3.807 \(\int (d+e x) (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx\)
Optimal. Leaf size=146 \[ \frac{(f+g x)^{n+2} \left (a e g^2+c \left (2 d^2 g^2-6 d e f g+3 e^2 f^2\right )\right )}{g^4 (n+2)}-\frac{(e f-d g) (f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^4 (n+1)}-\frac{3 c e (e f-d g) (f+g x)^{n+3}}{g^4 (n+3)}+\frac{c e^2 (f+g x)^{n+4}}{g^4 (n+4)} \]
[Out]
-(((e*f - d*g)*(a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^4*(1 + n))) + (
(a*e*g^2 + c*(3*e^2*f^2 - 6*d*e*f*g + 2*d^2*g^2))*(f + g*x)^(2 + n))/(g^4*(2 + n
)) - (3*c*e*(e*f - d*g)*(f + g*x)^(3 + n))/(g^4*(3 + n)) + (c*e^2*(f + g*x)^(4 +
n))/(g^4*(4 + n))
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Rubi [A] time = 0.258078, antiderivative size = 146, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038
\[ \frac{(f+g x)^{n+2} \left (a e g^2+c \left (2 d^2 g^2-6 d e f g+3 e^2 f^2\right )\right )}{g^4 (n+2)}-\frac{(e f-d g) (f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^4 (n+1)}-\frac{3 c e (e f-d g) (f+g x)^{n+3}}{g^4 (n+3)}+\frac{c e^2 (f+g x)^{n+4}}{g^4 (n+4)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]
[Out]
-(((e*f - d*g)*(a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^4*(1 + n))) + (
(a*e*g^2 + c*(3*e^2*f^2 - 6*d*e*f*g + 2*d^2*g^2))*(f + g*x)^(2 + n))/(g^4*(2 + n
)) - (3*c*e*(e*f - d*g)*(f + g*x)^(3 + n))/(g^4*(3 + n)) + (c*e^2*(f + g*x)^(4 +
n))/(g^4*(4 + n))
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Rubi in Sympy [A] time = 54.5194, size = 141, normalized size = 0.97 \[ \frac{c e^{2} \left (f + g x\right )^{n + 4}}{g^{4} \left (n + 4\right )} + \frac{3 c e \left (f + g x\right )^{n + 3} \left (d g - e f\right )}{g^{4} \left (n + 3\right )} + \frac{\left (f + g x\right )^{n + 1} \left (d g - e f\right ) \left (a g^{2} - 2 c d f g + c e f^{2}\right )}{g^{4} \left (n + 1\right )} + \frac{\left (f + g x\right )^{n + 2} \left (a e g^{2} + 2 c d^{2} g^{2} - 6 c d e f g + 3 c e^{2} f^{2}\right )}{g^{4} \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(g*x+f)**n*(c*e*x**2+2*c*d*x+a),x)
[Out]
c*e**2*(f + g*x)**(n + 4)/(g**4*(n + 4)) + 3*c*e*(f + g*x)**(n + 3)*(d*g - e*f)/
(g**4*(n + 3)) + (f + g*x)**(n + 1)*(d*g - e*f)*(a*g**2 - 2*c*d*f*g + c*e*f**2)/
(g**4*(n + 1)) + (f + g*x)**(n + 2)*(a*e*g**2 + 2*c*d**2*g**2 - 6*c*d*e*f*g + 3*
c*e**2*f**2)/(g**4*(n + 2))
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Mathematica [A] time = 0.307104, size = 190, normalized size = 1.3 \[ \frac{(f+g x)^{n+1} \left (a g^2 \left (n^2+7 n+12\right ) (d g (n+2)-e f+e g (n+1) x)+c \left (2 d^2 g^2 \left (n^2+7 n+12\right ) (g (n+1) x-f)+3 d e g (n+4) \left (2 f^2-2 f g (n+1) x+g^2 \left (n^2+3 n+2\right ) x^2\right )+e^2 \left (-\left (6 f^3-6 f^2 g (n+1) x+3 f g^2 \left (n^2+3 n+2\right ) x^2-g^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )\right )\right )\right )}{g^4 (n+1) (n+2) (n+3) (n+4)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]
[Out]
((f + g*x)^(1 + n)*(a*g^2*(12 + 7*n + n^2)*(-(e*f) + d*g*(2 + n) + e*g*(1 + n)*x
) + c*(2*d^2*g^2*(12 + 7*n + n^2)*(-f + g*(1 + n)*x) + 3*d*e*g*(4 + n)*(2*f^2 -
2*f*g*(1 + n)*x + g^2*(2 + 3*n + n^2)*x^2) - e^2*(6*f^3 - 6*f^2*g*(1 + n)*x + 3*
f*g^2*(2 + 3*n + n^2)*x^2 - g^3*(6 + 11*n + 6*n^2 + n^3)*x^3))))/(g^4*(1 + n)*(2
+ n)*(3 + n)*(4 + n))
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Maple [B] time = 0.009, size = 449, normalized size = 3.1 \[{\frac{ \left ( gx+f \right ) ^{1+n} \left ( c{e}^{2}{g}^{3}{n}^{3}{x}^{3}+3\,cde{g}^{3}{n}^{3}{x}^{2}+6\,c{e}^{2}{g}^{3}{n}^{2}{x}^{3}+2\,c{d}^{2}{g}^{3}{n}^{3}x+21\,cde{g}^{3}{n}^{2}{x}^{2}-3\,c{e}^{2}f{g}^{2}{n}^{2}{x}^{2}+11\,c{e}^{2}{g}^{3}n{x}^{3}+ae{g}^{3}{n}^{3}x+16\,c{d}^{2}{g}^{3}{n}^{2}x-6\,cdef{g}^{2}{n}^{2}x+42\,cde{g}^{3}n{x}^{2}-9\,c{e}^{2}f{g}^{2}n{x}^{2}+6\,c{e}^{2}{x}^{3}{g}^{3}+ad{g}^{3}{n}^{3}+8\,ae{g}^{3}{n}^{2}x-2\,c{d}^{2}f{g}^{2}{n}^{2}+38\,c{d}^{2}{g}^{3}nx-30\,cdef{g}^{2}nx+24\,cde{g}^{3}{x}^{2}+6\,c{e}^{2}{f}^{2}gnx-6\,c{e}^{2}f{g}^{2}{x}^{2}+9\,ad{g}^{3}{n}^{2}-aef{g}^{2}{n}^{2}+19\,ae{g}^{3}nx-14\,c{d}^{2}f{g}^{2}n+24\,c{d}^{2}{g}^{3}x+6\,cde{f}^{2}gn-24\,cdef{g}^{2}x+6\,c{e}^{2}{f}^{2}gx+26\,ad{g}^{3}n-7\,aef{g}^{2}n+12\,ae{g}^{3}x-24\,c{d}^{2}f{g}^{2}+24\,cde{f}^{2}g-6\,c{e}^{2}{f}^{3}+24\,ad{g}^{3}-12\,aef{g}^{2} \right ) }{{g}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x)
[Out]
(g*x+f)^(1+n)*(c*e^2*g^3*n^3*x^3+3*c*d*e*g^3*n^3*x^2+6*c*e^2*g^3*n^2*x^3+2*c*d^2
*g^3*n^3*x+21*c*d*e*g^3*n^2*x^2-3*c*e^2*f*g^2*n^2*x^2+11*c*e^2*g^3*n*x^3+a*e*g^3
*n^3*x+16*c*d^2*g^3*n^2*x-6*c*d*e*f*g^2*n^2*x+42*c*d*e*g^3*n*x^2-9*c*e^2*f*g^2*n
*x^2+6*c*e^2*g^3*x^3+a*d*g^3*n^3+8*a*e*g^3*n^2*x-2*c*d^2*f*g^2*n^2+38*c*d^2*g^3*
n*x-30*c*d*e*f*g^2*n*x+24*c*d*e*g^3*x^2+6*c*e^2*f^2*g*n*x-6*c*e^2*f*g^2*x^2+9*a*
d*g^3*n^2-a*e*f*g^2*n^2+19*a*e*g^3*n*x-14*c*d^2*f*g^2*n+24*c*d^2*g^3*x+6*c*d*e*f
^2*g*n-24*c*d*e*f*g^2*x+6*c*e^2*f^2*g*x+26*a*d*g^3*n-7*a*e*f*g^2*n+12*a*e*g^3*x-
24*c*d^2*f*g^2+24*c*d*e*f^2*g-6*c*e^2*f^3+24*a*d*g^3-12*a*e*f*g^2)/g^4/(n^4+10*n
^3+35*n^2+50*n+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*e*x^2 + 2*c*d*x + a)*(e*x + d)*(g*x + f)^n,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.301461, size = 741, normalized size = 5.08 \[ \frac{{\left (a d f g^{3} n^{3} - 6 \, c e^{2} f^{4} + 24 \, c d e f^{3} g + 24 \, a d f g^{3} - 12 \,{\left (2 \, c d^{2} + a e\right )} f^{2} g^{2} +{\left (c e^{2} g^{4} n^{3} + 6 \, c e^{2} g^{4} n^{2} + 11 \, c e^{2} g^{4} n + 6 \, c e^{2} g^{4}\right )} x^{4} +{\left (24 \, c d e g^{4} +{\left (c e^{2} f g^{3} + 3 \, c d e g^{4}\right )} n^{3} + 3 \,{\left (c e^{2} f g^{3} + 7 \, c d e g^{4}\right )} n^{2} + 2 \,{\left (c e^{2} f g^{3} + 21 \, c d e g^{4}\right )} n\right )} x^{3} +{\left (9 \, a d f g^{3} -{\left (2 \, c d^{2} + a e\right )} f^{2} g^{2}\right )} n^{2} +{\left (12 \,{\left (2 \, c d^{2} + a e\right )} g^{4} +{\left (3 \, c d e f g^{3} +{\left (2 \, c d^{2} + a e\right )} g^{4}\right )} n^{3} -{\left (3 \, c e^{2} f^{2} g^{2} - 15 \, c d e f g^{3} - 8 \,{\left (2 \, c d^{2} + a e\right )} g^{4}\right )} n^{2} -{\left (3 \, c e^{2} f^{2} g^{2} - 12 \, c d e f g^{3} - 19 \,{\left (2 \, c d^{2} + a e\right )} g^{4}\right )} n\right )} x^{2} +{\left (6 \, c d e f^{3} g + 26 \, a d f g^{3} - 7 \,{\left (2 \, c d^{2} + a e\right )} f^{2} g^{2}\right )} n +{\left (24 \, a d g^{4} +{\left (a d g^{4} +{\left (2 \, c d^{2} + a e\right )} f g^{3}\right )} n^{3} -{\left (6 \, c d e f^{2} g^{2} - 9 \, a d g^{4} - 7 \,{\left (2 \, c d^{2} + a e\right )} f g^{3}\right )} n^{2} + 2 \,{\left (3 \, c e^{2} f^{3} g - 12 \, c d e f^{2} g^{2} + 13 \, a d g^{4} + 6 \,{\left (2 \, c d^{2} + a e\right )} f g^{3}\right )} n\right )} x\right )}{\left (g x + f\right )}^{n}}{g^{4} n^{4} + 10 \, g^{4} n^{3} + 35 \, g^{4} n^{2} + 50 \, g^{4} n + 24 \, g^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e*x^2 + 2*c*d*x + a)*(e*x + d)*(g*x + f)^n,x, algorithm="fricas")
[Out]
(a*d*f*g^3*n^3 - 6*c*e^2*f^4 + 24*c*d*e*f^3*g + 24*a*d*f*g^3 - 12*(2*c*d^2 + a*e
)*f^2*g^2 + (c*e^2*g^4*n^3 + 6*c*e^2*g^4*n^2 + 11*c*e^2*g^4*n + 6*c*e^2*g^4)*x^4
+ (24*c*d*e*g^4 + (c*e^2*f*g^3 + 3*c*d*e*g^4)*n^3 + 3*(c*e^2*f*g^3 + 7*c*d*e*g^
4)*n^2 + 2*(c*e^2*f*g^3 + 21*c*d*e*g^4)*n)*x^3 + (9*a*d*f*g^3 - (2*c*d^2 + a*e)*
f^2*g^2)*n^2 + (12*(2*c*d^2 + a*e)*g^4 + (3*c*d*e*f*g^3 + (2*c*d^2 + a*e)*g^4)*n
^3 - (3*c*e^2*f^2*g^2 - 15*c*d*e*f*g^3 - 8*(2*c*d^2 + a*e)*g^4)*n^2 - (3*c*e^2*f
^2*g^2 - 12*c*d*e*f*g^3 - 19*(2*c*d^2 + a*e)*g^4)*n)*x^2 + (6*c*d*e*f^3*g + 26*a
*d*f*g^3 - 7*(2*c*d^2 + a*e)*f^2*g^2)*n + (24*a*d*g^4 + (a*d*g^4 + (2*c*d^2 + a*
e)*f*g^3)*n^3 - (6*c*d*e*f^2*g^2 - 9*a*d*g^4 - 7*(2*c*d^2 + a*e)*f*g^3)*n^2 + 2*
(3*c*e^2*f^3*g - 12*c*d*e*f^2*g^2 + 13*a*d*g^4 + 6*(2*c*d^2 + a*e)*f*g^3)*n)*x)*
(g*x + f)^n/(g^4*n^4 + 10*g^4*n^3 + 35*g^4*n^2 + 50*g^4*n + 24*g^4)
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Sympy [A] time = 14.9814, size = 4935, normalized size = 33.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(g*x+f)**n*(c*e*x**2+2*c*d*x+a),x)
[Out]
Piecewise((f**n*(a*d*x + a*e*x**2/2 + c*d**2*x**2 + c*d*e*x**3 + c*e**2*x**4/4),
Eq(g, 0)), (-2*a*d*f**2*g**3/(6*f**5*g**4 + 18*f**4*g**5*x + 18*f**3*g**6*x**2
+ 6*f**2*g**7*x**3) + 3*a*e*f*g**4*x**2/(6*f**5*g**4 + 18*f**4*g**5*x + 18*f**3*
g**6*x**2 + 6*f**2*g**7*x**3) + a*e*g**5*x**3/(6*f**5*g**4 + 18*f**4*g**5*x + 18
*f**3*g**6*x**2 + 6*f**2*g**7*x**3) + 6*c*d**2*f*g**4*x**2/(6*f**5*g**4 + 18*f**
4*g**5*x + 18*f**3*g**6*x**2 + 6*f**2*g**7*x**3) + 2*c*d**2*g**5*x**3/(6*f**5*g*
*4 + 18*f**4*g**5*x + 18*f**3*g**6*x**2 + 6*f**2*g**7*x**3) + 6*c*d*e*f*g**4*x**
3/(6*f**5*g**4 + 18*f**4*g**5*x + 18*f**3*g**6*x**2 + 6*f**2*g**7*x**3) + 6*c*e*
*2*f**5*log(f/g + x)/(6*f**5*g**4 + 18*f**4*g**5*x + 18*f**3*g**6*x**2 + 6*f**2*
g**7*x**3) + 2*c*e**2*f**5/(6*f**5*g**4 + 18*f**4*g**5*x + 18*f**3*g**6*x**2 + 6
*f**2*g**7*x**3) + 18*c*e**2*f**4*g*x*log(f/g + x)/(6*f**5*g**4 + 18*f**4*g**5*x
+ 18*f**3*g**6*x**2 + 6*f**2*g**7*x**3) + 18*c*e**2*f**3*g**2*x**2*log(f/g + x)
/(6*f**5*g**4 + 18*f**4*g**5*x + 18*f**3*g**6*x**2 + 6*f**2*g**7*x**3) - 9*c*e**
2*f**3*g**2*x**2/(6*f**5*g**4 + 18*f**4*g**5*x + 18*f**3*g**6*x**2 + 6*f**2*g**7
*x**3) + 6*c*e**2*f**2*g**3*x**3*log(f/g + x)/(6*f**5*g**4 + 18*f**4*g**5*x + 18
*f**3*g**6*x**2 + 6*f**2*g**7*x**3) - 9*c*e**2*f**2*g**3*x**3/(6*f**5*g**4 + 18*
f**4*g**5*x + 18*f**3*g**6*x**2 + 6*f**2*g**7*x**3), Eq(n, -4)), (-a*d*f*g**3/(2
*f**3*g**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) + a*e*g**4*x**2/(2*f**3*g**4 + 4*f**
2*g**5*x + 2*f*g**6*x**2) + 2*c*d**2*g**4*x**2/(2*f**3*g**4 + 4*f**2*g**5*x + 2*
f*g**6*x**2) + 6*c*d*e*f**3*g*log(f/g + x)/(2*f**3*g**4 + 4*f**2*g**5*x + 2*f*g*
*6*x**2) + 3*c*d*e*f**3*g/(2*f**3*g**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) + 12*c*d
*e*f**2*g**2*x*log(f/g + x)/(2*f**3*g**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) + 6*c*
d*e*f*g**3*x**2*log(f/g + x)/(2*f**3*g**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) - 6*c
*d*e*f*g**3*x**2/(2*f**3*g**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) - 6*c*e**2*f**4*l
og(f/g + x)/(2*f**3*g**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) - 3*c*e**2*f**4/(2*f**
3*g**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) - 12*c*e**2*f**3*g*x*log(f/g + x)/(2*f**
3*g**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) - 6*c*e**2*f**2*g**2*x**2*log(f/g + x)/(
2*f**3*g**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) + 6*c*e**2*f**2*g**2*x**2/(2*f**3*g
**4 + 4*f**2*g**5*x + 2*f*g**6*x**2) + 2*c*e**2*f*g**3*x**3/(2*f**3*g**4 + 4*f**
2*g**5*x + 2*f*g**6*x**2), Eq(n, -3)), (-2*a*d*g**3/(2*f*g**4 + 2*g**5*x) + 2*a*
e*f*g**2*log(f/g + x)/(2*f*g**4 + 2*g**5*x) + 2*a*e*f*g**2/(2*f*g**4 + 2*g**5*x)
+ 2*a*e*g**3*x*log(f/g + x)/(2*f*g**4 + 2*g**5*x) + 4*c*d**2*f*g**2*log(f/g + x
)/(2*f*g**4 + 2*g**5*x) + 4*c*d**2*f*g**2/(2*f*g**4 + 2*g**5*x) + 4*c*d**2*g**3*
x*log(f/g + x)/(2*f*g**4 + 2*g**5*x) - 12*c*d*e*f**2*g*log(f/g + x)/(2*f*g**4 +
2*g**5*x) - 12*c*d*e*f**2*g/(2*f*g**4 + 2*g**5*x) - 12*c*d*e*f*g**2*x*log(f/g +
x)/(2*f*g**4 + 2*g**5*x) + 6*c*d*e*g**3*x**2/(2*f*g**4 + 2*g**5*x) + 6*c*e**2*f*
*3*log(f/g + x)/(2*f*g**4 + 2*g**5*x) + 6*c*e**2*f**3/(2*f*g**4 + 2*g**5*x) + 6*
c*e**2*f**2*g*x*log(f/g + x)/(2*f*g**4 + 2*g**5*x) - 3*c*e**2*f*g**2*x**2/(2*f*g
**4 + 2*g**5*x) + c*e**2*g**3*x**3/(2*f*g**4 + 2*g**5*x), Eq(n, -2)), (a*d*log(f
/g + x)/g - a*e*f*log(f/g + x)/g**2 + a*e*x/g - 2*c*d**2*f*log(f/g + x)/g**2 + 2
*c*d**2*x/g + 3*c*d*e*f**2*log(f/g + x)/g**3 - 3*c*d*e*f*x/g**2 + 3*c*d*e*x**2/(
2*g) - c*e**2*f**3*log(f/g + x)/g**4 + c*e**2*f**2*x/g**3 - c*e**2*f*x**2/(2*g**
2) + c*e**2*x**3/(3*g), Eq(n, -1)), (a*d*f*g**3*n**3*(f + g*x)**n/(g**4*n**4 + 1
0*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 9*a*d*f*g**3*n**2*(f + g*x)*
*n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 26*a*d*f*g*
*3*n*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4
) + 24*a*d*f*g**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**
4*n + 24*g**4) + a*d*g**4*n**3*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**
4*n**2 + 50*g**4*n + 24*g**4) + 9*a*d*g**4*n**2*x*(f + g*x)**n/(g**4*n**4 + 10*g
**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 26*a*d*g**4*n*x*(f + g*x)**n/(g
**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 24*a*d*g**4*x*(f
+ g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) - a*e
*f**2*g**2*n**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*
n + 24*g**4) - 7*a*e*f**2*g**2*n*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**
4*n**2 + 50*g**4*n + 24*g**4) - 12*a*e*f**2*g**2*(f + g*x)**n/(g**4*n**4 + 10*g*
*4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + a*e*f*g**3*n**3*x*(f + g*x)**n/(
g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 7*a*e*f*g**3*n*
*2*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4
) + 12*a*e*f*g**3*n*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50
*g**4*n + 24*g**4) + a*e*g**4*n**3*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 +
35*g**4*n**2 + 50*g**4*n + 24*g**4) + 8*a*e*g**4*n**2*x**2*(f + g*x)**n/(g**4*n
**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 19*a*e*g**4*n*x**2*(f
+ g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 12*
a*e*g**4*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n
+ 24*g**4) - 2*c*d**2*f**2*g**2*n**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35
*g**4*n**2 + 50*g**4*n + 24*g**4) - 14*c*d**2*f**2*g**2*n*(f + g*x)**n/(g**4*n**
4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) - 24*c*d**2*f**2*g**2*(f
+ g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 2*c*
d**2*f*g**3*n**3*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g*
*4*n + 24*g**4) + 14*c*d**2*f*g**3*n**2*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3
+ 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 24*c*d**2*f*g**3*n*x*(f + g*x)**n/(g**4
*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 2*c*d**2*g**4*n**3*
x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4
) + 16*c*d**2*g**4*n**2*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n*
*2 + 50*g**4*n + 24*g**4) + 38*c*d**2*g**4*n*x**2*(f + g*x)**n/(g**4*n**4 + 10*g
**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 24*c*d**2*g**4*x**2*(f + g*x)**
n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 6*c*d*e*f**3
*g*n*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4
) + 24*c*d*e*f**3*g*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g
**4*n + 24*g**4) - 6*c*d*e*f**2*g**2*n**2*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n*
*3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) - 24*c*d*e*f**2*g**2*n*x*(f + g*x)**n/(
g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 3*c*d*e*f*g**3*
n**3*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24
*g**4) + 15*c*d*e*f*g**3*n**2*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g
**4*n**2 + 50*g**4*n + 24*g**4) + 12*c*d*e*f*g**3*n*x**2*(f + g*x)**n/(g**4*n**4
+ 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 3*c*d*e*g**4*n**3*x**3*(
f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 21
*c*d*e*g**4*n**2*x**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50
*g**4*n + 24*g**4) + 42*c*d*e*g**4*n*x**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3
+ 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 24*c*d*e*g**4*x**3*(f + g*x)**n/(g**4*n
**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) - 6*c*e**2*f**4*(f + g*
x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 6*c*e**2
*f**3*g*n*x*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n +
24*g**4) - 3*c*e**2*f**2*g**2*n**2*x**2*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 +
35*g**4*n**2 + 50*g**4*n + 24*g**4) - 3*c*e**2*f**2*g**2*n*x**2*(f + g*x)**n/(g
**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + c*e**2*f*g**3*n*
*3*x**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g
**4) + 3*c*e**2*f*g**3*n**2*x**3*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**
4*n**2 + 50*g**4*n + 24*g**4) + 2*c*e**2*f*g**3*n*x**3*(f + g*x)**n/(g**4*n**4 +
10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + c*e**2*g**4*n**3*x**4*(f +
g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4) + 6*c*e
**2*g**4*n**2*x**4*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g*
*4*n + 24*g**4) + 11*c*e**2*g**4*n*x**4*(f + g*x)**n/(g**4*n**4 + 10*g**4*n**3 +
35*g**4*n**2 + 50*g**4*n + 24*g**4) + 6*c*e**2*g**4*x**4*(f + g*x)**n/(g**4*n**
4 + 10*g**4*n**3 + 35*g**4*n**2 + 50*g**4*n + 24*g**4), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.266955, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e*x^2 + 2*c*d*x + a)*(e*x + d)*(g*x + f)^n,x, algorithm="giac")
[Out]
Done