3.28 \(\int (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx\)
Optimal. Leaf size=126 \[ \frac{(c+d x)^{n+1} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^4 (n+1)}-\frac{(c+d x)^{n+2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4 (n+2)}+\frac{(C d-3 c D) (c+d x)^{n+3}}{d^4 (n+3)}+\frac{D (c+d x)^{n+4}}{d^4 (n+4)} \]
[Out]
((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^4*(1 + n)) - ((2*c*C*
d - B*d^2 - 3*c^2*D)*(c + d*x)^(2 + n))/(d^4*(2 + n)) + ((C*d - 3*c*D)*(c + d*x)
^(3 + n))/(d^4*(3 + n)) + (D*(c + d*x)^(4 + n))/(d^4*(4 + n))
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Rubi [A] time = 0.1585, antiderivative size = 126, 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
\[ \frac{(c+d x)^{n+1} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^4 (n+1)}-\frac{(c+d x)^{n+2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4 (n+2)}+\frac{(C d-3 c D) (c+d x)^{n+3}}{d^4 (n+3)}+\frac{D (c+d x)^{n+4}}{d^4 (n+4)} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]
[Out]
((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^4*(1 + n)) - ((2*c*C*
d - B*d^2 - 3*c^2*D)*(c + d*x)^(2 + n))/(d^4*(2 + n)) + ((C*d - 3*c*D)*(c + d*x)
^(3 + n))/(d^4*(3 + n)) + (D*(c + d*x)^(4 + n))/(d^4*(4 + n))
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Rubi in Sympy [A] time = 36.2376, size = 112, normalized size = 0.89 \[ \frac{D \left (c + d x\right )^{n + 4}}{d^{4} \left (n + 4\right )} + \frac{\left (c + d x\right )^{n + 1} \left (A d^{3} - B c d^{2} + C c^{2} d - D c^{3}\right )}{d^{4} \left (n + 1\right )} + \frac{\left (c + d x\right )^{n + 2} \left (B d^{2} - 2 C c d + 3 D c^{2}\right )}{d^{4} \left (n + 2\right )} + \frac{\left (c + d x\right )^{n + 3} \left (C d - 3 D c\right )}{d^{4} \left (n + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**n*(D*x**3+C*x**2+B*x+A),x)
[Out]
D*(c + d*x)**(n + 4)/(d**4*(n + 4)) + (c + d*x)**(n + 1)*(A*d**3 - B*c*d**2 + C*
c**2*d - D*c**3)/(d**4*(n + 1)) + (c + d*x)**(n + 2)*(B*d**2 - 2*C*c*d + 3*D*c**
2)/(d**4*(n + 2)) + (c + d*x)**(n + 3)*(C*d - 3*D*c)/(d**4*(n + 3))
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Mathematica [A] time = 0.211243, size = 148, normalized size = 1.17 \[ \frac{(c+d x)^{n+1} \left (d^3 \left (A \left (n^3+9 n^2+26 n+24\right )+(n+1) x \left (B \left (n^2+7 n+12\right )+(n+2) x (C (n+4)+D (n+3) x)\right )\right )-c d^2 \left (B \left (n^2+7 n+12\right )+(n+1) x (2 C (n+4)+3 D (n+2) x)\right )-6 c^3 D+2 c^2 d (C (n+4)+3 D (n+1) x)\right )}{d^4 (n+1) (n+2) (n+3) (n+4)} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]
[Out]
((c + d*x)^(1 + n)*(-6*c^3*D + 2*c^2*d*(C*(4 + n) + 3*D*(1 + n)*x) - c*d^2*(B*(1
2 + 7*n + n^2) + (1 + n)*x*(2*C*(4 + n) + 3*D*(2 + n)*x)) + d^3*(A*(24 + 26*n +
9*n^2 + n^3) + (1 + n)*x*(B*(12 + 7*n + n^2) + (2 + n)*x*(C*(4 + n) + D*(3 + n)*
x)))))/(d^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))
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Maple [B] time = 0.01, size = 308, normalized size = 2.4 \[{\frac{ \left ( dx+c \right ) ^{1+n} \left ( D{d}^{3}{n}^{3}{x}^{3}+C{d}^{3}{n}^{3}{x}^{2}+6\,D{d}^{3}{n}^{2}{x}^{3}+B{d}^{3}{n}^{3}x+7\,C{d}^{3}{n}^{2}{x}^{2}-3\,Dc{d}^{2}{n}^{2}{x}^{2}+11\,D{d}^{3}n{x}^{3}+A{d}^{3}{n}^{3}+8\,B{d}^{3}{n}^{2}x-2\,Cc{d}^{2}{n}^{2}x+14\,C{d}^{3}n{x}^{2}-9\,Dc{d}^{2}n{x}^{2}+6\,D{x}^{3}{d}^{3}+9\,A{d}^{3}{n}^{2}-Bc{d}^{2}{n}^{2}+19\,B{d}^{3}nx-10\,Cc{d}^{2}nx+8\,C{d}^{3}{x}^{2}+6\,D{c}^{2}dnx-6\,Dc{d}^{2}{x}^{2}+26\,A{d}^{3}n-7\,Bc{d}^{2}n+12\,B{d}^{3}x+2\,C{c}^{2}dn-8\,Cc{d}^{2}x+6\,D{c}^{2}dx+24\,A{d}^{3}-12\,Bc{d}^{2}+8\,C{c}^{2}d-6\,D{c}^{3} \right ) }{{d}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^n*(D*x^3+C*x^2+B*x+A),x)
[Out]
(d*x+c)^(1+n)*(D*d^3*n^3*x^3+C*d^3*n^3*x^2+6*D*d^3*n^2*x^3+B*d^3*n^3*x+7*C*d^3*n
^2*x^2-3*D*c*d^2*n^2*x^2+11*D*d^3*n*x^3+A*d^3*n^3+8*B*d^3*n^2*x-2*C*c*d^2*n^2*x+
14*C*d^3*n*x^2-9*D*c*d^2*n*x^2+6*D*d^3*x^3+9*A*d^3*n^2-B*c*d^2*n^2+19*B*d^3*n*x-
10*C*c*d^2*n*x+8*C*d^3*x^2+6*D*c^2*d*n*x-6*D*c*d^2*x^2+26*A*d^3*n-7*B*c*d^2*n+12
*B*d^3*x+2*C*c^2*d*n-8*C*c*d^2*x+6*D*c^2*d*x+24*A*d^3-12*B*c*d^2+8*C*c^2*d-6*D*c
^3)/d^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((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^n,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.229822, size = 532, normalized size = 4.22 \[ \frac{{\left (A c d^{3} n^{3} - 6 \, D c^{4} + 8 \, C c^{3} d - 12 \, B c^{2} d^{2} + 24 \, A c d^{3} +{\left (D d^{4} n^{3} + 6 \, D d^{4} n^{2} + 11 \, D d^{4} n + 6 \, D d^{4}\right )} x^{4} +{\left (8 \, C d^{4} +{\left (D c d^{3} + C d^{4}\right )} n^{3} +{\left (3 \, D c d^{3} + 7 \, C d^{4}\right )} n^{2} + 2 \,{\left (D c d^{3} + 7 \, C d^{4}\right )} n\right )} x^{3} -{\left (B c^{2} d^{2} - 9 \, A c d^{3}\right )} n^{2} +{\left (12 \, B d^{4} +{\left (C c d^{3} + B d^{4}\right )} n^{3} -{\left (3 \, D c^{2} d^{2} - 5 \, C c d^{3} - 8 \, B d^{4}\right )} n^{2} -{\left (3 \, D c^{2} d^{2} - 4 \, C c d^{3} - 19 \, B d^{4}\right )} n\right )} x^{2} +{\left (2 \, C c^{3} d - 7 \, B c^{2} d^{2} + 26 \, A c d^{3}\right )} n +{\left (24 \, A d^{4} +{\left (B c d^{3} + A d^{4}\right )} n^{3} -{\left (2 \, C c^{2} d^{2} - 7 \, B c d^{3} - 9 \, A d^{4}\right )} n^{2} + 2 \,{\left (3 \, D c^{3} d - 4 \, C c^{2} d^{2} + 6 \, B c d^{3} + 13 \, A d^{4}\right )} n\right )} x\right )}{\left (d x + c\right )}^{n}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^n,x, algorithm="fricas")
[Out]
(A*c*d^3*n^3 - 6*D*c^4 + 8*C*c^3*d - 12*B*c^2*d^2 + 24*A*c*d^3 + (D*d^4*n^3 + 6*
D*d^4*n^2 + 11*D*d^4*n + 6*D*d^4)*x^4 + (8*C*d^4 + (D*c*d^3 + C*d^4)*n^3 + (3*D*
c*d^3 + 7*C*d^4)*n^2 + 2*(D*c*d^3 + 7*C*d^4)*n)*x^3 - (B*c^2*d^2 - 9*A*c*d^3)*n^
2 + (12*B*d^4 + (C*c*d^3 + B*d^4)*n^3 - (3*D*c^2*d^2 - 5*C*c*d^3 - 8*B*d^4)*n^2
- (3*D*c^2*d^2 - 4*C*c*d^3 - 19*B*d^4)*n)*x^2 + (2*C*c^3*d - 7*B*c^2*d^2 + 26*A*
c*d^3)*n + (24*A*d^4 + (B*c*d^3 + A*d^4)*n^3 - (2*C*c^2*d^2 - 7*B*c*d^3 - 9*A*d^
4)*n^2 + 2*(3*D*c^3*d - 4*C*c^2*d^2 + 6*B*c*d^3 + 13*A*d^4)*n)*x)*(d*x + c)^n/(d
^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n + 24*d^4)
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Sympy [A] time = 15.6643, size = 3822, normalized size = 30.33 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**n*(D*x**3+C*x**2+B*x+A),x)
[Out]
Piecewise((c**n*(A*x + B*x**2/2 + C*x**3/3 + D*x**4/4), Eq(d, 0)), (-2*A*d**3/(6
*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - B*c*d**2/(6*c**3*d
**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 3*B*d**3*x/(6*c**3*d**4 +
18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 2*C*c**2*d/(6*c**3*d**4 + 18*c
**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 6*C*c*d**2*x/(6*c**3*d**4 + 18*c**2
*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 6*C*d**3*x**2/(6*c**3*d**4 + 18*c**2*d
**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 6*D*c**3*log(c/d + x)/(6*c**3*d**4 + 18*
c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 11*D*c**3/(6*c**3*d**4 + 18*c**2*d
**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 18*D*c**2*d*x*log(c/d + x)/(6*c**3*d**4
+ 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 27*D*c**2*d*x/(6*c**3*d**4 +
18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 18*D*c*d**2*x**2*log(c/d + x)/(
6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 18*D*c*d**2*x**2/
(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 6*D*d**3*x**3*lo
g(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3), Eq(n,
-4)), (-A*c*d**3/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**2) + B*d**4*x**2/(2*
c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**2) + 2*C*c**3*d*log(c/d + x)/(2*c**3*d**
4 + 4*c**2*d**5*x + 2*c*d**6*x**2) + C*c**3*d/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c
*d**6*x**2) + 4*C*c**2*d**2*x*log(c/d + x)/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d*
*6*x**2) + 2*C*c*d**3*x**2*log(c/d + x)/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*
x**2) - 2*C*c*d**3*x**2/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**2) - 6*D*c**4
*log(c/d + x)/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**2) - 3*D*c**4/(2*c**3*d
**4 + 4*c**2*d**5*x + 2*c*d**6*x**2) - 12*D*c**3*d*x*log(c/d + x)/(2*c**3*d**4 +
4*c**2*d**5*x + 2*c*d**6*x**2) - 6*D*c**2*d**2*x**2*log(c/d + x)/(2*c**3*d**4 +
4*c**2*d**5*x + 2*c*d**6*x**2) + 6*D*c**2*d**2*x**2/(2*c**3*d**4 + 4*c**2*d**5*
x + 2*c*d**6*x**2) + 2*D*c*d**3*x**3/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**
2), Eq(n, -3)), (-2*A*d**3/(2*c*d**4 + 2*d**5*x) + 2*B*c*d**2*log(c/d + x)/(2*c*
d**4 + 2*d**5*x) + 2*B*c*d**2/(2*c*d**4 + 2*d**5*x) + 2*B*d**3*x*log(c/d + x)/(2
*c*d**4 + 2*d**5*x) - 4*C*c**2*d*log(c/d + x)/(2*c*d**4 + 2*d**5*x) - 4*C*c**2*d
/(2*c*d**4 + 2*d**5*x) - 4*C*c*d**2*x*log(c/d + x)/(2*c*d**4 + 2*d**5*x) + 2*C*d
**3*x**2/(2*c*d**4 + 2*d**5*x) + 6*D*c**3*log(c/d + x)/(2*c*d**4 + 2*d**5*x) + 6
*D*c**3/(2*c*d**4 + 2*d**5*x) + 6*D*c**2*d*x*log(c/d + x)/(2*c*d**4 + 2*d**5*x)
- 3*D*c*d**2*x**2/(2*c*d**4 + 2*d**5*x) + D*d**3*x**3/(2*c*d**4 + 2*d**5*x), Eq(
n, -2)), (A*log(c/d + x)/d - B*c*log(c/d + x)/d**2 + B*x/d + C*c**2*log(c/d + x)
/d**3 - C*c*x/d**2 + C*x**2/(2*d) - D*c**3*log(c/d + x)/d**4 + D*c**2*x/d**3 - D
*c*x**2/(2*d**2) + D*x**3/(3*d), Eq(n, -1)), (A*c*d**3*n**3*(c + d*x)**n/(d**4*n
**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 9*A*c*d**3*n**2*(c +
d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 26*A*c
*d**3*n*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d
**4) + 24*A*c*d**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d*
*4*n + 24*d**4) + A*d**4*n**3*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4
*n**2 + 50*d**4*n + 24*d**4) + 9*A*d**4*n**2*x*(c + d*x)**n/(d**4*n**4 + 10*d**4
*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 26*A*d**4*n*x*(c + d*x)**n/(d**4*n
**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 24*A*d**4*x*(c + d*x)
**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - B*c**2*d**
2*n**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d*
*4) - 7*B*c**2*d**2*n*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50
*d**4*n + 24*d**4) - 12*B*c**2*d**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*
d**4*n**2 + 50*d**4*n + 24*d**4) + B*c*d**3*n**3*x*(c + d*x)**n/(d**4*n**4 + 10*
d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 7*B*c*d**3*n**2*x*(c + d*x)**n
/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 12*B*c*d**3*n
*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4)
+ B*d**4*n**3*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d*
*4*n + 24*d**4) + 8*B*d**4*n**2*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35
*d**4*n**2 + 50*d**4*n + 24*d**4) + 19*B*d**4*n*x**2*(c + d*x)**n/(d**4*n**4 + 1
0*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 12*B*d**4*x**2*(c + d*x)**n/
(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 2*C*c**3*d*n*(
c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 8*
C*c**3*d*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*
d**4) - 2*C*c**2*d**2*n**2*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n*
*2 + 50*d**4*n + 24*d**4) - 8*C*c**2*d**2*n*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*
n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + C*c*d**3*n**3*x**2*(c + d*x)**n/(d*
*4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 5*C*c*d**3*n**2*x
**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4)
+ 4*C*c*d**3*n*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*
d**4*n + 24*d**4) + C*d**4*n**3*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35
*d**4*n**2 + 50*d**4*n + 24*d**4) + 7*C*d**4*n**2*x**3*(c + d*x)**n/(d**4*n**4 +
10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 14*C*d**4*n*x**3*(c + d*x)
**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 8*C*d**4*x
**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4)
- 6*D*c**4*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n +
24*d**4) + 6*D*c**3*d*n*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2
+ 50*d**4*n + 24*d**4) - 3*D*c**2*d**2*n**2*x**2*(c + d*x)**n/(d**4*n**4 + 10*d*
*4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 3*D*c**2*d**2*n*x**2*(c + d*x)**
n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + D*c*d**3*n**
3*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d*
*4) + 3*D*c*d**3*n**2*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2
+ 50*d**4*n + 24*d**4) + 2*D*c*d**3*n*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n*
*3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + D*d**4*n**3*x**4*(c + d*x)**n/(d**4*n
**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 6*D*d**4*n**2*x**4*(c
+ d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 11*
D*d**4*n*x**4*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n
+ 24*d**4) + 6*D*d**4*x**4*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2
+ 50*d**4*n + 24*d**4), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.215109, size = 1091, normalized size = 8.66 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^n,x, algorithm="giac")
[Out]
(D*d^4*n^3*x^4*e^(n*ln(d*x + c)) + D*c*d^3*n^3*x^3*e^(n*ln(d*x + c)) + C*d^4*n^3
*x^3*e^(n*ln(d*x + c)) + 6*D*d^4*n^2*x^4*e^(n*ln(d*x + c)) + C*c*d^3*n^3*x^2*e^(
n*ln(d*x + c)) + B*d^4*n^3*x^2*e^(n*ln(d*x + c)) + 3*D*c*d^3*n^2*x^3*e^(n*ln(d*x
+ c)) + 7*C*d^4*n^2*x^3*e^(n*ln(d*x + c)) + 11*D*d^4*n*x^4*e^(n*ln(d*x + c)) +
B*c*d^3*n^3*x*e^(n*ln(d*x + c)) + A*d^4*n^3*x*e^(n*ln(d*x + c)) - 3*D*c^2*d^2*n^
2*x^2*e^(n*ln(d*x + c)) + 5*C*c*d^3*n^2*x^2*e^(n*ln(d*x + c)) + 8*B*d^4*n^2*x^2*
e^(n*ln(d*x + c)) + 2*D*c*d^3*n*x^3*e^(n*ln(d*x + c)) + 14*C*d^4*n*x^3*e^(n*ln(d
*x + c)) + 6*D*d^4*x^4*e^(n*ln(d*x + c)) + A*c*d^3*n^3*e^(n*ln(d*x + c)) - 2*C*c
^2*d^2*n^2*x*e^(n*ln(d*x + c)) + 7*B*c*d^3*n^2*x*e^(n*ln(d*x + c)) + 9*A*d^4*n^2
*x*e^(n*ln(d*x + c)) - 3*D*c^2*d^2*n*x^2*e^(n*ln(d*x + c)) + 4*C*c*d^3*n*x^2*e^(
n*ln(d*x + c)) + 19*B*d^4*n*x^2*e^(n*ln(d*x + c)) + 8*C*d^4*x^3*e^(n*ln(d*x + c)
) - B*c^2*d^2*n^2*e^(n*ln(d*x + c)) + 9*A*c*d^3*n^2*e^(n*ln(d*x + c)) + 6*D*c^3*
d*n*x*e^(n*ln(d*x + c)) - 8*C*c^2*d^2*n*x*e^(n*ln(d*x + c)) + 12*B*c*d^3*n*x*e^(
n*ln(d*x + c)) + 26*A*d^4*n*x*e^(n*ln(d*x + c)) + 12*B*d^4*x^2*e^(n*ln(d*x + c))
+ 2*C*c^3*d*n*e^(n*ln(d*x + c)) - 7*B*c^2*d^2*n*e^(n*ln(d*x + c)) + 26*A*c*d^3*
n*e^(n*ln(d*x + c)) + 24*A*d^4*x*e^(n*ln(d*x + c)) - 6*D*c^4*e^(n*ln(d*x + c)) +
8*C*c^3*d*e^(n*ln(d*x + c)) - 12*B*c^2*d^2*e^(n*ln(d*x + c)) + 24*A*c*d^3*e^(n*
ln(d*x + c)))/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n + 24*d^4)