3.32.80 \(\int (a+b x)^4 (A+B x) (d+e x)^m \, dx\) [3180]

Optimal. Leaf size=234 \[ -\frac {(b d-a e)^4 (B d-A e) (d+e x)^{1+m}}{e^6 (1+m)}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{2+m}}{e^6 (2+m)}-\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{3+m}}{e^6 (3+m)}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{4+m}}{e^6 (4+m)}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac {b^4 B (d+e x)^{6+m}}{e^6 (6+m)} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 234, 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 = {78} \begin {gather*} -\frac {b^3 (d+e x)^{m+5} (-4 a B e-A b e+5 b B d)}{e^6 (m+5)}+\frac {2 b^2 (b d-a e) (d+e x)^{m+4} (-3 a B e-2 A b e+5 b B d)}{e^6 (m+4)}-\frac {(b d-a e)^4 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac {(b d-a e)^3 (d+e x)^{m+2} (-a B e-4 A b e+5 b B d)}{e^6 (m+2)}-\frac {2 b (b d-a e)^2 (d+e x)^{m+3} (-2 a B e-3 A b e+5 b B d)}{e^6 (m+3)}+\frac {b^4 B (d+e x)^{m+6}}{e^6 (m+6)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*(A + B*x)*(d + e*x)^m,x]

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]
time = 0.28, size = 208, normalized size = 0.89 \begin {gather*} \frac {(d+e x)^{1+m} \left (-\frac {(b d-a e)^4 (B d-A e)}{1+m}+\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)}{2+m}-\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^2}{3+m}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^3}{4+m}-\frac {b^3 (5 b B d-A b e-4 a B e) (d+e x)^4}{5+m}+\frac {b^4 B (d+e x)^5}{6+m}\right )}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*(A + B*x)*(d + e*x)^m,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1957\) vs. \(2(234)=468\).
time = 0.12, size = 1958, normalized size = 8.37

method	result	size
norman	\(\text {Expression too large to display}\)	\(1958\)
gosper	\(\text {Expression too large to display}\)	\(2355\)
risch	\(\text {Expression too large to display}\)	\(3035\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*(B*x+A)*(e*x+d)^m,x,method=_RETURNVERBOSE)

[Out]

B*b^4/(6+m)*x^6*exp(m*ln(e*x+d))+d*(A*a^4*e^5*m^5+20*A*a^4*e^5*m^4-4*A*a^3*b*d*e^4*m^4-B*a^4*d*e^4*m^4+155*A*a
^4*e^5*m^3-72*A*a^3*b*d*e^4*m^3+12*A*a^2*b^2*d^2*e^3*m^3-18*B*a^4*d*e^4*m^3+8*B*a^3*b*d^2*e^3*m^3+580*A*a^4*e^
5*m^2-476*A*a^3*b*d*e^4*m^2+180*A*a^2*b^2*d^2*e^3*m^2-24*A*a*b^3*d^3*e^2*m^2-119*B*a^4*d*e^4*m^2+120*B*a^3*b*d
^2*e^3*m^2-36*B*a^2*b^2*d^3*e^2*m^2+1044*A*a^4*e^5*m-1368*A*a^3*b*d*e^4*m+888*A*a^2*b^2*d^2*e^3*m-264*A*a*b^3*
d^3*e^2*m+24*A*b^4*d^4*e*m-342*B*a^4*d*e^4*m+592*B*a^3*b*d^2*e^3*m-396*B*a^2*b^2*d^3*e^2*m+96*B*a*b^3*d^4*e*m+
720*A*a^4*e^5-1440*A*a^3*b*d*e^4+1440*A*a^2*b^2*d^2*e^3-720*A*a*b^3*d^3*e^2+144*A*b^4*d^4*e-360*B*a^4*d*e^4+96
0*B*a^3*b*d^2*e^3-1080*B*a^2*b^2*d^3*e^2+576*B*a*b^3*d^4*e-120*B*b^4*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))+(4*A*a^3*b*e^4*m^4+6*A*a^2*b^2*d*e^3*m^4+B*a^4*e^4*m^4+4*B*a^3*b*d*e^3*m^4+7
2*A*a^3*b*e^4*m^3+90*A*a^2*b^2*d*e^3*m^3-12*A*a*b^3*d^2*e^2*m^3+18*B*a^4*e^4*m^3+60*B*a^3*b*d*e^3*m^3-18*B*a^2
*b^2*d^2*e^2*m^3+476*A*a^3*b*e^4*m^2+444*A*a^2*b^2*d*e^3*m^2-132*A*a*b^3*d^2*e^2*m^2+12*A*b^4*d^3*e*m^2+119*B*
a^4*e^4*m^2+296*B*a^3*b*d*e^3*m^2-198*B*a^2*b^2*d^2*e^2*m^2+48*B*a*b^3*d^3*e*m^2+1368*A*a^3*b*e^4*m+720*A*a^2*
b^2*d*e^3*m-360*A*a*b^3*d^2*e^2*m+72*A*b^4*d^3*e*m+342*B*a^4*e^4*m+480*B*a^3*b*d*e^3*m-540*B*a^2*b^2*d^2*e^2*m
+288*B*a*b^3*d^3*e*m-60*B*b^4*d^4*m+1440*A*a^3*b*e^4+360*B*a^4*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*a^4*e^5*m^5+4*A*a^3*b*d*e^4*m^5+B*a^4*d*e^4*m^5+20*A*a^4*e^5*m^4+72*A*a^3*b*d*e^4*m^
4-12*A*a^2*b^2*d^2*e^3*m^4+18*B*a^4*d*e^4*m^4-8*B*a^3*b*d^2*e^3*m^4+155*A*a^4*e^5*m^3+476*A*a^3*b*d*e^4*m^3-18
0*A*a^2*b^2*d^2*e^3*m^3+24*A*a*b^3*d^3*e^2*m^3+119*B*a^4*d*e^4*m^3-120*B*a^3*b*d^2*e^3*m^3+36*B*a^2*b^2*d^3*e^
2*m^3+580*A*a^4*e^5*m^2+1368*A*a^3*b*d*e^4*m^2-888*A*a^2*b^2*d^2*e^3*m^2+264*A*a*b^3*d^3*e^2*m^2-24*A*b^4*d^4*
e*m^2+342*B*a^4*d*e^4*m^2-592*B*a^3*b*d^2*e^3*m^2+396*B*a^2*b^2*d^3*e^2*m^2-96*B*a*b^3*d^4*e*m^2+1044*A*a^4*e^
5*m+1440*A*a^3*b*d*e^4*m-1440*A*a^2*b^2*d^2*e^3*m+720*A*a*b^3*d^3*e^2*m-144*A*b^4*d^4*e*m+360*B*a^4*d*e^4*m-96
0*B*a^3*b*d^2*e^3*m+1080*B*a^2*b^2*d^3*e^2*m-576*B*a*b^3*d^4*e*m+120*B*b^4*d^5*m+720*A*a^4*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))+(A*b*e*m+4*B*a*e*m+B*b*d*m+6*A*b*e+24*B*a*e)*b^3/e/(
m^2+11*m+30)*x^5*exp(m*ln(e*x+d))+(4*A*a*b*e^2*m^2+A*b^2*d*e*m^2+6*B*a^2*e^2*m^2+4*B*a*b*d*e*m^2+44*A*a*b*e^2*
m+6*A*b^2*d*e*m+66*B*a^2*e^2*m+24*B*a*b*d*e*m-5*B*b^2*d^2*m+120*A*a*b*e^2+180*B*a^2*e^2)/e^2*b^2/(m^3+15*m^2+7
4*m+120)*x^4*exp(m*ln(e*x+d))+2*(3*A*a^2*b*e^3*m^3+2*A*a*b^2*d*e^2*m^3+2*B*a^3*e^3*m^3+3*B*a^2*b*d*e^2*m^3+45*
A*a^2*b*e^3*m^2+22*A*a*b^2*d*e^2*m^2-2*A*b^3*d^2*e*m^2+30*B*a^3*e^3*m^2+33*B*a^2*b*d*e^2*m^2-8*B*a*b^2*d^2*e*m
^2+222*A*a^2*b*e^3*m+60*A*a*b^2*d*e^2*m-12*A*b^3*d^2*e*m+148*B*a^3*e^3*m+90*B*a^2*b*d*e^2*m-48*B*a*b^2*d^2*e*m
+10*B*b^3*d^3*m+360*A*a^2*b*e^3+240*B*a^3*e^3)*b/e^3/(m^4+18*m^3+119*m^2+342*m+360)*x^3*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. 959 vs. \(2 (247) = 494\).
time = 0.35, size = 959, normalized size = 4.10 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} A a^{4} e^{\left (-1\right )}}{m + 1} + \frac {{\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} B a^{4} e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {4 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} A a^{3} b e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {4 \, {\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 a^{3} b 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 a^{2} b^{2} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {6 \, {\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 a^{2} b^{2} 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 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 {4 \, {\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 a b^{3} 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 {{\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 )} A 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} + \frac {{\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 )} B b^{4} 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((b*x+a)^4*(B*x+A)*(e*x+d)^m,x, algorithm="maxima")

[Out]

(x*e + d)^(m + 1)*A*a^4*e^(-1)/(m + 1) + ((m + 1)*x^2*e^2 + d*m*x*e - d^2)*B*a^4*e^(m*log(x*e + d) - 2)/(m^2 +
3*m + 2) + 4*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*A*a^3*b*e^(m*log(x*e + d) - 2)/(m^2 + 3*m + 2) + 4*((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*a^3*b*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*a^2*b^2*e^(m*log(x*e + d) - 3
)/(m^3 + 6*m^2 + 11*m + 6) + 6*((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*a^2*b^2*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*a*b^3*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) + 4*((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*a*b^3*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) +
((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)*A*b^4*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 8
5*m^3 + 225*m^2 + 274*m + 120) + ((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)*B*b^4*e^(m*log(x*e + d) - 6)/(m^6 + 21*m^5 + 175*m^4
+ 735*m^3 + 1624*m^2 + 1764*m + 720)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2088 vs. \(2 (247) = 494\).
time = 1.12, size = 2088, normalized size = 8.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(B*x+A)*(e*x+d)^m,x, algorithm="fricas")

[Out]

-(120*B*b^4*d^6 - ((B*b^4*m^5 + 15*B*b^4*m^4 + 85*B*b^4*m^3 + 225*B*b^4*m^2 + 274*B*b^4*m + 120*B*b^4)*x^6 + (
(4*B*a*b^3 + A*b^4)*m^5 + 576*B*a*b^3 + 144*A*b^4 + 16*(4*B*a*b^3 + A*b^4)*m^4 + 95*(4*B*a*b^3 + A*b^4)*m^3 +
260*(4*B*a*b^3 + A*b^4)*m^2 + 324*(4*B*a*b^3 + A*b^4)*m)*x^5 + 2*((3*B*a^2*b^2 + 2*A*a*b^3)*m^5 + 540*B*a^2*b^
2 + 360*A*a*b^3 + 17*(3*B*a^2*b^2 + 2*A*a*b^3)*m^4 + 107*(3*B*a^2*b^2 + 2*A*a*b^3)*m^3 + 307*(3*B*a^2*b^2 + 2*
A*a*b^3)*m^2 + 396*(3*B*a^2*b^2 + 2*A*a*b^3)*m)*x^4 + 2*((2*B*a^3*b + 3*A*a^2*b^2)*m^5 + 480*B*a^3*b + 720*A*a
^2*b^2 + 18*(2*B*a^3*b + 3*A*a^2*b^2)*m^4 + 121*(2*B*a^3*b + 3*A*a^2*b^2)*m^3 + 372*(2*B*a^3*b + 3*A*a^2*b^2)*
m^2 + 508*(2*B*a^3*b + 3*A*a^2*b^2)*m)*x^3 + ((B*a^4 + 4*A*a^3*b)*m^5 + 360*B*a^4 + 1440*A*a^3*b + 19*(B*a^4 +
4*A*a^3*b)*m^4 + 137*(B*a^4 + 4*A*a^3*b)*m^3 + 461*(B*a^4 + 4*A*a^3*b)*m^2 + 702*(B*a^4 + 4*A*a^3*b)*m)*x^2 +
(A*a^4*m^5 + 20*A*a^4*m^4 + 155*A*a^4*m^3 + 580*A*a^4*m^2 + 1044*A*a^4*m + 720*A*a^4)*x)*e^6 - (A*a^4*d*m^5 +
20*A*a^4*d*m^4 + 155*A*a^4*d*m^3 + 580*A*a^4*d*m^2 + 1044*A*a^4*d*m + 720*A*a^4*d + (B*b^4*d*m^5 + 10*B*b^4*d
*m^4 + 35*B*b^4*d*m^3 + 50*B*b^4*d*m^2 + 24*B*b^4*d*m)*x^5 + ((4*B*a*b^3 + A*b^4)*d*m^5 + 12*(4*B*a*b^3 + A*b^
4)*d*m^4 + 47*(4*B*a*b^3 + A*b^4)*d*m^3 + 72*(4*B*a*b^3 + A*b^4)*d*m^2 + 36*(4*B*a*b^3 + A*b^4)*d*m)*x^4 + 2*(
(3*B*a^2*b^2 + 2*A*a*b^3)*d*m^5 + 14*(3*B*a^2*b^2 + 2*A*a*b^3)*d*m^4 + 65*(3*B*a^2*b^2 + 2*A*a*b^3)*d*m^3 + 11
2*(3*B*a^2*b^2 + 2*A*a*b^3)*d*m^2 + 60*(3*B*a^2*b^2 + 2*A*a*b^3)*d*m)*x^3 + 2*((2*B*a^3*b + 3*A*a^2*b^2)*d*m^5
+ 16*(2*B*a^3*b + 3*A*a^2*b^2)*d*m^4 + 89*(2*B*a^3*b + 3*A*a^2*b^2)*d*m^3 + 194*(2*B*a^3*b + 3*A*a^2*b^2)*d*m
^2 + 120*(2*B*a^3*b + 3*A*a^2*b^2)*d*m)*x^2 + ((B*a^4 + 4*A*a^3*b)*d*m^5 + 18*(B*a^4 + 4*A*a^3*b)*d*m^4 + 119*
(B*a^4 + 4*A*a^3*b)*d*m^3 + 342*(B*a^4 + 4*A*a^3*b)*d*m^2 + 360*(B*a^4 + 4*A*a^3*b)*d*m)*x)*e^5 + ((B*a^4 + 4*
A*a^3*b)*d^2*m^4 + 18*(B*a^4 + 4*A*a^3*b)*d^2*m^3 + 119*(B*a^4 + 4*A*a^3*b)*d^2*m^2 + 5*(B*b^4*d^2*m^4 + 6*B*b
^4*d^2*m^3 + 11*B*b^4*d^2*m^2 + 6*B*b^4*d^2*m)*x^4 + 342*(B*a^4 + 4*A*a^3*b)*d^2*m + 4*((4*B*a*b^3 + A*b^4)*d^
2*m^4 + 9*(4*B*a*b^3 + A*b^4)*d^2*m^3 + 20*(4*B*a*b^3 + A*b^4)*d^2*m^2 + 12*(4*B*a*b^3 + A*b^4)*d^2*m)*x^3 + 3
60*(B*a^4 + 4*A*a^3*b)*d^2 + 6*((3*B*a^2*b^2 + 2*A*a*b^3)*d^2*m^4 + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*m^3 + 41*
(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*m^2 + 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*m)*x^2 + 4*((2*B*a^3*b + 3*A*a^2*b^2)*d^2
*m^4 + 15*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*m^3 + 74*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*m^2 + 120*(2*B*a^3*b + 3*A*a^2*
b^2)*d^2*m)*x)*e^4 - 4*((2*B*a^3*b + 3*A*a^2*b^2)*d^3*m^3 + 15*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*m^2 + 74*(2*B*a^3
*b + 3*A*a^2*b^2)*d^3*m + 120*(2*B*a^3*b + 3*A*a^2*b^2)*d^3 + 5*(B*b^4*d^3*m^3 + 3*B*b^4*d^3*m^2 + 2*B*b^4*d^3
*m)*x^3 + 3*((4*B*a*b^3 + A*b^4)*d^3*m^3 + 7*(4*B*a*b^3 + A*b^4)*d^3*m^2 + 6*(4*B*a*b^3 + A*b^4)*d^3*m)*x^2 +
3*((3*B*a^2*b^2 + 2*A*a*b^3)*d^3*m^3 + 11*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*m^2 + 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3
*m)*x)*e^3 + 12*((3*B*a^2*b^2 + 2*A*a*b^3)*d^4*m^2 + 11*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*m + 30*(3*B*a^2*b^2 + 2*
A*a*b^3)*d^4 + 5*(B*b^4*d^4*m^2 + B*b^4*d^4*m)*x^2 + 2*((4*B*a*b^3 + A*b^4)*d^4*m^2 + 6*(4*B*a*b^3 + A*b^4)*d^
4*m)*x)*e^2 - 24*(5*B*b^4*d^5*m*x + (4*B*a*b^3 + A*b^4)*d^5*m + 6*(4*B*a*b^3 + A*b^4)*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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 28048 vs. \(2 (224) = 448\).
time = 5.84, size = 28048, normalized size = 119.86 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4*(B*x+A)*(e*x+d)**m,x)

[Out]

Piecewise((d**m*(A*a**4*x + 2*A*a**3*b*x**2 + 2*A*a**2*b**2*x**3 + A*a*b**3*x**4 + A*b**4*x**5/5 + B*a**4*x**2
/2 + 4*B*a**3*b*x**3/3 + 3*B*a**2*b**2*x**4/2 + 4*B*a*b**3*x**5/5 + B*b**4*x**6/6), Eq(e, 0)), (-12*A*a**4*e**
5/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5
) - 12*A*a**3*b*d*e**4/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10
*x**4 + 60*e**11*x**5) - 60*A*a**3*b*e**5*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e*
*9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 12*A*a**2*b**2*d**2*e**3/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d
**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 60*A*a**2*b**2*d*e**4*x/(60*d**5*e**6
+ 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 120*A*a**2*
b**2*e**5*x**2/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 +
60*e**11*x**5) - 12*A*a*b**3*d**3*e**2/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x*
*3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 60*A*a*b**3*d**2*e**3*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e*
*8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 120*A*a*b**3*d*e**4*x**2/(60*d**5*e**6 + 30
0*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 120*A*a*b**3*e**
5*x**3/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11
*x**5) - 12*A*b**4*d**4*e/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e*
*10*x**4 + 60*e**11*x**5) - 60*A*b**4*d**3*e**2*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d
**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 120*A*b**4*d**2*e**3*x**2/(60*d**5*e**6 + 300*d**4*e**7*x
+ 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 120*A*b**4*d*e**4*x**3/(60*d**
5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 60*A*
b**4*e**5*x**4/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 +
60*e**11*x**5) - 3*B*a**4*d*e**4/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 3
00*d*e**10*x**4 + 60*e**11*x**5) - 15*B*a**4*e**5*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600
*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 8*B*a**3*b*d**2*e**3/(60*d**5*e**6 + 300*d**4*e**7*x + 6
00*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 40*B*a**3*b*d*e**4*x/(60*d**5*e**
6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 80*B*a**3*
b*e**5*x**2/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*
e**11*x**5) - 18*B*a**2*b**2*d**3*e**2/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x*
*3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 90*B*a**2*b**2*d**2*e**3*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3
*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 180*B*a**2*b**2*d*e**4*x**2/(60*d**5*e**
6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 180*B*a**2
*b**2*e**5*x**3/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 +
60*e**11*x**5) - 48*B*a*b**3*d**4*e/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3
+ 300*d*e**10*x**4 + 60*e**11*x**5) - 240*B*a*b**3*d**3*e**2*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**
8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 480*B*a*b**3*d**2*e**3*x**2/(60*d**5*e**6 +
300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) - 480*B*a*b**3*d
*e**4*x**3/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e
**11*x**5) - 240*B*a*b**3*e**5*x**4/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3
+ 300*d*e**10*x**4 + 60*e**11*x**5) + 60*B*b**4*d**5*log(d/e + x)/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e
**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) + 137*B*b**4*d**5/(60*d**5*e**6 + 300*d**4*e
**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) + 300*B*b**4*d**4*e*x*log(
d/e + x)/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**
11*x**5) + 625*B*b**4*d**4*e*x/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300
*d*e**10*x**4 + 60*e**11*x**5) + 600*B*b**4*d**3*e**2*x**2*log(d/e + x)/(60*d**5*e**6 + 300*d**4*e**7*x + 600*
d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) + 1100*B*b**4*d**3*e**2*x**2/(60*d**5*
e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x**5) + 600*B*b
**4*d**2*e**3*x**3*log(d/e + x)/(60*d**5*e**6 +...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4381 vs. \(2 (247) = 494\).
time = 1.06, size = 4381, normalized size = 18.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(B*x+A)*(e*x+d)^m,x, algorithm="giac")

[Out]

((x*e + d)^m*B*b^4*m^5*x^6*e^6 + (x*e + d)^m*B*b^4*d*m^5*x^5*e^5 + 4*(x*e + d)^m*B*a*b^3*m^5*x^5*e^6 + (x*e +
d)^m*A*b^4*m^5*x^5*e^6 + 15*(x*e + d)^m*B*b^4*m^4*x^6*e^6 + 4*(x*e + d)^m*B*a*b^3*d*m^5*x^4*e^5 + (x*e + d)^m*
A*b^4*d*m^5*x^4*e^5 + 10*(x*e + d)^m*B*b^4*d*m^4*x^5*e^5 - 5*(x*e + d)^m*B*b^4*d^2*m^4*x^4*e^4 + 6*(x*e + d)^m
*B*a^2*b^2*m^5*x^4*e^6 + 4*(x*e + d)^m*A*a*b^3*m^5*x^4*e^6 + 64*(x*e + d)^m*B*a*b^3*m^4*x^5*e^6 + 16*(x*e + d)
^m*A*b^4*m^4*x^5*e^6 + 85*(x*e + d)^m*B*b^4*m^3*x^6*e^6 + 6*(x*e + d)^m*B*a^2*b^2*d*m^5*x^3*e^5 + 4*(x*e + d)^
m*A*a*b^3*d*m^5*x^3*e^5 + 48*(x*e + d)^m*B*a*b^3*d*m^4*x^4*e^5 + 12*(x*e + d)^m*A*b^4*d*m^4*x^4*e^5 + 35*(x*e
+ d)^m*B*b^4*d*m^3*x^5*e^5 - 16*(x*e + d)^m*B*a*b^3*d^2*m^4*x^3*e^4 - 4*(x*e + d)^m*A*b^4*d^2*m^4*x^3*e^4 - 30
*(x*e + d)^m*B*b^4*d^2*m^3*x^4*e^4 + 20*(x*e + d)^m*B*b^4*d^3*m^3*x^3*e^3 + 4*(x*e + d)^m*B*a^3*b*m^5*x^3*e^6
+ 6*(x*e + d)^m*A*a^2*b^2*m^5*x^3*e^6 + 102*(x*e + d)^m*B*a^2*b^2*m^4*x^4*e^6 + 68*(x*e + d)^m*A*a*b^3*m^4*x^4
*e^6 + 380*(x*e + d)^m*B*a*b^3*m^3*x^5*e^6 + 95*(x*e + d)^m*A*b^4*m^3*x^5*e^6 + 225*(x*e + d)^m*B*b^4*m^2*x^6*
e^6 + 4*(x*e + d)^m*B*a^3*b*d*m^5*x^2*e^5 + 6*(x*e + d)^m*A*a^2*b^2*d*m^5*x^2*e^5 + 84*(x*e + d)^m*B*a^2*b^2*d
*m^4*x^3*e^5 + 56*(x*e + d)^m*A*a*b^3*d*m^4*x^3*e^5 + 188*(x*e + d)^m*B*a*b^3*d*m^3*x^4*e^5 + 47*(x*e + d)^m*A
*b^4*d*m^3*x^4*e^5 + 50*(x*e + d)^m*B*b^4*d*m^2*x^5*e^5 - 18*(x*e + d)^m*B*a^2*b^2*d^2*m^4*x^2*e^4 - 12*(x*e +
d)^m*A*a*b^3*d^2*m^4*x^2*e^4 - 144*(x*e + d)^m*B*a*b^3*d^2*m^3*x^3*e^4 - 36*(x*e + d)^m*A*b^4*d^2*m^3*x^3*e^4
- 55*(x*e + d)^m*B*b^4*d^2*m^2*x^4*e^4 + 48*(x*e + d)^m*B*a*b^3*d^3*m^3*x^2*e^3 + 12*(x*e + d)^m*A*b^4*d^3*m^
3*x^2*e^3 + 60*(x*e + d)^m*B*b^4*d^3*m^2*x^3*e^3 - 60*(x*e + d)^m*B*b^4*d^4*m^2*x^2*e^2 + (x*e + d)^m*B*a^4*m^
5*x^2*e^6 + 4*(x*e + d)^m*A*a^3*b*m^5*x^2*e^6 + 72*(x*e + d)^m*B*a^3*b*m^4*x^3*e^6 + 108*(x*e + d)^m*A*a^2*b^2
*m^4*x^3*e^6 + 642*(x*e + d)^m*B*a^2*b^2*m^3*x^4*e^6 + 428*(x*e + d)^m*A*a*b^3*m^3*x^4*e^6 + 1040*(x*e + d)^m*
B*a*b^3*m^2*x^5*e^6 + 260*(x*e + d)^m*A*b^4*m^2*x^5*e^6 + 274*(x*e + d)^m*B*b^4*m*x^6*e^6 + (x*e + d)^m*B*a^4*
d*m^5*x*e^5 + 4*(x*e + d)^m*A*a^3*b*d*m^5*x*e^5 + 64*(x*e + d)^m*B*a^3*b*d*m^4*x^2*e^5 + 96*(x*e + d)^m*A*a^2*
b^2*d*m^4*x^2*e^5 + 390*(x*e + d)^m*B*a^2*b^2*d*m^3*x^3*e^5 + 260*(x*e + d)^m*A*a*b^3*d*m^3*x^3*e^5 + 288*(x*e
+ d)^m*B*a*b^3*d*m^2*x^4*e^5 + 72*(x*e + d)^m*A*b^4*d*m^2*x^4*e^5 + 24*(x*e + d)^m*B*b^4*d*m*x^5*e^5 - 8*(x*e
+ d)^m*B*a^3*b*d^2*m^4*x*e^4 - 12*(x*e + d)^m*A*a^2*b^2*d^2*m^4*x*e^4 - 216*(x*e + d)^m*B*a^2*b^2*d^2*m^3*x^2
*e^4 - 144*(x*e + d)^m*A*a*b^3*d^2*m^3*x^2*e^4 - 320*(x*e + d)^m*B*a*b^3*d^2*m^2*x^3*e^4 - 80*(x*e + d)^m*A*b^
4*d^2*m^2*x^3*e^4 - 30*(x*e + d)^m*B*b^4*d^2*m*x^4*e^4 + 36*(x*e + d)^m*B*a^2*b^2*d^3*m^3*x*e^3 + 24*(x*e + d)
^m*A*a*b^3*d^3*m^3*x*e^3 + 336*(x*e + d)^m*B*a*b^3*d^3*m^2*x^2*e^3 + 84*(x*e + d)^m*A*b^4*d^3*m^2*x^2*e^3 + 40
*(x*e + d)^m*B*b^4*d^3*m*x^3*e^3 - 96*(x*e + d)^m*B*a*b^3*d^4*m^2*x*e^2 - 24*(x*e + d)^m*A*b^4*d^4*m^2*x*e^2 -
60*(x*e + d)^m*B*b^4*d^4*m*x^2*e^2 + 120*(x*e + d)^m*B*b^4*d^5*m*x*e + (x*e + d)^m*A*a^4*m^5*x*e^6 + 19*(x*e
+ d)^m*B*a^4*m^4*x^2*e^6 + 76*(x*e + d)^m*A*a^3*b*m^4*x^2*e^6 + 484*(x*e + d)^m*B*a^3*b*m^3*x^3*e^6 + 726*(x*e
+ d)^m*A*a^2*b^2*m^3*x^3*e^6 + 1842*(x*e + d)^m*B*a^2*b^2*m^2*x^4*e^6 + 1228*(x*e + d)^m*A*a*b^3*m^2*x^4*e^6
+ 1296*(x*e + d)^m*B*a*b^3*m*x^5*e^6 + 324*(x*e + d)^m*A*b^4*m*x^5*e^6 + 120*(x*e + d)^m*B*b^4*x^6*e^6 + (x*e
+ d)^m*A*a^4*d*m^5*e^5 + 18*(x*e + d)^m*B*a^4*d*m^4*x*e^5 + 72*(x*e + d)^m*A*a^3*b*d*m^4*x*e^5 + 356*(x*e + d)
^m*B*a^3*b*d*m^3*x^2*e^5 + 534*(x*e + d)^m*A*a^2*b^2*d*m^3*x^2*e^5 + 672*(x*e + d)^m*B*a^2*b^2*d*m^2*x^3*e^5 +
448*(x*e + d)^m*A*a*b^3*d*m^2*x^3*e^5 + 144*(x*e + d)^m*B*a*b^3*d*m*x^4*e^5 + 36*(x*e + d)^m*A*b^4*d*m*x^4*e^
5 - (x*e + d)^m*B*a^4*d^2*m^4*e^4 - 4*(x*e + d)^m*A*a^3*b*d^2*m^4*e^4 - 120*(x*e + d)^m*B*a^3*b*d^2*m^3*x*e^4
- 180*(x*e + d)^m*A*a^2*b^2*d^2*m^3*x*e^4 - 738*(x*e + d)^m*B*a^2*b^2*d^2*m^2*x^2*e^4 - 492*(x*e + d)^m*A*a*b^
3*d^2*m^2*x^2*e^4 - 192*(x*e + d)^m*B*a*b^3*d^2*m*x^3*e^4 - 48*(x*e + d)^m*A*b^4*d^2*m*x^3*e^4 + 8*(x*e + d)^m
*B*a^3*b*d^3*m^3*e^3 + 12*(x*e + d)^m*A*a^2*b^2*d^3*m^3*e^3 + 396*(x*e + d)^m*B*a^2*b^2*d^3*m^2*x*e^3 + 264*(x
*e + d)^m*A*a*b^3*d^3*m^2*x*e^3 + 288*(x*e + d)^m*B*a*b^3*d^3*m*x^2*e^3 + 72*(x*e + d)^m*A*b^4*d^3*m*x^2*e^3 -
36*(x*e + d)^m*B*a^2*b^2*d^4*m^2*e^2 - 24*(x*e + d)^m*A*a*b^3*d^4*m^2*e^2 - 576*(x*e + d)^m*B*a*b^3*d^4*m*x*e
^2 - 144*(x*e + d)^m*A*b^4*d^4*m*x*e^2 + 96*(x*e + d)^m*B*a*b^3*d^5*m*e + 24*(x*e + d)^m*A*b^4*d^5*m*e - 120*(
x*e + d)^m*B*b^4*d^6 + 20*(x*e + d)^m*A*a^4*m^4*x*e^6 + 137*(x*e + d)^m*B*a^4*m^3*x^2*e^6 + 548*(x*e + d)^m*A*
a^3*b*m^3*x^2*e^6 + 1488*(x*e + d)^m*B*a^3*b*m^2*x^3*e^6 + 2232*(x*e + d)^m*A*a^2*b^2*m^2*x^3*e^6 + 2376*(x*e
+ d)^m*B*a^2*b^2*m*x^4*e^6 + 1584*(x*e + d)^m*A*a*b^3*m*x^4*e^6 + 576*(x*e + d)^m*B*a*b^3*x^5*e^6 + 144*(x*e +
d)^m*A*b^4*x^5*e^6 + 20*(x*e + d)^m*A*a^4*d*m^4*e^5 + 119*(x*e + d)^m*B*a^4*d*m^3*x*e^5 + 476*(x*e + d)^m*A*a
^3*b*d*m^3*x*e^5 + 776*(x*e + d)^m*B*a^3*b*d*m^...

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Mupad [B]
time = 3.54, size = 2117, normalized size = 9.05 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (-B\,a^4\,d^2\,e^4\,m^4-18\,B\,a^4\,d^2\,e^4\,m^3-119\,B\,a^4\,d^2\,e^4\,m^2-342\,B\,a^4\,d^2\,e^4\,m-360\,B\,a^4\,d^2\,e^4+A\,a^4\,d\,e^5\,m^5+20\,A\,a^4\,d\,e^5\,m^4+155\,A\,a^4\,d\,e^5\,m^3+580\,A\,a^4\,d\,e^5\,m^2+1044\,A\,a^4\,d\,e^5\,m+720\,A\,a^4\,d\,e^5+8\,B\,a^3\,b\,d^3\,e^3\,m^3+120\,B\,a^3\,b\,d^3\,e^3\,m^2+592\,B\,a^3\,b\,d^3\,e^3\,m+960\,B\,a^3\,b\,d^3\,e^3-4\,A\,a^3\,b\,d^2\,e^4\,m^4-72\,A\,a^3\,b\,d^2\,e^4\,m^3-476\,A\,a^3\,b\,d^2\,e^4\,m^2-1368\,A\,a^3\,b\,d^2\,e^4\,m-1440\,A\,a^3\,b\,d^2\,e^4-36\,B\,a^2\,b^2\,d^4\,e^2\,m^2-396\,B\,a^2\,b^2\,d^4\,e^2\,m-1080\,B\,a^2\,b^2\,d^4\,e^2+12\,A\,a^2\,b^2\,d^3\,e^3\,m^3+180\,A\,a^2\,b^2\,d^3\,e^3\,m^2+888\,A\,a^2\,b^2\,d^3\,e^3\,m+1440\,A\,a^2\,b^2\,d^3\,e^3+96\,B\,a\,b^3\,d^5\,e\,m+576\,B\,a\,b^3\,d^5\,e-24\,A\,a\,b^3\,d^4\,e^2\,m^2-264\,A\,a\,b^3\,d^4\,e^2\,m-720\,A\,a\,b^3\,d^4\,e^2-120\,B\,b^4\,d^6+24\,A\,b^4\,d^5\,e\,m+144\,A\,b^4\,d^5\,e\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 (B\,a^4\,d\,e^5\,m^5+18\,B\,a^4\,d\,e^5\,m^4+119\,B\,a^4\,d\,e^5\,m^3+342\,B\,a^4\,d\,e^5\,m^2+360\,B\,a^4\,d\,e^5\,m+A\,a^4\,e^6\,m^5+20\,A\,a^4\,e^6\,m^4+155\,A\,a^4\,e^6\,m^3+580\,A\,a^4\,e^6\,m^2+1044\,A\,a^4\,e^6\,m+720\,A\,a^4\,e^6-8\,B\,a^3\,b\,d^2\,e^4\,m^4-120\,B\,a^3\,b\,d^2\,e^4\,m^3-592\,B\,a^3\,b\,d^2\,e^4\,m^2-960\,B\,a^3\,b\,d^2\,e^4\,m+4\,A\,a^3\,b\,d\,e^5\,m^5+72\,A\,a^3\,b\,d\,e^5\,m^4+476\,A\,a^3\,b\,d\,e^5\,m^3+1368\,A\,a^3\,b\,d\,e^5\,m^2+1440\,A\,a^3\,b\,d\,e^5\,m+36\,B\,a^2\,b^2\,d^3\,e^3\,m^3+396\,B\,a^2\,b^2\,d^3\,e^3\,m^2+1080\,B\,a^2\,b^2\,d^3\,e^3\,m-12\,A\,a^2\,b^2\,d^2\,e^4\,m^4-180\,A\,a^2\,b^2\,d^2\,e^4\,m^3-888\,A\,a^2\,b^2\,d^2\,e^4\,m^2-1440\,A\,a^2\,b^2\,d^2\,e^4\,m-96\,B\,a\,b^3\,d^4\,e^2\,m^2-576\,B\,a\,b^3\,d^4\,e^2\,m+24\,A\,a\,b^3\,d^3\,e^3\,m^3+264\,A\,a\,b^3\,d^3\,e^3\,m^2+720\,A\,a\,b^3\,d^3\,e^3\,m+120\,B\,b^4\,d^5\,e\,m-24\,A\,b^4\,d^4\,e^2\,m^2-144\,A\,b^4\,d^4\,e^2\,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 (B\,a^4\,e^4\,m^4+18\,B\,a^4\,e^4\,m^3+119\,B\,a^4\,e^4\,m^2+342\,B\,a^4\,e^4\,m+360\,B\,a^4\,e^4+4\,B\,a^3\,b\,d\,e^3\,m^4+60\,B\,a^3\,b\,d\,e^3\,m^3+296\,B\,a^3\,b\,d\,e^3\,m^2+480\,B\,a^3\,b\,d\,e^3\,m+4\,A\,a^3\,b\,e^4\,m^4+72\,A\,a^3\,b\,e^4\,m^3+476\,A\,a^3\,b\,e^4\,m^2+1368\,A\,a^3\,b\,e^4\,m+1440\,A\,a^3\,b\,e^4-18\,B\,a^2\,b^2\,d^2\,e^2\,m^3-198\,B\,a^2\,b^2\,d^2\,e^2\,m^2-540\,B\,a^2\,b^2\,d^2\,e^2\,m+6\,A\,a^2\,b^2\,d\,e^3\,m^4+90\,A\,a^2\,b^2\,d\,e^3\,m^3+444\,A\,a^2\,b^2\,d\,e^3\,m^2+720\,A\,a^2\,b^2\,d\,e^3\,m+48\,B\,a\,b^3\,d^3\,e\,m^2+288\,B\,a\,b^3\,d^3\,e\,m-12\,A\,a\,b^3\,d^2\,e^2\,m^3-132\,A\,a\,b^3\,d^2\,e^2\,m^2-360\,A\,a\,b^3\,d^2\,e^2\,m-60\,B\,b^4\,d^4\,m+12\,A\,b^4\,d^3\,e\,m^2+72\,A\,b^4\,d^3\,e\,m\right )}{e^4\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {B\,b^4\,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 {b^2\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (6\,B\,a^2\,e^2\,m^2+66\,B\,a^2\,e^2\,m+180\,B\,a^2\,e^2+4\,B\,a\,b\,d\,e\,m^2+24\,B\,a\,b\,d\,e\,m+4\,A\,a\,b\,e^2\,m^2+44\,A\,a\,b\,e^2\,m+120\,A\,a\,b\,e^2-5\,B\,b^2\,d^2\,m+A\,b^2\,d\,e\,m^2+6\,A\,b^2\,d\,e\,m\right )}{e^2\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {b^3\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )\,\left (6\,A\,b\,e+24\,B\,a\,e+A\,b\,e\,m+4\,B\,a\,e\,m+B\,b\,d\,m\right )}{e\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}+\frac {2\,b\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (2\,B\,a^3\,e^3\,m^3+30\,B\,a^3\,e^3\,m^2+148\,B\,a^3\,e^3\,m+240\,B\,a^3\,e^3+3\,B\,a^2\,b\,d\,e^2\,m^3+33\,B\,a^2\,b\,d\,e^2\,m^2+90\,B\,a^2\,b\,d\,e^2\,m+3\,A\,a^2\,b\,e^3\,m^3+45\,A\,a^2\,b\,e^3\,m^2+222\,A\,a^2\,b\,e^3\,m+360\,A\,a^2\,b\,e^3-8\,B\,a\,b^2\,d^2\,e\,m^2-48\,B\,a\,b^2\,d^2\,e\,m+2\,A\,a\,b^2\,d\,e^2\,m^3+22\,A\,a\,b^2\,d\,e^2\,m^2+60\,A\,a\,b^2\,d\,e^2\,m+10\,B\,b^3\,d^3\,m-2\,A\,b^3\,d^2\,e\,m^2-12\,A\,b^3\,d^2\,e\,m\right )}{e^3\,\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((A + B*x)*(a + b*x)^4*(d + e*x)^m,x)

[Out]

((d + e*x)^m*(720*A*a^4*d*e^5 - 120*B*b^4*d^6 + 144*A*b^4*d^5*e - 360*B*a^4*d^2*e^4 - 720*A*a*b^3*d^4*e^2 - 14
40*A*a^3*b*d^2*e^4 + 960*B*a^3*b*d^3*e^3 + 580*A*a^4*d*e^5*m^2 + 155*A*a^4*d*e^5*m^3 + 20*A*a^4*d*e^5*m^4 + A*
a^4*d*e^5*m^5 - 342*B*a^4*d^2*e^4*m + 1440*A*a^2*b^2*d^3*e^3 - 1080*B*a^2*b^2*d^4*e^2 - 119*B*a^4*d^2*e^4*m^2
- 18*B*a^4*d^2*e^4*m^3 - B*a^4*d^2*e^4*m^4 + 576*B*a*b^3*d^5*e + 1044*A*a^4*d*e^5*m + 24*A*b^4*d^5*e*m + 888*A
*a^2*b^2*d^3*e^3*m - 24*A*a*b^3*d^4*e^2*m^2 - 476*A*a^3*b*d^2*e^4*m^2 - 72*A*a^3*b*d^2*e^4*m^3 - 4*A*a^3*b*d^2
*e^4*m^4 - 396*B*a^2*b^2*d^4*e^2*m + 120*B*a^3*b*d^3*e^3*m^2 + 8*B*a^3*b*d^3*e^3*m^3 + 96*B*a*b^3*d^5*e*m + 18
0*A*a^2*b^2*d^3*e^3*m^2 + 12*A*a^2*b^2*d^3*e^3*m^3 - 36*B*a^2*b^2*d^4*e^2*m^2 - 264*A*a*b^3*d^4*e^2*m - 1368*A
*a^3*b*d^2*e^4*m + 592*B*a^3*b*d^3*e^3*m))/(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*a^4*e^6 + 1044*A*a^4*e^6*m + 580*A*a^4*e^6*m^2 + 155*A*a^4*e^6*m^3 + 20*A*a^4*e^6*m^4
+ A*a^4*e^6*m^5 - 144*A*b^4*d^4*e^2*m + 342*B*a^4*d*e^5*m^2 + 119*B*a^4*d*e^5*m^3 + 18*B*a^4*d*e^5*m^4 + B*a^4
*d*e^5*m^5 - 24*A*b^4*d^4*e^2*m^2 + 360*B*a^4*d*e^5*m + 120*B*b^4*d^5*e*m - 1440*A*a^2*b^2*d^2*e^4*m + 264*A*a
*b^3*d^3*e^3*m^2 + 24*A*a*b^3*d^3*e^3*m^3 + 1080*B*a^2*b^2*d^3*e^3*m - 96*B*a*b^3*d^4*e^2*m^2 - 592*B*a^3*b*d^
2*e^4*m^2 - 120*B*a^3*b*d^2*e^4*m^3 - 8*B*a^3*b*d^2*e^4*m^4 + 1440*A*a^3*b*d*e^5*m - 888*A*a^2*b^2*d^2*e^4*m^2
- 180*A*a^2*b^2*d^2*e^4*m^3 - 12*A*a^2*b^2*d^2*e^4*m^4 + 396*B*a^2*b^2*d^3*e^3*m^2 + 36*B*a^2*b^2*d^3*e^3*m^3
+ 720*A*a*b^3*d^3*e^3*m + 1368*A*a^3*b*d*e^5*m^2 + 476*A*a^3*b*d*e^5*m^3 + 72*A*a^3*b*d*e^5*m^4 + 4*A*a^3*b*d
*e^5*m^5 - 576*B*a*b^3*d^4*e^2*m - 960*B*a^3*b*d^2*e^4*m))/(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*(360*B*a^4*e^4 + 1440*A*a^3*b*e^4 + 342*B*a^4*e^4*m - 60*B*b^4*d^4*
m + 119*B*a^4*e^4*m^2 + 18*B*a^4*e^4*m^3 + B*a^4*e^4*m^4 + 476*A*a^3*b*e^4*m^2 + 72*A*a^3*b*e^4*m^3 + 4*A*a^3*
b*e^4*m^4 + 12*A*b^4*d^3*e*m^2 + 1368*A*a^3*b*e^4*m + 72*A*b^4*d^3*e*m - 132*A*a*b^3*d^2*e^2*m^2 + 444*A*a^2*b
^2*d*e^3*m^2 - 12*A*a*b^3*d^2*e^2*m^3 + 90*A*a^2*b^2*d*e^3*m^3 + 6*A*a^2*b^2*d*e^3*m^4 - 540*B*a^2*b^2*d^2*e^2
*m + 288*B*a*b^3*d^3*e*m + 480*B*a^3*b*d*e^3*m - 198*B*a^2*b^2*d^2*e^2*m^2 - 18*B*a^2*b^2*d^2*e^2*m^3 - 360*A*
a*b^3*d^2*e^2*m + 720*A*a^2*b^2*d*e^3*m + 48*B*a*b^3*d^3*e*m^2 + 296*B*a^3*b*d*e^3*m^2 + 60*B*a^3*b*d*e^3*m^3
+ 4*B*a^3*b*d*e^3*m^4))/(e^4*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (B*b^4*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) + (b^2*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(180*B*a^2*e^2 + 120*A*a*b*e^2 + 66*B*a^2*e^2*m - 5*B*b^
2*d^2*m + 6*B*a^2*e^2*m^2 + 44*A*a*b*e^2*m + 6*A*b^2*d*e*m + 4*A*a*b*e^2*m^2 + A*b^2*d*e*m^2 + 24*B*a*b*d*e*m
+ 4*B*a*b*d*e*m^2))/(e^2*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (b^3*x^5*(d + e*x)^m*
(50*m + 35*m^2 + 10*m^3 + m^4 + 24)*(6*A*b*e + 24*B*a*e + A*b*e*m + 4*B*a*e*m + B*b*d*m))/(e*(1764*m + 1624*m^
2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)) + (2*b*x^3*(d + e*x)^m*(3*m + m^2 + 2)*(240*B*a^3*e^3 + 360*A*a^2
*b*e^3 + 148*B*a^3*e^3*m + 10*B*b^3*d^3*m + 30*B*a^3*e^3*m^2 + 2*B*a^3*e^3*m^3 + 45*A*a^2*b*e^3*m^2 + 3*A*a^2*
b*e^3*m^3 - 2*A*b^3*d^2*e*m^2 + 222*A*a^2*b*e^3*m - 12*A*b^3*d^2*e*m + 60*A*a*b^2*d*e^2*m - 48*B*a*b^2*d^2*e*m
+ 90*B*a^2*b*d*e^2*m + 22*A*a*b^2*d*e^2*m^2 + 2*A*a*b^2*d*e^2*m^3 - 8*B*a*b^2*d^2*e*m^2 + 33*B*a^2*b*d*e^2*m^
2 + 3*B*a^2*b*d*e^2*m^3))/(e^3*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))

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