3.1084 \(\int (e x)^m (A+B x) \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=155 \[ \frac{a^2 A (e x)^{m+1}}{e (m+1)}+\frac{(e x)^{m+4} \left (2 a B c+2 A b c+b^2 B\right )}{e^4 (m+4)}+\frac{(e x)^{m+3} \left (A \left (2 a c+b^2\right )+2 a b B\right )}{e^3 (m+3)}+\frac{a (e x)^{m+2} (a B+2 A b)}{e^2 (m+2)}+\frac{c (e x)^{m+5} (A c+2 b B)}{e^5 (m+5)}+\frac{B c^2 (e x)^{m+6}}{e^6 (m+6)} \]

[Out]

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

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Rubi [A]  time = 0.24377, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{a^2 A (e x)^{m+1}}{e (m+1)}+\frac{(e x)^{m+4} \left (2 a B c+2 A b c+b^2 B\right )}{e^4 (m+4)}+\frac{(e x)^{m+3} \left (A \left (2 a c+b^2\right )+2 a b B\right )}{e^3 (m+3)}+\frac{a (e x)^{m+2} (a B+2 A b)}{e^2 (m+2)}+\frac{c (e x)^{m+5} (A c+2 b B)}{e^5 (m+5)}+\frac{B c^2 (e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.4836, size = 146, normalized size = 0.94 \[ \frac{A a^{2} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B c^{2} \left (e x\right )^{m + 6}}{e^{6} \left (m + 6\right )} + \frac{a \left (e x\right )^{m + 2} \left (2 A b + B a\right )}{e^{2} \left (m + 2\right )} + \frac{c \left (e x\right )^{m + 5} \left (A c + 2 B b\right )}{e^{5} \left (m + 5\right )} + \frac{\left (e x\right )^{m + 3} \left (2 A a c + A b^{2} + 2 B a b\right )}{e^{3} \left (m + 3\right )} + \frac{\left (e x\right )^{m + 4} \left (2 A b c + 2 B a c + B b^{2}\right )}{e^{4} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(B*x+A)*(c*x**2+b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.368084, size = 117, normalized size = 0.75 \[ (e x)^m \left (\frac{a^2 A x}{m+1}+\frac{x^4 \left (2 a B c+2 A b c+b^2 B\right )}{m+4}+\frac{x^3 \left (2 a A c+2 a b B+A b^2\right )}{m+3}+\frac{a x^2 (a B+2 A b)}{m+2}+\frac{c x^5 (A c+2 b B)}{m+5}+\frac{B c^2 x^6}{m+6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 759, normalized size = 4.9 \[{\frac{ \left ( B{c}^{2}{m}^{5}{x}^{5}+A{c}^{2}{m}^{5}{x}^{4}+2\,Bbc{m}^{5}{x}^{4}+15\,B{c}^{2}{m}^{4}{x}^{5}+2\,Abc{m}^{5}{x}^{3}+16\,A{c}^{2}{m}^{4}{x}^{4}+2\,Bac{m}^{5}{x}^{3}+B{b}^{2}{m}^{5}{x}^{3}+32\,Bbc{m}^{4}{x}^{4}+85\,B{c}^{2}{m}^{3}{x}^{5}+2\,Aac{m}^{5}{x}^{2}+A{b}^{2}{m}^{5}{x}^{2}+34\,Abc{m}^{4}{x}^{3}+95\,A{c}^{2}{m}^{3}{x}^{4}+2\,Bab{m}^{5}{x}^{2}+34\,Bac{m}^{4}{x}^{3}+17\,B{b}^{2}{m}^{4}{x}^{3}+190\,Bbc{m}^{3}{x}^{4}+225\,B{c}^{2}{m}^{2}{x}^{5}+2\,Aab{m}^{5}x+36\,Aac{m}^{4}{x}^{2}+18\,A{b}^{2}{m}^{4}{x}^{2}+214\,Abc{m}^{3}{x}^{3}+260\,A{c}^{2}{m}^{2}{x}^{4}+B{a}^{2}{m}^{5}x+36\,Bab{m}^{4}{x}^{2}+214\,Bac{m}^{3}{x}^{3}+107\,B{b}^{2}{m}^{3}{x}^{3}+520\,Bbc{m}^{2}{x}^{4}+274\,B{c}^{2}m{x}^{5}+A{a}^{2}{m}^{5}+38\,Aab{m}^{4}x+242\,Aac{m}^{3}{x}^{2}+121\,A{b}^{2}{m}^{3}{x}^{2}+614\,Abc{m}^{2}{x}^{3}+324\,A{c}^{2}m{x}^{4}+19\,B{a}^{2}{m}^{4}x+242\,Bab{m}^{3}{x}^{2}+614\,Bac{m}^{2}{x}^{3}+307\,B{b}^{2}{m}^{2}{x}^{3}+648\,Bbcm{x}^{4}+120\,B{c}^{2}{x}^{5}+20\,A{a}^{2}{m}^{4}+274\,Aab{m}^{3}x+744\,Aac{m}^{2}{x}^{2}+372\,A{b}^{2}{m}^{2}{x}^{2}+792\,Abcm{x}^{3}+144\,A{c}^{2}{x}^{4}+137\,B{a}^{2}{m}^{3}x+744\,Bab{m}^{2}{x}^{2}+792\,Bacm{x}^{3}+396\,B{b}^{2}m{x}^{3}+288\,Bbc{x}^{4}+155\,A{a}^{2}{m}^{3}+922\,Aab{m}^{2}x+1016\,Aacm{x}^{2}+508\,A{b}^{2}m{x}^{2}+360\,Abc{x}^{3}+461\,B{a}^{2}{m}^{2}x+1016\,Babm{x}^{2}+360\,aBc{x}^{3}+180\,B{b}^{2}{x}^{3}+580\,A{a}^{2}{m}^{2}+1404\,Aabmx+480\,aAc{x}^{2}+240\,A{b}^{2}{x}^{2}+702\,B{a}^{2}mx+480\,Bab{x}^{2}+1044\,A{a}^{2}m+720\,aAbx+360\,{a}^{2}Bx+720\,A{a}^{2} \right ) x \left ( ex \right ) ^{m}}{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^2,x)

[Out]

x*(B*c^2*m^5*x^5+A*c^2*m^5*x^4+2*B*b*c*m^5*x^4+15*B*c^2*m^4*x^5+2*A*b*c*m^5*x^3+
16*A*c^2*m^4*x^4+2*B*a*c*m^5*x^3+B*b^2*m^5*x^3+32*B*b*c*m^4*x^4+85*B*c^2*m^3*x^5
+2*A*a*c*m^5*x^2+A*b^2*m^5*x^2+34*A*b*c*m^4*x^3+95*A*c^2*m^3*x^4+2*B*a*b*m^5*x^2
+34*B*a*c*m^4*x^3+17*B*b^2*m^4*x^3+190*B*b*c*m^3*x^4+225*B*c^2*m^2*x^5+2*A*a*b*m
^5*x+36*A*a*c*m^4*x^2+18*A*b^2*m^4*x^2+214*A*b*c*m^3*x^3+260*A*c^2*m^2*x^4+B*a^2
*m^5*x+36*B*a*b*m^4*x^2+214*B*a*c*m^3*x^3+107*B*b^2*m^3*x^3+520*B*b*c*m^2*x^4+27
4*B*c^2*m*x^5+A*a^2*m^5+38*A*a*b*m^4*x+242*A*a*c*m^3*x^2+121*A*b^2*m^3*x^2+614*A
*b*c*m^2*x^3+324*A*c^2*m*x^4+19*B*a^2*m^4*x+242*B*a*b*m^3*x^2+614*B*a*c*m^2*x^3+
307*B*b^2*m^2*x^3+648*B*b*c*m*x^4+120*B*c^2*x^5+20*A*a^2*m^4+274*A*a*b*m^3*x+744
*A*a*c*m^2*x^2+372*A*b^2*m^2*x^2+792*A*b*c*m*x^3+144*A*c^2*x^4+137*B*a^2*m^3*x+7
44*B*a*b*m^2*x^2+792*B*a*c*m*x^3+396*B*b^2*m*x^3+288*B*b*c*x^4+155*A*a^2*m^3+922
*A*a*b*m^2*x+1016*A*a*c*m*x^2+508*A*b^2*m*x^2+360*A*b*c*x^3+461*B*a^2*m^2*x+1016
*B*a*b*m*x^2+360*B*a*c*x^3+180*B*b^2*x^3+580*A*a^2*m^2+1404*A*a*b*m*x+480*A*a*c*
x^2+240*A*b^2*x^2+702*B*a^2*m*x+480*B*a*b*x^2+1044*A*a^2*m+720*A*a*b*x+360*B*a^2
*x+720*A*a^2)*(e*x)^m/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)

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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*(B*x + A)*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.346046, size = 774, normalized size = 4.99 \[ \frac{{\left ({\left (B c^{2} m^{5} + 15 \, B c^{2} m^{4} + 85 \, B c^{2} m^{3} + 225 \, B c^{2} m^{2} + 274 \, B c^{2} m + 120 \, B c^{2}\right )} x^{6} +{\left ({\left (2 \, B b c + A c^{2}\right )} m^{5} + 16 \,{\left (2 \, B b c + A c^{2}\right )} m^{4} + 95 \,{\left (2 \, B b c + A c^{2}\right )} m^{3} + 288 \, B b c + 144 \, A c^{2} + 260 \,{\left (2 \, B b c + A c^{2}\right )} m^{2} + 324 \,{\left (2 \, B b c + A c^{2}\right )} m\right )} x^{5} +{\left ({\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} m^{5} + 17 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} m^{4} + 107 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} m^{3} + 180 \, B b^{2} + 307 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} m^{2} + 360 \,{\left (B a + A b\right )} c + 396 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} m\right )} x^{4} +{\left ({\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{5} + 18 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{4} + 121 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{3} + 480 \, B a b + 240 \, A b^{2} + 480 \, A a c + 372 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m^{2} + 508 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} m\right )} x^{3} +{\left ({\left (B a^{2} + 2 \, A a b\right )} m^{5} + 19 \,{\left (B a^{2} + 2 \, A a b\right )} m^{4} + 137 \,{\left (B a^{2} + 2 \, A a b\right )} m^{3} + 360 \, B a^{2} + 720 \, A a b + 461 \,{\left (B a^{2} + 2 \, A a b\right )} m^{2} + 702 \,{\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{2} +{\left (A a^{2} m^{5} + 20 \, A a^{2} m^{4} + 155 \, A a^{2} m^{3} + 580 \, A a^{2} m^{2} + 1044 \, A a^{2} m + 720 \, A a^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*c^2*m^5 + 15*B*c^2*m^4 + 85*B*c^2*m^3 + 225*B*c^2*m^2 + 274*B*c^2*m + 120*B*
c^2)*x^6 + ((2*B*b*c + A*c^2)*m^5 + 16*(2*B*b*c + A*c^2)*m^4 + 95*(2*B*b*c + A*c
^2)*m^3 + 288*B*b*c + 144*A*c^2 + 260*(2*B*b*c + A*c^2)*m^2 + 324*(2*B*b*c + A*c
^2)*m)*x^5 + ((B*b^2 + 2*(B*a + A*b)*c)*m^5 + 17*(B*b^2 + 2*(B*a + A*b)*c)*m^4 +
107*(B*b^2 + 2*(B*a + A*b)*c)*m^3 + 180*B*b^2 + 307*(B*b^2 + 2*(B*a + A*b)*c)*m
^2 + 360*(B*a + A*b)*c + 396*(B*b^2 + 2*(B*a + A*b)*c)*m)*x^4 + ((2*B*a*b + A*b^
2 + 2*A*a*c)*m^5 + 18*(2*B*a*b + A*b^2 + 2*A*a*c)*m^4 + 121*(2*B*a*b + A*b^2 + 2
*A*a*c)*m^3 + 480*B*a*b + 240*A*b^2 + 480*A*a*c + 372*(2*B*a*b + A*b^2 + 2*A*a*c
)*m^2 + 508*(2*B*a*b + A*b^2 + 2*A*a*c)*m)*x^3 + ((B*a^2 + 2*A*a*b)*m^5 + 19*(B*
a^2 + 2*A*a*b)*m^4 + 137*(B*a^2 + 2*A*a*b)*m^3 + 360*B*a^2 + 720*A*a*b + 461*(B*
a^2 + 2*A*a*b)*m^2 + 702*(B*a^2 + 2*A*a*b)*m)*x^2 + (A*a^2*m^5 + 20*A*a^2*m^4 +
155*A*a^2*m^3 + 580*A*a^2*m^2 + 1044*A*a^2*m + 720*A*a^2)*x)*(e*x)^m/(m^6 + 21*m
^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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Sympy [A]  time = 7.24901, size = 4150, normalized size = 26.77 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(B*x+A)*(c*x**2+b*x+a)**2,x)

[Out]

Piecewise(((-A*a**2/(5*x**5) - A*a*b/(2*x**4) - 2*A*a*c/(3*x**3) - A*b**2/(3*x**
3) - A*b*c/x**2 - A*c**2/x - B*a**2/(4*x**4) - 2*B*a*b/(3*x**3) - B*a*c/x**2 - B
*b**2/(2*x**2) - 2*B*b*c/x + B*c**2*log(x))/e**6, Eq(m, -6)), ((-A*a**2/(4*x**4)
- 2*A*a*b/(3*x**3) - A*a*c/x**2 - A*b**2/(2*x**2) - 2*A*b*c/x + A*c**2*log(x) -
B*a**2/(3*x**3) - B*a*b/x**2 - 2*B*a*c/x - B*b**2/x + 2*B*b*c*log(x) + B*c**2*x
)/e**5, Eq(m, -5)), ((-A*a**2/(3*x**3) - A*a*b/x**2 - 2*A*a*c/x - A*b**2/x + 2*A
*b*c*log(x) + A*c**2*x - B*a**2/(2*x**2) - 2*B*a*b/x + 2*B*a*c*log(x) + B*b**2*l
og(x) + 2*B*b*c*x + B*c**2*x**2/2)/e**4, Eq(m, -4)), ((-A*a**2/(2*x**2) - 2*A*a*
b/x + 2*A*a*c*log(x) + A*b**2*log(x) + 2*A*b*c*x + A*c**2*x**2/2 - B*a**2/x + 2*
B*a*b*log(x) + 2*B*a*c*x + B*b**2*x + B*b*c*x**2 + B*c**2*x**3/3)/e**3, Eq(m, -3
)), ((-A*a**2/x + 2*A*a*b*log(x) + 2*A*a*c*x + A*b**2*x + A*b*c*x**2 + A*c**2*x*
*3/3 + B*a**2*log(x) + 2*B*a*b*x + B*a*c*x**2 + B*b**2*x**2/2 + 2*B*b*c*x**3/3 +
B*c**2*x**4/4)/e**2, Eq(m, -2)), ((A*a**2*log(x) + 2*A*a*b*x + A*a*c*x**2 + A*b
**2*x**2/2 + 2*A*b*c*x**3/3 + A*c**2*x**4/4 + B*a**2*x + B*a*b*x**2 + 2*B*a*c*x*
*3/3 + B*b**2*x**3/3 + B*b*c*x**4/2 + B*c**2*x**5/5)/e, Eq(m, -1)), (A*a**2*e**m
*m**5*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) +
20*A*a**2*e**m*m**4*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 +
1764*m + 720) + 155*A*a**2*e**m*m**3*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**
3 + 1624*m**2 + 1764*m + 720) + 580*A*a**2*e**m*m**2*x*x**m/(m**6 + 21*m**5 + 17
5*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1044*A*a**2*e**m*m*x*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 720*A*a**2*e**m*x*
x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*A*a*b
*e**m*m**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m
+ 720) + 38*A*a*b*e**m*m**4*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 16
24*m**2 + 1764*m + 720) + 274*A*a*b*e**m*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m*
*4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 922*A*a*b*e**m*m**2*x**2*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1404*A*a*b*e**m*m*
x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 72
0*A*a*b*e**m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*
m + 720) + 2*A*a*c*e**m*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1
624*m**2 + 1764*m + 720) + 36*A*a*c*e**m*m**4*x**3*x**m/(m**6 + 21*m**5 + 175*m*
*4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 242*A*a*c*e**m*m**3*x**3*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 744*A*a*c*e**m*m**
2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) +
1016*A*a*c*e**m*m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 +
1764*m + 720) + 480*A*a*c*e**m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 +
1624*m**2 + 1764*m + 720) + A*b**2*e**m*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m*
*4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 18*A*b**2*e**m*m**4*x**3*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 121*A*b**2*e**m*m*
*3*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) +
372*A*b**2*e**m*m**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**
2 + 1764*m + 720) + 508*A*b**2*e**m*m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735
*m**3 + 1624*m**2 + 1764*m + 720) + 240*A*b**2*e**m*x**3*x**m/(m**6 + 21*m**5 +
175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*A*b*c*e**m*m**5*x**4*x**m/(m
**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 34*A*b*c*e**m*
m**4*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720)
+ 214*A*b*c*e**m*m**3*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m*
*2 + 1764*m + 720) + 614*A*b*c*e**m*m**2*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) + 792*A*b*c*e**m*m*x**4*x**m/(m**6 + 21*m**
5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 360*A*b*c*e**m*x**4*x**m/(
m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + A*c**2*e**m*m
**5*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720)
+ 16*A*c**2*e**m*m**4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**
2 + 1764*m + 720) + 95*A*c**2*e**m*m**3*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 7
35*m**3 + 1624*m**2 + 1764*m + 720) + 260*A*c**2*e**m*m**2*x**5*x**m/(m**6 + 21*
m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 324*A*c**2*e**m*m*x**5*
x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 144*A*c
**2*e**m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m +
720) + B*a**2*e**m*m**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m
**2 + 1764*m + 720) + 19*B*a**2*e**m*m**4*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) + 137*B*a**2*e**m*m**3*x**2*x**m/(m**6 + 2
1*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 461*B*a**2*e**m*m**2*
x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 70
2*B*a**2*e**m*m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 17
64*m + 720) + 360*B*a**2*e**m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 +
1624*m**2 + 1764*m + 720) + 2*B*a*b*e**m*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m*
*4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 36*B*a*b*e**m*m**4*x**3*x**m/(m**6 +
21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 242*B*a*b*e**m*m**3
*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 7
44*B*a*b*e**m*m**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 +
1764*m + 720) + 1016*B*a*b*e**m*m*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m*
*3 + 1624*m**2 + 1764*m + 720) + 480*B*a*b*e**m*x**3*x**m/(m**6 + 21*m**5 + 175*
m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*B*a*c*e**m*m**5*x**4*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 34*B*a*c*e**m*m**4
*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2
14*B*a*c*e**m*m**3*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 +
1764*m + 720) + 614*B*a*c*e**m*m**2*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*
m**3 + 1624*m**2 + 1764*m + 720) + 792*B*a*c*e**m*m*x**4*x**m/(m**6 + 21*m**5 +
175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 360*B*a*c*e**m*x**4*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + B*b**2*e**m*m**5*
x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 17
*B*b**2*e**m*m**4*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 +
1764*m + 720) + 107*B*b**2*e**m*m**3*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*
m**3 + 1624*m**2 + 1764*m + 720) + 307*B*b**2*e**m*m**2*x**4*x**m/(m**6 + 21*m**
5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 396*B*b**2*e**m*m*x**4*x**
m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 180*B*b**2
*e**m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720
) + 2*B*b*c*e**m*m**5*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**
2 + 1764*m + 720) + 32*B*b*c*e**m*m**4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 73
5*m**3 + 1624*m**2 + 1764*m + 720) + 190*B*b*c*e**m*m**3*x**5*x**m/(m**6 + 21*m*
*5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 520*B*b*c*e**m*m**2*x**5*
x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 648*B*b
*c*e**m*m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m +
720) + 288*B*b*c*e**m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m*
*2 + 1764*m + 720) + B*c**2*e**m*m**5*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735
*m**3 + 1624*m**2 + 1764*m + 720) + 15*B*c**2*e**m*m**4*x**6*x**m/(m**6 + 21*m**
5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 85*B*c**2*e**m*m**3*x**6*x
**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 225*B*c*
*2*e**m*m**2*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*
m + 720) + 274*B*c**2*e**m*m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1
624*m**2 + 1764*m + 720) + 120*B*c**2*e**m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4
+ 735*m**3 + 1624*m**2 + 1764*m + 720), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.285386, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done