3.1884 \(\int (A+B x) (d+e x)^m (a^2+2 a b x+b^2 x^2)^2 \, dx\)
Optimal. Leaf size=234 \[ \frac{2 b^2 (b d-a e) (d+e x)^{m+4} (-3 a B e-2 A b e+5 b B d)}{e^6 (m+4)}-\frac{b^3 (d+e x)^{m+5} (-4 a B e-A b e+5 b B d)}{e^6 (m+5)}-\frac{(b d-a e)^4 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac{(b d-a e)^3 (d+e x)^{m+2} (-a B e-4 A b e+5 b B d)}{e^6 (m+2)}-\frac{2 b (b d-a e)^2 (d+e x)^{m+3} (-2 a B e-3 A b e+5 b B d)}{e^6 (m+3)}+\frac{b^4 B (d+e x)^{m+6}}{e^6 (m+6)} \]
[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))
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Rubi [A] time = 0.166595, antiderivative size = 234, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used =
{27, 77} \[ \frac{2 b^2 (b d-a e) (d+e x)^{m+4} (-3 a B e-2 A b e+5 b B d)}{e^6 (m+4)}-\frac{b^3 (d+e x)^{m+5} (-4 a B e-A b e+5 b B d)}{e^6 (m+5)}-\frac{(b d-a e)^4 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac{(b d-a e)^3 (d+e x)^{m+2} (-a B e-4 A b e+5 b B d)}{e^6 (m+2)}-\frac{2 b (b d-a e)^2 (d+e x)^{m+3} (-2 a B e-3 A b e+5 b B d)}{e^6 (m+3)}+\frac{b^4 B (d+e x)^{m+6}}{e^6 (m+6)} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^2,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 27
Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Rule 77
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) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\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*}
Mathematica [A] time = 0.253736, size = 208, normalized size = 0.89 \[ \frac{(d+e x)^{m+1} \left (-\frac{b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)}{m+5}+\frac{2 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{m+4}-\frac{2 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{m+3}+\frac{(d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{m+2}-\frac{(b d-a e)^4 (B d-A e)}{m+1}+\frac{b^4 B (d+e x)^5}{m+6}\right )}{e^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*(-(((b*d - a*e)^4*(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] time = 0.013, size = 2355, normalized size = 10.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
(e*x+d)^(1+m)*(B*b^4*e^5*m^5*x^5+A*b^4*e^5*m^5*x^4+4*B*a*b^3*e^5*m^5*x^4+15*B*b^4*e^5*m^4*x^5+4*A*a*b^3*e^5*m^
5*x^3+16*A*b^4*e^5*m^4*x^4+6*B*a^2*b^2*e^5*m^5*x^3+64*B*a*b^3*e^5*m^4*x^4-5*B*b^4*d*e^4*m^4*x^4+85*B*b^4*e^5*m
^3*x^5+6*A*a^2*b^2*e^5*m^5*x^2+68*A*a*b^3*e^5*m^4*x^3-4*A*b^4*d*e^4*m^4*x^3+95*A*b^4*e^5*m^3*x^4+4*B*a^3*b*e^5
*m^5*x^2+102*B*a^2*b^2*e^5*m^4*x^3-16*B*a*b^3*d*e^4*m^4*x^3+380*B*a*b^3*e^5*m^3*x^4-50*B*b^4*d*e^4*m^3*x^4+225
*B*b^4*e^5*m^2*x^5+4*A*a^3*b*e^5*m^5*x+108*A*a^2*b^2*e^5*m^4*x^2-12*A*a*b^3*d*e^4*m^4*x^2+428*A*a*b^3*e^5*m^3*
x^3-48*A*b^4*d*e^4*m^3*x^3+260*A*b^4*e^5*m^2*x^4+B*a^4*e^5*m^5*x+72*B*a^3*b*e^5*m^4*x^2-18*B*a^2*b^2*d*e^4*m^4
*x^2+642*B*a^2*b^2*e^5*m^3*x^3-192*B*a*b^3*d*e^4*m^3*x^3+1040*B*a*b^3*e^5*m^2*x^4+20*B*b^4*d^2*e^3*m^3*x^3-175
*B*b^4*d*e^4*m^2*x^4+274*B*b^4*e^5*m*x^5+A*a^4*e^5*m^5+76*A*a^3*b*e^5*m^4*x-12*A*a^2*b^2*d*e^4*m^4*x+726*A*a^2
*b^2*e^5*m^3*x^2-168*A*a*b^3*d*e^4*m^3*x^2+1228*A*a*b^3*e^5*m^2*x^3+12*A*b^4*d^2*e^3*m^3*x^2-188*A*b^4*d*e^4*m
^2*x^3+324*A*b^4*e^5*m*x^4+19*B*a^4*e^5*m^4*x-8*B*a^3*b*d*e^4*m^4*x+484*B*a^3*b*e^5*m^3*x^2-252*B*a^2*b^2*d*e^
4*m^3*x^2+1842*B*a^2*b^2*e^5*m^2*x^3+48*B*a*b^3*d^2*e^3*m^3*x^2-752*B*a*b^3*d*e^4*m^2*x^3+1296*B*a*b^3*e^5*m*x
^4+120*B*b^4*d^2*e^3*m^2*x^3-250*B*b^4*d*e^4*m*x^4+120*B*b^4*e^5*x^5+20*A*a^4*e^5*m^4-4*A*a^3*b*d*e^4*m^4+548*
A*a^3*b*e^5*m^3*x-192*A*a^2*b^2*d*e^4*m^3*x+2232*A*a^2*b^2*e^5*m^2*x^2+24*A*a*b^3*d^2*e^3*m^3*x-780*A*a*b^3*d*
e^4*m^2*x^2+1584*A*a*b^3*e^5*m*x^3+108*A*b^4*d^2*e^3*m^2*x^2-288*A*b^4*d*e^4*m*x^3+144*A*b^4*e^5*x^4-B*a^4*d*e
^4*m^4+137*B*a^4*e^5*m^3*x-128*B*a^3*b*d*e^4*m^3*x+1488*B*a^3*b*e^5*m^2*x^2+36*B*a^2*b^2*d^2*e^3*m^3*x-1170*B*
a^2*b^2*d*e^4*m^2*x^2+2376*B*a^2*b^2*e^5*m*x^3+432*B*a*b^3*d^2*e^3*m^2*x^2-1152*B*a*b^3*d*e^4*m*x^3+576*B*a*b^
3*e^5*x^4-60*B*b^4*d^3*e^2*m^2*x^2+220*B*b^4*d^2*e^3*m*x^3-120*B*b^4*d*e^4*x^4+155*A*a^4*e^5*m^3-72*A*a^3*b*d*
e^4*m^3+1844*A*a^3*b*e^5*m^2*x+12*A*a^2*b^2*d^2*e^3*m^3-1068*A*a^2*b^2*d*e^4*m^2*x+3048*A*a^2*b^2*e^5*m*x^2+28
8*A*a*b^3*d^2*e^3*m^2*x-1344*A*a*b^3*d*e^4*m*x^2+720*A*a*b^3*e^5*x^3-24*A*b^4*d^3*e^2*m^2*x+240*A*b^4*d^2*e^3*
m*x^2-144*A*b^4*d*e^4*x^3-18*B*a^4*d*e^4*m^3+461*B*a^4*e^5*m^2*x+8*B*a^3*b*d^2*e^3*m^3-712*B*a^3*b*d*e^4*m^2*x
+2032*B*a^3*b*e^5*m*x^2+432*B*a^2*b^2*d^2*e^3*m^2*x-2016*B*a^2*b^2*d*e^4*m*x^2+1080*B*a^2*b^2*e^5*x^3-96*B*a*b
^3*d^3*e^2*m^2*x+960*B*a*b^3*d^2*e^3*m*x^2-576*B*a*b^3*d*e^4*x^3-180*B*b^4*d^3*e^2*m*x^2+120*B*b^4*d^2*e^3*x^3
+580*A*a^4*e^5*m^2-476*A*a^3*b*d*e^4*m^2+2808*A*a^3*b*e^5*m*x+180*A*a^2*b^2*d^2*e^3*m^2-2328*A*a^2*b^2*d*e^4*m
*x+1440*A*a^2*b^2*e^5*x^2-24*A*a*b^3*d^3*e^2*m^2+984*A*a*b^3*d^2*e^3*m*x-720*A*a*b^3*d*e^4*x^2-168*A*b^4*d^3*e
^2*m*x+144*A*b^4*d^2*e^3*x^2-119*B*a^4*d*e^4*m^2+702*B*a^4*e^5*m*x+120*B*a^3*b*d^2*e^3*m^2-1552*B*a^3*b*d*e^4*
m*x+960*B*a^3*b*e^5*x^2-36*B*a^2*b^2*d^3*e^2*m^2+1476*B*a^2*b^2*d^2*e^3*m*x-1080*B*a^2*b^2*d*e^4*x^2-672*B*a*b
^3*d^3*e^2*m*x+576*B*a*b^3*d^2*e^3*x^2+120*B*b^4*d^4*e*m*x-120*B*b^4*d^3*e^2*x^2+1044*A*a^4*e^5*m-1368*A*a^3*b
*d*e^4*m+1440*A*a^3*b*e^5*x+888*A*a^2*b^2*d^2*e^3*m-1440*A*a^2*b^2*d*e^4*x-264*A*a*b^3*d^3*e^2*m+720*A*a*b^3*d
^2*e^3*x+24*A*b^4*d^4*e*m-144*A*b^4*d^3*e^2*x-342*B*a^4*d*e^4*m+360*B*a^4*e^5*x+592*B*a^3*b*d^2*e^3*m-960*B*a^
3*b*d*e^4*x-396*B*a^2*b^2*d^3*e^2*m+1080*B*a^2*b^2*d^2*e^3*x+96*B*a*b^3*d^4*e*m-576*B*a*b^3*d^3*e^2*x+120*B*b^
4*d^4*e*x+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+960*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+73
5*m^3+1624*m^2+1764*m+720)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.84522, size = 4830, normalized size = 20.64 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
[Out]
(A*a^4*d*e^5*m^5 - 120*B*b^4*d^6 + 720*A*a^4*d*e^5 + 144*(4*B*a*b^3 + A*b^4)*d^5*e - 360*(3*B*a^2*b^2 + 2*A*a*
b^3)*d^4*e^2 + 480*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^3 - 360*(B*a^4 + 4*A*a^3*b)*d^2*e^4 + (B*b^4*e^6*m^5 + 15*B
*b^4*e^6*m^4 + 85*B*b^4*e^6*m^3 + 225*B*b^4*e^6*m^2 + 274*B*b^4*e^6*m + 120*B*b^4*e^6)*x^6 + (144*(4*B*a*b^3 +
A*b^4)*e^6 + (B*b^4*d*e^5 + (4*B*a*b^3 + A*b^4)*e^6)*m^5 + 2*(5*B*b^4*d*e^5 + 8*(4*B*a*b^3 + A*b^4)*e^6)*m^4
+ 5*(7*B*b^4*d*e^5 + 19*(4*B*a*b^3 + A*b^4)*e^6)*m^3 + 10*(5*B*b^4*d*e^5 + 26*(4*B*a*b^3 + A*b^4)*e^6)*m^2 + 1
2*(2*B*b^4*d*e^5 + 27*(4*B*a*b^3 + A*b^4)*e^6)*m)*x^5 + (20*A*a^4*d*e^5 - (B*a^4 + 4*A*a^3*b)*d^2*e^4)*m^4 + (
360*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6 + ((4*B*a*b^3 + A*b^4)*d*e^5 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^5 - (5*B*b
^4*d^2*e^4 - 12*(4*B*a*b^3 + A*b^4)*d*e^5 - 34*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^4 - (30*B*b^4*d^2*e^4 - 47*(4*
B*a*b^3 + A*b^4)*d*e^5 - 214*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^3 - (55*B*b^4*d^2*e^4 - 72*(4*B*a*b^3 + A*b^4)*d
*e^5 - 614*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^2 - 6*(5*B*b^4*d^2*e^4 - 6*(4*B*a*b^3 + A*b^4)*d*e^5 - 132*(3*B*a^
2*b^2 + 2*A*a*b^3)*e^6)*m)*x^4 + (155*A*a^4*d*e^5 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^3 - 18*(B*a^4 + 4*A*a^3*
b)*d^2*e^4)*m^3 + 2*(240*(2*B*a^3*b + 3*A*a^2*b^2)*e^6 + ((3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 + (2*B*a^3*b + 3*A*a
^2*b^2)*e^6)*m^5 - 2*((4*B*a*b^3 + A*b^4)*d^2*e^4 - 7*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 - 9*(2*B*a^3*b + 3*A*a^2
*b^2)*e^6)*m^4 + (10*B*b^4*d^3*e^3 - 18*(4*B*a*b^3 + A*b^4)*d^2*e^4 + 65*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 + 121
*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*m^3 + 2*(15*B*b^4*d^3*e^3 - 20*(4*B*a*b^3 + A*b^4)*d^2*e^4 + 56*(3*B*a^2*b^2 +
2*A*a*b^3)*d*e^5 + 186*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*m^2 + 4*(5*B*b^4*d^3*e^3 - 6*(4*B*a*b^3 + A*b^4)*d^2*e^
4 + 15*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 + 127*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*m)*x^3 + (580*A*a^4*d*e^5 - 12*(3*
B*a^2*b^2 + 2*A*a*b^3)*d^4*e^2 + 60*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^3 - 119*(B*a^4 + 4*A*a^3*b)*d^2*e^4)*m^2 +
(360*(B*a^4 + 4*A*a^3*b)*e^6 + (2*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 + (B*a^4 + 4*A*a^3*b)*e^6)*m^5 - (6*(3*B*a^
2*b^2 + 2*A*a*b^3)*d^2*e^4 - 32*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 - 19*(B*a^4 + 4*A*a^3*b)*e^6)*m^4 + (12*(4*B*a
*b^3 + A*b^4)*d^3*e^3 - 72*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^4 + 178*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 + 137*(B*a^
4 + 4*A*a^3*b)*e^6)*m^3 - (60*B*b^4*d^4*e^2 - 84*(4*B*a*b^3 + A*b^4)*d^3*e^3 + 246*(3*B*a^2*b^2 + 2*A*a*b^3)*d
^2*e^4 - 388*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 - 461*(B*a^4 + 4*A*a^3*b)*e^6)*m^2 - 6*(10*B*b^4*d^4*e^2 - 12*(4*
B*a*b^3 + A*b^4)*d^3*e^3 + 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^4 - 40*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 - 117*(B*
a^4 + 4*A*a^3*b)*e^6)*m)*x^2 + 2*(522*A*a^4*d*e^5 + 12*(4*B*a*b^3 + A*b^4)*d^5*e - 66*(3*B*a^2*b^2 + 2*A*a*b^3
)*d^4*e^2 + 148*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^3 - 171*(B*a^4 + 4*A*a^3*b)*d^2*e^4)*m + (720*A*a^4*e^6 + (A*a
^4*e^6 + (B*a^4 + 4*A*a^3*b)*d*e^5)*m^5 + 2*(10*A*a^4*e^6 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 + 9*(B*a^4 + 4
*A*a^3*b)*d*e^5)*m^4 + (155*A*a^4*e^6 + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^3 - 60*(2*B*a^3*b + 3*A*a^2*b^2)*d^
2*e^4 + 119*(B*a^4 + 4*A*a^3*b)*d*e^5)*m^3 + 2*(290*A*a^4*e^6 - 12*(4*B*a*b^3 + A*b^4)*d^4*e^2 + 66*(3*B*a^2*b
^2 + 2*A*a*b^3)*d^3*e^3 - 148*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 + 171*(B*a^4 + 4*A*a^3*b)*d*e^5)*m^2 + 12*(10*
B*b^4*d^5*e + 87*A*a^4*e^6 - 12*(4*B*a*b^3 + A*b^4)*d^4*e^2 + 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^3 - 40*(2*B*a
^3*b + 3*A*a^2*b^2)*d^2*e^4 + 30*(B*a^4 + 4*A*a^3*b)*d*e^5)*m)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 + 175*e^6*
m^4 + 735*e^6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*e^6)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.23854, size = 5914, normalized size = 25.27 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^2,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^2*x^2*e^5 + 1164*(x*e + d)^m*A*a^2*b^2*d*m^2*x^2*e^5 + 360*(x*e
+ d)^m*B*a^2*b^2*d*m*x^3*e^5 + 240*(x*e + d)^m*A*a*b^3*d*m*x^3*e^5 - 18*(x*e + d)^m*B*a^4*d^2*m^3*e^4 - 72*(x
*e + d)^m*A*a^3*b*d^2*m^3*e^4 - 592*(x*e + d)^m*B*a^3*b*d^2*m^2*x*e^4 - 888*(x*e + d)^m*A*a^2*b^2*d^2*m^2*x*e^
4 - 540*(x*e + d)^m*B*a^2*b^2*d^2*m*x^2*e^4 - 360*(x*e + d)^m*A*a*b^3*d^2*m*x^2*e^4 + 120*(x*e + d)^m*B*a^3*b*
d^3*m^2*e^3 + 180*(x*e + d)^m*A*a^2*b^2*d^3*m^2*e^3 + 1080*(x*e + d)^m*B*a^2*b^2*d^3*m*x*e^3 + 720*(x*e + d)^m
*A*a*b^3*d^3*m*x*e^3 - 396*(x*e + d)^m*B*a^2*b^2*d^4*m*e^2 - 264*(x*e + d)^m*A*a*b^3*d^4*m*e^2 + 576*(x*e + d)
^m*B*a*b^3*d^5*e + 144*(x*e + d)^m*A*b^4*d^5*e + 155*(x*e + d)^m*A*a^4*m^3*x*e^6 + 461*(x*e + d)^m*B*a^4*m^2*x
^2*e^6 + 1844*(x*e + d)^m*A*a^3*b*m^2*x^2*e^6 + 2032*(x*e + d)^m*B*a^3*b*m*x^3*e^6 + 3048*(x*e + d)^m*A*a^2*b^
2*m*x^3*e^6 + 1080*(x*e + d)^m*B*a^2*b^2*x^4*e^6 + 720*(x*e + d)^m*A*a*b^3*x^4*e^6 + 155*(x*e + d)^m*A*a^4*d*m
^3*e^5 + 342*(x*e + d)^m*B*a^4*d*m^2*x*e^5 + 1368*(x*e + d)^m*A*a^3*b*d*m^2*x*e^5 + 480*(x*e + d)^m*B*a^3*b*d*
m*x^2*e^5 + 720*(x*e + d)^m*A*a^2*b^2*d*m*x^2*e^5 - 119*(x*e + d)^m*B*a^4*d^2*m^2*e^4 - 476*(x*e + d)^m*A*a^3*
b*d^2*m^2*e^4 - 960*(x*e + d)^m*B*a^3*b*d^2*m*x*e^4 - 1440*(x*e + d)^m*A*a^2*b^2*d^2*m*x*e^4 + 592*(x*e + d)^m
*B*a^3*b*d^3*m*e^3 + 888*(x*e + d)^m*A*a^2*b^2*d^3*m*e^3 - 1080*(x*e + d)^m*B*a^2*b^2*d^4*e^2 - 720*(x*e + d)^
m*A*a*b^3*d^4*e^2 + 580*(x*e + d)^m*A*a^4*m^2*x*e^6 + 702*(x*e + d)^m*B*a^4*m*x^2*e^6 + 2808*(x*e + d)^m*A*a^3
*b*m*x^2*e^6 + 960*(x*e + d)^m*B*a^3*b*x^3*e^6 + 1440*(x*e + d)^m*A*a^2*b^2*x^3*e^6 + 580*(x*e + d)^m*A*a^4*d*
m^2*e^5 + 360*(x*e + d)^m*B*a^4*d*m*x*e^5 + 1440*(x*e + d)^m*A*a^3*b*d*m*x*e^5 - 342*(x*e + d)^m*B*a^4*d^2*m*e
^4 - 1368*(x*e + d)^m*A*a^3*b*d^2*m*e^4 + 960*(x*e + d)^m*B*a^3*b*d^3*e^3 + 1440*(x*e + d)^m*A*a^2*b^2*d^3*e^3
+ 1044*(x*e + d)^m*A*a^4*m*x*e^6 + 360*(x*e + d)^m*B*a^4*x^2*e^6 + 1440*(x*e + d)^m*A*a^3*b*x^2*e^6 + 1044*(x
*e + d)^m*A*a^4*d*m*e^5 - 360*(x*e + d)^m*B*a^4*d^2*e^4 - 1440*(x*e + d)^m*A*a^3*b*d^2*e^4 + 720*(x*e + d)^m*A
*a^4*x*e^6 + 720*(x*e + d)^m*A*a^4*d*e^5)/(m^6*e^6 + 21*m^5*e^6 + 175*m^4*e^6 + 735*m^3*e^6 + 1624*m^2*e^6 + 1
764*m*e^6 + 720*e^6)