3.926 \(\int (d+e x)^m (f+g x) \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=311 \[ \frac{(d+e x)^{m+4} \left (2 c e (a e g-4 b d g+b e f)+b^2 e^2 g-2 c^2 d (2 e f-5 d g)\right )}{e^6 (m+4)}+\frac{(d+e x)^{m+3} \left (2 c e (a e (e f-3 d g)-3 b d (e f-2 d g))+b e^2 (2 a e g-3 b d g+b e f)+2 c^2 d^2 (3 e f-5 d g)\right )}{e^6 (m+3)}+\frac{(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) (c d (4 e f-5 d g)-e (a e g-3 b d g+2 b e f))}{e^6 (m+2)}+\frac{c (d+e x)^{m+5} (2 b e g-5 c d g+c e f)}{e^6 (m+5)}+\frac{c^2 g (d+e x)^{m+6}}{e^6 (m+6)} \]

[Out]

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

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Rubi [A]  time = 0.983522, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{(d+e x)^{m+4} \left (2 c e (a e g-4 b d g+b e f)+b^2 e^2 g-2 c^2 d (2 e f-5 d g)\right )}{e^6 (m+4)}+\frac{(d+e x)^{m+3} \left (2 c e (a e (e f-3 d g)-3 b d (e f-2 d g))+b e^2 (2 a e g-3 b d g+b e f)+2 c^2 d^2 (3 e f-5 d g)\right )}{e^6 (m+3)}+\frac{(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) (c d (4 e f-5 d g)-e (a e g-3 b d g+2 b e f))}{e^6 (m+2)}+\frac{c (d+e x)^{m+5} (2 b e g-5 c d g+c e f)}{e^6 (m+5)}+\frac{c^2 g (d+e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(g*x+f)*(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.70122, size = 593, normalized size = 1.91 \[ \frac{(d+e x)^{m+1} \left (e^2 \left (m^2+11 m+30\right ) \left (a^2 e^2 \left (m^2+7 m+12\right ) (-d g+e f (m+2)+e g (m+1) x)+2 a b e (m+4) \left (2 d^2 g-d e (f (m+3)+2 g (m+1) x)+e^2 (m+1) x (f (m+3)+g (m+2) x)\right )+b^2 \left (-6 d^3 g+2 d^2 e (f (m+4)+3 g (m+1) x)-d e^2 (m+1) x (2 f (m+4)+3 g (m+2) x)+e^3 \left (m^2+3 m+2\right ) x^2 (f (m+4)+g (m+3) x)\right )\right )+2 c e (m+6) \left (a e (m+5) \left (-6 d^3 g+2 d^2 e (f (m+4)+3 g (m+1) x)-d e^2 (m+1) x (2 f (m+4)+3 g (m+2) x)+e^3 \left (m^2+3 m+2\right ) x^2 (f (m+4)+g (m+3) x)\right )+b \left (24 d^4 g-6 d^3 e (f (m+5)+4 g (m+1) x)+6 d^2 e^2 (m+1) x (f (m+5)+2 g (m+2) x)-d e^3 \left (m^2+3 m+2\right ) x^2 (3 f (m+5)+4 g (m+3) x)+e^4 \left (m^3+6 m^2+11 m+6\right ) x^3 (f (m+5)+g (m+4) x)\right )\right )+c^2 \left (-\left (120 d^5 g-24 d^4 e (f (m+6)+5 g (m+1) x)+12 d^3 e^2 (m+1) x (2 f (m+6)+5 g (m+2) x)-4 d^2 e^3 \left (m^2+3 m+2\right ) x^2 (3 f (m+6)+5 g (m+3) x)+d e^4 \left (m^3+6 m^2+11 m+6\right ) x^3 (4 f (m+6)+5 g (m+4) x)-e^5 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 (f (m+6)+g (m+5) x)\right )\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(-(c^2*(120*d^5*g - 24*d^4*e*(f*(6 + m) + 5*g*(1 + m)*x) + 12
*d^3*e^2*(1 + m)*x*(2*f*(6 + m) + 5*g*(2 + m)*x) - 4*d^2*e^3*(2 + 3*m + m^2)*x^2
*(3*f*(6 + m) + 5*g*(3 + m)*x) + d*e^4*(6 + 11*m + 6*m^2 + m^3)*x^3*(4*f*(6 + m)
+ 5*g*(4 + m)*x) - e^5*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4*(f*(6 + m) + g*(
5 + m)*x))) + e^2*(30 + 11*m + m^2)*(a^2*e^2*(12 + 7*m + m^2)*(-(d*g) + e*f*(2 +
m) + e*g*(1 + m)*x) + 2*a*b*e*(4 + m)*(2*d^2*g - d*e*(f*(3 + m) + 2*g*(1 + m)*x
) + e^2*(1 + m)*x*(f*(3 + m) + g*(2 + m)*x)) + b^2*(-6*d^3*g + 2*d^2*e*(f*(4 + m
) + 3*g*(1 + m)*x) - d*e^2*(1 + m)*x*(2*f*(4 + m) + 3*g*(2 + m)*x) + e^3*(2 + 3*
m + m^2)*x^2*(f*(4 + m) + g*(3 + m)*x))) + 2*c*e*(6 + m)*(a*e*(5 + m)*(-6*d^3*g
+ 2*d^2*e*(f*(4 + m) + 3*g*(1 + m)*x) - d*e^2*(1 + m)*x*(2*f*(4 + m) + 3*g*(2 +
m)*x) + e^3*(2 + 3*m + m^2)*x^2*(f*(4 + m) + g*(3 + m)*x)) + b*(24*d^4*g - 6*d^3
*e*(f*(5 + m) + 4*g*(1 + m)*x) + 6*d^2*e^2*(1 + m)*x*(f*(5 + m) + 2*g*(2 + m)*x)
- d*e^3*(2 + 3*m + m^2)*x^2*(3*f*(5 + m) + 4*g*(3 + m)*x) + e^4*(6 + 11*m + 6*m
^2 + m^3)*x^3*(f*(5 + m) + g*(4 + m)*x)))))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 + m)
*(5 + m)*(6 + m))

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Maple [B]  time = 0.019, size = 2563, normalized size = 8.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a)^2,x)

[Out]

-(e*x+d)^(1+m)*(-c^2*e^5*g*m^5*x^5-2*b*c*e^5*g*m^5*x^4-c^2*e^5*f*m^5*x^4-15*c^2*
e^5*g*m^4*x^5-2*a*c*e^5*g*m^5*x^3-b^2*e^5*g*m^5*x^3-2*b*c*e^5*f*m^5*x^3-32*b*c*e
^5*g*m^4*x^4+5*c^2*d*e^4*g*m^4*x^4-16*c^2*e^5*f*m^4*x^4-85*c^2*e^5*g*m^3*x^5-2*a
*b*e^5*g*m^5*x^2-2*a*c*e^5*f*m^5*x^2-34*a*c*e^5*g*m^4*x^3-b^2*e^5*f*m^5*x^2-17*b
^2*e^5*g*m^4*x^3+8*b*c*d*e^4*g*m^4*x^3-34*b*c*e^5*f*m^4*x^3-190*b*c*e^5*g*m^3*x^
4+4*c^2*d*e^4*f*m^4*x^3+50*c^2*d*e^4*g*m^3*x^4-95*c^2*e^5*f*m^3*x^4-225*c^2*e^5*
g*m^2*x^5-a^2*e^5*g*m^5*x-2*a*b*e^5*f*m^5*x-36*a*b*e^5*g*m^4*x^2+6*a*c*d*e^4*g*m
^4*x^2-36*a*c*e^5*f*m^4*x^2-214*a*c*e^5*g*m^3*x^3+3*b^2*d*e^4*g*m^4*x^2-18*b^2*e
^5*f*m^4*x^2-107*b^2*e^5*g*m^3*x^3+6*b*c*d*e^4*f*m^4*x^2+96*b*c*d*e^4*g*m^3*x^3-
214*b*c*e^5*f*m^3*x^3-520*b*c*e^5*g*m^2*x^4-20*c^2*d^2*e^3*g*m^3*x^3+48*c^2*d*e^
4*f*m^3*x^3+175*c^2*d*e^4*g*m^2*x^4-260*c^2*e^5*f*m^2*x^4-274*c^2*e^5*g*m*x^5-a^
2*e^5*f*m^5-19*a^2*e^5*g*m^4*x+4*a*b*d*e^4*g*m^4*x-38*a*b*e^5*f*m^4*x-242*a*b*e^
5*g*m^3*x^2+4*a*c*d*e^4*f*m^4*x+84*a*c*d*e^4*g*m^3*x^2-242*a*c*e^5*f*m^3*x^2-614
*a*c*e^5*g*m^2*x^3+2*b^2*d*e^4*f*m^4*x+42*b^2*d*e^4*g*m^3*x^2-121*b^2*e^5*f*m^3*
x^2-307*b^2*e^5*g*m^2*x^3-24*b*c*d^2*e^3*g*m^3*x^2+84*b*c*d*e^4*f*m^3*x^2+376*b*
c*d*e^4*g*m^2*x^3-614*b*c*e^5*f*m^2*x^3-648*b*c*e^5*g*m*x^4-12*c^2*d^2*e^3*f*m^3
*x^2-120*c^2*d^2*e^3*g*m^2*x^3+188*c^2*d*e^4*f*m^2*x^3+250*c^2*d*e^4*g*m*x^4-324
*c^2*e^5*f*m*x^4-120*c^2*e^5*g*x^5+a^2*d*e^4*g*m^4-20*a^2*e^5*f*m^4-137*a^2*e^5*
g*m^3*x+2*a*b*d*e^4*f*m^4+64*a*b*d*e^4*g*m^3*x-274*a*b*e^5*f*m^3*x-744*a*b*e^5*g
*m^2*x^2-12*a*c*d^2*e^3*g*m^3*x+64*a*c*d*e^4*f*m^3*x+390*a*c*d*e^4*g*m^2*x^2-744
*a*c*e^5*f*m^2*x^2-792*a*c*e^5*g*m*x^3-6*b^2*d^2*e^3*g*m^3*x+32*b^2*d*e^4*f*m^3*
x+195*b^2*d*e^4*g*m^2*x^2-372*b^2*e^5*f*m^2*x^2-396*b^2*e^5*g*m*x^3-12*b*c*d^2*e
^3*f*m^3*x-216*b*c*d^2*e^3*g*m^2*x^2+390*b*c*d*e^4*f*m^2*x^2+576*b*c*d*e^4*g*m*x
^3-792*b*c*e^5*f*m*x^3-288*b*c*e^5*g*x^4+60*c^2*d^3*e^2*g*m^2*x^2-108*c^2*d^2*e^
3*f*m^2*x^2-220*c^2*d^2*e^3*g*m*x^3+288*c^2*d*e^4*f*m*x^3+120*c^2*d*e^4*g*x^4-14
4*c^2*e^5*f*x^4+18*a^2*d*e^4*g*m^3-155*a^2*e^5*f*m^3-461*a^2*e^5*g*m^2*x-4*a*b*d
^2*e^3*g*m^3+36*a*b*d*e^4*f*m^3+356*a*b*d*e^4*g*m^2*x-922*a*b*e^5*f*m^2*x-1016*a
*b*e^5*g*m*x^2-4*a*c*d^2*e^3*f*m^3-144*a*c*d^2*e^3*g*m^2*x+356*a*c*d*e^4*f*m^2*x
+672*a*c*d*e^4*g*m*x^2-1016*a*c*e^5*f*m*x^2-360*a*c*e^5*g*x^3-2*b^2*d^2*e^3*f*m^
3-72*b^2*d^2*e^3*g*m^2*x+178*b^2*d*e^4*f*m^2*x+336*b^2*d*e^4*g*m*x^2-508*b^2*e^5
*f*m*x^2-180*b^2*e^5*g*x^3+48*b*c*d^3*e^2*g*m^2*x-144*b*c*d^2*e^3*f*m^2*x-480*b*
c*d^2*e^3*g*m*x^2+672*b*c*d*e^4*f*m*x^2+288*b*c*d*e^4*g*x^3-360*b*c*e^5*f*x^3+24
*c^2*d^3*e^2*f*m^2*x+180*c^2*d^3*e^2*g*m*x^2-240*c^2*d^2*e^3*f*m*x^2-120*c^2*d^2
*e^3*g*x^3+144*c^2*d*e^4*f*x^3+119*a^2*d*e^4*g*m^2-580*a^2*e^5*f*m^2-702*a^2*e^5
*g*m*x-60*a*b*d^2*e^3*g*m^2+238*a*b*d*e^4*f*m^2+776*a*b*d*e^4*g*m*x-1404*a*b*e^5
*f*m*x-480*a*b*e^5*g*x^2+12*a*c*d^3*e^2*g*m^2-60*a*c*d^2*e^3*f*m^2-492*a*c*d^2*e
^3*g*m*x+776*a*c*d*e^4*f*m*x+360*a*c*d*e^4*g*x^2-480*a*c*e^5*f*x^2+6*b^2*d^3*e^2
*g*m^2-30*b^2*d^2*e^3*f*m^2-246*b^2*d^2*e^3*g*m*x+388*b^2*d*e^4*f*m*x+180*b^2*d*
e^4*g*x^2-240*b^2*e^5*f*x^2+12*b*c*d^3*e^2*f*m^2+336*b*c*d^3*e^2*g*m*x-492*b*c*d
^2*e^3*f*m*x-288*b*c*d^2*e^3*g*x^2+360*b*c*d*e^4*f*x^2-120*c^2*d^4*e*g*m*x+168*c
^2*d^3*e^2*f*m*x+120*c^2*d^3*e^2*g*x^2-144*c^2*d^2*e^3*f*x^2+342*a^2*d*e^4*g*m-1
044*a^2*e^5*f*m-360*a^2*e^5*g*x-296*a*b*d^2*e^3*g*m+684*a*b*d*e^4*f*m+480*a*b*d*
e^4*g*x-720*a*b*e^5*f*x+132*a*c*d^3*e^2*g*m-296*a*c*d^2*e^3*f*m-360*a*c*d^2*e^3*
g*x+480*a*c*d*e^4*f*x+66*b^2*d^3*e^2*g*m-148*b^2*d^2*e^3*f*m-180*b^2*d^2*e^3*g*x
+240*b^2*d*e^4*f*x-48*b*c*d^4*e*g*m+132*b*c*d^3*e^2*f*m+288*b*c*d^3*e^2*g*x-360*
b*c*d^2*e^3*f*x-24*c^2*d^4*e*f*m-120*c^2*d^4*e*g*x+144*c^2*d^3*e^2*f*x+360*a^2*d
*e^4*g-720*a^2*e^5*f-480*a*b*d^2*e^3*g+720*a*b*d*e^4*f+360*a*c*d^3*e^2*g-480*a*c
*d^2*e^3*f+180*b^2*d^3*e^2*g-240*b^2*d^2*e^3*f-288*b*c*d^4*e*g+360*b*c*d^3*e^2*f
+120*c^2*d^5*g-144*c^2*d^4*e*f)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+
720)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(g*x + f)*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320257, size = 3197, normalized size = 10.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(g*x + f)*(e*x + d)^m,x, algorithm="fricas")

[Out]

(a^2*d*e^5*f*m^5 + (c^2*e^6*g*m^5 + 15*c^2*e^6*g*m^4 + 85*c^2*e^6*g*m^3 + 225*c^
2*e^6*g*m^2 + 274*c^2*e^6*g*m + 120*c^2*e^6*g)*x^6 + (144*c^2*e^6*f + 288*b*c*e^
6*g + (c^2*e^6*f + (c^2*d*e^5 + 2*b*c*e^6)*g)*m^5 + 2*(8*c^2*e^6*f + (5*c^2*d*e^
5 + 16*b*c*e^6)*g)*m^4 + 5*(19*c^2*e^6*f + (7*c^2*d*e^5 + 38*b*c*e^6)*g)*m^3 + 1
0*(26*c^2*e^6*f + (5*c^2*d*e^5 + 52*b*c*e^6)*g)*m^2 + 12*(27*c^2*e^6*f + 2*(c^2*
d*e^5 + 27*b*c*e^6)*g)*m)*x^5 - (a^2*d^2*e^4*g + 2*(a*b*d^2*e^4 - 10*a^2*d*e^5)*
f)*m^4 + (360*b*c*e^6*f + 180*(b^2 + 2*a*c)*e^6*g + ((c^2*d*e^5 + 2*b*c*e^6)*f +
(2*b*c*d*e^5 + (b^2 + 2*a*c)*e^6)*g)*m^5 + (2*(6*c^2*d*e^5 + 17*b*c*e^6)*f - (5
*c^2*d^2*e^4 - 24*b*c*d*e^5 - 17*(b^2 + 2*a*c)*e^6)*g)*m^4 + ((47*c^2*d*e^5 + 21
4*b*c*e^6)*f - (30*c^2*d^2*e^4 - 94*b*c*d*e^5 - 107*(b^2 + 2*a*c)*e^6)*g)*m^3 +
(2*(36*c^2*d*e^5 + 307*b*c*e^6)*f - (55*c^2*d^2*e^4 - 144*b*c*d*e^5 - 307*(b^2 +
2*a*c)*e^6)*g)*m^2 + 6*(6*(c^2*d*e^5 + 22*b*c*e^6)*f - (5*c^2*d^2*e^4 - 12*b*c*
d*e^5 - 66*(b^2 + 2*a*c)*e^6)*g)*m)*x^4 - ((36*a*b*d^2*e^4 - 155*a^2*d*e^5 - 2*(
b^2 + 2*a*c)*d^3*e^3)*f - 2*(2*a*b*d^3*e^3 - 9*a^2*d^2*e^4)*g)*m^3 + (480*a*b*e^
6*g + 240*(b^2 + 2*a*c)*e^6*f + ((2*b*c*d*e^5 + (b^2 + 2*a*c)*e^6)*f + (2*a*b*e^
6 + (b^2 + 2*a*c)*d*e^5)*g)*m^5 - 2*((2*c^2*d^2*e^4 - 14*b*c*d*e^5 - 9*(b^2 + 2*
a*c)*e^6)*f + (4*b*c*d^2*e^4 - 18*a*b*e^6 - 7*(b^2 + 2*a*c)*d*e^5)*g)*m^4 - ((36
*c^2*d^2*e^4 - 130*b*c*d*e^5 - 121*(b^2 + 2*a*c)*e^6)*f - (20*c^2*d^3*e^3 - 72*b
*c*d^2*e^4 + 242*a*b*e^6 + 65*(b^2 + 2*a*c)*d*e^5)*g)*m^3 - 4*((20*c^2*d^2*e^4 -
56*b*c*d*e^5 - 93*(b^2 + 2*a*c)*e^6)*f - (15*c^2*d^3*e^3 - 40*b*c*d^2*e^4 + 186
*a*b*e^6 + 28*(b^2 + 2*a*c)*d*e^5)*g)*m^2 - 4*((12*c^2*d^2*e^4 - 30*b*c*d*e^5 -
127*(b^2 + 2*a*c)*e^6)*f - (10*c^2*d^3*e^3 - 24*b*c*d^2*e^4 + 254*a*b*e^6 + 15*(
b^2 + 2*a*c)*d*e^5)*g)*m)*x^3 - (2*(6*b*c*d^4*e^2 + 119*a*b*d^2*e^4 - 290*a^2*d*
e^5 - 15*(b^2 + 2*a*c)*d^3*e^3)*f - (60*a*b*d^3*e^3 - 119*a^2*d^2*e^4 - 6*(b^2 +
2*a*c)*d^4*e^2)*g)*m^2 + (720*a*b*e^6*f + 360*a^2*e^6*g + ((2*a*b*e^6 + (b^2 +
2*a*c)*d*e^5)*f + (2*a*b*d*e^5 + a^2*e^6)*g)*m^5 - (2*(3*b*c*d^2*e^4 - 19*a*b*e^
6 - 8*(b^2 + 2*a*c)*d*e^5)*f - (32*a*b*d*e^5 + 19*a^2*e^6 - 3*(b^2 + 2*a*c)*d^2*
e^4)*g)*m^4 + ((12*c^2*d^3*e^3 - 72*b*c*d^2*e^4 + 274*a*b*e^6 + 89*(b^2 + 2*a*c)
*d*e^5)*f + (24*b*c*d^3*e^3 + 178*a*b*d*e^5 + 137*a^2*e^6 - 36*(b^2 + 2*a*c)*d^2
*e^4)*g)*m^3 + (2*(42*c^2*d^3*e^3 - 123*b*c*d^2*e^4 + 461*a*b*e^6 + 97*(b^2 + 2*
a*c)*d*e^5)*f - (60*c^2*d^4*e^2 - 168*b*c*d^3*e^3 - 388*a*b*d*e^5 - 461*a^2*e^6
+ 123*(b^2 + 2*a*c)*d^2*e^4)*g)*m^2 + 6*(2*(6*c^2*d^3*e^3 - 15*b*c*d^2*e^4 + 117
*a*b*e^6 + 10*(b^2 + 2*a*c)*d*e^5)*f - (10*c^2*d^4*e^2 - 24*b*c*d^3*e^3 - 40*a*b
*d*e^5 - 117*a^2*e^6 + 15*(b^2 + 2*a*c)*d^2*e^4)*g)*m)*x^2 + 24*(6*c^2*d^5*e - 1
5*b*c*d^4*e^2 - 30*a*b*d^2*e^4 + 30*a^2*d*e^5 + 10*(b^2 + 2*a*c)*d^3*e^3)*f - 12
*(10*c^2*d^6 - 24*b*c*d^5*e - 40*a*b*d^3*e^3 + 30*a^2*d^2*e^4 + 15*(b^2 + 2*a*c)
*d^4*e^2)*g + 2*(2*(6*c^2*d^5*e - 33*b*c*d^4*e^2 - 171*a*b*d^2*e^4 + 261*a^2*d*e
^5 + 37*(b^2 + 2*a*c)*d^3*e^3)*f + (24*b*c*d^5*e + 148*a*b*d^3*e^3 - 171*a^2*d^2
*e^4 - 33*(b^2 + 2*a*c)*d^4*e^2)*g)*m + (720*a^2*e^6*f + (a^2*d*e^5*g + (2*a*b*d
*e^5 + a^2*e^6)*f)*m^5 + 2*((18*a*b*d*e^5 + 10*a^2*e^6 - (b^2 + 2*a*c)*d^2*e^4)*
f - (2*a*b*d^2*e^4 - 9*a^2*d*e^5)*g)*m^4 + ((12*b*c*d^3*e^3 + 238*a*b*d*e^5 + 15
5*a^2*e^6 - 30*(b^2 + 2*a*c)*d^2*e^4)*f - (60*a*b*d^2*e^4 - 119*a^2*d*e^5 - 6*(b
^2 + 2*a*c)*d^3*e^3)*g)*m^3 - 2*(2*(6*c^2*d^4*e^2 - 33*b*c*d^3*e^3 - 171*a*b*d*e
^5 - 145*a^2*e^6 + 37*(b^2 + 2*a*c)*d^2*e^4)*f + (24*b*c*d^4*e^2 + 148*a*b*d^2*e
^4 - 171*a^2*d*e^5 - 33*(b^2 + 2*a*c)*d^3*e^3)*g)*m^2 - 12*((12*c^2*d^4*e^2 - 30
*b*c*d^3*e^3 - 60*a*b*d*e^5 - 87*a^2*e^6 + 20*(b^2 + 2*a*c)*d^2*e^4)*f - (10*c^2
*d^5*e - 24*b*c*d^4*e^2 - 40*a*b*d^2*e^4 + 30*a^2*d*e^5 + 15*(b^2 + 2*a*c)*d^3*e
^3)*g)*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 + 162
4*e^6*m^2 + 1764*e^6*m + 720*e^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(g*x+f)*(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280473, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(g*x + f)*(e*x + d)^m,x, algorithm="giac")

[Out]

Done