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3.17.55 \(\int (b+2 c x) (d+e x)^m (a+b x+c x^2)^2 \, dx\) [1655]

Optimal. Leaf size=270 \[ -\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{2+m}}{e^6 (2+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{3+m}}{e^6 (3+m)}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {5 c^2 (2 c d-b e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {2 c^3 (d+e x)^{6+m}}{e^6 (6+m)} \]

[Out]

-(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(1+m)/e^6/(1+m)+2*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e
+5*b*d))*(e*x+d)^(2+m)/e^6/(2+m)-(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*(e*x+d)^(3+m)/e^6/(3+m
)+4*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(4+m)/e^6/(4+m)-5*c^2*(-b*e+2*c*d)*(e*x+d)^(5+m)/e^6/(5+m)+
2*c^3*(e*x+d)^(6+m)/e^6/(6+m)

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Rubi [A]
time = 0.13, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {785} \begin {gather*} \frac {2 (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (m+2)}-\frac {(2 c d-b e) (d+e x)^{m+3} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (m+3)}+\frac {4 c (d+e x)^{m+4} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (m+4)}-\frac {(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac {5 c^2 (2 c d-b e) (d+e x)^{m+5}}{e^6 (m+5)}+\frac {2 c^3 (d+e x)^{m+6}}{e^6 (m+6)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (2*(c*d^2 - b*d*e + a*e^2)*(5*c
^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(2 + m))/(e^6*(2 + m)) - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2
- 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(3 + m))/(e^6*(3 + m)) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d
+ e*x)^(4 + m))/(e^6*(4 + m)) - (5*c^2*(2*c*d - b*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (2*c^3*(d + e*x)^(6 +
m))/(e^6*(6 + m))

Rule 785

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

Rubi steps

\begin {align*} \int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^m}{e^5}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{1+m}}{e^5}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{2+m}}{e^5}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3+m}}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{4+m}}{e^5}+\frac {2 c^3 (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{2+m}}{e^6 (2+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{3+m}}{e^6 (3+m)}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {5 c^2 (2 c d-b e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {2 c^3 (d+e x)^{6+m}}{e^6 (6+m)}\\ \end {align*}

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Mathematica [A]
time = 0.90, size = 408, normalized size = 1.51 \begin {gather*} \frac {(d+e x)^{1+m} \left (c (a+x (b+c x))^2 (b e (10+m)+2 c (-5 d+e (5+m) x))+\frac {2 \left (-\frac {(2 c d-b e) \left (c d^2+e (-b d+a e)\right ) \left (60 c^2 d^2+b^2 e^2 m (1+m)-4 c e \left (15 b d+a e \left (-15+m+m^2\right )\right )\right )}{e^2 (1+m)}+\frac {\left (120 c^4 d^4+b^4 e^4 m (2+m)-2 b^2 c e^3 m (b d (-4+m)+3 a e (4+m))-8 c^3 d^2 e \left (30 b d+a e \left (-30-4 m+m^2\right )\right )+2 c^2 e^2 \left (4 a b d e \left (-30-4 m+m^2\right )+b^2 d^2 \left (60-4 m+m^2\right )+4 a^2 e^2 \left (15+8 m+m^2\right )\right )\right ) (d+e x)}{e^2 (2+m)}-(a+x (b+c x)) \left (b^3 e^3 m+20 c^3 d^2 (3 d-e (3+m) x)+b c e^2 \left (-2 a e (30+7 m)+b d \left (60+11 m+m^2\right )+b e m (3+m) x\right )-2 c^2 e \left (5 b d (d (12+m)-2 e (3+m) x)+2 a e \left (d \left (-15+m+m^2\right )+e \left (15+8 m+m^2\right ) x\right )\right )\right )\right )}{e^2 (3+m) (4+m)}\right )}{c e^2 (5+m) (6+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(c*(a + x*(b + c*x))^2*(b*e*(10 + m) + 2*c*(-5*d + e*(5 + m)*x)) + (2*(-(((2*c*d - b*e)*(c*
d^2 + e*(-(b*d) + a*e))*(60*c^2*d^2 + b^2*e^2*m*(1 + m) - 4*c*e*(15*b*d + a*e*(-15 + m + m^2))))/(e^2*(1 + m))
) + ((120*c^4*d^4 + b^4*e^4*m*(2 + m) - 2*b^2*c*e^3*m*(b*d*(-4 + m) + 3*a*e*(4 + m)) - 8*c^3*d^2*e*(30*b*d + a
*e*(-30 - 4*m + m^2)) + 2*c^2*e^2*(4*a*b*d*e*(-30 - 4*m + m^2) + b^2*d^2*(60 - 4*m + m^2) + 4*a^2*e^2*(15 + 8*
m + m^2)))*(d + e*x))/(e^2*(2 + m)) - (a + x*(b + c*x))*(b^3*e^3*m + 20*c^3*d^2*(3*d - e*(3 + m)*x) + b*c*e^2*
(-2*a*e*(30 + 7*m) + b*d*(60 + 11*m + m^2) + b*e*m*(3 + m)*x) - 2*c^2*e*(5*b*d*(d*(12 + m) - 2*e*(3 + m)*x) +
2*a*e*(d*(-15 + m + m^2) + e*(15 + 8*m + m^2)*x)))))/(e^2*(3 + m)*(4 + m))))/(c*e^2*(5 + m)*(6 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1667\) vs. \(2(270)=540\).
time = 1.04, size = 1668, normalized size = 6.18

method result size
norman \(\text {Expression too large to display}\) \(1668\)
gosper \(\text {Expression too large to display}\) \(1852\)
risch \(\text {Expression too large to display}\) \(2445\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(a^2*b*e^5*m^5+20*a^2*b*e^5*m^4-2*a^2*c*d*e^4*m^4-2*a*b^2*d*e^4*m^4+155*a^2*b*e^5*m^3-36*a^2*c*d*e^4*m^3-36*
a*b^2*d*e^4*m^3+12*a*b*c*d^2*e^3*m^3+2*b^3*d^2*e^3*m^3+580*a^2*b*e^5*m^2-238*a^2*c*d*e^4*m^2-238*a*b^2*d*e^4*m
^2+180*a*b*c*d^2*e^3*m^2-24*a*c^2*d^3*e^2*m^2+30*b^3*d^2*e^3*m^2-24*b^2*c*d^3*e^2*m^2+1044*a^2*b*e^5*m-684*a^2
*c*d*e^4*m-684*a*b^2*d*e^4*m+888*a*b*c*d^2*e^3*m-264*a*c^2*d^3*e^2*m+148*b^3*d^2*e^3*m-264*b^2*c*d^3*e^2*m+120
*b*c^2*d^4*e*m+720*a^2*b*e^5-720*a^2*c*d*e^4-720*a*b^2*d*e^4+1440*a*b*c*d^2*e^3-720*a*c^2*d^3*e^2+240*b^3*d^2*
e^3-720*b^2*c*d^3*e^2+720*b*c^2*d^4*e-240*c^3*d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)*exp(m*
ln(e*x+d))+(6*a*b*c*e^3*m^3+4*a*c^2*d*e^2*m^3+b^3*e^3*m^3+4*b^2*c*d*e^2*m^3+90*a*b*c*e^3*m^2+44*a*c^2*d*e^2*m^
2+15*b^3*e^3*m^2+44*b^2*c*d*e^2*m^2-20*b*c^2*d^2*e*m^2+444*a*b*c*e^3*m+120*a*c^2*d*e^2*m+74*b^3*e^3*m+120*b^2*
c*d*e^2*m-120*b*c^2*d^2*e*m+40*c^3*d^3*m+720*a*b*c*e^3+120*b^3*e^3)/e^3/(m^4+18*m^3+119*m^2+342*m+360)*x^3*exp
(m*ln(e*x+d))+(2*a^2*c*e^4*m^4+2*a*b^2*e^4*m^4+6*a*b*c*d*e^3*m^4+b^3*d*e^3*m^4+36*a^2*c*e^4*m^3+36*a*b^2*e^4*m
^3+90*a*b*c*d*e^3*m^3-12*a*c^2*d^2*e^2*m^3+15*b^3*d*e^3*m^3-12*b^2*c*d^2*e^2*m^3+238*a^2*c*e^4*m^2+238*a*b^2*e
^4*m^2+444*a*b*c*d*e^3*m^2-132*a*c^2*d^2*e^2*m^2+74*b^3*d*e^3*m^2-132*b^2*c*d^2*e^2*m^2+60*b*c^2*d^3*e*m^2+684
*a^2*c*e^4*m+684*a*b^2*e^4*m+720*a*b*c*d*e^3*m-360*a*c^2*d^2*e^2*m+120*b^3*d*e^3*m-360*b^2*c*d^2*e^2*m+360*b*c
^2*d^3*e*m-120*c^3*d^4*m+720*a^2*c*e^4+720*a*b^2*e^4)/e^4/(m^5+20*m^4+155*m^3+580*m^2+1044*m+720)*x^2*exp(m*ln
(e*x+d))+(a^2*b*e^5*m^5+2*a^2*c*d*e^4*m^5+2*a*b^2*d*e^4*m^5+20*a^2*b*e^5*m^4+36*a^2*c*d*e^4*m^4+36*a*b^2*d*e^4
*m^4-12*a*b*c*d^2*e^3*m^4-2*b^3*d^2*e^3*m^4+155*a^2*b*e^5*m^3+238*a^2*c*d*e^4*m^3+238*a*b^2*d*e^4*m^3-180*a*b*
c*d^2*e^3*m^3+24*a*c^2*d^3*e^2*m^3-30*b^3*d^2*e^3*m^3+24*b^2*c*d^3*e^2*m^3+580*a^2*b*e^5*m^2+684*a^2*c*d*e^4*m
^2+684*a*b^2*d*e^4*m^2-888*a*b*c*d^2*e^3*m^2+264*a*c^2*d^3*e^2*m^2-148*b^3*d^2*e^3*m^2+264*b^2*c*d^3*e^2*m^2-1
20*b*c^2*d^4*e*m^2+1044*a^2*b*e^5*m+720*a^2*c*d*e^4*m+720*a*b^2*d*e^4*m-1440*a*b*c*d^2*e^3*m+720*a*c^2*d^3*e^2
*m-240*b^3*d^2*e^3*m+720*b^2*c*d^3*e^2*m-720*b*c^2*d^4*e*m+240*c^3*d^5*m+720*a^2*b*e^5)/e^5/(m^6+21*m^5+175*m^
4+735*m^3+1624*m^2+1764*m+720)*x*exp(m*ln(e*x+d))+(5*b*e*m+2*c*d*m+30*b*e)*c^2/e/(m^2+11*m+30)*x^5*exp(m*ln(e*
x+d))+(4*a*c*e^2*m^2+4*b^2*e^2*m^2+5*b*c*d*e*m^2+44*a*c*e^2*m+44*b^2*e^2*m+30*b*c*d*e*m-10*c^2*d^2*m+120*a*c*e
^2+120*b^2*e^2)*c/e^2/(m^3+15*m^2+74*m+120)*x^4*exp(m*ln(e*x+d))+2*c^3/(6+m)*x^6*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. 805 vs. \(2 (276) = 552\).
time = 0.32, size = 805, normalized size = 2.98 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} a^{2} b 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^{2} e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {2 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} a^{2} c 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^{3} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \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 b c 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 )} b^{2} c e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \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 c^{2} e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {5 \, {\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 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} + \frac {2 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} x^{6} e^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d x^{5} e^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} x^{4} e^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} x^{3} e^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} x^{2} e^{2} + 120 \, d^{5} m x e - 120 \, d^{6}\right )} c^{3} e^{\left (m \log \left (x e + d\right ) - 6\right )}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e + d)^(m + 1)*a^2*b*e^(-1)/(m + 1) + 2*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*a*b^2*e^(m*log(x*e + d) - 2)/(m^2
 + 3*m + 2) + 2*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*a^2*c*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^3*e^(m*log(x*e + d) - 3)/(m^3 + 6*m^2 + 11*m + 6) +
 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*b*c*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)*b^2*c*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + 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*c^2*e^(m
*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + 5*((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*c^2*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + 2*((m^5 + 15*m^4 +
 85*m^3 + 225*m^2 + 274*m + 120)*x^6*e^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*x^5*e^5 - 5*(m^4 + 6*m^3
+ 11*m^2 + 6*m)*d^2*x^4*e^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*x^3*e^3 - 60*(m^2 + m)*d^4*x^2*e^2 + 120*d^5*m*x*e -
120*d^6)*c^3*e^(m*log(x*e + d) - 6)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1569 vs. \(2 (276) = 552\).
time = 2.60, size = 1569, normalized size = 5.81 \begin {gather*} -\frac {{\left (240 \, c^{3} d^{6} - {\left (2 \, {\left (c^{3} m^{5} + 15 \, c^{3} m^{4} + 85 \, c^{3} m^{3} + 225 \, c^{3} m^{2} + 274 \, c^{3} m + 120 \, c^{3}\right )} x^{6} + 5 \, {\left (b c^{2} m^{5} + 16 \, b c^{2} m^{4} + 95 \, b c^{2} m^{3} + 260 \, b c^{2} m^{2} + 324 \, b c^{2} m + 144 \, b c^{2}\right )} x^{5} + 4 \, {\left ({\left (b^{2} c + a c^{2}\right )} m^{5} + 17 \, {\left (b^{2} c + a c^{2}\right )} m^{4} + 107 \, {\left (b^{2} c + a c^{2}\right )} m^{3} + 180 \, b^{2} c + 180 \, a c^{2} + 307 \, {\left (b^{2} c + a c^{2}\right )} m^{2} + 396 \, {\left (b^{2} c + a c^{2}\right )} m\right )} x^{4} + {\left ({\left (b^{3} + 6 \, a b c\right )} m^{5} + 18 \, {\left (b^{3} + 6 \, a b c\right )} m^{4} + 121 \, {\left (b^{3} + 6 \, a b c\right )} m^{3} + 240 \, b^{3} + 1440 \, a b c + 372 \, {\left (b^{3} + 6 \, a b c\right )} m^{2} + 508 \, {\left (b^{3} + 6 \, a b c\right )} m\right )} x^{3} + 2 \, {\left ({\left (a b^{2} + a^{2} c\right )} m^{5} + 19 \, {\left (a b^{2} + a^{2} c\right )} m^{4} + 137 \, {\left (a b^{2} + a^{2} c\right )} m^{3} + 360 \, a b^{2} + 360 \, a^{2} c + 461 \, {\left (a b^{2} + a^{2} c\right )} m^{2} + 702 \, {\left (a b^{2} + a^{2} c\right )} m\right )} x^{2} + {\left (a^{2} b m^{5} + 20 \, a^{2} b m^{4} + 155 \, a^{2} b m^{3} + 580 \, a^{2} b m^{2} + 1044 \, a^{2} b m + 720 \, a^{2} b\right )} x\right )} e^{6} - {\left (a^{2} b d m^{5} + 20 \, a^{2} b d m^{4} + 155 \, a^{2} b d m^{3} + 580 \, a^{2} b d m^{2} + 2 \, {\left (c^{3} d m^{5} + 10 \, c^{3} d m^{4} + 35 \, c^{3} d m^{3} + 50 \, c^{3} d m^{2} + 24 \, c^{3} d m\right )} x^{5} + 1044 \, a^{2} b d m + 5 \, {\left (b c^{2} d m^{5} + 12 \, b c^{2} d m^{4} + 47 \, b c^{2} d m^{3} + 72 \, b c^{2} d m^{2} + 36 \, b c^{2} d m\right )} x^{4} + 720 \, a^{2} b d + 4 \, {\left ({\left (b^{2} c + a c^{2}\right )} d m^{5} + 14 \, {\left (b^{2} c + a c^{2}\right )} d m^{4} + 65 \, {\left (b^{2} c + a c^{2}\right )} d m^{3} + 112 \, {\left (b^{2} c + a c^{2}\right )} d m^{2} + 60 \, {\left (b^{2} c + a c^{2}\right )} d m\right )} x^{3} + {\left ({\left (b^{3} + 6 \, a b c\right )} d m^{5} + 16 \, {\left (b^{3} + 6 \, a b c\right )} d m^{4} + 89 \, {\left (b^{3} + 6 \, a b c\right )} d m^{3} + 194 \, {\left (b^{3} + 6 \, a b c\right )} d m^{2} + 120 \, {\left (b^{3} + 6 \, a b c\right )} d m\right )} x^{2} + 2 \, {\left ({\left (a b^{2} + a^{2} c\right )} d m^{5} + 18 \, {\left (a b^{2} + a^{2} c\right )} d m^{4} + 119 \, {\left (a b^{2} + a^{2} c\right )} d m^{3} + 342 \, {\left (a b^{2} + a^{2} c\right )} d m^{2} + 360 \, {\left (a b^{2} + a^{2} c\right )} d m\right )} x\right )} e^{5} + 2 \, {\left ({\left (a b^{2} + a^{2} c\right )} d^{2} m^{4} + 18 \, {\left (a b^{2} + a^{2} c\right )} d^{2} m^{3} + 119 \, {\left (a b^{2} + a^{2} c\right )} d^{2} m^{2} + 5 \, {\left (c^{3} d^{2} m^{4} + 6 \, c^{3} d^{2} m^{3} + 11 \, c^{3} d^{2} m^{2} + 6 \, c^{3} d^{2} m\right )} x^{4} + 342 \, {\left (a b^{2} + a^{2} c\right )} d^{2} m + 10 \, {\left (b c^{2} d^{2} m^{4} + 9 \, b c^{2} d^{2} m^{3} + 20 \, b c^{2} d^{2} m^{2} + 12 \, b c^{2} d^{2} m\right )} x^{3} + 360 \, {\left (a b^{2} + a^{2} c\right )} d^{2} + 6 \, {\left ({\left (b^{2} c + a c^{2}\right )} d^{2} m^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} m^{3} + 41 \, {\left (b^{2} c + a c^{2}\right )} d^{2} m^{2} + 30 \, {\left (b^{2} c + a c^{2}\right )} d^{2} m\right )} x^{2} + {\left ({\left (b^{3} + 6 \, a b c\right )} d^{2} m^{4} + 15 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} m^{3} + 74 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} m^{2} + 120 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} m\right )} x\right )} e^{4} - 2 \, {\left ({\left (b^{3} + 6 \, a b c\right )} d^{3} m^{3} + 15 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} m^{2} + 74 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} m + 120 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} + 20 \, {\left (c^{3} d^{3} m^{3} + 3 \, c^{3} d^{3} m^{2} + 2 \, c^{3} d^{3} m\right )} x^{3} + 30 \, {\left (b c^{2} d^{3} m^{3} + 7 \, b c^{2} d^{3} m^{2} + 6 \, b c^{2} d^{3} m\right )} x^{2} + 12 \, {\left ({\left (b^{2} c + a c^{2}\right )} d^{3} m^{3} + 11 \, {\left (b^{2} c + a c^{2}\right )} d^{3} m^{2} + 30 \, {\left (b^{2} c + a c^{2}\right )} d^{3} m\right )} x\right )} e^{3} + 24 \, {\left ({\left (b^{2} c + a c^{2}\right )} d^{4} m^{2} + 11 \, {\left (b^{2} c + a c^{2}\right )} d^{4} m + 30 \, {\left (b^{2} c + a c^{2}\right )} d^{4} + 5 \, {\left (c^{3} d^{4} m^{2} + c^{3} d^{4} m\right )} x^{2} + 5 \, {\left (b c^{2} d^{4} m^{2} + 6 \, b c^{2} d^{4} m\right )} x\right )} e^{2} - 120 \, {\left (2 \, c^{3} d^{5} m x + b c^{2} d^{5} m + 6 \, b c^{2} d^{5}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-6\right )}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(240*c^3*d^6 - (2*(c^3*m^5 + 15*c^3*m^4 + 85*c^3*m^3 + 225*c^3*m^2 + 274*c^3*m + 120*c^3)*x^6 + 5*(b*c^2*m^5
+ 16*b*c^2*m^4 + 95*b*c^2*m^3 + 260*b*c^2*m^2 + 324*b*c^2*m + 144*b*c^2)*x^5 + 4*((b^2*c + a*c^2)*m^5 + 17*(b^
2*c + a*c^2)*m^4 + 107*(b^2*c + a*c^2)*m^3 + 180*b^2*c + 180*a*c^2 + 307*(b^2*c + a*c^2)*m^2 + 396*(b^2*c + a*
c^2)*m)*x^4 + ((b^3 + 6*a*b*c)*m^5 + 18*(b^3 + 6*a*b*c)*m^4 + 121*(b^3 + 6*a*b*c)*m^3 + 240*b^3 + 1440*a*b*c +
 372*(b^3 + 6*a*b*c)*m^2 + 508*(b^3 + 6*a*b*c)*m)*x^3 + 2*((a*b^2 + a^2*c)*m^5 + 19*(a*b^2 + a^2*c)*m^4 + 137*
(a*b^2 + a^2*c)*m^3 + 360*a*b^2 + 360*a^2*c + 461*(a*b^2 + a^2*c)*m^2 + 702*(a*b^2 + a^2*c)*m)*x^2 + (a^2*b*m^
5 + 20*a^2*b*m^4 + 155*a^2*b*m^3 + 580*a^2*b*m^2 + 1044*a^2*b*m + 720*a^2*b)*x)*e^6 - (a^2*b*d*m^5 + 20*a^2*b*
d*m^4 + 155*a^2*b*d*m^3 + 580*a^2*b*d*m^2 + 2*(c^3*d*m^5 + 10*c^3*d*m^4 + 35*c^3*d*m^3 + 50*c^3*d*m^2 + 24*c^3
*d*m)*x^5 + 1044*a^2*b*d*m + 5*(b*c^2*d*m^5 + 12*b*c^2*d*m^4 + 47*b*c^2*d*m^3 + 72*b*c^2*d*m^2 + 36*b*c^2*d*m)
*x^4 + 720*a^2*b*d + 4*((b^2*c + a*c^2)*d*m^5 + 14*(b^2*c + a*c^2)*d*m^4 + 65*(b^2*c + a*c^2)*d*m^3 + 112*(b^2
*c + a*c^2)*d*m^2 + 60*(b^2*c + a*c^2)*d*m)*x^3 + ((b^3 + 6*a*b*c)*d*m^5 + 16*(b^3 + 6*a*b*c)*d*m^4 + 89*(b^3
+ 6*a*b*c)*d*m^3 + 194*(b^3 + 6*a*b*c)*d*m^2 + 120*(b^3 + 6*a*b*c)*d*m)*x^2 + 2*((a*b^2 + a^2*c)*d*m^5 + 18*(a
*b^2 + a^2*c)*d*m^4 + 119*(a*b^2 + a^2*c)*d*m^3 + 342*(a*b^2 + a^2*c)*d*m^2 + 360*(a*b^2 + a^2*c)*d*m)*x)*e^5
+ 2*((a*b^2 + a^2*c)*d^2*m^4 + 18*(a*b^2 + a^2*c)*d^2*m^3 + 119*(a*b^2 + a^2*c)*d^2*m^2 + 5*(c^3*d^2*m^4 + 6*c
^3*d^2*m^3 + 11*c^3*d^2*m^2 + 6*c^3*d^2*m)*x^4 + 342*(a*b^2 + a^2*c)*d^2*m + 10*(b*c^2*d^2*m^4 + 9*b*c^2*d^2*m
^3 + 20*b*c^2*d^2*m^2 + 12*b*c^2*d^2*m)*x^3 + 360*(a*b^2 + a^2*c)*d^2 + 6*((b^2*c + a*c^2)*d^2*m^4 + 12*(b^2*c
 + a*c^2)*d^2*m^3 + 41*(b^2*c + a*c^2)*d^2*m^2 + 30*(b^2*c + a*c^2)*d^2*m)*x^2 + ((b^3 + 6*a*b*c)*d^2*m^4 + 15
*(b^3 + 6*a*b*c)*d^2*m^3 + 74*(b^3 + 6*a*b*c)*d^2*m^2 + 120*(b^3 + 6*a*b*c)*d^2*m)*x)*e^4 - 2*((b^3 + 6*a*b*c)
*d^3*m^3 + 15*(b^3 + 6*a*b*c)*d^3*m^2 + 74*(b^3 + 6*a*b*c)*d^3*m + 120*(b^3 + 6*a*b*c)*d^3 + 20*(c^3*d^3*m^3 +
 3*c^3*d^3*m^2 + 2*c^3*d^3*m)*x^3 + 30*(b*c^2*d^3*m^3 + 7*b*c^2*d^3*m^2 + 6*b*c^2*d^3*m)*x^2 + 12*((b^2*c + a*
c^2)*d^3*m^3 + 11*(b^2*c + a*c^2)*d^3*m^2 + 30*(b^2*c + a*c^2)*d^3*m)*x)*e^3 + 24*((b^2*c + a*c^2)*d^4*m^2 + 1
1*(b^2*c + a*c^2)*d^4*m + 30*(b^2*c + a*c^2)*d^4 + 5*(c^3*d^4*m^2 + c^3*d^4*m)*x^2 + 5*(b*c^2*d^4*m^2 + 6*b*c^
2*d^4*m)*x)*e^2 - 120*(2*c^3*d^5*m*x + b*c^2*d^5*m + 6*b*c^2*d^5)*e)*(x*e + d)^m*e^(-6)/(m^6 + 21*m^5 + 175*m^
4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 23525 vs. \(2 (252) = 504\).
time = 4.84, size = 23525, normalized size = 87.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(a**2*b*x + a**2*c*x**2 + a*b**2*x**2 + 2*a*b*c*x**3 + a*c**2*x**4 + b**3*x**3/3 + b**2*c*x**4
 + b*c**2*x**5 + c**3*x**6/3), Eq(e, 0)), (-6*a**2*b*e**5/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2
 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 3*a**2*c*d*e**4/(30*d**5*e**6 + 150*d**4*e**7*x +
300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 15*a**2*c*e**5*x/(30*d**5*e**6 +
 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 3*a*b**2*d*e*
*4/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**
5) - 15*a*b**2*e**5*x/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*
x**4 + 30*e**11*x**5) - 6*a*b*c*d**2*e**3/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9
*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 30*a*b*c*d*e**4*x/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*
x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 60*a*b*c*e**5*x**2/(30*d**5*e**6 + 150*d**4*e*
*7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 6*a*c**2*d**3*e**2/(30*d*
*5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 30*a
*c**2*d**2*e**3*x/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4
 + 30*e**11*x**5) - 60*a*c**2*d*e**4*x**2/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9
*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 60*a*c**2*e**5*x**3/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**
8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - b**3*d**2*e**3/(30*d**5*e**6 + 150*d**4*e**7
*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 5*b**3*d*e**4*x/(30*d**5*e*
*6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 10*b**3*e
**5*x**2/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**
11*x**5) - 6*b**2*c*d**3*e**2/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*
d*e**10*x**4 + 30*e**11*x**5) - 30*b**2*c*d**2*e**3*x/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 3
00*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 60*b**2*c*d*e**4*x**2/(30*d**5*e**6 + 150*d**4*e**7*x
+ 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 60*b**2*c*e**5*x**3/(30*d**5*e
**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 30*b*c**
2*d**4*e/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**
11*x**5) - 150*b*c**2*d**3*e**2*x/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 +
150*d*e**10*x**4 + 30*e**11*x**5) - 300*b*c**2*d**2*e**3*x**2/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*
x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 300*b*c**2*d*e**4*x**3/(30*d**5*e**6 + 150*d**
4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) - 150*b*c**2*e**5*x**4/
(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5)
+ 60*c**3*d**5*log(d/e + x)/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*
e**10*x**4 + 30*e**11*x**5) + 137*c**3*d**5/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e*
*9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) + 300*c**3*d**4*e*x*log(d/e + x)/(30*d**5*e**6 + 150*d**4*e**7*x +
 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) + 625*c**3*d**4*e*x/(30*d**5*e**6
 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) + 600*c**3*d*
*3*e**2*x**2*log(d/e + x)/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e*
*10*x**4 + 30*e**11*x**5) + 1100*c**3*d**3*e**2*x**2/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 30
0*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) + 600*c**3*d**2*e**3*x**3*log(d/e + x)/(30*d**5*e**6 + 15
0*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) + 900*c**3*d**2*e*
*3*x**3/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**1
1*x**5) + 300*c**3*d*e**4*x**4*log(d/e + x)/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e*
*9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) + 300*c**3*d*e**4*x**4/(30*d**5*e**6 + 150*d**4*e**7*x + 300*d**3*
e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5) + 60*c**3*e**5*x**5*log(d/e + x)/(30*d**5*e
**6 + 150*d**4*e**7*x + 300*d**3*e**8*x**2 + 300*d**2*e**9*x**3 + 150*d*e**10*x**4 + 30*e**11*x**5), Eq(m, -6)
), (-3*a**2*b*e**5/(12*d**4*e**6 + 48*d**3*e**7*x + 72*d**2*e**8*x**2 + 48*d*e**9*x**3 + 12*e**10*x**4) - 2*a*
*2*c*d*e**4/(12*d**4*e**6 + 48*d**3*e**7*x + 72...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3633 vs. \(2 (276) = 552\).
time = 1.47, size = 3633, normalized size = 13.46 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(x*e + d)^m*c^3*m^5*x^6*e^6 + 2*(x*e + d)^m*c^3*d*m^5*x^5*e^5 + 5*(x*e + d)^m*b*c^2*m^5*x^5*e^6 + 30*(x*e +
 d)^m*c^3*m^4*x^6*e^6 + 5*(x*e + d)^m*b*c^2*d*m^5*x^4*e^5 + 20*(x*e + d)^m*c^3*d*m^4*x^5*e^5 - 10*(x*e + d)^m*
c^3*d^2*m^4*x^4*e^4 + 4*(x*e + d)^m*b^2*c*m^5*x^4*e^6 + 4*(x*e + d)^m*a*c^2*m^5*x^4*e^6 + 80*(x*e + d)^m*b*c^2
*m^4*x^5*e^6 + 170*(x*e + d)^m*c^3*m^3*x^6*e^6 + 4*(x*e + d)^m*b^2*c*d*m^5*x^3*e^5 + 4*(x*e + d)^m*a*c^2*d*m^5
*x^3*e^5 + 60*(x*e + d)^m*b*c^2*d*m^4*x^4*e^5 + 70*(x*e + d)^m*c^3*d*m^3*x^5*e^5 - 20*(x*e + d)^m*b*c^2*d^2*m^
4*x^3*e^4 - 60*(x*e + d)^m*c^3*d^2*m^3*x^4*e^4 + 40*(x*e + d)^m*c^3*d^3*m^3*x^3*e^3 + (x*e + d)^m*b^3*m^5*x^3*
e^6 + 6*(x*e + d)^m*a*b*c*m^5*x^3*e^6 + 68*(x*e + d)^m*b^2*c*m^4*x^4*e^6 + 68*(x*e + d)^m*a*c^2*m^4*x^4*e^6 +
475*(x*e + d)^m*b*c^2*m^3*x^5*e^6 + 450*(x*e + d)^m*c^3*m^2*x^6*e^6 + (x*e + d)^m*b^3*d*m^5*x^2*e^5 + 6*(x*e +
 d)^m*a*b*c*d*m^5*x^2*e^5 + 56*(x*e + d)^m*b^2*c*d*m^4*x^3*e^5 + 56*(x*e + d)^m*a*c^2*d*m^4*x^3*e^5 + 235*(x*e
 + d)^m*b*c^2*d*m^3*x^4*e^5 + 100*(x*e + d)^m*c^3*d*m^2*x^5*e^5 - 12*(x*e + d)^m*b^2*c*d^2*m^4*x^2*e^4 - 12*(x
*e + d)^m*a*c^2*d^2*m^4*x^2*e^4 - 180*(x*e + d)^m*b*c^2*d^2*m^3*x^3*e^4 - 110*(x*e + d)^m*c^3*d^2*m^2*x^4*e^4
+ 60*(x*e + d)^m*b*c^2*d^3*m^3*x^2*e^3 + 120*(x*e + d)^m*c^3*d^3*m^2*x^3*e^3 - 120*(x*e + d)^m*c^3*d^4*m^2*x^2
*e^2 + 2*(x*e + d)^m*a*b^2*m^5*x^2*e^6 + 2*(x*e + d)^m*a^2*c*m^5*x^2*e^6 + 18*(x*e + d)^m*b^3*m^4*x^3*e^6 + 10
8*(x*e + d)^m*a*b*c*m^4*x^3*e^6 + 428*(x*e + d)^m*b^2*c*m^3*x^4*e^6 + 428*(x*e + d)^m*a*c^2*m^3*x^4*e^6 + 1300
*(x*e + d)^m*b*c^2*m^2*x^5*e^6 + 548*(x*e + d)^m*c^3*m*x^6*e^6 + 2*(x*e + d)^m*a*b^2*d*m^5*x*e^5 + 2*(x*e + d)
^m*a^2*c*d*m^5*x*e^5 + 16*(x*e + d)^m*b^3*d*m^4*x^2*e^5 + 96*(x*e + d)^m*a*b*c*d*m^4*x^2*e^5 + 260*(x*e + d)^m
*b^2*c*d*m^3*x^3*e^5 + 260*(x*e + d)^m*a*c^2*d*m^3*x^3*e^5 + 360*(x*e + d)^m*b*c^2*d*m^2*x^4*e^5 + 48*(x*e + d
)^m*c^3*d*m*x^5*e^5 - 2*(x*e + d)^m*b^3*d^2*m^4*x*e^4 - 12*(x*e + d)^m*a*b*c*d^2*m^4*x*e^4 - 144*(x*e + d)^m*b
^2*c*d^2*m^3*x^2*e^4 - 144*(x*e + d)^m*a*c^2*d^2*m^3*x^2*e^4 - 400*(x*e + d)^m*b*c^2*d^2*m^2*x^3*e^4 - 60*(x*e
 + d)^m*c^3*d^2*m*x^4*e^4 + 24*(x*e + d)^m*b^2*c*d^3*m^3*x*e^3 + 24*(x*e + d)^m*a*c^2*d^3*m^3*x*e^3 + 420*(x*e
 + d)^m*b*c^2*d^3*m^2*x^2*e^3 + 80*(x*e + d)^m*c^3*d^3*m*x^3*e^3 - 120*(x*e + d)^m*b*c^2*d^4*m^2*x*e^2 - 120*(
x*e + d)^m*c^3*d^4*m*x^2*e^2 + 240*(x*e + d)^m*c^3*d^5*m*x*e + (x*e + d)^m*a^2*b*m^5*x*e^6 + 38*(x*e + d)^m*a*
b^2*m^4*x^2*e^6 + 38*(x*e + d)^m*a^2*c*m^4*x^2*e^6 + 121*(x*e + d)^m*b^3*m^3*x^3*e^6 + 726*(x*e + d)^m*a*b*c*m
^3*x^3*e^6 + 1228*(x*e + d)^m*b^2*c*m^2*x^4*e^6 + 1228*(x*e + d)^m*a*c^2*m^2*x^4*e^6 + 1620*(x*e + d)^m*b*c^2*
m*x^5*e^6 + 240*(x*e + d)^m*c^3*x^6*e^6 + (x*e + d)^m*a^2*b*d*m^5*e^5 + 36*(x*e + d)^m*a*b^2*d*m^4*x*e^5 + 36*
(x*e + d)^m*a^2*c*d*m^4*x*e^5 + 89*(x*e + d)^m*b^3*d*m^3*x^2*e^5 + 534*(x*e + d)^m*a*b*c*d*m^3*x^2*e^5 + 448*(
x*e + d)^m*b^2*c*d*m^2*x^3*e^5 + 448*(x*e + d)^m*a*c^2*d*m^2*x^3*e^5 + 180*(x*e + d)^m*b*c^2*d*m*x^4*e^5 - 2*(
x*e + d)^m*a*b^2*d^2*m^4*e^4 - 2*(x*e + d)^m*a^2*c*d^2*m^4*e^4 - 30*(x*e + d)^m*b^3*d^2*m^3*x*e^4 - 180*(x*e +
 d)^m*a*b*c*d^2*m^3*x*e^4 - 492*(x*e + d)^m*b^2*c*d^2*m^2*x^2*e^4 - 492*(x*e + d)^m*a*c^2*d^2*m^2*x^2*e^4 - 24
0*(x*e + d)^m*b*c^2*d^2*m*x^3*e^4 + 2*(x*e + d)^m*b^3*d^3*m^3*e^3 + 12*(x*e + d)^m*a*b*c*d^3*m^3*e^3 + 264*(x*
e + d)^m*b^2*c*d^3*m^2*x*e^3 + 264*(x*e + d)^m*a*c^2*d^3*m^2*x*e^3 + 360*(x*e + d)^m*b*c^2*d^3*m*x^2*e^3 - 24*
(x*e + d)^m*b^2*c*d^4*m^2*e^2 - 24*(x*e + d)^m*a*c^2*d^4*m^2*e^2 - 720*(x*e + d)^m*b*c^2*d^4*m*x*e^2 + 120*(x*
e + d)^m*b*c^2*d^5*m*e - 240*(x*e + d)^m*c^3*d^6 + 20*(x*e + d)^m*a^2*b*m^4*x*e^6 + 274*(x*e + d)^m*a*b^2*m^3*
x^2*e^6 + 274*(x*e + d)^m*a^2*c*m^3*x^2*e^6 + 372*(x*e + d)^m*b^3*m^2*x^3*e^6 + 2232*(x*e + d)^m*a*b*c*m^2*x^3
*e^6 + 1584*(x*e + d)^m*b^2*c*m*x^4*e^6 + 1584*(x*e + d)^m*a*c^2*m*x^4*e^6 + 720*(x*e + d)^m*b*c^2*x^5*e^6 + 2
0*(x*e + d)^m*a^2*b*d*m^4*e^5 + 238*(x*e + d)^m*a*b^2*d*m^3*x*e^5 + 238*(x*e + d)^m*a^2*c*d*m^3*x*e^5 + 194*(x
*e + d)^m*b^3*d*m^2*x^2*e^5 + 1164*(x*e + d)^m*a*b*c*d*m^2*x^2*e^5 + 240*(x*e + d)^m*b^2*c*d*m*x^3*e^5 + 240*(
x*e + d)^m*a*c^2*d*m*x^3*e^5 - 36*(x*e + d)^m*a*b^2*d^2*m^3*e^4 - 36*(x*e + d)^m*a^2*c*d^2*m^3*e^4 - 148*(x*e
+ d)^m*b^3*d^2*m^2*x*e^4 - 888*(x*e + d)^m*a*b*c*d^2*m^2*x*e^4 - 360*(x*e + d)^m*b^2*c*d^2*m*x^2*e^4 - 360*(x*
e + d)^m*a*c^2*d^2*m*x^2*e^4 + 30*(x*e + d)^m*b^3*d^3*m^2*e^3 + 180*(x*e + d)^m*a*b*c*d^3*m^2*e^3 + 720*(x*e +
 d)^m*b^2*c*d^3*m*x*e^3 + 720*(x*e + d)^m*a*c^2*d^3*m*x*e^3 - 264*(x*e + d)^m*b^2*c*d^4*m*e^2 - 264*(x*e + d)^
m*a*c^2*d^4*m*e^2 + 720*(x*e + d)^m*b*c^2*d^5*e + 155*(x*e + d)^m*a^2*b*m^3*x*e^6 + 922*(x*e + d)^m*a*b^2*m^2*
x^2*e^6 + 922*(x*e + d)^m*a^2*c*m^2*x^2*e^6 + 508*(x*e + d)^m*b^3*m*x^3*e^6 + 3048*(x*e + d)^m*a*b*c*m*x^3*e^6
 + 720*(x*e + d)^m*b^2*c*x^4*e^6 + 720*(x*e + d)^m*a*c^2*x^4*e^6 + 155*(x*e + d)^m*a^2*b*d*m^3*e^5 + 684*(x*e
+ d)^m*a*b^2*d*m^2*x*e^5 + 684*(x*e + d)^m*a^2*c*d*m^2*x*e^5 + 120*(x*e + d)^m*b^3*d*m*x^2*e^5 + 720*(x*e + d)
^m*a*b*c*d*m*x^2*e^5 - 238*(x*e + d)^m*a*b^2*d^...

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Mupad [B]
time = 2.82, size = 1825, normalized size = 6.76 \begin {gather*} \frac {2\,c^3\,x^6\,{\left (d+e\,x\right )}^m\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {{\left (d+e\,x\right )}^m\,\left (-a^2\,b\,d\,e^5\,m^5-20\,a^2\,b\,d\,e^5\,m^4-155\,a^2\,b\,d\,e^5\,m^3-580\,a^2\,b\,d\,e^5\,m^2-1044\,a^2\,b\,d\,e^5\,m-720\,a^2\,b\,d\,e^5+2\,a^2\,c\,d^2\,e^4\,m^4+36\,a^2\,c\,d^2\,e^4\,m^3+238\,a^2\,c\,d^2\,e^4\,m^2+684\,a^2\,c\,d^2\,e^4\,m+720\,a^2\,c\,d^2\,e^4+2\,a\,b^2\,d^2\,e^4\,m^4+36\,a\,b^2\,d^2\,e^4\,m^3+238\,a\,b^2\,d^2\,e^4\,m^2+684\,a\,b^2\,d^2\,e^4\,m+720\,a\,b^2\,d^2\,e^4-12\,a\,b\,c\,d^3\,e^3\,m^3-180\,a\,b\,c\,d^3\,e^3\,m^2-888\,a\,b\,c\,d^3\,e^3\,m-1440\,a\,b\,c\,d^3\,e^3+24\,a\,c^2\,d^4\,e^2\,m^2+264\,a\,c^2\,d^4\,e^2\,m+720\,a\,c^2\,d^4\,e^2-2\,b^3\,d^3\,e^3\,m^3-30\,b^3\,d^3\,e^3\,m^2-148\,b^3\,d^3\,e^3\,m-240\,b^3\,d^3\,e^3+24\,b^2\,c\,d^4\,e^2\,m^2+264\,b^2\,c\,d^4\,e^2\,m+720\,b^2\,c\,d^4\,e^2-120\,b\,c^2\,d^5\,e\,m-720\,b\,c^2\,d^5\,e+240\,c^3\,d^6\right )}{e^6\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (a^2\,b\,e^6\,m^5+20\,a^2\,b\,e^6\,m^4+155\,a^2\,b\,e^6\,m^3+580\,a^2\,b\,e^6\,m^2+1044\,a^2\,b\,e^6\,m+720\,a^2\,b\,e^6+2\,a^2\,c\,d\,e^5\,m^5+36\,a^2\,c\,d\,e^5\,m^4+238\,a^2\,c\,d\,e^5\,m^3+684\,a^2\,c\,d\,e^5\,m^2+720\,a^2\,c\,d\,e^5\,m+2\,a\,b^2\,d\,e^5\,m^5+36\,a\,b^2\,d\,e^5\,m^4+238\,a\,b^2\,d\,e^5\,m^3+684\,a\,b^2\,d\,e^5\,m^2+720\,a\,b^2\,d\,e^5\,m-12\,a\,b\,c\,d^2\,e^4\,m^4-180\,a\,b\,c\,d^2\,e^4\,m^3-888\,a\,b\,c\,d^2\,e^4\,m^2-1440\,a\,b\,c\,d^2\,e^4\,m+24\,a\,c^2\,d^3\,e^3\,m^3+264\,a\,c^2\,d^3\,e^3\,m^2+720\,a\,c^2\,d^3\,e^3\,m-2\,b^3\,d^2\,e^4\,m^4-30\,b^3\,d^2\,e^4\,m^3-148\,b^3\,d^2\,e^4\,m^2-240\,b^3\,d^2\,e^4\,m+24\,b^2\,c\,d^3\,e^3\,m^3+264\,b^2\,c\,d^3\,e^3\,m^2+720\,b^2\,c\,d^3\,e^3\,m-120\,b\,c^2\,d^4\,e^2\,m^2-720\,b\,c^2\,d^4\,e^2\,m+240\,c^3\,d^5\,e\,m\right )}{e^6\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (2\,a^2\,c\,e^4\,m^4+36\,a^2\,c\,e^4\,m^3+238\,a^2\,c\,e^4\,m^2+684\,a^2\,c\,e^4\,m+720\,a^2\,c\,e^4+2\,a\,b^2\,e^4\,m^4+36\,a\,b^2\,e^4\,m^3+238\,a\,b^2\,e^4\,m^2+684\,a\,b^2\,e^4\,m+720\,a\,b^2\,e^4+6\,a\,b\,c\,d\,e^3\,m^4+90\,a\,b\,c\,d\,e^3\,m^3+444\,a\,b\,c\,d\,e^3\,m^2+720\,a\,b\,c\,d\,e^3\,m-12\,a\,c^2\,d^2\,e^2\,m^3-132\,a\,c^2\,d^2\,e^2\,m^2-360\,a\,c^2\,d^2\,e^2\,m+b^3\,d\,e^3\,m^4+15\,b^3\,d\,e^3\,m^3+74\,b^3\,d\,e^3\,m^2+120\,b^3\,d\,e^3\,m-12\,b^2\,c\,d^2\,e^2\,m^3-132\,b^2\,c\,d^2\,e^2\,m^2-360\,b^2\,c\,d^2\,e^2\,m+60\,b\,c^2\,d^3\,e\,m^2+360\,b\,c^2\,d^3\,e\,m-120\,c^3\,d^4\,m\right )}{e^4\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (b^3\,e^3\,m^3+15\,b^3\,e^3\,m^2+74\,b^3\,e^3\,m+120\,b^3\,e^3+4\,b^2\,c\,d\,e^2\,m^3+44\,b^2\,c\,d\,e^2\,m^2+120\,b^2\,c\,d\,e^2\,m-20\,b\,c^2\,d^2\,e\,m^2-120\,b\,c^2\,d^2\,e\,m+6\,a\,b\,c\,e^3\,m^3+90\,a\,b\,c\,e^3\,m^2+444\,a\,b\,c\,e^3\,m+720\,a\,b\,c\,e^3+40\,c^3\,d^3\,m+4\,a\,c^2\,d\,e^2\,m^3+44\,a\,c^2\,d\,e^2\,m^2+120\,a\,c^2\,d\,e^2\,m\right )}{e^3\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {c^2\,x^5\,{\left (d+e\,x\right )}^m\,\left (30\,b\,e+5\,b\,e\,m+2\,c\,d\,m\right )\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{e\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {c\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (4\,b^2\,e^2\,m^2+44\,b^2\,e^2\,m+120\,b^2\,e^2+5\,b\,c\,d\,e\,m^2+30\,b\,c\,d\,e\,m-10\,c^2\,d^2\,m+4\,a\,c\,e^2\,m^2+44\,a\,c\,e^2\,m+120\,a\,c\,e^2\right )}{e^2\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^3*x^6*(d + e*x)^m*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4
 + 21*m^5 + m^6 + 720) - ((d + e*x)^m*(240*c^3*d^6 - 240*b^3*d^3*e^3 + 720*a*b^2*d^2*e^4 + 720*a*c^2*d^4*e^2 +
 720*a^2*c*d^2*e^4 + 720*b^2*c*d^4*e^2 - 148*b^3*d^3*e^3*m - 30*b^3*d^3*e^3*m^2 - 2*b^3*d^3*e^3*m^3 - 720*a^2*
b*d*e^5 - 720*b*c^2*d^5*e - 1440*a*b*c*d^3*e^3 - 1044*a^2*b*d*e^5*m - 120*b*c^2*d^5*e*m + 684*a*b^2*d^2*e^4*m
- 580*a^2*b*d*e^5*m^2 - 155*a^2*b*d*e^5*m^3 - 20*a^2*b*d*e^5*m^4 - a^2*b*d*e^5*m^5 + 264*a*c^2*d^4*e^2*m + 684
*a^2*c*d^2*e^4*m + 264*b^2*c*d^4*e^2*m + 238*a*b^2*d^2*e^4*m^2 + 36*a*b^2*d^2*e^4*m^3 + 2*a*b^2*d^2*e^4*m^4 +
24*a*c^2*d^4*e^2*m^2 + 238*a^2*c*d^2*e^4*m^2 + 36*a^2*c*d^2*e^4*m^3 + 2*a^2*c*d^2*e^4*m^4 + 24*b^2*c*d^4*e^2*m
^2 - 888*a*b*c*d^3*e^3*m - 180*a*b*c*d^3*e^3*m^2 - 12*a*b*c*d^3*e^3*m^3))/(e^6*(1764*m + 1624*m^2 + 735*m^3 +
175*m^4 + 21*m^5 + m^6 + 720)) + (x*(d + e*x)^m*(720*a^2*b*e^6 + 580*a^2*b*e^6*m^2 + 155*a^2*b*e^6*m^3 + 20*a^
2*b*e^6*m^4 + a^2*b*e^6*m^5 - 240*b^3*d^2*e^4*m - 148*b^3*d^2*e^4*m^2 - 30*b^3*d^2*e^4*m^3 - 2*b^3*d^2*e^4*m^4
 + 1044*a^2*b*e^6*m + 240*c^3*d^5*e*m + 720*a*b^2*d*e^5*m + 720*a^2*c*d*e^5*m + 684*a*b^2*d*e^5*m^2 + 238*a*b^
2*d*e^5*m^3 + 36*a*b^2*d*e^5*m^4 + 2*a*b^2*d*e^5*m^5 + 720*a*c^2*d^3*e^3*m + 684*a^2*c*d*e^5*m^2 + 238*a^2*c*d
*e^5*m^3 + 36*a^2*c*d*e^5*m^4 + 2*a^2*c*d*e^5*m^5 - 720*b*c^2*d^4*e^2*m + 720*b^2*c*d^3*e^3*m + 264*a*c^2*d^3*
e^3*m^2 + 24*a*c^2*d^3*e^3*m^3 - 120*b*c^2*d^4*e^2*m^2 + 264*b^2*c*d^3*e^3*m^2 + 24*b^2*c*d^3*e^3*m^3 - 1440*a
*b*c*d^2*e^4*m - 888*a*b*c*d^2*e^4*m^2 - 180*a*b*c*d^2*e^4*m^3 - 12*a*b*c*d^2*e^4*m^4))/(e^6*(1764*m + 1624*m^
2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (x^2*(m + 1)*(d + e*x)^m*(720*a*b^2*e^4 + 720*a^2*c*e^4 - 120*c
^3*d^4*m + 238*a*b^2*e^4*m^2 + 36*a*b^2*e^4*m^3 + 2*a*b^2*e^4*m^4 + 238*a^2*c*e^4*m^2 + 36*a^2*c*e^4*m^3 + 2*a
^2*c*e^4*m^4 + 74*b^3*d*e^3*m^2 + 15*b^3*d*e^3*m^3 + b^3*d*e^3*m^4 + 684*a*b^2*e^4*m + 684*a^2*c*e^4*m + 120*b
^3*d*e^3*m + 360*b*c^2*d^3*e*m - 360*a*c^2*d^2*e^2*m - 360*b^2*c*d^2*e^2*m + 60*b*c^2*d^3*e*m^2 - 132*a*c^2*d^
2*e^2*m^2 - 12*a*c^2*d^2*e^2*m^3 - 132*b^2*c*d^2*e^2*m^2 - 12*b^2*c*d^2*e^2*m^3 + 720*a*b*c*d*e^3*m + 444*a*b*
c*d*e^3*m^2 + 90*a*b*c*d*e^3*m^3 + 6*a*b*c*d*e^3*m^4))/(e^4*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 +
m^6 + 720)) + (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(120*b^3*e^3 + 74*b^3*e^3*m + 40*c^3*d^3*m + 15*b^3*e^3*m^2 + b
^3*e^3*m^3 + 720*a*b*c*e^3 + 90*a*b*c*e^3*m^2 + 6*a*b*c*e^3*m^3 + 120*a*c^2*d*e^2*m - 120*b*c^2*d^2*e*m + 120*
b^2*c*d*e^2*m + 44*a*c^2*d*e^2*m^2 + 4*a*c^2*d*e^2*m^3 - 20*b*c^2*d^2*e*m^2 + 44*b^2*c*d*e^2*m^2 + 4*b^2*c*d*e
^2*m^3 + 444*a*b*c*e^3*m))/(e^3*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (c^2*x^5*(d +
e*x)^m*(30*b*e + 5*b*e*m + 2*c*d*m)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(e*(1764*m + 1624*m^2 + 735*m^3 + 175
*m^4 + 21*m^5 + m^6 + 720)) + (c*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(120*b^2*e^2 + 44*b^2*e^2*m - 10*c^2
*d^2*m + 4*b^2*e^2*m^2 + 120*a*c*e^2 + 44*a*c*e^2*m + 4*a*c*e^2*m^2 + 30*b*c*d*e*m + 5*b*c*d*e*m^2))/(e^2*(176
4*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))

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