 1
88*a^3*b*d^2*e^3*m + 108*a^2*b^2*d^3*e^2*m - 48*a^3*b*d^2*e^3*m^2 - 4*a^3*b*d^2*e^3*m^3))/(e^5*(274*m + 225*m^
2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x*(d + e*x)^m*(120*a^4*e^5 + 154*a^4*e^5*m + 71*a^4*e^5*m^2 + 14*a^4*e^5*
m^3 + a^4*e^5*m^4 - 24*b^4*d^4*e*m + 240*a^3*b*d*e^4*m - 108*a^2*b^2*d^2*e^3*m^2 - 12*a^2*b^2*d^2*e^3*m^3 + 12
0*a*b^3*d^3*e^2*m + 188*a^3*b*d*e^4*m^2 + 48*a^3*b*d*e^4*m^3 + 4*a^3*b*d*e^4*m^4 - 240*a^2*b^2*d^2*e^3*m + 24*
a*b^3*d^3*e^2*m^2))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (b^4*x^5*(d + e*x)^m*(50*m + 35*m^
2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (2*b^2*x^3*(d + e*x)^m*(3*m + m^2 +
2)*(60*a^2*e^2 + 27*a^2*e^2*m - 2*b^2*d^2*m + 3*a^2*e^2*m^2 + 10*a*b*d*e*m + 2*a*b*d*e*m^2))/(e^2*(274*m + 225
*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (b^3*x^4*(d + e*x)^m*(20*a*e + 4*a*e*m + b*d*m)*(11*m + 6*m^2 + m^3 + 6
))/(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (2*b*x^2*(m + 1)*(d + e*x)^m*(120*a^3*e^3 + 94*a^3*e^
3*m + 6*b^3*d^3*m + 24*a^3*e^3*m^2 + 2*a^3*e^3*m^3 - 30*a*b^2*d^2*e*m + 60*a^2*b*d*e^2*m - 6*a*b^2*d^2*e*m^2 +
27*a^2*b*d*e^2*m^2 + 3*a^2*b*d*e^2*m^3))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))

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