3.1656 \(\int (b+2 c x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx\)
Optimal. Leaf size=143 \[ \frac{(d+e x)^{m+2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 (m+2)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{3 c (2 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac{2 c^2 (d+e x)^{m+4}}{e^4 (m+4)} \]
[Out]
-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((6
*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (3*
c*(2*c*d - b*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (2*c^2*(d + e*x)^(4 + m))/(e^
4*(4 + m))
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Rubi [A] time = 0.210851, antiderivative size = 143, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042
\[ \frac{(d+e x)^{m+2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 (m+2)}-\frac{(2 c d-b e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{3 c (2 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac{2 c^2 (d+e x)^{m+4}}{e^4 (m+4)} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2),x]
[Out]
-(((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((6
*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (3*
c*(2*c*d - b*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (2*c^2*(d + e*x)^(4 + m))/(e^
4*(4 + m))
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Rubi in Sympy [A] time = 46.223, size = 133, normalized size = 0.93 \[ \frac{2 c^{2} \left (d + e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} + \frac{3 c \left (d + e x\right )^{m + 3} \left (b e - 2 c d\right )}{e^{4} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 1} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{4} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{e^{4} \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)*(e*x+d)**m*(c*x**2+b*x+a),x)
[Out]
2*c**2*(d + e*x)**(m + 4)/(e**4*(m + 4)) + 3*c*(d + e*x)**(m + 3)*(b*e - 2*c*d)/
(e**4*(m + 3)) + (d + e*x)**(m + 1)*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)/(e**
4*(m + 1)) + (d + e*x)**(m + 2)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**
2)/(e**4*(m + 2))
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Mathematica [A] time = 0.231576, size = 180, normalized size = 1.26 \[ \frac{(d+e x)^{m+1} \left (c e (m+4) \left (2 a e (m+3) (e (m+1) x-d)+3 b \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )+b e^2 \left (m^2+7 m+12\right ) (a e (m+2)-b d+b e (m+1) x)-2 c^2 \left (6 d^3-6 d^2 e (m+1) x+3 d e^2 \left (m^2+3 m+2\right ) x^2-e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )}{e^4 (m+1) (m+2) (m+3) (m+4)} \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x)*(d + e*x)^m*(a + b*x + c*x^2),x]
[Out]
((d + e*x)^(1 + m)*(b*e^2*(12 + 7*m + m^2)*(-(b*d) + a*e*(2 + m) + b*e*(1 + m)*x
) - 2*c^2*(6*d^3 - 6*d^2*e*(1 + m)*x + 3*d*e^2*(2 + 3*m + m^2)*x^2 - e^3*(6 + 11
*m + 6*m^2 + m^3)*x^3) + c*e*(4 + m)*(2*a*e*(3 + m)*(-d + e*(1 + m)*x) + 3*b*(2*
d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2))))/(e^4*(1 + m)*(2 + m)*(3 + m)
*(4 + m))
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Maple [B] time = 0.009, size = 424, normalized size = 3. \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( 2\,{c}^{2}{e}^{3}{m}^{3}{x}^{3}+3\,bc{e}^{3}{m}^{3}{x}^{2}+12\,{c}^{2}{e}^{3}{m}^{2}{x}^{3}+2\,ac{e}^{3}{m}^{3}x+{b}^{2}{e}^{3}{m}^{3}x+21\,bc{e}^{3}{m}^{2}{x}^{2}-6\,{c}^{2}d{e}^{2}{m}^{2}{x}^{2}+22\,{c}^{2}{e}^{3}m{x}^{3}+ab{e}^{3}{m}^{3}+16\,ac{e}^{3}{m}^{2}x+8\,{b}^{2}{e}^{3}{m}^{2}x-6\,bcd{e}^{2}{m}^{2}x+42\,bc{e}^{3}m{x}^{2}-18\,{c}^{2}d{e}^{2}m{x}^{2}+12\,{x}^{3}{c}^{2}{e}^{3}+9\,ab{e}^{3}{m}^{2}-2\,acd{e}^{2}{m}^{2}+38\,ac{e}^{3}mx-{b}^{2}d{e}^{2}{m}^{2}+19\,{b}^{2}{e}^{3}mx-30\,bcd{e}^{2}mx+24\,{x}^{2}bc{e}^{3}+12\,{c}^{2}{d}^{2}emx-12\,{x}^{2}{c}^{2}d{e}^{2}+26\,ab{e}^{3}m-14\,acd{e}^{2}m+24\,xac{e}^{3}-7\,{b}^{2}d{e}^{2}m+12\,x{b}^{2}{e}^{3}+6\,bc{d}^{2}em-24\,xbcd{e}^{2}+12\,x{c}^{2}{d}^{2}e+24\,ab{e}^{3}-24\,ad{e}^{2}c-12\,{b}^{2}d{e}^{2}+24\,bc{d}^{2}e-12\,{c}^{2}{d}^{3} \right ) }{{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(e*x+d)^m*(c*x^2+b*x+a),x)
[Out]
(e*x+d)^(1+m)*(2*c^2*e^3*m^3*x^3+3*b*c*e^3*m^3*x^2+12*c^2*e^3*m^2*x^3+2*a*c*e^3*
m^3*x+b^2*e^3*m^3*x+21*b*c*e^3*m^2*x^2-6*c^2*d*e^2*m^2*x^2+22*c^2*e^3*m*x^3+a*b*
e^3*m^3+16*a*c*e^3*m^2*x+8*b^2*e^3*m^2*x-6*b*c*d*e^2*m^2*x+42*b*c*e^3*m*x^2-18*c
^2*d*e^2*m*x^2+12*c^2*e^3*x^3+9*a*b*e^3*m^2-2*a*c*d*e^2*m^2+38*a*c*e^3*m*x-b^2*d
*e^2*m^2+19*b^2*e^3*m*x-30*b*c*d*e^2*m*x+24*b*c*e^3*x^2+12*c^2*d^2*e*m*x-12*c^2*
d*e^2*x^2+26*a*b*e^3*m-14*a*c*d*e^2*m+24*a*c*e^3*x-7*b^2*d*e^2*m+12*b^2*e^3*x+6*
b*c*d^2*e*m-24*b*c*d*e^2*x+12*c^2*d^2*e*x+24*a*b*e^3-24*a*c*d*e^2-12*b^2*d*e^2+2
4*b*c*d^2*e-12*c^2*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)
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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*c*x + b)*(e*x + d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.289631, size = 690, normalized size = 4.83 \[ \frac{{\left (a b d e^{3} m^{3} - 12 \, c^{2} d^{4} + 24 \, b c d^{3} e + 24 \, a b d e^{3} - 12 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 2 \,{\left (c^{2} e^{4} m^{3} + 6 \, c^{2} e^{4} m^{2} + 11 \, c^{2} e^{4} m + 6 \, c^{2} e^{4}\right )} x^{4} +{\left (24 \, b c e^{4} +{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} m^{3} + 3 \,{\left (2 \, c^{2} d e^{3} + 7 \, b c e^{4}\right )} m^{2} + 2 \,{\left (2 \, c^{2} d e^{3} + 21 \, b c e^{4}\right )} m\right )} x^{3} +{\left (9 \, a b d e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} m^{2} +{\left (12 \,{\left (b^{2} + 2 \, a c\right )} e^{4} +{\left (3 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} m^{3} -{\left (6 \, c^{2} d^{2} e^{2} - 15 \, b c d e^{3} - 8 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} m^{2} -{\left (6 \, c^{2} d^{2} e^{2} - 12 \, b c d e^{3} - 19 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} m\right )} x^{2} +{\left (6 \, b c d^{3} e + 26 \, a b d e^{3} - 7 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} m +{\left (24 \, a b e^{4} +{\left (a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} m^{3} -{\left (6 \, b c d^{2} e^{2} - 9 \, a b e^{4} - 7 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} m^{2} + 2 \,{\left (6 \, c^{2} d^{3} e - 12 \, b c d^{2} e^{2} + 13 \, a b e^{4} + 6 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(2*c*x + b)*(e*x + d)^m,x, algorithm="fricas")
[Out]
(a*b*d*e^3*m^3 - 12*c^2*d^4 + 24*b*c*d^3*e + 24*a*b*d*e^3 - 12*(b^2 + 2*a*c)*d^2
*e^2 + 2*(c^2*e^4*m^3 + 6*c^2*e^4*m^2 + 11*c^2*e^4*m + 6*c^2*e^4)*x^4 + (24*b*c*
e^4 + (2*c^2*d*e^3 + 3*b*c*e^4)*m^3 + 3*(2*c^2*d*e^3 + 7*b*c*e^4)*m^2 + 2*(2*c^2
*d*e^3 + 21*b*c*e^4)*m)*x^3 + (9*a*b*d*e^3 - (b^2 + 2*a*c)*d^2*e^2)*m^2 + (12*(b
^2 + 2*a*c)*e^4 + (3*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*m^3 - (6*c^2*d^2*e^2 - 15*b*
c*d*e^3 - 8*(b^2 + 2*a*c)*e^4)*m^2 - (6*c^2*d^2*e^2 - 12*b*c*d*e^3 - 19*(b^2 + 2
*a*c)*e^4)*m)*x^2 + (6*b*c*d^3*e + 26*a*b*d*e^3 - 7*(b^2 + 2*a*c)*d^2*e^2)*m + (
24*a*b*e^4 + (a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*m^3 - (6*b*c*d^2*e^2 - 9*a*b*e^4 -
7*(b^2 + 2*a*c)*d*e^3)*m^2 + 2*(6*c^2*d^3*e - 12*b*c*d^2*e^2 + 13*a*b*e^4 + 6*(b
^2 + 2*a*c)*d*e^3)*m)*x)*(e*x + d)^m/(e^4*m^4 + 10*e^4*m^3 + 35*e^4*m^2 + 50*e^4
*m + 24*e^4)
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Sympy [A] time = 13.8671, size = 4748, normalized size = 33.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(e*x+d)**m*(c*x**2+b*x+a),x)
[Out]
Piecewise((d**m*(a*b*x + a*c*x**2 + b**2*x**2/2 + b*c*x**3 + c**2*x**4/2), Eq(e,
0)), (-2*a*b*d**2*e**3/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d*
*2*e**7*x**3) + 6*a*c*d*e**4*x**2/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x
**2 + 6*d**2*e**7*x**3) + 2*a*c*e**5*x**3/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**
3*e**6*x**2 + 6*d**2*e**7*x**3) + 3*b**2*d*e**4*x**2/(6*d**5*e**4 + 18*d**4*e**5
*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + b**2*e**5*x**3/(6*d**5*e**4 + 18*d*
*4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 6*b*c*d*e**4*x**3/(6*d**5*e*
*4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 12*c**2*d**5*log(d
/e + x)/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) +
4*c**2*d**5/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3
) + 36*c**2*d**4*e*x*log(d/e + x)/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x
**2 + 6*d**2*e**7*x**3) + 36*c**2*d**3*e**2*x**2*log(d/e + x)/(6*d**5*e**4 + 18*
d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) - 18*c**2*d**3*e**2*x**2/(6*
d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 12*c**2*d**
2*e**3*x**3*log(d/e + x)/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d
**2*e**7*x**3) - 18*c**2*d**2*e**3*x**3/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*
e**6*x**2 + 6*d**2*e**7*x**3), Eq(m, -4)), (-a*b*d*e**3/(2*d**3*e**4 + 4*d**2*e*
*5*x + 2*d*e**6*x**2) + 2*a*c*e**4*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*
x**2) + b**2*e**4*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 6*b*c*d**
3*e*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 3*b*c*d**3*e/(2
*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 12*b*c*d**2*e**2*x*log(d/e + x)/(2
*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 6*b*c*d*e**3*x**2*log(d/e + x)/(2*
d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 6*b*c*d*e**3*x**2/(2*d**3*e**4 + 4*
d**2*e**5*x + 2*d*e**6*x**2) - 12*c**2*d**4*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e
**5*x + 2*d*e**6*x**2) - 6*c**2*d**4/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**
2) - 24*c**2*d**3*e*x*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2)
- 12*c**2*d**2*e**2*x**2*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x
**2) + 12*c**2*d**2*e**2*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 4*
c**2*d*e**3*x**3/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2), Eq(m, -3)), (-a*
b*e**3/(d*e**4 + e**5*x) + 2*a*c*d*e**2*log(d/e + x)/(d*e**4 + e**5*x) + 2*a*c*d
*e**2/(d*e**4 + e**5*x) + 2*a*c*e**3*x*log(d/e + x)/(d*e**4 + e**5*x) + b**2*d*e
**2*log(d/e + x)/(d*e**4 + e**5*x) + b**2*d*e**2/(d*e**4 + e**5*x) + b**2*e**3*x
*log(d/e + x)/(d*e**4 + e**5*x) - 6*b*c*d**2*e*log(d/e + x)/(d*e**4 + e**5*x) -
6*b*c*d**2*e/(d*e**4 + e**5*x) - 6*b*c*d*e**2*x*log(d/e + x)/(d*e**4 + e**5*x) +
3*b*c*e**3*x**2/(d*e**4 + e**5*x) + 6*c**2*d**3*log(d/e + x)/(d*e**4 + e**5*x)
+ 6*c**2*d**3/(d*e**4 + e**5*x) + 6*c**2*d**2*e*x*log(d/e + x)/(d*e**4 + e**5*x)
- 3*c**2*d*e**2*x**2/(d*e**4 + e**5*x) + c**2*e**3*x**3/(d*e**4 + e**5*x), Eq(m
, -2)), (a*b*log(d/e + x)/e - 2*a*c*d*log(d/e + x)/e**2 + 2*a*c*x/e - b**2*d*log
(d/e + x)/e**2 + b**2*x/e + 3*b*c*d**2*log(d/e + x)/e**3 - 3*b*c*d*x/e**2 + 3*b*
c*x**2/(2*e) - 2*c**2*d**3*log(d/e + x)/e**4 + 2*c**2*d**2*x/e**3 - c**2*d*x**2/
e**2 + 2*c**2*x**3/(3*e), Eq(m, -1)), (a*b*d*e**3*m**3*(d + e*x)**m/(e**4*m**4 +
10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 9*a*b*d*e**3*m**2*(d + e*x
)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*a*b*d*
e**3*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e*
*4) + 24*a*b*d*e**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) + a*b*e**4*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e
**4*m**2 + 50*e**4*m + 24*e**4) + 9*a*b*e**4*m**2*x*(d + e*x)**m/(e**4*m**4 + 10
*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*a*b*e**4*m*x*(d + e*x)**m/
(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*a*b*e**4*x*
(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 2
*a*c*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) - 14*a*c*d**2*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 3
5*e**4*m**2 + 50*e**4*m + 24*e**4) - 24*a*c*d**2*e**2*(d + e*x)**m/(e**4*m**4 +
10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*a*c*d*e**3*m**3*x*(d + e*
x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 14*a*c*d
*e**3*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m +
24*e**4) + 24*a*c*d*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m
**2 + 50*e**4*m + 24*e**4) + 2*a*c*e**4*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e
**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 16*a*c*e**4*m**2*x**2*(d + e*x)
**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 38*a*c*e**
4*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*
e**4) + 24*a*c*e**4*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 +
50*e**4*m + 24*e**4) - b**2*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m*
*3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 7*b**2*d**2*e**2*m*(d + e*x)**m/(e**4
*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 12*b**2*d**2*e**2*(
d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + b*
*2*d*e**3*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4
*m + 24*e**4) + 7*b**2*d*e**3*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35
*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*b**2*d*e**3*m*x*(d + e*x)**m/(e**4*m**4 +
10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + b**2*e**4*m**3*x**2*(d + e
*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*b**2*
e**4*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m
+ 24*e**4) + 19*b**2*e**4*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e*
*4*m**2 + 50*e**4*m + 24*e**4) + 12*b**2*e**4*x**2*(d + e*x)**m/(e**4*m**4 + 10*
e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*b*c*d**3*e*m*(d + e*x)**m/(e
**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*b*c*d**3*e*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 6*b
*c*d**2*e**2*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) - 24*b*c*d**2*e**2*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 +
35*e**4*m**2 + 50*e**4*m + 24*e**4) + 3*b*c*d*e**3*m**3*x**2*(d + e*x)**m/(e**4
*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 15*b*c*d*e**3*m**2*
x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4
) + 12*b*c*d*e**3*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 +
50*e**4*m + 24*e**4) + 3*b*c*e**4*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m
**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 21*b*c*e**4*m**2*x**3*(d + e*x)**m/(
e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 42*b*c*e**4*m*x
**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4)
+ 24*b*c*e**4*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) - 12*c**2*d**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4
*m**2 + 50*e**4*m + 24*e**4) + 12*c**2*d**3*e*m*x*(d + e*x)**m/(e**4*m**4 + 10*e
**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 6*c**2*d**2*e**2*m**2*x**2*(d +
e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 6*c**
2*d**2*e**2*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e*
*4*m + 24*e**4) + 2*c**2*d*e**3*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3
+ 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*c**2*d*e**3*m**2*x**3*(d + e*x)**m/(e
**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 4*c**2*d*e**3*m*
x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4
) + 2*c**2*e**4*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2
+ 50*e**4*m + 24*e**4) + 12*c**2*e**4*m**2*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**
4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 22*c**2*e**4*m*x**4*(d + e*x)**m/
(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*c**2*e**4*x
**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4)
, True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.281209, 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*c*x + b)*(e*x + d)^m,x, algorithm="giac")
[Out]
Done