3.26.51 \(\int (d+e x)^m (a+b x+c x^2)^2 \, dx\) [2551]

Optimal. Leaf size=178 \[ \frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {2 c (2 c d-b e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \begin {gather*} \frac {(d+e x)^{m+3} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 (m+3)}+\frac {(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^5 (m+1)}-\frac {2 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )}{e^5 (m+2)}-\frac {2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]
time = 0.35, size = 178, normalized size = 1.00 \begin {gather*} \frac {(d+e x)^{1+m} \left (-\frac {2 (d+e x) (-6 c d+b e (7+m)+2 c e (4+m) x) (a+x (b+c x))}{e^2 (4+m) (5+m)}+(a+x (b+c x))^2+\frac {2 (d+e x) \left (\frac {6 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )}{2+m}-\frac {\left (12 c^2 d^2-b^2 e^2 (1+m)+4 c e (-3 b d+a e (4+m))\right ) (d+e x)}{3+m}\right )}{e^4 (4+m) (5+m)}\right )}{e (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*((-2*(d + e*x)*(-6*c*d + b*e*(7 + m) + 2*c*e*(4 + m)*x)*(a + x*(b + c*x)))/(e^2*(4 + m)*(5
+ m)) + (a + x*(b + c*x))^2 + (2*(d + e*x)*((6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(2 + m) - ((12*c^2*d^
2 - b^2*e^2*(1 + m) + 4*c*e*(-3*b*d + a*e*(4 + m)))*(d + e*x))/(3 + m)))/(e^4*(4 + m)*(5 + m))))/(e*(1 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(798\) vs. \(2(178)=356\).
time = 0.81, size = 799, normalized size = 4.49 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2/(5+m)*x^5*exp(m*ln(e*x+d))+d*(a^2*e^4*m^4+14*a^2*e^4*m^3-2*a*b*d*e^3*m^3+71*a^2*e^4*m^2-24*a*b*d*e^3*m^2+4
*a*c*d^2*e^2*m^2+2*b^2*d^2*e^2*m^2+154*a^2*e^4*m-94*a*b*d*e^3*m+36*a*c*d^2*e^2*m+18*b^2*d^2*e^2*m-12*b*c*d^3*e
*m+120*a^2*e^4-120*a*b*d*e^3+80*a*c*d^2*e^2+40*b^2*d^2*e^2-60*b*c*d^3*e+24*c^2*d^4)/e^5/(m^5+15*m^4+85*m^3+225
*m^2+274*m+120)*exp(m*ln(e*x+d))+(2*a*c*e^2*m^2+b^2*e^2*m^2+2*b*c*d*e*m^2+18*a*c*e^2*m+9*b^2*e^2*m+10*b*c*d*e*
m-4*c^2*d^2*m+40*a*c*e^2+20*b^2*e^2)/e^2/(m^3+12*m^2+47*m+60)*x^3*exp(m*ln(e*x+d))+(2*a*b*e^3*m^3+2*a*c*d*e^2*
m^3+b^2*d*e^2*m^3+24*a*b*e^3*m^2+18*a*c*d*e^2*m^2+9*b^2*d*e^2*m^2-6*b*c*d^2*e*m^2+94*a*b*e^3*m+40*a*c*d*e^2*m+
20*b^2*d*e^2*m-30*b*c*d^2*e*m+12*c^2*d^3*m+120*a*b*e^3)/e^3/(m^4+14*m^3+71*m^2+154*m+120)*x^2*exp(m*ln(e*x+d))
+(a^2*e^4*m^4+2*a*b*d*e^3*m^4+14*a^2*e^4*m^3+24*a*b*d*e^3*m^3-4*a*c*d^2*e^2*m^3-2*b^2*d^2*e^2*m^3+71*a^2*e^4*m
^2+94*a*b*d*e^3*m^2-36*a*c*d^2*e^2*m^2-18*b^2*d^2*e^2*m^2+12*b*c*d^3*e*m^2+154*a^2*e^4*m+120*a*b*d*e^3*m-80*a*
c*d^2*e^2*m-40*b^2*d^2*e^2*m+60*b*c*d^3*e*m-24*c^2*d^4*m+120*a^2*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))+(2*b*e*m+c*d*m+10*b*e)*c/e/(m^2+9*m+20)*x^4*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. 458 vs. \(2 (181) = 362\).
time = 0.29, size = 458, normalized size = 2.57 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} a^{2} e^{\left (-1\right )}}{m + 1} + \frac {2 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} a b e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {{\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 )} b^{2} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {2 \, {\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 c e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {2 \, {\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 )} b c 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 )} c^{2} 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*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

(x*e + d)^(m + 1)*a^2*e^(-1)/(m + 1) + 2*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*a*b*e^(m*log(x*e + d) - 2)/(m^2 + 3
*m + 2) + ((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)*b^2*e^(m*log(x*e + d) - 3)/(m^
3 + 6*m^2 + 11*m + 6) + 2*((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*c*e^(m*log(x
*e + d) - 3)/(m^3 + 6*m^2 + 11*m + 6) + 2*((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)*b*c*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)*c^2*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 8
5*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. 742 vs. \(2 (181) = 362\).
time = 2.91, size = 742, normalized size = 4.17 \begin {gather*} \frac {{\left (24 \, c^{2} d^{5} + {\left ({\left (c^{2} m^{4} + 10 \, c^{2} m^{3} + 35 \, c^{2} m^{2} + 50 \, c^{2} m + 24 \, c^{2}\right )} x^{5} + 2 \, {\left (b c m^{4} + 11 \, b c m^{3} + 41 \, b c m^{2} + 61 \, b c m + 30 \, b c\right )} x^{4} + {\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 12 \, {\left (b^{2} + 2 \, a c\right )} m^{3} + 49 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + 40 \, b^{2} + 80 \, a c + 78 \, {\left (b^{2} + 2 \, a c\right )} m\right )} x^{3} + 2 \, {\left (a b m^{4} + 13 \, a b m^{3} + 59 \, a b m^{2} + 107 \, a b m + 60 \, a b\right )} x^{2} + {\left (a^{2} m^{4} + 14 \, a^{2} m^{3} + 71 \, a^{2} m^{2} + 154 \, a^{2} m + 120 \, a^{2}\right )} x\right )} e^{5} + {\left (a^{2} d m^{4} + 14 \, a^{2} d m^{3} + 71 \, a^{2} d m^{2} + {\left (c^{2} d m^{4} + 6 \, c^{2} d m^{3} + 11 \, c^{2} d m^{2} + 6 \, c^{2} d m\right )} x^{4} + 154 \, a^{2} d m + 2 \, {\left (b c d m^{4} + 8 \, b c d m^{3} + 17 \, b c d m^{2} + 10 \, b c d m\right )} x^{3} + 120 \, a^{2} d + {\left ({\left (b^{2} + 2 \, a c\right )} d m^{4} + 10 \, {\left (b^{2} + 2 \, a c\right )} d m^{3} + 29 \, {\left (b^{2} + 2 \, a c\right )} d m^{2} + 20 \, {\left (b^{2} + 2 \, a c\right )} d m\right )} x^{2} + 2 \, {\left (a b d m^{4} + 12 \, a b d m^{3} + 47 \, a b d m^{2} + 60 \, a b d m\right )} x\right )} e^{4} - 2 \, {\left (a b d^{2} m^{3} + 12 \, a b d^{2} m^{2} + 47 \, a b d^{2} m + 60 \, a b d^{2} + 2 \, {\left (c^{2} d^{2} m^{3} + 3 \, c^{2} d^{2} m^{2} + 2 \, c^{2} d^{2} m\right )} x^{3} + 3 \, {\left (b c d^{2} m^{3} + 6 \, b c d^{2} m^{2} + 5 \, b c d^{2} m\right )} x^{2} + {\left ({\left (b^{2} + 2 \, a c\right )} d^{2} m^{3} + 9 \, {\left (b^{2} + 2 \, a c\right )} d^{2} m^{2} + 20 \, {\left (b^{2} + 2 \, a c\right )} d^{2} m\right )} x\right )} e^{3} + 2 \, {\left ({\left (b^{2} + 2 \, a c\right )} d^{3} m^{2} + 9 \, {\left (b^{2} + 2 \, a c\right )} d^{3} m + 20 \, {\left (b^{2} + 2 \, a c\right )} d^{3} + 6 \, {\left (c^{2} d^{3} m^{2} + c^{2} d^{3} m\right )} x^{2} + 6 \, {\left (b c d^{3} m^{2} + 5 \, b c d^{3} m\right )} x\right )} e^{2} - 12 \, {\left (2 \, c^{2} d^{4} m x + b c d^{4} m + 5 \, b c 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*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

(24*c^2*d^5 + ((c^2*m^4 + 10*c^2*m^3 + 35*c^2*m^2 + 50*c^2*m + 24*c^2)*x^5 + 2*(b*c*m^4 + 11*b*c*m^3 + 41*b*c*
m^2 + 61*b*c*m + 30*b*c)*x^4 + ((b^2 + 2*a*c)*m^4 + 12*(b^2 + 2*a*c)*m^3 + 49*(b^2 + 2*a*c)*m^2 + 40*b^2 + 80*
a*c + 78*(b^2 + 2*a*c)*m)*x^3 + 2*(a*b*m^4 + 13*a*b*m^3 + 59*a*b*m^2 + 107*a*b*m + 60*a*b)*x^2 + (a^2*m^4 + 14
*a^2*m^3 + 71*a^2*m^2 + 154*a^2*m + 120*a^2)*x)*e^5 + (a^2*d*m^4 + 14*a^2*d*m^3 + 71*a^2*d*m^2 + (c^2*d*m^4 +
6*c^2*d*m^3 + 11*c^2*d*m^2 + 6*c^2*d*m)*x^4 + 154*a^2*d*m + 2*(b*c*d*m^4 + 8*b*c*d*m^3 + 17*b*c*d*m^2 + 10*b*c
*d*m)*x^3 + 120*a^2*d + ((b^2 + 2*a*c)*d*m^4 + 10*(b^2 + 2*a*c)*d*m^3 + 29*(b^2 + 2*a*c)*d*m^2 + 20*(b^2 + 2*a
*c)*d*m)*x^2 + 2*(a*b*d*m^4 + 12*a*b*d*m^3 + 47*a*b*d*m^2 + 60*a*b*d*m)*x)*e^4 - 2*(a*b*d^2*m^3 + 12*a*b*d^2*m
^2 + 47*a*b*d^2*m + 60*a*b*d^2 + 2*(c^2*d^2*m^3 + 3*c^2*d^2*m^2 + 2*c^2*d^2*m)*x^3 + 3*(b*c*d^2*m^3 + 6*b*c*d^
2*m^2 + 5*b*c*d^2*m)*x^2 + ((b^2 + 2*a*c)*d^2*m^3 + 9*(b^2 + 2*a*c)*d^2*m^2 + 20*(b^2 + 2*a*c)*d^2*m)*x)*e^3 +
2*((b^2 + 2*a*c)*d^3*m^2 + 9*(b^2 + 2*a*c)*d^3*m + 20*(b^2 + 2*a*c)*d^3 + 6*(c^2*d^3*m^2 + c^2*d^3*m)*x^2 + 6
*(b*c*d^3*m^2 + 5*b*c*d^3*m)*x)*e^2 - 12*(2*c^2*d^4*m*x + b*c*d^4*m + 5*b*c*d^4)*e)*(x*e + d)^m*e^(-5)/(m^5 +
15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 10171 vs. \(2 (162) = 324\).
time = 2.40, size = 10171, normalized size = 57.14 \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*(c*x**2+b*x+a)**2,x)

[Out]

Piecewise((d**m*(a**2*x + a*b*x**2 + 2*a*c*x**3/3 + b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), Eq(e, 0)), (-3*a*
*2*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) - 2*a*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) - 8*a*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) - 2*a*c*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) - 8*a*c*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) - 12*a*c*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) - 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) - 4*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) - 6*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) - 6*b*c*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) - 24*b*c*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) - 36*b*c*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)
- 24*b*c*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*c
**2*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
*c**2*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*c**2*d**3*
e*x*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) + 88*c**2
*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*c**2*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) + 10
8*c**2*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
*c**2*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*c**2*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*c**2*e**4*x**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), Eq(m, -5)), (-a**2*e**4/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - a*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) - 3*a*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) - 2*a*c*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) - 6*a*c*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) - 6*a*c*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) - 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)
- 3*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) - 3*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) + 6*b*c*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) + 11*b*c*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) + 18*b*c
*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) + 27*b*c*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) + 18*b*c*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) + 18*b*c*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) + 6*b*c*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*c*
*2*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*c**2*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*c**2*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*c**2*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) - 3
6*c**2*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*c**2*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*c**2*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*c**2*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**2*e**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 2*a*b*d*e**3/(2*d*
*2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*a*b*e**4*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*a*c*d**2*e**2*
log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 6*a*c*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**
2) + 8*a*c*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 8*a*c*d*e**3*x/(2*d**2*e**5 + 4*d*
e**6*x + 2*e**7*x**2) + 4*a*c*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*b**2*d**2*e*
*2*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. 1717 vs. \(2 (181) = 362\).
time = 1.41, size = 1717, normalized size = 9.65 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c^{2} m^{4} x^{5} e^{5} + {\left (x e + d\right )}^{m} c^{2} d m^{4} x^{4} e^{4} + 2 \, {\left (x e + d\right )}^{m} b c m^{4} x^{4} e^{5} + 10 \, {\left (x e + d\right )}^{m} c^{2} m^{3} x^{5} e^{5} + 2 \, {\left (x e + d\right )}^{m} b c d m^{4} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m^{3} x^{4} e^{4} - 4 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{3} x^{3} e^{3} + {\left (x e + d\right )}^{m} b^{2} m^{4} x^{3} e^{5} + 2 \, {\left (x e + d\right )}^{m} a c m^{4} x^{3} e^{5} + 22 \, {\left (x e + d\right )}^{m} b c m^{3} x^{4} e^{5} + 35 \, {\left (x e + d\right )}^{m} c^{2} m^{2} x^{5} e^{5} + {\left (x e + d\right )}^{m} b^{2} d m^{4} x^{2} e^{4} + 2 \, {\left (x e + d\right )}^{m} a c d m^{4} x^{2} e^{4} + 16 \, {\left (x e + d\right )}^{m} b c d m^{3} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} c^{2} d m^{2} x^{4} e^{4} - 6 \, {\left (x e + d\right )}^{m} b c d^{2} m^{3} x^{2} e^{3} - 12 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m^{2} x^{2} e^{2} + 2 \, {\left (x e + d\right )}^{m} a b m^{4} x^{2} e^{5} + 12 \, {\left (x e + d\right )}^{m} b^{2} m^{3} x^{3} e^{5} + 24 \, {\left (x e + d\right )}^{m} a c m^{3} x^{3} e^{5} + 82 \, {\left (x e + d\right )}^{m} b c m^{2} x^{4} e^{5} + 50 \, {\left (x e + d\right )}^{m} c^{2} m x^{5} e^{5} + 2 \, {\left (x e + d\right )}^{m} a b d m^{4} x e^{4} + 10 \, {\left (x e + d\right )}^{m} b^{2} d m^{3} x^{2} e^{4} + 20 \, {\left (x e + d\right )}^{m} a c d m^{3} x^{2} e^{4} + 34 \, {\left (x e + d\right )}^{m} b c d m^{2} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m x^{4} e^{4} - 2 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m^{3} x e^{3} - 4 \, {\left (x e + d\right )}^{m} a c d^{2} m^{3} x e^{3} - 36 \, {\left (x e + d\right )}^{m} b c d^{2} m^{2} x^{2} e^{3} - 8 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} b c d^{3} m^{2} x e^{2} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m x^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m x e + {\left (x e + d\right )}^{m} a^{2} m^{4} x e^{5} + 26 \, {\left (x e + d\right )}^{m} a b m^{3} x^{2} e^{5} + 49 \, {\left (x e + d\right )}^{m} b^{2} m^{2} x^{3} e^{5} + 98 \, {\left (x e + d\right )}^{m} a c m^{2} x^{3} e^{5} + 122 \, {\left (x e + d\right )}^{m} b c m x^{4} e^{5} + 24 \, {\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + {\left (x e + d\right )}^{m} a^{2} d m^{4} e^{4} + 24 \, {\left (x e + d\right )}^{m} a b d m^{3} x e^{4} + 29 \, {\left (x e + d\right )}^{m} b^{2} d m^{2} x^{2} e^{4} + 58 \, {\left (x e + d\right )}^{m} a c d m^{2} x^{2} e^{4} + 20 \, {\left (x e + d\right )}^{m} b c d m x^{3} e^{4} - 2 \, {\left (x e + d\right )}^{m} a b d^{2} m^{3} e^{3} - 18 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m^{2} x e^{3} - 36 \, {\left (x e + d\right )}^{m} a c d^{2} m^{2} x e^{3} - 30 \, {\left (x e + d\right )}^{m} b c d^{2} m x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} b^{2} d^{3} m^{2} e^{2} + 4 \, {\left (x e + d\right )}^{m} a c d^{3} m^{2} e^{2} + 60 \, {\left (x e + d\right )}^{m} b c d^{3} m x e^{2} - 12 \, {\left (x e + d\right )}^{m} b c d^{4} m e + 24 \, {\left (x e + d\right )}^{m} c^{2} d^{5} + 14 \, {\left (x e + d\right )}^{m} a^{2} m^{3} x e^{5} + 118 \, {\left (x e + d\right )}^{m} a b m^{2} x^{2} e^{5} + 78 \, {\left (x e + d\right )}^{m} b^{2} m x^{3} e^{5} + 156 \, {\left (x e + d\right )}^{m} a c m x^{3} e^{5} + 60 \, {\left (x e + d\right )}^{m} b c x^{4} e^{5} + 14 \, {\left (x e + d\right )}^{m} a^{2} d m^{3} e^{4} + 94 \, {\left (x e + d\right )}^{m} a b d m^{2} x e^{4} + 20 \, {\left (x e + d\right )}^{m} b^{2} d m x^{2} e^{4} + 40 \, {\left (x e + d\right )}^{m} a c d m x^{2} e^{4} - 24 \, {\left (x e + d\right )}^{m} a b d^{2} m^{2} e^{3} - 40 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m x e^{3} - 80 \, {\left (x e + d\right )}^{m} a c d^{2} m x e^{3} + 18 \, {\left (x e + d\right )}^{m} b^{2} d^{3} m e^{2} + 36 \, {\left (x e + d\right )}^{m} a c d^{3} m e^{2} - 60 \, {\left (x e + d\right )}^{m} b c d^{4} e + 71 \, {\left (x e + d\right )}^{m} a^{2} m^{2} x e^{5} + 214 \, {\left (x e + d\right )}^{m} a b m x^{2} e^{5} + 40 \, {\left (x e + d\right )}^{m} b^{2} x^{3} e^{5} + 80 \, {\left (x e + d\right )}^{m} a c x^{3} e^{5} + 71 \, {\left (x e + d\right )}^{m} a^{2} d m^{2} e^{4} + 120 \, {\left (x e + d\right )}^{m} a b d m x e^{4} - 94 \, {\left (x e + d\right )}^{m} a b d^{2} m e^{3} + 40 \, {\left (x e + d\right )}^{m} b^{2} d^{3} e^{2} + 80 \, {\left (x e + d\right )}^{m} a c d^{3} e^{2} + 154 \, {\left (x e + d\right )}^{m} a^{2} m x e^{5} + 120 \, {\left (x e + d\right )}^{m} a b x^{2} e^{5} + 154 \, {\left (x e + d\right )}^{m} a^{2} d m e^{4} - 120 \, {\left (x e + d\right )}^{m} a b d^{2} e^{3} + 120 \, {\left (x e + d\right )}^{m} a^{2} x e^{5} + 120 \, {\left (x e + d\right )}^{m} a^{2} 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*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*c^2*m^4*x^5*e^5 + (x*e + d)^m*c^2*d*m^4*x^4*e^4 + 2*(x*e + d)^m*b*c*m^4*x^4*e^5 + 10*(x*e + d)^m*
c^2*m^3*x^5*e^5 + 2*(x*e + d)^m*b*c*d*m^4*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m^3*x^4*e^4 - 4*(x*e + d)^m*c^2*d^2*m^
3*x^3*e^3 + (x*e + d)^m*b^2*m^4*x^3*e^5 + 2*(x*e + d)^m*a*c*m^4*x^3*e^5 + 22*(x*e + d)^m*b*c*m^3*x^4*e^5 + 35*
(x*e + d)^m*c^2*m^2*x^5*e^5 + (x*e + d)^m*b^2*d*m^4*x^2*e^4 + 2*(x*e + d)^m*a*c*d*m^4*x^2*e^4 + 16*(x*e + d)^m
*b*c*d*m^3*x^3*e^4 + 11*(x*e + d)^m*c^2*d*m^2*x^4*e^4 - 6*(x*e + d)^m*b*c*d^2*m^3*x^2*e^3 - 12*(x*e + d)^m*c^2
*d^2*m^2*x^3*e^3 + 12*(x*e + d)^m*c^2*d^3*m^2*x^2*e^2 + 2*(x*e + d)^m*a*b*m^4*x^2*e^5 + 12*(x*e + d)^m*b^2*m^3
*x^3*e^5 + 24*(x*e + d)^m*a*c*m^3*x^3*e^5 + 82*(x*e + d)^m*b*c*m^2*x^4*e^5 + 50*(x*e + d)^m*c^2*m*x^5*e^5 + 2*
(x*e + d)^m*a*b*d*m^4*x*e^4 + 10*(x*e + d)^m*b^2*d*m^3*x^2*e^4 + 20*(x*e + d)^m*a*c*d*m^3*x^2*e^4 + 34*(x*e +
d)^m*b*c*d*m^2*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m*x^4*e^4 - 2*(x*e + d)^m*b^2*d^2*m^3*x*e^3 - 4*(x*e + d)^m*a*c*d
^2*m^3*x*e^3 - 36*(x*e + d)^m*b*c*d^2*m^2*x^2*e^3 - 8*(x*e + d)^m*c^2*d^2*m*x^3*e^3 + 12*(x*e + d)^m*b*c*d^3*m
^2*x*e^2 + 12*(x*e + d)^m*c^2*d^3*m*x^2*e^2 - 24*(x*e + d)^m*c^2*d^4*m*x*e + (x*e + d)^m*a^2*m^4*x*e^5 + 26*(x
*e + d)^m*a*b*m^3*x^2*e^5 + 49*(x*e + d)^m*b^2*m^2*x^3*e^5 + 98*(x*e + d)^m*a*c*m^2*x^3*e^5 + 122*(x*e + d)^m*
b*c*m*x^4*e^5 + 24*(x*e + d)^m*c^2*x^5*e^5 + (x*e + d)^m*a^2*d*m^4*e^4 + 24*(x*e + d)^m*a*b*d*m^3*x*e^4 + 29*(
x*e + d)^m*b^2*d*m^2*x^2*e^4 + 58*(x*e + d)^m*a*c*d*m^2*x^2*e^4 + 20*(x*e + d)^m*b*c*d*m*x^3*e^4 - 2*(x*e + d)
^m*a*b*d^2*m^3*e^3 - 18*(x*e + d)^m*b^2*d^2*m^2*x*e^3 - 36*(x*e + d)^m*a*c*d^2*m^2*x*e^3 - 30*(x*e + d)^m*b*c*
d^2*m*x^2*e^3 + 2*(x*e + d)^m*b^2*d^3*m^2*e^2 + 4*(x*e + d)^m*a*c*d^3*m^2*e^2 + 60*(x*e + d)^m*b*c*d^3*m*x*e^2
- 12*(x*e + d)^m*b*c*d^4*m*e + 24*(x*e + d)^m*c^2*d^5 + 14*(x*e + d)^m*a^2*m^3*x*e^5 + 118*(x*e + d)^m*a*b*m^
2*x^2*e^5 + 78*(x*e + d)^m*b^2*m*x^3*e^5 + 156*(x*e + d)^m*a*c*m*x^3*e^5 + 60*(x*e + d)^m*b*c*x^4*e^5 + 14*(x*
e + d)^m*a^2*d*m^3*e^4 + 94*(x*e + d)^m*a*b*d*m^2*x*e^4 + 20*(x*e + d)^m*b^2*d*m*x^2*e^4 + 40*(x*e + d)^m*a*c*
d*m*x^2*e^4 - 24*(x*e + d)^m*a*b*d^2*m^2*e^3 - 40*(x*e + d)^m*b^2*d^2*m*x*e^3 - 80*(x*e + d)^m*a*c*d^2*m*x*e^3
+ 18*(x*e + d)^m*b^2*d^3*m*e^2 + 36*(x*e + d)^m*a*c*d^3*m*e^2 - 60*(x*e + d)^m*b*c*d^4*e + 71*(x*e + d)^m*a^2
*m^2*x*e^5 + 214*(x*e + d)^m*a*b*m*x^2*e^5 + 40*(x*e + d)^m*b^2*x^3*e^5 + 80*(x*e + d)^m*a*c*x^3*e^5 + 71*(x*e
+ d)^m*a^2*d*m^2*e^4 + 120*(x*e + d)^m*a*b*d*m*x*e^4 - 94*(x*e + d)^m*a*b*d^2*m*e^3 + 40*(x*e + d)^m*b^2*d^3*
e^2 + 80*(x*e + d)^m*a*c*d^3*e^2 + 154*(x*e + d)^m*a^2*m*x*e^5 + 120*(x*e + d)^m*a*b*x^2*e^5 + 154*(x*e + d)^m
*a^2*d*m*e^4 - 120*(x*e + d)^m*a*b*d^2*e^3 + 120*(x*e + d)^m*a^2*x*e^5 + 120*(x*e + d)^m*a^2*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 = 1.55, size = 895, normalized size = 5.03 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (a^2\,d\,e^4\,m^4+14\,a^2\,d\,e^4\,m^3+71\,a^2\,d\,e^4\,m^2+154\,a^2\,d\,e^4\,m+120\,a^2\,d\,e^4-2\,a\,b\,d^2\,e^3\,m^3-24\,a\,b\,d^2\,e^3\,m^2-94\,a\,b\,d^2\,e^3\,m-120\,a\,b\,d^2\,e^3+4\,a\,c\,d^3\,e^2\,m^2+36\,a\,c\,d^3\,e^2\,m+80\,a\,c\,d^3\,e^2+2\,b^2\,d^3\,e^2\,m^2+18\,b^2\,d^3\,e^2\,m+40\,b^2\,d^3\,e^2-12\,b\,c\,d^4\,e\,m-60\,b\,c\,d^4\,e+24\,c^2\,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^2\,e^5\,m^4+14\,a^2\,e^5\,m^3+71\,a^2\,e^5\,m^2+154\,a^2\,e^5\,m+120\,a^2\,e^5+2\,a\,b\,d\,e^4\,m^4+24\,a\,b\,d\,e^4\,m^3+94\,a\,b\,d\,e^4\,m^2+120\,a\,b\,d\,e^4\,m-4\,a\,c\,d^2\,e^3\,m^3-36\,a\,c\,d^2\,e^3\,m^2-80\,a\,c\,d^2\,e^3\,m-2\,b^2\,d^2\,e^3\,m^3-18\,b^2\,d^2\,e^3\,m^2-40\,b^2\,d^2\,e^3\,m+12\,b\,c\,d^3\,e^2\,m^2+60\,b\,c\,d^3\,e^2\,m-24\,c^2\,d^4\,e\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c^2\,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 {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2+2\,b\,c\,d\,e\,m^2+10\,b\,c\,d\,e\,m-4\,c^2\,d^2\,m+2\,a\,c\,e^2\,m^2+18\,a\,c\,e^2\,m+40\,a\,c\,e^2\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (b^2\,d\,e^2\,m^3+9\,b^2\,d\,e^2\,m^2+20\,b^2\,d\,e^2\,m-6\,b\,c\,d^2\,e\,m^2-30\,b\,c\,d^2\,e\,m+2\,a\,b\,e^3\,m^3+24\,a\,b\,e^3\,m^2+94\,a\,b\,e^3\,m+120\,a\,b\,e^3+12\,c^2\,d^3\,m+2\,a\,c\,d\,e^2\,m^3+18\,a\,c\,d\,e^2\,m^2+40\,a\,c\,d\,e^2\,m\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c\,x^4\,{\left (d+e\,x\right )}^m\,\left (10\,b\,e+2\,b\,e\,m+c\,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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^m*(24*c^2*d^5 + 120*a^2*d*e^4 + 40*b^2*d^3*e^2 + 71*a^2*d*e^4*m^2 + 14*a^2*d*e^4*m^3 + a^2*d*e^4*m^
4 + 18*b^2*d^3*e^2*m - 60*b*c*d^4*e + 2*b^2*d^3*e^2*m^2 - 120*a*b*d^2*e^3 + 80*a*c*d^3*e^2 + 154*a^2*d*e^4*m -
94*a*b*d^2*e^3*m + 36*a*c*d^3*e^2*m - 24*a*b*d^2*e^3*m^2 - 2*a*b*d^2*e^3*m^3 + 4*a*c*d^3*e^2*m^2 - 12*b*c*d^4
*e*m))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x*(d + e*x)^m*(120*a^2*e^5 + 154*a^2*e^5*m + 7
1*a^2*e^5*m^2 + 14*a^2*e^5*m^3 + a^2*e^5*m^4 - 40*b^2*d^2*e^3*m - 18*b^2*d^2*e^3*m^2 - 2*b^2*d^2*e^3*m^3 - 24*
c^2*d^4*e*m + 94*a*b*d*e^4*m^2 + 24*a*b*d*e^4*m^3 + 2*a*b*d*e^4*m^4 - 80*a*c*d^2*e^3*m + 60*b*c*d^3*e^2*m - 36
*a*c*d^2*e^3*m^2 - 4*a*c*d^2*e^3*m^3 + 12*b*c*d^3*e^2*m^2 + 120*a*b*d*e^4*m))/(e^5*(274*m + 225*m^2 + 85*m^3 +
15*m^4 + m^5 + 120)) + (c^2*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) + (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(20*b^2*e^2 + 9*b^2*e^2*m - 4*c^2*d^2*m + b^2*e^2*m^2 +
40*a*c*e^2 + 18*a*c*e^2*m + 2*a*c*e^2*m^2 + 10*b*c*d*e*m + 2*b*c*d*e*m^2))/(e^2*(274*m + 225*m^2 + 85*m^3 + 1
5*m^4 + m^5 + 120)) + (x^2*(m + 1)*(d + e*x)^m*(12*c^2*d^3*m + 120*a*b*e^3 + 9*b^2*d*e^2*m^2 + b^2*d*e^2*m^3 +
94*a*b*e^3*m + 24*a*b*e^3*m^2 + 2*a*b*e^3*m^3 + 20*b^2*d*e^2*m + 18*a*c*d*e^2*m^2 + 2*a*c*d*e^2*m^3 - 6*b*c*d
^2*e*m^2 + 40*a*c*d*e^2*m - 30*b*c*d^2*e*m))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (c*x^4*(d
+ e*x)^m*(10*b*e + 2*b*e*m + c*d*m)*(11*m + 6*m^2 + m^3 + 6))/(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 1
20))

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