3.921 \(\int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx\)
Optimal. Leaf size=144 \[ \frac{(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{(d+e x)^{m+2} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac{(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac{c g (d+e x)^{m+4}}{e^4 (m+4)} \]
[Out]
((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)^(1 + m))/(e^4*(1 + m)) - ((c*d*(2
*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^4*(2 + m)) +
((c*e*f - 3*c*d*g + b*e*g)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (c*g*(d + e*x)^(4
+ m))/(e^4*(4 + m))
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Rubi [A] time = 0.290246, antiderivative size = 144, 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
\[ \frac{(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{(d+e x)^{m+2} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac{(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac{c g (d+e x)^{m+4}}{e^4 (m+4)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2),x]
[Out]
((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)^(1 + m))/(e^4*(1 + m)) - ((c*d*(2
*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^4*(2 + m)) +
((c*e*f - 3*c*d*g + b*e*g)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (c*g*(d + e*x)^(4
+ m))/(e^4*(4 + m))
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Rubi in Sympy [A] time = 57.3127, size = 141, normalized size = 0.98 \[ \frac{c g \left (d + e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} - \frac{\left (d + e x\right )^{m + 1} \left (d g - e f\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{4} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (a e^{2} g - 2 b d e g + b e^{2} f + 3 c d^{2} g - 2 c d e f\right )}{e^{4} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 3} \left (b e g - 3 c d g + c e f\right )}{e^{4} \left (m + 3\right )} \]
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),x)
[Out]
c*g*(d + e*x)**(m + 4)/(e**4*(m + 4)) - (d + e*x)**(m + 1)*(d*g - e*f)*(a*e**2 -
b*d*e + c*d**2)/(e**4*(m + 1)) + (d + e*x)**(m + 2)*(a*e**2*g - 2*b*d*e*g + b*e
**2*f + 3*c*d**2*g - 2*c*d*e*f)/(e**4*(m + 2)) + (d + e*x)**(m + 3)*(b*e*g - 3*c
*d*g + c*e*f)/(e**4*(m + 3))
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Mathematica [A] time = 0.363793, size = 190, normalized size = 1.32 \[ \frac{(d+e x)^{m+1} \left (e (m+4) \left (a e (m+3) (-d g+e f (m+2)+e g (m+1) x)+b \left (2 d^2 g-d e (f (m+3)+2 g (m+1) x)+e^2 (m+1) x (f (m+3)+g (m+2) x)\right )\right )-c \left (6 d^3 g-2 d^2 e (f (m+4)+3 g (m+1) x)+d e^2 (m+1) x (2 f (m+4)+3 g (m+2) x)-e^3 \left (m^2+3 m+2\right ) x^2 (f (m+4)+g (m+3) x)\right )\right )}{e^4 (m+1) (m+2) (m+3) (m+4)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2),x]
[Out]
((d + e*x)^(1 + m)*(-(c*(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))) + e*(4 + m)*(a*e*(3 + m)*(-(d*g) + e*f*(2 + m) + e*g*(1 + m)*x) + b
*(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)))))/(e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m))
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Maple [B] time = 0.009, size = 503, normalized size = 3.5 \[ -{\frac{ \left ( ex+d \right ) ^{1+m} \left ( -c{e}^{3}g{m}^{3}{x}^{3}-b{e}^{3}g{m}^{3}{x}^{2}-c{e}^{3}f{m}^{3}{x}^{2}-6\,c{e}^{3}g{m}^{2}{x}^{3}-a{e}^{3}g{m}^{3}x-b{e}^{3}f{m}^{3}x-7\,b{e}^{3}g{m}^{2}{x}^{2}+3\,cd{e}^{2}g{m}^{2}{x}^{2}-7\,c{e}^{3}f{m}^{2}{x}^{2}-11\,c{e}^{3}gm{x}^{3}-a{e}^{3}f{m}^{3}-8\,a{e}^{3}g{m}^{2}x+2\,bd{e}^{2}g{m}^{2}x-8\,b{e}^{3}f{m}^{2}x-14\,b{e}^{3}gm{x}^{2}+2\,cd{e}^{2}f{m}^{2}x+9\,cd{e}^{2}gm{x}^{2}-14\,c{e}^{3}fm{x}^{2}-6\,cg{x}^{3}{e}^{3}+ad{e}^{2}g{m}^{2}-9\,a{e}^{3}f{m}^{2}-19\,a{e}^{3}gmx+bd{e}^{2}f{m}^{2}+10\,bd{e}^{2}gmx-19\,b{e}^{3}fmx-8\,b{e}^{3}g{x}^{2}-6\,c{d}^{2}egmx+10\,cd{e}^{2}fmx+6\,cd{e}^{2}g{x}^{2}-8\,c{e}^{3}f{x}^{2}+7\,ad{e}^{2}gm-26\,a{e}^{3}fm-12\,a{e}^{3}gx-2\,b{d}^{2}egm+7\,bd{e}^{2}fm+8\,bd{e}^{2}gx-12\,b{e}^{3}fx-2\,c{d}^{2}efm-6\,c{d}^{2}egx+8\,cd{e}^{2}fx+12\,ad{e}^{2}g-24\,a{e}^{3}f-8\,b{d}^{2}eg+12\,bd{e}^{2}f+6\,c{d}^{3}g-8\,c{d}^{2}ef \right ) }{{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a),x)
[Out]
-(e*x+d)^(1+m)*(-c*e^3*g*m^3*x^3-b*e^3*g*m^3*x^2-c*e^3*f*m^3*x^2-6*c*e^3*g*m^2*x
^3-a*e^3*g*m^3*x-b*e^3*f*m^3*x-7*b*e^3*g*m^2*x^2+3*c*d*e^2*g*m^2*x^2-7*c*e^3*f*m
^2*x^2-11*c*e^3*g*m*x^3-a*e^3*f*m^3-8*a*e^3*g*m^2*x+2*b*d*e^2*g*m^2*x-8*b*e^3*f*
m^2*x-14*b*e^3*g*m*x^2+2*c*d*e^2*f*m^2*x+9*c*d*e^2*g*m*x^2-14*c*e^3*f*m*x^2-6*c*
e^3*g*x^3+a*d*e^2*g*m^2-9*a*e^3*f*m^2-19*a*e^3*g*m*x+b*d*e^2*f*m^2+10*b*d*e^2*g*
m*x-19*b*e^3*f*m*x-8*b*e^3*g*x^2-6*c*d^2*e*g*m*x+10*c*d*e^2*f*m*x+6*c*d*e^2*g*x^
2-8*c*e^3*f*x^2+7*a*d*e^2*g*m-26*a*e^3*f*m-12*a*e^3*g*x-2*b*d^2*e*g*m+7*b*d*e^2*
f*m+8*b*d*e^2*g*x-12*b*e^3*f*x-2*c*d^2*e*f*m-6*c*d^2*e*g*x+8*c*d*e^2*f*x+12*a*d*
e^2*g-24*a*e^3*f-8*b*d^2*e*g+12*b*d*e^2*f+6*c*d^3*g-8*c*d^2*e*f)/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)*(g*x + f)*(e*x + d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.312078, size = 828, normalized size = 5.75 \[ \frac{{\left (a d e^{3} f m^{3} +{\left (c e^{4} g m^{3} + 6 \, c e^{4} g m^{2} + 11 \, c e^{4} g m + 6 \, c e^{4} g\right )} x^{4} +{\left (8 \, c e^{4} f + 8 \, b e^{4} g +{\left (c e^{4} f +{\left (c d e^{3} + b e^{4}\right )} g\right )} m^{3} +{\left (7 \, c e^{4} f +{\left (3 \, c d e^{3} + 7 \, b e^{4}\right )} g\right )} m^{2} + 2 \,{\left (7 \, c e^{4} f +{\left (c d e^{3} + 7 \, b e^{4}\right )} g\right )} m\right )} x^{3} -{\left (a d^{2} e^{2} g +{\left (b d^{2} e^{2} - 9 \, a d e^{3}\right )} f\right )} m^{2} +{\left (12 \, b e^{4} f + 12 \, a e^{4} g +{\left ({\left (c d e^{3} + b e^{4}\right )} f +{\left (b d e^{3} + a e^{4}\right )} g\right )} m^{3} +{\left ({\left (5 \, c d e^{3} + 8 \, b e^{4}\right )} f -{\left (3 \, c d^{2} e^{2} - 5 \, b d e^{3} - 8 \, a e^{4}\right )} g\right )} m^{2} +{\left ({\left (4 \, c d e^{3} + 19 \, b e^{4}\right )} f -{\left (3 \, c d^{2} e^{2} - 4 \, b d e^{3} - 19 \, a e^{4}\right )} g\right )} m\right )} x^{2} + 4 \,{\left (2 \, c d^{3} e - 3 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} f - 2 \,{\left (3 \, c d^{4} - 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} g +{\left ({\left (2 \, c d^{3} e - 7 \, b d^{2} e^{2} + 26 \, a d e^{3}\right )} f +{\left (2 \, b d^{3} e - 7 \, a d^{2} e^{2}\right )} g\right )} m +{\left (24 \, a e^{4} f +{\left (a d e^{3} g +{\left (b d e^{3} + a e^{4}\right )} f\right )} m^{3} -{\left ({\left (2 \, c d^{2} e^{2} - 7 \, b d e^{3} - 9 \, a e^{4}\right )} f +{\left (2 \, b d^{2} e^{2} - 7 \, a d e^{3}\right )} g\right )} m^{2} - 2 \,{\left ({\left (4 \, c d^{2} e^{2} - 6 \, b d e^{3} - 13 \, a e^{4}\right )} f -{\left (3 \, c d^{3} e - 4 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} g\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(g*x + f)*(e*x + d)^m,x, algorithm="fricas")
[Out]
(a*d*e^3*f*m^3 + (c*e^4*g*m^3 + 6*c*e^4*g*m^2 + 11*c*e^4*g*m + 6*c*e^4*g)*x^4 +
(8*c*e^4*f + 8*b*e^4*g + (c*e^4*f + (c*d*e^3 + b*e^4)*g)*m^3 + (7*c*e^4*f + (3*c
*d*e^3 + 7*b*e^4)*g)*m^2 + 2*(7*c*e^4*f + (c*d*e^3 + 7*b*e^4)*g)*m)*x^3 - (a*d^2
*e^2*g + (b*d^2*e^2 - 9*a*d*e^3)*f)*m^2 + (12*b*e^4*f + 12*a*e^4*g + ((c*d*e^3 +
b*e^4)*f + (b*d*e^3 + a*e^4)*g)*m^3 + ((5*c*d*e^3 + 8*b*e^4)*f - (3*c*d^2*e^2 -
5*b*d*e^3 - 8*a*e^4)*g)*m^2 + ((4*c*d*e^3 + 19*b*e^4)*f - (3*c*d^2*e^2 - 4*b*d*
e^3 - 19*a*e^4)*g)*m)*x^2 + 4*(2*c*d^3*e - 3*b*d^2*e^2 + 6*a*d*e^3)*f - 2*(3*c*d
^4 - 4*b*d^3*e + 6*a*d^2*e^2)*g + ((2*c*d^3*e - 7*b*d^2*e^2 + 26*a*d*e^3)*f + (2
*b*d^3*e - 7*a*d^2*e^2)*g)*m + (24*a*e^4*f + (a*d*e^3*g + (b*d*e^3 + a*e^4)*f)*m
^3 - ((2*c*d^2*e^2 - 7*b*d*e^3 - 9*a*e^4)*f + (2*b*d^2*e^2 - 7*a*d*e^3)*g)*m^2 -
2*((4*c*d^2*e^2 - 6*b*d*e^3 - 13*a*e^4)*f - (3*c*d^3*e - 4*b*d^2*e^2 + 6*a*d*e^
3)*g)*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 = 16.9795, size = 5846, normalized size = 40.6 \[ \text{result too large to display} \]
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),x)
[Out]
Piecewise((d**m*(a*f*x + a*g*x**2/2 + b*f*x**2/2 + b*g*x**3/3 + c*f*x**3/3 + c*g
*x**4/4), Eq(e, 0)), (-2*a*d**2*e**3*f/(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*a*d*e**4*g*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) + a*e**5*g*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*d*e**4*f*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*b*d*e**4*g*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) + b*e**5*f*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*c*d**
5*g*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) + 2*c*d**5*g/(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*d**4*e*g*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) + 18*c*d**3*e**2*g*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) - 9*c*d**3*e**2*g*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) + 6*c*d**
2*e**3*g*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) - 9*c*d**2*e**3*g*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) + 2*c*d*e**4*f*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*d*e**3*f/(2*d**3*e**4 +
4*d**2*e**5*x + 2*d*e**6*x**2) + a*e**4*g*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2
*d*e**6*x**2) + 2*b*d**3*e*g*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**
6*x**2) + b*d**3*e*g/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 4*b*d**2*e*
*2*g*x*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 2*b*d*e**3*g
*x**2*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 2*b*d*e**3*g*
x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + b*e**4*f*x**2/(2*d**3*e**4
+ 4*d**2*e**5*x + 2*d*e**6*x**2) - 6*c*d**4*g*log(d/e + x)/(2*d**3*e**4 + 4*d**2
*e**5*x + 2*d*e**6*x**2) - 3*c*d**4*g/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x*
*2) + 2*c*d**3*e*f*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) +
c*d**3*e*f/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 12*c*d**3*e*g*x*log(d
/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 4*c*d**2*e**2*f*x*log(d/
e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 6*c*d**2*e**2*g*x**2*log(
d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 6*c*d**2*e**2*g*x**2/(2
*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 2*c*d*e**3*f*x**2*log(d/e + x)/(2*
d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 2*c*d*e**3*f*x**2/(2*d**3*e**4 + 4*
d**2*e**5*x + 2*d*e**6*x**2) + 2*c*d*e**3*g*x**3/(2*d**3*e**4 + 4*d**2*e**5*x +
2*d*e**6*x**2), Eq(m, -3)), (2*a*d*e**2*g*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2
*a*d*e**2*g/(2*d*e**4 + 2*e**5*x) - 2*a*e**3*f/(2*d*e**4 + 2*e**5*x) + 2*a*e**3*
g*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*b*d**2*e*g*log(d/e + x)/(2*d*e**4 + 2
*e**5*x) - 4*b*d**2*e*g/(2*d*e**4 + 2*e**5*x) + 2*b*d*e**2*f*log(d/e + x)/(2*d*e
**4 + 2*e**5*x) + 2*b*d*e**2*f/(2*d*e**4 + 2*e**5*x) - 4*b*d*e**2*g*x*log(d/e +
x)/(2*d*e**4 + 2*e**5*x) + 2*b*e**3*f*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*b
*e**3*g*x**2/(2*d*e**4 + 2*e**5*x) + 6*c*d**3*g*log(d/e + x)/(2*d*e**4 + 2*e**5*
x) + 6*c*d**3*g/(2*d*e**4 + 2*e**5*x) - 4*c*d**2*e*f*log(d/e + x)/(2*d*e**4 + 2*
e**5*x) - 4*c*d**2*e*f/(2*d*e**4 + 2*e**5*x) + 6*c*d**2*e*g*x*log(d/e + x)/(2*d*
e**4 + 2*e**5*x) - 4*c*d*e**2*f*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 3*c*d*e**
2*g*x**2/(2*d*e**4 + 2*e**5*x) + 2*c*e**3*f*x**2/(2*d*e**4 + 2*e**5*x) + c*e**3*
g*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (-a*d*g*log(d/e + x)/e**2 + a*f*log(d/
e + x)/e + a*g*x/e + b*d**2*g*log(d/e + x)/e**3 - b*d*f*log(d/e + x)/e**2 - b*d*
g*x/e**2 + b*f*x/e + b*g*x**2/(2*e) - c*d**3*g*log(d/e + x)/e**4 + c*d**2*f*log(
d/e + x)/e**3 + c*d**2*g*x/e**3 - c*d*f*x/e**2 - c*d*g*x**2/(2*e**2) + c*f*x**2/
(2*e) + c*g*x**3/(3*e), Eq(m, -1)), (-a*d**2*e**2*g*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*a*d**2*e**2*g*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*a*d*
*2*e**2*g*(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*d*e**3*f*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*d*e**3*f*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*d*e**3*f*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*d*e**3*f*(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*d*e**3*g*
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*a*d*e**3*g*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*a*d*e**3*g*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) + a*e**4*f*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*e**4*f*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*e
**4*f*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*e**4*f*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) + a*e**4*g*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*a*e**4*g*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*a*e**4*g*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
*a*e**4*g*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*b*d**3*e*g*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) + 8*b*d**3*e*g*(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*d**2*e**2*f*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*d**2*e**2*f*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*
d**2*e**2*f*(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*b*d**2*e**2*g*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) - 8*b*d**2*e**2*g*m*x*(d + e*x)**m/(e**4*m**4 + 1
0*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + b*d*e**3*f*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*d*e**3*
f*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*d*e**3*f*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*d*e**3*g*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) + 5*b*d*e**3*g*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) + 4*b*d*e**3*g*
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) + b*e**4*f*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*e**4*f*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*e**4*f*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*e**4*f*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*
e**4*g*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) + 7*b*e**4*g*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) + 14*b*e**4*g*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) + 8*b*e**4*g*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*d**4*g*
(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*d**3*e*f*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) + 8*c*d**3*e*f*(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*d**3*e*g*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*c*d**2*e**2*f*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) - 8*c*d**2*e**2*f*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*c*d**2*e**2*g*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) - 3*c*d**2*e**2*g*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) + c*d*e**3*f*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) + 5*c*d*e**3*
f*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) + 4*c*d*e**3*f*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) + c*d*e**3*g*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) + 3*c*d*e**3*g*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) + 2*c*d*e*
*3*g*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) + c*e**4*f*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) + 7*c*e**4*f*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) + 14*c*e**4*f*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) + 8*c*e**4*f*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)
+ c*e**4*g*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) + 6*c*e**4*g*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) + 11*c*e**4*g*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) + 6*c*e**4*g*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 = 0.265818, 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)*(g*x + f)*(e*x + d)^m,x, algorithm="giac")
[Out]
Done