3.920 \(\int (a+b x)^n (c+d x)^3 \, dx\)
Optimal. Leaf size=110 \[ \frac{3 d^2 (b c-a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac{(b c-a d)^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 d (b c-a d)^2 (a+b x)^{n+2}}{b^4 (n+2)}+\frac{d^3 (a+b x)^{n+4}}{b^4 (n+4)} \]
[Out]
((b*c - a*d)^3*(a + b*x)^(1 + n))/(b^4*(1 + n)) + (3*d*(b*c - a*d)^2*(a + b*x)^(
2 + n))/(b^4*(2 + n)) + (3*d^2*(b*c - a*d)*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d
^3*(a + b*x)^(4 + n))/(b^4*(4 + n))
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Rubi [A] time = 0.111553, antiderivative size = 110, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067
\[ \frac{3 d^2 (b c-a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac{(b c-a d)^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 d (b c-a d)^2 (a+b x)^{n+2}}{b^4 (n+2)}+\frac{d^3 (a+b x)^{n+4}}{b^4 (n+4)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n*(c + d*x)^3,x]
[Out]
((b*c - a*d)^3*(a + b*x)^(1 + n))/(b^4*(1 + n)) + (3*d*(b*c - a*d)^2*(a + b*x)^(
2 + n))/(b^4*(2 + n)) + (3*d^2*(b*c - a*d)*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d
^3*(a + b*x)^(4 + n))/(b^4*(4 + n))
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Rubi in Sympy [A] time = 29.6995, size = 95, normalized size = 0.86 \[ \frac{d^{3} \left (a + b x\right )^{n + 4}}{b^{4} \left (n + 4\right )} - \frac{3 d^{2} \left (a + b x\right )^{n + 3} \left (a d - b c\right )}{b^{4} \left (n + 3\right )} + \frac{3 d \left (a + b x\right )^{n + 2} \left (a d - b c\right )^{2}}{b^{4} \left (n + 2\right )} - \frac{\left (a + b x\right )^{n + 1} \left (a d - b c\right )^{3}}{b^{4} \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(d*x+c)**3,x)
[Out]
d**3*(a + b*x)**(n + 4)/(b**4*(n + 4)) - 3*d**2*(a + b*x)**(n + 3)*(a*d - b*c)/(
b**4*(n + 3)) + 3*d*(a + b*x)**(n + 2)*(a*d - b*c)**2/(b**4*(n + 2)) - (a + b*x)
**(n + 1)*(a*d - b*c)**3/(b**4*(n + 1))
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Mathematica [A] time = 0.264451, size = 195, normalized size = 1.77 \[ \frac{(a+b x)^{n+1} \left (-6 a^3 d^3+6 a^2 b d^2 (c (n+4)+d (n+1) x)-3 a b^2 d \left (c^2 \left (n^2+7 n+12\right )+2 c d \left (n^2+5 n+4\right ) x+d^2 \left (n^2+3 n+2\right ) x^2\right )+b^3 \left (c^3 \left (n^3+9 n^2+26 n+24\right )+3 c^2 d \left (n^3+8 n^2+19 n+12\right ) x+3 c d^2 \left (n^3+7 n^2+14 n+8\right ) x^2+d^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )\right )}{b^4 (n+1) (n+2) (n+3) (n+4)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^n*(c + d*x)^3,x]
[Out]
((a + b*x)^(1 + n)*(-6*a^3*d^3 + 6*a^2*b*d^2*(c*(4 + n) + d*(1 + n)*x) - 3*a*b^2
*d*(c^2*(12 + 7*n + n^2) + 2*c*d*(4 + 5*n + n^2)*x + d^2*(2 + 3*n + n^2)*x^2) +
b^3*(c^3*(24 + 26*n + 9*n^2 + n^3) + 3*c^2*d*(12 + 19*n + 8*n^2 + n^3)*x + 3*c*d
^2*(8 + 14*n + 7*n^2 + n^3)*x^2 + d^3*(6 + 11*n + 6*n^2 + n^3)*x^3)))/(b^4*(1 +
n)*(2 + n)*(3 + n)*(4 + n))
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Maple [B] time = 0.013, size = 389, normalized size = 3.5 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}{d}^{3}{n}^{3}{x}^{3}-3\,{b}^{3}c{d}^{2}{n}^{3}{x}^{2}-6\,{b}^{3}{d}^{3}{n}^{2}{x}^{3}+3\,a{b}^{2}{d}^{3}{n}^{2}{x}^{2}-3\,{b}^{3}{c}^{2}d{n}^{3}x-21\,{b}^{3}c{d}^{2}{n}^{2}{x}^{2}-11\,{b}^{3}{d}^{3}n{x}^{3}+6\,a{b}^{2}c{d}^{2}{n}^{2}x+9\,a{b}^{2}{d}^{3}n{x}^{2}-{b}^{3}{c}^{3}{n}^{3}-24\,{b}^{3}{c}^{2}d{n}^{2}x-42\,{b}^{3}c{d}^{2}n{x}^{2}-6\,{x}^{3}{b}^{3}{d}^{3}-6\,{a}^{2}b{d}^{3}nx+3\,a{b}^{2}{c}^{2}d{n}^{2}+30\,a{b}^{2}c{d}^{2}nx+6\,a{b}^{2}{d}^{3}{x}^{2}-9\,{b}^{3}{c}^{3}{n}^{2}-57\,{b}^{3}{c}^{2}dnx-24\,{b}^{3}c{d}^{2}{x}^{2}-6\,{a}^{2}bc{d}^{2}n-6\,{a}^{2}b{d}^{3}x+21\,a{b}^{2}{c}^{2}dn+24\,a{b}^{2}c{d}^{2}x-26\,{b}^{3}{c}^{3}n-36\,{b}^{3}{c}^{2}dx+6\,{a}^{3}{d}^{3}-24\,{a}^{2}cb{d}^{2}+36\,a{b}^{2}{c}^{2}d-24\,{b}^{3}{c}^{3} \right ) }{{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(d*x+c)^3,x)
[Out]
-(b*x+a)^(1+n)*(-b^3*d^3*n^3*x^3-3*b^3*c*d^2*n^3*x^2-6*b^3*d^3*n^2*x^3+3*a*b^2*d
^3*n^2*x^2-3*b^3*c^2*d*n^3*x-21*b^3*c*d^2*n^2*x^2-11*b^3*d^3*n*x^3+6*a*b^2*c*d^2
*n^2*x+9*a*b^2*d^3*n*x^2-b^3*c^3*n^3-24*b^3*c^2*d*n^2*x-42*b^3*c*d^2*n*x^2-6*b^3
*d^3*x^3-6*a^2*b*d^3*n*x+3*a*b^2*c^2*d*n^2+30*a*b^2*c*d^2*n*x+6*a*b^2*d^3*x^2-9*
b^3*c^3*n^2-57*b^3*c^2*d*n*x-24*b^3*c*d^2*x^2-6*a^2*b*c*d^2*n-6*a^2*b*d^3*x+21*a
*b^2*c^2*d*n+24*a*b^2*c*d^2*x-26*b^3*c^3*n-36*b^3*c^2*d*x+6*a^3*d^3-24*a^2*b*c*d
^2+36*a*b^2*c^2*d-24*b^3*c^3)/b^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 + c)^3*(b*x + a)^n,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.226152, size = 671, normalized size = 6.1 \[ \frac{{\left (a b^{3} c^{3} n^{3} + 24 \, a b^{3} c^{3} - 36 \, a^{2} b^{2} c^{2} d + 24 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} +{\left (b^{4} d^{3} n^{3} + 6 \, b^{4} d^{3} n^{2} + 11 \, b^{4} d^{3} n + 6 \, b^{4} d^{3}\right )} x^{4} +{\left (24 \, b^{4} c d^{2} +{\left (3 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n^{3} + 3 \,{\left (7 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n^{2} + 2 \,{\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n\right )} x^{3} + 3 \,{\left (3 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )} n^{2} + 3 \,{\left (12 \, b^{4} c^{2} d +{\left (b^{4} c^{2} d + a b^{3} c d^{2}\right )} n^{3} +{\left (8 \, b^{4} c^{2} d + 5 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} n^{2} +{\left (19 \, b^{4} c^{2} d + 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} n\right )} x^{2} +{\left (26 \, a b^{3} c^{3} - 21 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2}\right )} n +{\left (24 \, b^{4} c^{3} +{\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d\right )} n^{3} + 3 \,{\left (3 \, b^{4} c^{3} + 7 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2}\right )} n^{2} + 2 \,{\left (13 \, b^{4} c^{3} + 18 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3}\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*(b*x + a)^n,x, algorithm="fricas")
[Out]
(a*b^3*c^3*n^3 + 24*a*b^3*c^3 - 36*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 - 6*a^4*d^3 +
(b^4*d^3*n^3 + 6*b^4*d^3*n^2 + 11*b^4*d^3*n + 6*b^4*d^3)*x^4 + (24*b^4*c*d^2 + (
3*b^4*c*d^2 + a*b^3*d^3)*n^3 + 3*(7*b^4*c*d^2 + a*b^3*d^3)*n^2 + 2*(21*b^4*c*d^2
+ a*b^3*d^3)*n)*x^3 + 3*(3*a*b^3*c^3 - a^2*b^2*c^2*d)*n^2 + 3*(12*b^4*c^2*d + (
b^4*c^2*d + a*b^3*c*d^2)*n^3 + (8*b^4*c^2*d + 5*a*b^3*c*d^2 - a^2*b^2*d^3)*n^2 +
(19*b^4*c^2*d + 4*a*b^3*c*d^2 - a^2*b^2*d^3)*n)*x^2 + (26*a*b^3*c^3 - 21*a^2*b^
2*c^2*d + 6*a^3*b*c*d^2)*n + (24*b^4*c^3 + (b^4*c^3 + 3*a*b^3*c^2*d)*n^3 + 3*(3*
b^4*c^3 + 7*a*b^3*c^2*d - 2*a^2*b^2*c*d^2)*n^2 + 2*(13*b^4*c^3 + 18*a*b^3*c^2*d
- 12*a^2*b^2*c*d^2 + 3*a^3*b*d^3)*n)*x)*(b*x + a)^n/(b^4*n^4 + 10*b^4*n^3 + 35*b
^4*n^2 + 50*b^4*n + 24*b^4)
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Sympy [A] time = 11.7419, size = 4056, normalized size = 36.87 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**n*(d*x+c)**3,x)
[Out]
Piecewise((a**n*(c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4), Eq(b, 0)
), (6*a**5*d**3*log(a/b + x)/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 +
6*a**2*b**7*x**3) + 2*a**5*d**3/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x*
*2 + 6*a**2*b**7*x**3) + 18*a**4*b*d**3*x*log(a/b + x)/(6*a**5*b**4 + 18*a**4*b*
*5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3) + 18*a**3*b**2*d**3*x**2*log(a/b +
x)/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3) - 9*a**
3*b**2*d**3*x**2/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7
*x**3) - 2*a**2*b**3*c**3/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*
a**2*b**7*x**3) + 6*a**2*b**3*d**3*x**3*log(a/b + x)/(6*a**5*b**4 + 18*a**4*b**5
*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3) - 9*a**2*b**3*d**3*x**3/(6*a**5*b**4
+ 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3) + 9*a*b**4*c**2*d*x**2/
(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3) + 6*a*b**4
*c*d**2*x**3/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**
3) + 3*b**5*c**2*d*x**3/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a*
*2*b**7*x**3), Eq(n, -4)), (-6*a**4*d**3*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5
*x + 2*a*b**6*x**2) - 3*a**4*d**3/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2)
+ 6*a**3*b*c*d**2*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 3
*a**3*b*c*d**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - 12*a**3*b*d**3*x*
log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 12*a**2*b**2*c*d**2
*x*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - 6*a**2*b**2*d**3
*x**2*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 6*a**2*b**2*d
**3*x**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - a*b**3*c**3/(2*a**3*b**
4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 6*a*b**3*c*d**2*x**2*log(a/b + x)/(2*a**3*b
**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - 6*a*b**3*c*d**2*x**2/(2*a**3*b**4 + 4*a**
2*b**5*x + 2*a*b**6*x**2) + 2*a*b**3*d**3*x**3/(2*a**3*b**4 + 4*a**2*b**5*x + 2*
a*b**6*x**2) + 3*b**4*c**2*d*x**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2),
Eq(n, -3)), (6*a**3*d**3*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3*d**3/(2*a*
b**4 + 2*b**5*x) - 12*a**2*b*c*d**2*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 12*a**2
*b*c*d**2/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*d**3*x*log(a/b + x)/(2*a*b**4 + 2*b**
5*x) + 6*a*b**2*c**2*d*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a*b**2*c**2*d/(2*a
*b**4 + 2*b**5*x) - 12*a*b**2*c*d**2*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*
b**2*d**3*x**2/(2*a*b**4 + 2*b**5*x) - 2*b**3*c**3/(2*a*b**4 + 2*b**5*x) + 6*b**
3*c**2*d*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*b**3*c*d**2*x**2/(2*a*b**4 + 2
*b**5*x) + b**3*d**3*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*d**3*log(a/b
+ x)/b**4 + 3*a**2*c*d**2*log(a/b + x)/b**3 + a**2*d**3*x/b**3 - 3*a*c**2*d*log
(a/b + x)/b**2 - 3*a*c*d**2*x/b**2 - a*d**3*x**2/(2*b**2) + c**3*log(a/b + x)/b
+ 3*c**2*d*x/b + 3*c*d**2*x**2/(2*b) + d**3*x**3/(3*b), Eq(n, -1)), (-6*a**4*d**
3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) +
6*a**3*b*c*d**2*n*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b*
*4*n + 24*b**4) + 24*a**3*b*c*d**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b
**4*n**2 + 50*b**4*n + 24*b**4) + 6*a**3*b*d**3*n*x*(a + b*x)**n/(b**4*n**4 + 10
*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*c**2*d*n**2*(a +
b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 21*a**
2*b**2*c**2*d*n*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*
n + 24*b**4) - 36*a**2*b**2*c**2*d*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b
**4*n**2 + 50*b**4*n + 24*b**4) - 6*a**2*b**2*c*d**2*n**2*x*(a + b*x)**n/(b**4*n
**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 24*a**2*b**2*c*d**2*n
*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4)
- 3*a**2*b**2*d**3*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n*
*2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*d**3*n*x**2*(a + b*x)**n/(b**4*n**4 + 10
*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + a*b**3*c**3*n**3*(a + b*x)**n
/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 9*a*b**3*c**3
*n**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**
4) + 26*a*b**3*c**3*n*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50
*b**4*n + 24*b**4) + 24*a*b**3*c**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*
b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*c**2*d*n**3*x*(a + b*x)**n/(b**4*n**
4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 21*a*b**3*c**2*d*n**2*x
*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) +
36*a*b**3*c**2*d*n*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*
b**4*n + 24*b**4) + 3*a*b**3*c*d**2*n**3*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*
n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 15*a*b**3*c*d**2*n**2*x**2*(a + b*x
)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 12*a*b**3
*c*d**2*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n
+ 24*b**4) + a*b**3*d**3*n**3*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*
b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*d**3*n**2*x**3*(a + b*x)**n/(b**4*n*
*4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 2*a*b**3*d**3*n*x**3*(
a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b*
*4*c**3*n**3*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n
+ 24*b**4) + 9*b**4*c**3*n**2*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**
4*n**2 + 50*b**4*n + 24*b**4) + 26*b**4*c**3*n*x*(a + b*x)**n/(b**4*n**4 + 10*b*
*4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*b**4*c**3*x*(a + b*x)**n/(b**
4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*b**4*c**2*d*n**3
*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**
4) + 24*b**4*c**2*d*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n
**2 + 50*b**4*n + 24*b**4) + 57*b**4*c**2*d*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*
b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 36*b**4*c**2*d*x**2*(a + b*x)*
*n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*b**4*c*d*
*2*n**3*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n +
24*b**4) + 21*b**4*c*d**2*n**2*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35
*b**4*n**2 + 50*b**4*n + 24*b**4) + 42*b**4*c*d**2*n*x**3*(a + b*x)**n/(b**4*n**
4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*b**4*c*d**2*x**3*(a
+ b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4
*d**3*n**3*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*
n + 24*b**4) + 6*b**4*d**3*n**2*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35
*b**4*n**2 + 50*b**4*n + 24*b**4) + 11*b**4*d**3*n*x**4*(a + b*x)**n/(b**4*n**4
+ 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d**3*x**4*(a + b*x
)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.231498, size = 1233, normalized size = 11.21 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^3*(b*x + a)^n,x, algorithm="giac")
[Out]
(b^4*d^3*n^3*x^4*e^(n*ln(b*x + a)) + 3*b^4*c*d^2*n^3*x^3*e^(n*ln(b*x + a)) + a*b
^3*d^3*n^3*x^3*e^(n*ln(b*x + a)) + 6*b^4*d^3*n^2*x^4*e^(n*ln(b*x + a)) + 3*b^4*c
^2*d*n^3*x^2*e^(n*ln(b*x + a)) + 3*a*b^3*c*d^2*n^3*x^2*e^(n*ln(b*x + a)) + 21*b^
4*c*d^2*n^2*x^3*e^(n*ln(b*x + a)) + 3*a*b^3*d^3*n^2*x^3*e^(n*ln(b*x + a)) + 11*b
^4*d^3*n*x^4*e^(n*ln(b*x + a)) + b^4*c^3*n^3*x*e^(n*ln(b*x + a)) + 3*a*b^3*c^2*d
*n^3*x*e^(n*ln(b*x + a)) + 24*b^4*c^2*d*n^2*x^2*e^(n*ln(b*x + a)) + 15*a*b^3*c*d
^2*n^2*x^2*e^(n*ln(b*x + a)) - 3*a^2*b^2*d^3*n^2*x^2*e^(n*ln(b*x + a)) + 42*b^4*
c*d^2*n*x^3*e^(n*ln(b*x + a)) + 2*a*b^3*d^3*n*x^3*e^(n*ln(b*x + a)) + 6*b^4*d^3*
x^4*e^(n*ln(b*x + a)) + a*b^3*c^3*n^3*e^(n*ln(b*x + a)) + 9*b^4*c^3*n^2*x*e^(n*l
n(b*x + a)) + 21*a*b^3*c^2*d*n^2*x*e^(n*ln(b*x + a)) - 6*a^2*b^2*c*d^2*n^2*x*e^(
n*ln(b*x + a)) + 57*b^4*c^2*d*n*x^2*e^(n*ln(b*x + a)) + 12*a*b^3*c*d^2*n*x^2*e^(
n*ln(b*x + a)) - 3*a^2*b^2*d^3*n*x^2*e^(n*ln(b*x + a)) + 24*b^4*c*d^2*x^3*e^(n*l
n(b*x + a)) + 9*a*b^3*c^3*n^2*e^(n*ln(b*x + a)) - 3*a^2*b^2*c^2*d*n^2*e^(n*ln(b*
x + a)) + 26*b^4*c^3*n*x*e^(n*ln(b*x + a)) + 36*a*b^3*c^2*d*n*x*e^(n*ln(b*x + a)
) - 24*a^2*b^2*c*d^2*n*x*e^(n*ln(b*x + a)) + 6*a^3*b*d^3*n*x*e^(n*ln(b*x + a)) +
36*b^4*c^2*d*x^2*e^(n*ln(b*x + a)) + 26*a*b^3*c^3*n*e^(n*ln(b*x + a)) - 21*a^2*
b^2*c^2*d*n*e^(n*ln(b*x + a)) + 6*a^3*b*c*d^2*n*e^(n*ln(b*x + a)) + 24*b^4*c^3*x
*e^(n*ln(b*x + a)) + 24*a*b^3*c^3*e^(n*ln(b*x + a)) - 36*a^2*b^2*c^2*d*e^(n*ln(b
*x + a)) + 24*a^3*b*c*d^2*e^(n*ln(b*x + a)) - 6*a^4*d^3*e^(n*ln(b*x + a)))/(b^4*
n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)