3.2641 \(\int (A+B x) (d+e x)^m (a+b x+c x^2)^2 \, dx\)
Optimal. Leaf size=333 \[ -\frac{(d+e x)^{m+3} \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^6 (m+3)}-\frac{(d+e x)^{m+4} \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6 (m+4)}-\frac{(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{e^6 (m+2)}-\frac{c (d+e x)^{m+5} (-A c e-2 b B e+5 B c d)}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]
[Out]
-(((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) - ((c*d^2 - b*d*e + a*e^2)*(2*A*e*(
2*c*d - b*e) - B*(5*c*d^2 - e*(3*b*d - a*e)))*(d + e*x)^(2 + m))/(e^6*(2 + m)) - ((B*(10*c^2*d^3 + b*e^2*(3*b*
d - 2*a*e) - 6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))*(d + e*x)^(3 + m))/(e^6
*(3 + m)) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))*(d + e*x)^(4 + m))/(e^6*
(4 + m)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6
+ m))
________________________________________________________________________________________
Rubi [A] time = 0.350569, antiderivative size = 330, normalized size of antiderivative =
0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules
used = {771} \[ -\frac{(d+e x)^{m+3} \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^6 (m+3)}-\frac{(d+e x)^{m+4} \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6 (m+4)}-\frac{(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}+\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^6 (m+2)}-\frac{c (d+e x)^{m+5} (-A c e-2 b B e+5 B c d)}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^m*(a + b*x + c*x^2)^2,x]
[Out]
-(((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + ((c*d^2 - b*d*e + a*e^2)*(5*B*c*d
^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e))*(d + e*x)^(2 + m))/(e^6*(2 + m)) - ((B*(10*c^2*d^3 + b*e^2*(3*b*
d - 2*a*e) - 6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))*(d + e*x)^(3 + m))/(e^6
*(3 + m)) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))*(d + e*x)^(4 + m))/(e^6*
(4 + m)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6
+ m))
Rule 771
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 (A+B x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^m}{e^5}+\frac{\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right ) (d+e x)^{1+m}}{e^5}+\frac{\left (-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^{2+m}}{e^5}+\frac{\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^{3+m}}{e^5}+\frac{c (-5 B c d+2 b B e+A c e) (d+e x)^{4+m}}{e^5}+\frac{B c^2 (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac{\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right ) (d+e x)^{2+m}}{e^6 (2+m)}-\frac{\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^{3+m}}{e^6 (3+m)}-\frac{\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac{c (5 B c d-2 b B e-A c e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac{B c^2 (d+e x)^{6+m}}{e^6 (6+m)}\\ \end{align*}
Mathematica [A] time = 1.64667, size = 633, normalized size = 1.9 \[ \frac{(d+e x)^{m+1} \left (\frac{2 \left (\frac{(d+e x) \left (B \left (c^2 e^2 \left (4 a^2 e^2 \left (m^2+8 m+15\right )+2 a b d e \left (4 m^2+11 m-18\right )+b^2 d^2 \left (m^2-13 m+6\right )\right )-b^2 c e^3 (m+2) (a e (5 m+21)+b d (2 m-3))+2 c^3 d^2 e \left (3 b d (m-14)-2 a e \left (m^2-4 m-30\right )\right )+b^4 e^4 \left (m^2+5 m+6\right )+60 c^4 d^4\right )-A c e (m+6) (2 c d-b e) \left (2 c e (a e (2 m+7)-3 b d)-b^2 e^2 (m+2)+6 c^2 d^2\right )\right )}{e^2 (m+2)}+\frac{\left (e (a e-b d)+c d^2\right ) \left (A c e (m+6) \left (4 c e (a e (m+4)-3 b d)-b^2 e^2 (m+1)+12 c^2 d^2\right )+B \left (2 c^2 d e \left (2 a e \left (m^2+m-15\right )-3 b d (m-9)\right )-b c e^2 (m+1) (2 a e (2 m+9)+b d (m-6))+b^3 e^3 \left (m^2+4 m+3\right )-60 c^3 d^3\right )\right )}{e^2 (m+1)}-(a+x (b+c x)) \left (c e (m+3) x \left (B \left (-c e (2 a e (m+5)+b d (m-4))+b^2 e^2 (m+3)-10 c^2 d^2\right )+A c e (m+6) (2 c d-b e)\right )-(3 c d-b e) \left (B \left (-c e (2 a e (m+5)+b d (m-4))+b^2 e^2 (m+3)-10 c^2 d^2\right )+A c e (m+6) (2 c d-b e)\right )+c e (m+4) \left (A c e (m+6) (b d-2 a e)+a b B e^2 (m+1)-2 a B c d e m+b B d (2 b e-5 c d)\right )\right )\right )}{c e^2 (m+3) (m+4)}+(a+x (b+c x))^2 (A c e (m+6)+B (2 b e-5 c d)+B c e (m+5) x)\right )}{c e^2 (m+5) (m+6)} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^m*(a + b*x + c*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*((B*(-5*c*d + 2*b*e) + A*c*e*(6 + m) + B*c*e*(5 + m)*x)*(a + x*(b + c*x))^2 + (2*(((c*d^2 +
e*(-(b*d) + a*e))*(A*c*e*(6 + m)*(12*c^2*d^2 - b^2*e^2*(1 + m) + 4*c*e*(-3*b*d + a*e*(4 + m))) + B*(-60*c^3*d
^3 + b^3*e^3*(3 + 4*m + m^2) - b*c*e^2*(1 + m)*(b*d*(-6 + m) + 2*a*e*(9 + 2*m)) + 2*c^2*d*e*(-3*b*d*(-9 + m) +
2*a*e*(-15 + m + m^2)))))/(e^2*(1 + m)) + ((-(A*c*e*(2*c*d - b*e)*(6 + m)*(6*c^2*d^2 - b^2*e^2*(2 + m) + 2*c*
e*(-3*b*d + a*e*(7 + 2*m)))) + B*(60*c^4*d^4 + b^4*e^4*(6 + 5*m + m^2) - b^2*c*e^3*(2 + m)*(b*d*(-3 + 2*m) + a
*e*(21 + 5*m)) + 2*c^3*d^2*e*(3*b*d*(-14 + m) - 2*a*e*(-30 - 4*m + m^2)) + c^2*e^2*(b^2*d^2*(6 - 13*m + m^2) +
4*a^2*e^2*(15 + 8*m + m^2) + 2*a*b*d*e*(-18 + 11*m + 4*m^2))))*(d + e*x))/(e^2*(2 + m)) - (c*e*(4 + m)*(b*B*d
*(-5*c*d + 2*b*e) - 2*a*B*c*d*e*m + a*b*B*e^2*(1 + m) + A*c*e*(b*d - 2*a*e)*(6 + m)) - (3*c*d - b*e)*(A*c*e*(2
*c*d - b*e)*(6 + m) + B*(-10*c^2*d^2 + b^2*e^2*(3 + m) - c*e*(b*d*(-4 + m) + 2*a*e*(5 + m)))) + c*e*(3 + m)*(A
*c*e*(2*c*d - b*e)*(6 + m) + B*(-10*c^2*d^2 + b^2*e^2*(3 + m) - c*e*(b*d*(-4 + m) + 2*a*e*(5 + m))))*x)*(a + x
*(b + c*x))))/(c*e^2*(3 + m)*(4 + m))))/(c*e^2*(5 + m)*(6 + m))
________________________________________________________________________________________
Maple [B] time = 0.01, size = 2557, normalized size = 7.7 \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*(c*x^2+b*x+a)^2,x)
[Out]
(e*x+d)^(1+m)*(B*c^2*e^5*m^5*x^5+A*c^2*e^5*m^5*x^4+2*B*b*c*e^5*m^5*x^4+15*B*c^2*e^5*m^4*x^5+2*A*b*c*e^5*m^5*x^
3+16*A*c^2*e^5*m^4*x^4+2*B*a*c*e^5*m^5*x^3+B*b^2*e^5*m^5*x^3+32*B*b*c*e^5*m^4*x^4-5*B*c^2*d*e^4*m^4*x^4+85*B*c
^2*e^5*m^3*x^5+2*A*a*c*e^5*m^5*x^2+A*b^2*e^5*m^5*x^2+34*A*b*c*e^5*m^4*x^3-4*A*c^2*d*e^4*m^4*x^3+95*A*c^2*e^5*m
^3*x^4+2*B*a*b*e^5*m^5*x^2+34*B*a*c*e^5*m^4*x^3+17*B*b^2*e^5*m^4*x^3-8*B*b*c*d*e^4*m^4*x^3+190*B*b*c*e^5*m^3*x
^4-50*B*c^2*d*e^4*m^3*x^4+225*B*c^2*e^5*m^2*x^5+2*A*a*b*e^5*m^5*x+36*A*a*c*e^5*m^4*x^2+18*A*b^2*e^5*m^4*x^2-6*
A*b*c*d*e^4*m^4*x^2+214*A*b*c*e^5*m^3*x^3-48*A*c^2*d*e^4*m^3*x^3+260*A*c^2*e^5*m^2*x^4+B*a^2*e^5*m^5*x+36*B*a*
b*e^5*m^4*x^2-6*B*a*c*d*e^4*m^4*x^2+214*B*a*c*e^5*m^3*x^3-3*B*b^2*d*e^4*m^4*x^2+107*B*b^2*e^5*m^3*x^3-96*B*b*c
*d*e^4*m^3*x^3+520*B*b*c*e^5*m^2*x^4+20*B*c^2*d^2*e^3*m^3*x^3-175*B*c^2*d*e^4*m^2*x^4+274*B*c^2*e^5*m*x^5+A*a^
2*e^5*m^5+38*A*a*b*e^5*m^4*x-4*A*a*c*d*e^4*m^4*x+242*A*a*c*e^5*m^3*x^2-2*A*b^2*d*e^4*m^4*x+121*A*b^2*e^5*m^3*x
^2-84*A*b*c*d*e^4*m^3*x^2+614*A*b*c*e^5*m^2*x^3+12*A*c^2*d^2*e^3*m^3*x^2-188*A*c^2*d*e^4*m^2*x^3+324*A*c^2*e^5
*m*x^4+19*B*a^2*e^5*m^4*x-4*B*a*b*d*e^4*m^4*x+242*B*a*b*e^5*m^3*x^2-84*B*a*c*d*e^4*m^3*x^2+614*B*a*c*e^5*m^2*x
^3-42*B*b^2*d*e^4*m^3*x^2+307*B*b^2*e^5*m^2*x^3+24*B*b*c*d^2*e^3*m^3*x^2-376*B*b*c*d*e^4*m^2*x^3+648*B*b*c*e^5
*m*x^4+120*B*c^2*d^2*e^3*m^2*x^3-250*B*c^2*d*e^4*m*x^4+120*B*c^2*e^5*x^5+20*A*a^2*e^5*m^4-2*A*a*b*d*e^4*m^4+27
4*A*a*b*e^5*m^3*x-64*A*a*c*d*e^4*m^3*x+744*A*a*c*e^5*m^2*x^2-32*A*b^2*d*e^4*m^3*x+372*A*b^2*e^5*m^2*x^2+12*A*b
*c*d^2*e^3*m^3*x-390*A*b*c*d*e^4*m^2*x^2+792*A*b*c*e^5*m*x^3+108*A*c^2*d^2*e^3*m^2*x^2-288*A*c^2*d*e^4*m*x^3+1
44*A*c^2*e^5*x^4-B*a^2*d*e^4*m^4+137*B*a^2*e^5*m^3*x-64*B*a*b*d*e^4*m^3*x+744*B*a*b*e^5*m^2*x^2+12*B*a*c*d^2*e
^3*m^3*x-390*B*a*c*d*e^4*m^2*x^2+792*B*a*c*e^5*m*x^3+6*B*b^2*d^2*e^3*m^3*x-195*B*b^2*d*e^4*m^2*x^2+396*B*b^2*e
^5*m*x^3+216*B*b*c*d^2*e^3*m^2*x^2-576*B*b*c*d*e^4*m*x^3+288*B*b*c*e^5*x^4-60*B*c^2*d^3*e^2*m^2*x^2+220*B*c^2*
d^2*e^3*m*x^3-120*B*c^2*d*e^4*x^4+155*A*a^2*e^5*m^3-36*A*a*b*d*e^4*m^3+922*A*a*b*e^5*m^2*x+4*A*a*c*d^2*e^3*m^3
-356*A*a*c*d*e^4*m^2*x+1016*A*a*c*e^5*m*x^2+2*A*b^2*d^2*e^3*m^3-178*A*b^2*d*e^4*m^2*x+508*A*b^2*e^5*m*x^2+144*
A*b*c*d^2*e^3*m^2*x-672*A*b*c*d*e^4*m*x^2+360*A*b*c*e^5*x^3-24*A*c^2*d^3*e^2*m^2*x+240*A*c^2*d^2*e^3*m*x^2-144
*A*c^2*d*e^4*x^3-18*B*a^2*d*e^4*m^3+461*B*a^2*e^5*m^2*x+4*B*a*b*d^2*e^3*m^3-356*B*a*b*d*e^4*m^2*x+1016*B*a*b*e
^5*m*x^2+144*B*a*c*d^2*e^3*m^2*x-672*B*a*c*d*e^4*m*x^2+360*B*a*c*e^5*x^3+72*B*b^2*d^2*e^3*m^2*x-336*B*b^2*d*e^
4*m*x^2+180*B*b^2*e^5*x^3-48*B*b*c*d^3*e^2*m^2*x+480*B*b*c*d^2*e^3*m*x^2-288*B*b*c*d*e^4*x^3-180*B*c^2*d^3*e^2
*m*x^2+120*B*c^2*d^2*e^3*x^3+580*A*a^2*e^5*m^2-238*A*a*b*d*e^4*m^2+1404*A*a*b*e^5*m*x+60*A*a*c*d^2*e^3*m^2-776
*A*a*c*d*e^4*m*x+480*A*a*c*e^5*x^2+30*A*b^2*d^2*e^3*m^2-388*A*b^2*d*e^4*m*x+240*A*b^2*e^5*x^2-12*A*b*c*d^3*e^2
*m^2+492*A*b*c*d^2*e^3*m*x-360*A*b*c*d*e^4*x^2-168*A*c^2*d^3*e^2*m*x+144*A*c^2*d^2*e^3*x^2-119*B*a^2*d*e^4*m^2
+702*B*a^2*e^5*m*x+60*B*a*b*d^2*e^3*m^2-776*B*a*b*d*e^4*m*x+480*B*a*b*e^5*x^2-12*B*a*c*d^3*e^2*m^2+492*B*a*c*d
^2*e^3*m*x-360*B*a*c*d*e^4*x^2-6*B*b^2*d^3*e^2*m^2+246*B*b^2*d^2*e^3*m*x-180*B*b^2*d*e^4*x^2-336*B*b*c*d^3*e^2
*m*x+288*B*b*c*d^2*e^3*x^2+120*B*c^2*d^4*e*m*x-120*B*c^2*d^3*e^2*x^2+1044*A*a^2*e^5*m-684*A*a*b*d*e^4*m+720*A*
a*b*e^5*x+296*A*a*c*d^2*e^3*m-480*A*a*c*d*e^4*x+148*A*b^2*d^2*e^3*m-240*A*b^2*d*e^4*x-132*A*b*c*d^3*e^2*m+360*
A*b*c*d^2*e^3*x+24*A*c^2*d^4*e*m-144*A*c^2*d^3*e^2*x-342*B*a^2*d*e^4*m+360*B*a^2*e^5*x+296*B*a*b*d^2*e^3*m-480
*B*a*b*d*e^4*x-132*B*a*c*d^3*e^2*m+360*B*a*c*d^2*e^3*x-66*B*b^2*d^3*e^2*m+180*B*b^2*d^2*e^3*x+48*B*b*c*d^4*e*m
-288*B*b*c*d^3*e^2*x+120*B*c^2*d^4*e*x+720*A*a^2*e^5-720*A*a*b*d*e^4+480*A*a*c*d^2*e^3+240*A*b^2*d^2*e^3-360*A
*b*c*d^3*e^2+144*A*c^2*d^4*e-360*B*a^2*d*e^4+480*B*a*b*d^2*e^3-360*B*a*c*d^3*e^2-180*B*b^2*d^3*e^2+288*B*b*c*d
^4*e-120*B*c^2*d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+720)
________________________________________________________________________________________
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*(c*x^2+b*x+a)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.94128, size = 4764, normalized size = 14.31 \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*(c*x^2+b*x+a)^2,x, algorithm="fricas")
[Out]
(A*a^2*d*e^5*m^5 - 120*B*c^2*d^6 + 720*A*a^2*d*e^5 + 144*(2*B*b*c + A*c^2)*d^5*e - 180*(B*b^2 + 2*(B*a + A*b)*
c)*d^4*e^2 + 240*(2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e^3 - 360*(B*a^2 + 2*A*a*b)*d^2*e^4 + (B*c^2*e^6*m^5 + 15*B*c
^2*e^6*m^4 + 85*B*c^2*e^6*m^3 + 225*B*c^2*e^6*m^2 + 274*B*c^2*e^6*m + 120*B*c^2*e^6)*x^6 + (144*(2*B*b*c + A*c
^2)*e^6 + (B*c^2*d*e^5 + (2*B*b*c + A*c^2)*e^6)*m^5 + 2*(5*B*c^2*d*e^5 + 8*(2*B*b*c + A*c^2)*e^6)*m^4 + 5*(7*B
*c^2*d*e^5 + 19*(2*B*b*c + A*c^2)*e^6)*m^3 + 10*(5*B*c^2*d*e^5 + 26*(2*B*b*c + A*c^2)*e^6)*m^2 + 12*(2*B*c^2*d
*e^5 + 27*(2*B*b*c + A*c^2)*e^6)*m)*x^5 + (20*A*a^2*d*e^5 - (B*a^2 + 2*A*a*b)*d^2*e^4)*m^4 + (180*(B*b^2 + 2*(
B*a + A*b)*c)*e^6 + ((2*B*b*c + A*c^2)*d*e^5 + (B*b^2 + 2*(B*a + A*b)*c)*e^6)*m^5 - (5*B*c^2*d^2*e^4 - 12*(2*B
*b*c + A*c^2)*d*e^5 - 17*(B*b^2 + 2*(B*a + A*b)*c)*e^6)*m^4 - (30*B*c^2*d^2*e^4 - 47*(2*B*b*c + A*c^2)*d*e^5 -
107*(B*b^2 + 2*(B*a + A*b)*c)*e^6)*m^3 - (55*B*c^2*d^2*e^4 - 72*(2*B*b*c + A*c^2)*d*e^5 - 307*(B*b^2 + 2*(B*a
+ A*b)*c)*e^6)*m^2 - 6*(5*B*c^2*d^2*e^4 - 6*(2*B*b*c + A*c^2)*d*e^5 - 66*(B*b^2 + 2*(B*a + A*b)*c)*e^6)*m)*x^
4 + (155*A*a^2*d*e^5 + 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e^3 - 18*(B*a^2 + 2*A*a*b)*d^2*e^4)*m^3 + (240*(2*B*a
*b + A*b^2 + 2*A*a*c)*e^6 + ((B*b^2 + 2*(B*a + A*b)*c)*d*e^5 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^6)*m^5 - 2*(2*(2*
B*b*c + A*c^2)*d^2*e^4 - 7*(B*b^2 + 2*(B*a + A*b)*c)*d*e^5 - 9*(2*B*a*b + A*b^2 + 2*A*a*c)*e^6)*m^4 + (20*B*c^
2*d^3*e^3 - 36*(2*B*b*c + A*c^2)*d^2*e^4 + 65*(B*b^2 + 2*(B*a + A*b)*c)*d*e^5 + 121*(2*B*a*b + A*b^2 + 2*A*a*c
)*e^6)*m^3 + 4*(15*B*c^2*d^3*e^3 - 20*(2*B*b*c + A*c^2)*d^2*e^4 + 28*(B*b^2 + 2*(B*a + A*b)*c)*d*e^5 + 93*(2*B
*a*b + A*b^2 + 2*A*a*c)*e^6)*m^2 + 4*(10*B*c^2*d^3*e^3 - 12*(2*B*b*c + A*c^2)*d^2*e^4 + 15*(B*b^2 + 2*(B*a + A
*b)*c)*d*e^5 + 127*(2*B*a*b + A*b^2 + 2*A*a*c)*e^6)*m)*x^3 + (580*A*a^2*d*e^5 - 6*(B*b^2 + 2*(B*a + A*b)*c)*d^
4*e^2 + 30*(2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e^3 - 119*(B*a^2 + 2*A*a*b)*d^2*e^4)*m^2 + (360*(B*a^2 + 2*A*a*b)*e
^6 + ((2*B*a*b + A*b^2 + 2*A*a*c)*d*e^5 + (B*a^2 + 2*A*a*b)*e^6)*m^5 - (3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^4 -
16*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^5 - 19*(B*a^2 + 2*A*a*b)*e^6)*m^4 + (12*(2*B*b*c + A*c^2)*d^3*e^3 - 36*(B*b
^2 + 2*(B*a + A*b)*c)*d^2*e^4 + 89*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^5 + 137*(B*a^2 + 2*A*a*b)*e^6)*m^3 - (60*B*
c^2*d^4*e^2 - 84*(2*B*b*c + A*c^2)*d^3*e^3 + 123*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^4 - 194*(2*B*a*b + A*b^2 + 2*
A*a*c)*d*e^5 - 461*(B*a^2 + 2*A*a*b)*e^6)*m^2 - 6*(10*B*c^2*d^4*e^2 - 12*(2*B*b*c + A*c^2)*d^3*e^3 + 15*(B*b^2
+ 2*(B*a + A*b)*c)*d^2*e^4 - 20*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^5 - 117*(B*a^2 + 2*A*a*b)*e^6)*m)*x^2 + 2*(52
2*A*a^2*d*e^5 + 12*(2*B*b*c + A*c^2)*d^5*e - 33*(B*b^2 + 2*(B*a + A*b)*c)*d^4*e^2 + 74*(2*B*a*b + A*b^2 + 2*A*
a*c)*d^3*e^3 - 171*(B*a^2 + 2*A*a*b)*d^2*e^4)*m + (720*A*a^2*e^6 + (A*a^2*e^6 + (B*a^2 + 2*A*a*b)*d*e^5)*m^5 +
2*(10*A*a^2*e^6 - (2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^4 + 9*(B*a^2 + 2*A*a*b)*d*e^5)*m^4 + (155*A*a^2*e^6 + 6*(
B*b^2 + 2*(B*a + A*b)*c)*d^3*e^3 - 30*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^4 + 119*(B*a^2 + 2*A*a*b)*d*e^5)*m^3 +
2*(290*A*a^2*e^6 - 12*(2*B*b*c + A*c^2)*d^4*e^2 + 33*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^3 - 74*(2*B*a*b + A*b^2
+ 2*A*a*c)*d^2*e^4 + 171*(B*a^2 + 2*A*a*b)*d*e^5)*m^2 + 12*(10*B*c^2*d^5*e + 87*A*a^2*e^6 - 12*(2*B*b*c + A*c^
2)*d^4*e^2 + 15*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^3 - 20*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^4 + 30*(B*a^2 + 2*A*a
*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 + 7
20*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*(c*x**2+b*x+a)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.445, size = 6669, normalized size = 20.03 \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*(c*x^2+b*x+a)^2,x, algorithm="giac")
[Out]
((x*e + d)^m*B*c^2*m^5*x^6*e^6 + (x*e + d)^m*B*c^2*d*m^5*x^5*e^5 + 2*(x*e + d)^m*B*b*c*m^5*x^5*e^6 + (x*e + d)
^m*A*c^2*m^5*x^5*e^6 + 15*(x*e + d)^m*B*c^2*m^4*x^6*e^6 + 2*(x*e + d)^m*B*b*c*d*m^5*x^4*e^5 + (x*e + d)^m*A*c^
2*d*m^5*x^4*e^5 + 10*(x*e + d)^m*B*c^2*d*m^4*x^5*e^5 - 5*(x*e + d)^m*B*c^2*d^2*m^4*x^4*e^4 + (x*e + d)^m*B*b^2
*m^5*x^4*e^6 + 2*(x*e + d)^m*B*a*c*m^5*x^4*e^6 + 2*(x*e + d)^m*A*b*c*m^5*x^4*e^6 + 32*(x*e + d)^m*B*b*c*m^4*x^
5*e^6 + 16*(x*e + d)^m*A*c^2*m^4*x^5*e^6 + 85*(x*e + d)^m*B*c^2*m^3*x^6*e^6 + (x*e + d)^m*B*b^2*d*m^5*x^3*e^5
+ 2*(x*e + d)^m*B*a*c*d*m^5*x^3*e^5 + 2*(x*e + d)^m*A*b*c*d*m^5*x^3*e^5 + 24*(x*e + d)^m*B*b*c*d*m^4*x^4*e^5 +
12*(x*e + d)^m*A*c^2*d*m^4*x^4*e^5 + 35*(x*e + d)^m*B*c^2*d*m^3*x^5*e^5 - 8*(x*e + d)^m*B*b*c*d^2*m^4*x^3*e^4
- 4*(x*e + d)^m*A*c^2*d^2*m^4*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m^3*x^4*e^4 + 20*(x*e + d)^m*B*c^2*d^3*m^3*x
^3*e^3 + 2*(x*e + d)^m*B*a*b*m^5*x^3*e^6 + (x*e + d)^m*A*b^2*m^5*x^3*e^6 + 2*(x*e + d)^m*A*a*c*m^5*x^3*e^6 + 1
7*(x*e + d)^m*B*b^2*m^4*x^4*e^6 + 34*(x*e + d)^m*B*a*c*m^4*x^4*e^6 + 34*(x*e + d)^m*A*b*c*m^4*x^4*e^6 + 190*(x
*e + d)^m*B*b*c*m^3*x^5*e^6 + 95*(x*e + d)^m*A*c^2*m^3*x^5*e^6 + 225*(x*e + d)^m*B*c^2*m^2*x^6*e^6 + 2*(x*e +
d)^m*B*a*b*d*m^5*x^2*e^5 + (x*e + d)^m*A*b^2*d*m^5*x^2*e^5 + 2*(x*e + d)^m*A*a*c*d*m^5*x^2*e^5 + 14*(x*e + d)^
m*B*b^2*d*m^4*x^3*e^5 + 28*(x*e + d)^m*B*a*c*d*m^4*x^3*e^5 + 28*(x*e + d)^m*A*b*c*d*m^4*x^3*e^5 + 94*(x*e + d)
^m*B*b*c*d*m^3*x^4*e^5 + 47*(x*e + d)^m*A*c^2*d*m^3*x^4*e^5 + 50*(x*e + d)^m*B*c^2*d*m^2*x^5*e^5 - 3*(x*e + d)
^m*B*b^2*d^2*m^4*x^2*e^4 - 6*(x*e + d)^m*B*a*c*d^2*m^4*x^2*e^4 - 6*(x*e + d)^m*A*b*c*d^2*m^4*x^2*e^4 - 72*(x*e
+ d)^m*B*b*c*d^2*m^3*x^3*e^4 - 36*(x*e + d)^m*A*c^2*d^2*m^3*x^3*e^4 - 55*(x*e + d)^m*B*c^2*d^2*m^2*x^4*e^4 +
24*(x*e + d)^m*B*b*c*d^3*m^3*x^2*e^3 + 12*(x*e + d)^m*A*c^2*d^3*m^3*x^2*e^3 + 60*(x*e + d)^m*B*c^2*d^3*m^2*x^3
*e^3 - 60*(x*e + d)^m*B*c^2*d^4*m^2*x^2*e^2 + (x*e + d)^m*B*a^2*m^5*x^2*e^6 + 2*(x*e + d)^m*A*a*b*m^5*x^2*e^6
+ 36*(x*e + d)^m*B*a*b*m^4*x^3*e^6 + 18*(x*e + d)^m*A*b^2*m^4*x^3*e^6 + 36*(x*e + d)^m*A*a*c*m^4*x^3*e^6 + 107
*(x*e + d)^m*B*b^2*m^3*x^4*e^6 + 214*(x*e + d)^m*B*a*c*m^3*x^4*e^6 + 214*(x*e + d)^m*A*b*c*m^3*x^4*e^6 + 520*(
x*e + d)^m*B*b*c*m^2*x^5*e^6 + 260*(x*e + d)^m*A*c^2*m^2*x^5*e^6 + 274*(x*e + d)^m*B*c^2*m*x^6*e^6 + (x*e + d)
^m*B*a^2*d*m^5*x*e^5 + 2*(x*e + d)^m*A*a*b*d*m^5*x*e^5 + 32*(x*e + d)^m*B*a*b*d*m^4*x^2*e^5 + 16*(x*e + d)^m*A
*b^2*d*m^4*x^2*e^5 + 32*(x*e + d)^m*A*a*c*d*m^4*x^2*e^5 + 65*(x*e + d)^m*B*b^2*d*m^3*x^3*e^5 + 130*(x*e + d)^m
*B*a*c*d*m^3*x^3*e^5 + 130*(x*e + d)^m*A*b*c*d*m^3*x^3*e^5 + 144*(x*e + d)^m*B*b*c*d*m^2*x^4*e^5 + 72*(x*e + d
)^m*A*c^2*d*m^2*x^4*e^5 + 24*(x*e + d)^m*B*c^2*d*m*x^5*e^5 - 4*(x*e + d)^m*B*a*b*d^2*m^4*x*e^4 - 2*(x*e + d)^m
*A*b^2*d^2*m^4*x*e^4 - 4*(x*e + d)^m*A*a*c*d^2*m^4*x*e^4 - 36*(x*e + d)^m*B*b^2*d^2*m^3*x^2*e^4 - 72*(x*e + d)
^m*B*a*c*d^2*m^3*x^2*e^4 - 72*(x*e + d)^m*A*b*c*d^2*m^3*x^2*e^4 - 160*(x*e + d)^m*B*b*c*d^2*m^2*x^3*e^4 - 80*(
x*e + d)^m*A*c^2*d^2*m^2*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m*x^4*e^4 + 6*(x*e + d)^m*B*b^2*d^3*m^3*x*e^3 + 12
*(x*e + d)^m*B*a*c*d^3*m^3*x*e^3 + 12*(x*e + d)^m*A*b*c*d^3*m^3*x*e^3 + 168*(x*e + d)^m*B*b*c*d^3*m^2*x^2*e^3
+ 84*(x*e + d)^m*A*c^2*d^3*m^2*x^2*e^3 + 40*(x*e + d)^m*B*c^2*d^3*m*x^3*e^3 - 48*(x*e + d)^m*B*b*c*d^4*m^2*x*e
^2 - 24*(x*e + d)^m*A*c^2*d^4*m^2*x*e^2 - 60*(x*e + d)^m*B*c^2*d^4*m*x^2*e^2 + 120*(x*e + d)^m*B*c^2*d^5*m*x*e
+ (x*e + d)^m*A*a^2*m^5*x*e^6 + 19*(x*e + d)^m*B*a^2*m^4*x^2*e^6 + 38*(x*e + d)^m*A*a*b*m^4*x^2*e^6 + 242*(x*
e + d)^m*B*a*b*m^3*x^3*e^6 + 121*(x*e + d)^m*A*b^2*m^3*x^3*e^6 + 242*(x*e + d)^m*A*a*c*m^3*x^3*e^6 + 307*(x*e
+ d)^m*B*b^2*m^2*x^4*e^6 + 614*(x*e + d)^m*B*a*c*m^2*x^4*e^6 + 614*(x*e + d)^m*A*b*c*m^2*x^4*e^6 + 648*(x*e +
d)^m*B*b*c*m*x^5*e^6 + 324*(x*e + d)^m*A*c^2*m*x^5*e^6 + 120*(x*e + d)^m*B*c^2*x^6*e^6 + (x*e + d)^m*A*a^2*d*m
^5*e^5 + 18*(x*e + d)^m*B*a^2*d*m^4*x*e^5 + 36*(x*e + d)^m*A*a*b*d*m^4*x*e^5 + 178*(x*e + d)^m*B*a*b*d*m^3*x^2
*e^5 + 89*(x*e + d)^m*A*b^2*d*m^3*x^2*e^5 + 178*(x*e + d)^m*A*a*c*d*m^3*x^2*e^5 + 112*(x*e + d)^m*B*b^2*d*m^2*
x^3*e^5 + 224*(x*e + d)^m*B*a*c*d*m^2*x^3*e^5 + 224*(x*e + d)^m*A*b*c*d*m^2*x^3*e^5 + 72*(x*e + d)^m*B*b*c*d*m
*x^4*e^5 + 36*(x*e + d)^m*A*c^2*d*m*x^4*e^5 - (x*e + d)^m*B*a^2*d^2*m^4*e^4 - 2*(x*e + d)^m*A*a*b*d^2*m^4*e^4
- 60*(x*e + d)^m*B*a*b*d^2*m^3*x*e^4 - 30*(x*e + d)^m*A*b^2*d^2*m^3*x*e^4 - 60*(x*e + d)^m*A*a*c*d^2*m^3*x*e^4
- 123*(x*e + d)^m*B*b^2*d^2*m^2*x^2*e^4 - 246*(x*e + d)^m*B*a*c*d^2*m^2*x^2*e^4 - 246*(x*e + d)^m*A*b*c*d^2*m
^2*x^2*e^4 - 96*(x*e + d)^m*B*b*c*d^2*m*x^3*e^4 - 48*(x*e + d)^m*A*c^2*d^2*m*x^3*e^4 + 4*(x*e + d)^m*B*a*b*d^3
*m^3*e^3 + 2*(x*e + d)^m*A*b^2*d^3*m^3*e^3 + 4*(x*e + d)^m*A*a*c*d^3*m^3*e^3 + 66*(x*e + d)^m*B*b^2*d^3*m^2*x*
e^3 + 132*(x*e + d)^m*B*a*c*d^3*m^2*x*e^3 + 132*(x*e + d)^m*A*b*c*d^3*m^2*x*e^3 + 144*(x*e + d)^m*B*b*c*d^3*m*
x^2*e^3 + 72*(x*e + d)^m*A*c^2*d^3*m*x^2*e^3 - 6*(x*e + d)^m*B*b^2*d^4*m^2*e^2 - 12*(x*e + d)^m*B*a*c*d^4*m^2*
e^2 - 12*(x*e + d)^m*A*b*c*d^4*m^2*e^2 - 288*(x*e + d)^m*B*b*c*d^4*m*x*e^2 - 144*(x*e + d)^m*A*c^2*d^4*m*x*e^2
+ 48*(x*e + d)^m*B*b*c*d^5*m*e + 24*(x*e + d)^m*A*c^2*d^5*m*e - 120*(x*e + d)^m*B*c^2*d^6 + 20*(x*e + d)^m*A*
a^2*m^4*x*e^6 + 137*(x*e + d)^m*B*a^2*m^3*x^2*e^6 + 274*(x*e + d)^m*A*a*b*m^3*x^2*e^6 + 744*(x*e + d)^m*B*a*b*
m^2*x^3*e^6 + 372*(x*e + d)^m*A*b^2*m^2*x^3*e^6 + 744*(x*e + d)^m*A*a*c*m^2*x^3*e^6 + 396*(x*e + d)^m*B*b^2*m*
x^4*e^6 + 792*(x*e + d)^m*B*a*c*m*x^4*e^6 + 792*(x*e + d)^m*A*b*c*m*x^4*e^6 + 288*(x*e + d)^m*B*b*c*x^5*e^6 +
144*(x*e + d)^m*A*c^2*x^5*e^6 + 20*(x*e + d)^m*A*a^2*d*m^4*e^5 + 119*(x*e + d)^m*B*a^2*d*m^3*x*e^5 + 238*(x*e
+ d)^m*A*a*b*d*m^3*x*e^5 + 388*(x*e + d)^m*B*a*b*d*m^2*x^2*e^5 + 194*(x*e + d)^m*A*b^2*d*m^2*x^2*e^5 + 388*(x*
e + d)^m*A*a*c*d*m^2*x^2*e^5 + 60*(x*e + d)^m*B*b^2*d*m*x^3*e^5 + 120*(x*e + d)^m*B*a*c*d*m*x^3*e^5 + 120*(x*e
+ d)^m*A*b*c*d*m*x^3*e^5 - 18*(x*e + d)^m*B*a^2*d^2*m^3*e^4 - 36*(x*e + d)^m*A*a*b*d^2*m^3*e^4 - 296*(x*e + d
)^m*B*a*b*d^2*m^2*x*e^4 - 148*(x*e + d)^m*A*b^2*d^2*m^2*x*e^4 - 296*(x*e + d)^m*A*a*c*d^2*m^2*x*e^4 - 90*(x*e
+ d)^m*B*b^2*d^2*m*x^2*e^4 - 180*(x*e + d)^m*B*a*c*d^2*m*x^2*e^4 - 180*(x*e + d)^m*A*b*c*d^2*m*x^2*e^4 + 60*(x
*e + d)^m*B*a*b*d^3*m^2*e^3 + 30*(x*e + d)^m*A*b^2*d^3*m^2*e^3 + 60*(x*e + d)^m*A*a*c*d^3*m^2*e^3 + 180*(x*e +
d)^m*B*b^2*d^3*m*x*e^3 + 360*(x*e + d)^m*B*a*c*d^3*m*x*e^3 + 360*(x*e + d)^m*A*b*c*d^3*m*x*e^3 - 66*(x*e + d)
^m*B*b^2*d^4*m*e^2 - 132*(x*e + d)^m*B*a*c*d^4*m*e^2 - 132*(x*e + d)^m*A*b*c*d^4*m*e^2 + 288*(x*e + d)^m*B*b*c
*d^5*e + 144*(x*e + d)^m*A*c^2*d^5*e + 155*(x*e + d)^m*A*a^2*m^3*x*e^6 + 461*(x*e + d)^m*B*a^2*m^2*x^2*e^6 + 9
22*(x*e + d)^m*A*a*b*m^2*x^2*e^6 + 1016*(x*e + d)^m*B*a*b*m*x^3*e^6 + 508*(x*e + d)^m*A*b^2*m*x^3*e^6 + 1016*(
x*e + d)^m*A*a*c*m*x^3*e^6 + 180*(x*e + d)^m*B*b^2*x^4*e^6 + 360*(x*e + d)^m*B*a*c*x^4*e^6 + 360*(x*e + d)^m*A
*b*c*x^4*e^6 + 155*(x*e + d)^m*A*a^2*d*m^3*e^5 + 342*(x*e + d)^m*B*a^2*d*m^2*x*e^5 + 684*(x*e + d)^m*A*a*b*d*m
^2*x*e^5 + 240*(x*e + d)^m*B*a*b*d*m*x^2*e^5 + 120*(x*e + d)^m*A*b^2*d*m*x^2*e^5 + 240*(x*e + d)^m*A*a*c*d*m*x
^2*e^5 - 119*(x*e + d)^m*B*a^2*d^2*m^2*e^4 - 238*(x*e + d)^m*A*a*b*d^2*m^2*e^4 - 480*(x*e + d)^m*B*a*b*d^2*m*x
*e^4 - 240*(x*e + d)^m*A*b^2*d^2*m*x*e^4 - 480*(x*e + d)^m*A*a*c*d^2*m*x*e^4 + 296*(x*e + d)^m*B*a*b*d^3*m*e^3
+ 148*(x*e + d)^m*A*b^2*d^3*m*e^3 + 296*(x*e + d)^m*A*a*c*d^3*m*e^3 - 180*(x*e + d)^m*B*b^2*d^4*e^2 - 360*(x*
e + d)^m*B*a*c*d^4*e^2 - 360*(x*e + d)^m*A*b*c*d^4*e^2 + 580*(x*e + d)^m*A*a^2*m^2*x*e^6 + 702*(x*e + d)^m*B*a
^2*m*x^2*e^6 + 1404*(x*e + d)^m*A*a*b*m*x^2*e^6 + 480*(x*e + d)^m*B*a*b*x^3*e^6 + 240*(x*e + d)^m*A*b^2*x^3*e^
6 + 480*(x*e + d)^m*A*a*c*x^3*e^6 + 580*(x*e + d)^m*A*a^2*d*m^2*e^5 + 360*(x*e + d)^m*B*a^2*d*m*x*e^5 + 720*(x
*e + d)^m*A*a*b*d*m*x*e^5 - 342*(x*e + d)^m*B*a^2*d^2*m*e^4 - 684*(x*e + d)^m*A*a*b*d^2*m*e^4 + 480*(x*e + d)^
m*B*a*b*d^3*e^3 + 240*(x*e + d)^m*A*b^2*d^3*e^3 + 480*(x*e + d)^m*A*a*c*d^3*e^3 + 1044*(x*e + d)^m*A*a^2*m*x*e
^6 + 360*(x*e + d)^m*B*a^2*x^2*e^6 + 720*(x*e + d)^m*A*a*b*x^2*e^6 + 1044*(x*e + d)^m*A*a^2*d*m*e^5 - 360*(x*e
+ d)^m*B*a^2*d^2*e^4 - 720*(x*e + d)^m*A*a*b*d^2*e^4 + 720*(x*e + d)^m*A*a^2*x*e^6 + 720*(x*e + d)^m*A*a^2*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 + 1764*m*e^6 + 720*e^6)