3.1852 \(\int (a+b x)^3 (c+d x)^n \, dx\)

Optimal. Leaf size=111 \[ -\frac{3 b^2 (b c-a d) (c+d x)^{n+3}}{d^4 (n+3)}-\frac{(b c-a d)^3 (c+d x)^{n+1}}{d^4 (n+1)}+\frac{3 b (b c-a d)^2 (c+d x)^{n+2}}{d^4 (n+2)}+\frac{b^3 (c+d x)^{n+4}}{d^4 (n+4)} \]

[Out]

-(((b*c - a*d)^3*(c + d*x)^(1 + n))/(d^4*(1 + n))) + (3*b*(b*c - a*d)^2*(c + d*x
)^(2 + n))/(d^4*(2 + n)) - (3*b^2*(b*c - a*d)*(c + d*x)^(3 + n))/(d^4*(3 + n)) +
(b^3*(c + d*x)^(4 + n))/(d^4*(4 + n))

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Rubi [A]  time = 0.127839, antiderivative size = 111, 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 \[ -\frac{3 b^2 (b c-a d) (c+d x)^{n+3}}{d^4 (n+3)}-\frac{(b c-a d)^3 (c+d x)^{n+1}}{d^4 (n+1)}+\frac{3 b (b c-a d)^2 (c+d x)^{n+2}}{d^4 (n+2)}+\frac{b^3 (c+d x)^{n+4}}{d^4 (n+4)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(c + d*x)^n,x]

[Out]

-(((b*c - a*d)^3*(c + d*x)^(1 + n))/(d^4*(1 + n))) + (3*b*(b*c - a*d)^2*(c + d*x
)^(2 + n))/(d^4*(2 + n)) - (3*b^2*(b*c - a*d)*(c + d*x)^(3 + n))/(d^4*(3 + n)) +
(b^3*(c + d*x)^(4 + n))/(d^4*(4 + n))

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Rubi in Sympy [A]  time = 28.3153, size = 95, normalized size = 0.86 \[ \frac{b^{3} \left (c + d x\right )^{n + 4}}{d^{4} \left (n + 4\right )} + \frac{3 b^{2} \left (c + d x\right )^{n + 3} \left (a d - b c\right )}{d^{4} \left (n + 3\right )} + \frac{3 b \left (c + d x\right )^{n + 2} \left (a d - b c\right )^{2}}{d^{4} \left (n + 2\right )} + \frac{\left (c + d x\right )^{n + 1} \left (a d - b c\right )^{3}}{d^{4} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(d*x+c)**n,x)

[Out]

b**3*(c + d*x)**(n + 4)/(d**4*(n + 4)) + 3*b**2*(c + d*x)**(n + 3)*(a*d - b*c)/(
d**4*(n + 3)) + 3*b*(c + d*x)**(n + 2)*(a*d - b*c)**2/(d**4*(n + 2)) + (c + d*x)
**(n + 1)*(a*d - b*c)**3/(d**4*(n + 1))

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Mathematica [A]  time = 0.168333, size = 178, normalized size = 1.6 \[ \frac{(c+d x)^{n+1} \left (a^3 d^3 \left (n^3+9 n^2+26 n+24\right )-3 a^2 b d^2 \left (n^2+7 n+12\right ) (c-d (n+1) x)+3 a b^2 d (n+4) \left (2 c^2-2 c d (n+1) x+d^2 \left (n^2+3 n+2\right ) x^2\right )+b^3 \left (-\left (6 c^3-6 c^2 d (n+1) x+3 c d^2 \left (n^2+3 n+2\right ) x^2-d^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )\right )\right )}{d^4 (n+1) (n+2) (n+3) (n+4)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3*(c + d*x)^n,x]

[Out]

((c + d*x)^(1 + n)*(a^3*d^3*(24 + 26*n + 9*n^2 + n^3) - 3*a^2*b*d^2*(12 + 7*n +
n^2)*(c - d*(1 + n)*x) + 3*a*b^2*d*(4 + n)*(2*c^2 - 2*c*d*(1 + n)*x + d^2*(2 + 3
*n + n^2)*x^2) - b^3*(6*c^3 - 6*c^2*d*(1 + n)*x + 3*c*d^2*(2 + 3*n + n^2)*x^2 -
d^3*(6 + 11*n + 6*n^2 + n^3)*x^3)))/(d^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))

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Maple [B]  time = 0.011, size = 386, normalized size = 3.5 \[{\frac{ \left ( dx+c \right ) ^{1+n} \left ({b}^{3}{d}^{3}{n}^{3}{x}^{3}+3\,a{b}^{2}{d}^{3}{n}^{3}{x}^{2}+6\,{b}^{3}{d}^{3}{n}^{2}{x}^{3}+3\,{a}^{2}b{d}^{3}{n}^{3}x+21\,a{b}^{2}{d}^{3}{n}^{2}{x}^{2}-3\,{b}^{3}c{d}^{2}{n}^{2}{x}^{2}+11\,{b}^{3}{d}^{3}n{x}^{3}+{a}^{3}{d}^{3}{n}^{3}+24\,{a}^{2}b{d}^{3}{n}^{2}x-6\,a{b}^{2}c{d}^{2}{n}^{2}x+42\,a{b}^{2}{d}^{3}n{x}^{2}-9\,{b}^{3}c{d}^{2}n{x}^{2}+6\,{x}^{3}{b}^{3}{d}^{3}+9\,{a}^{3}{d}^{3}{n}^{2}-3\,{a}^{2}bc{d}^{2}{n}^{2}+57\,{a}^{2}b{d}^{3}nx-30\,a{b}^{2}c{d}^{2}nx+24\,a{b}^{2}{d}^{3}{x}^{2}+6\,{b}^{3}{c}^{2}dnx-6\,{b}^{3}c{d}^{2}{x}^{2}+26\,{a}^{3}{d}^{3}n-21\,{a}^{2}bc{d}^{2}n+36\,{a}^{2}b{d}^{3}x+6\,a{b}^{2}{c}^{2}dn-24\,a{b}^{2}c{d}^{2}x+6\,{b}^{3}{c}^{2}dx+24\,{a}^{3}{d}^{3}-36\,{a}^{2}cb{d}^{2}+24\,a{b}^{2}{c}^{2}d-6\,{b}^{3}{c}^{3} \right ) }{{d}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(d*x+c)^n,x)

[Out]

(d*x+c)^(1+n)*(b^3*d^3*n^3*x^3+3*a*b^2*d^3*n^3*x^2+6*b^3*d^3*n^2*x^3+3*a^2*b*d^3
*n^3*x+21*a*b^2*d^3*n^2*x^2-3*b^3*c*d^2*n^2*x^2+11*b^3*d^3*n*x^3+a^3*d^3*n^3+24*
a^2*b*d^3*n^2*x-6*a*b^2*c*d^2*n^2*x+42*a*b^2*d^3*n*x^2-9*b^3*c*d^2*n*x^2+6*b^3*d
^3*x^3+9*a^3*d^3*n^2-3*a^2*b*c*d^2*n^2+57*a^2*b*d^3*n*x-30*a*b^2*c*d^2*n*x+24*a*
b^2*d^3*x^2+6*b^3*c^2*d*n*x-6*b^3*c*d^2*x^2+26*a^3*d^3*n-21*a^2*b*c*d^2*n+36*a^2
*b*d^3*x+6*a*b^2*c^2*d*n-24*a*b^2*c*d^2*x+6*b^3*c^2*d*x+24*a^3*d^3-36*a^2*b*c*d^
2+24*a*b^2*c^2*d-6*b^3*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((b*x + a)^3*(d*x + c)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228875, size = 670, normalized size = 6.04 \[ \frac{{\left (a^{3} c d^{3} n^{3} - 6 \, b^{3} c^{4} + 24 \, a b^{2} c^{3} d - 36 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3} +{\left (b^{3} d^{4} n^{3} + 6 \, b^{3} d^{4} n^{2} + 11 \, b^{3} d^{4} n + 6 \, b^{3} d^{4}\right )} x^{4} +{\left (24 \, a b^{2} d^{4} +{\left (b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} n^{3} + 3 \,{\left (b^{3} c d^{3} + 7 \, a b^{2} d^{4}\right )} n^{2} + 2 \,{\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} n\right )} x^{3} - 3 \,{\left (a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3}\right )} n^{2} + 3 \,{\left (12 \, a^{2} b d^{4} +{\left (a b^{2} c d^{3} + a^{2} b d^{4}\right )} n^{3} -{\left (b^{3} c^{2} d^{2} - 5 \, a b^{2} c d^{3} - 8 \, a^{2} b d^{4}\right )} n^{2} -{\left (b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} - 19 \, a^{2} b d^{4}\right )} n\right )} x^{2} +{\left (6 \, a b^{2} c^{3} d - 21 \, a^{2} b c^{2} d^{2} + 26 \, a^{3} c d^{3}\right )} n +{\left (24 \, a^{3} d^{4} +{\left (3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} n^{3} - 3 \,{\left (2 \, a b^{2} c^{2} d^{2} - 7 \, a^{2} b c d^{3} - 3 \, a^{3} d^{4}\right )} n^{2} + 2 \,{\left (3 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 18 \, a^{2} b c d^{3} + 13 \, a^{3} d^{4}\right )} n\right )} x\right )}{\left (d x + c\right )}^{n}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*(d*x + c)^n,x, algorithm="fricas")

[Out]

(a^3*c*d^3*n^3 - 6*b^3*c^4 + 24*a*b^2*c^3*d - 36*a^2*b*c^2*d^2 + 24*a^3*c*d^3 +
(b^3*d^4*n^3 + 6*b^3*d^4*n^2 + 11*b^3*d^4*n + 6*b^3*d^4)*x^4 + (24*a*b^2*d^4 + (
b^3*c*d^3 + 3*a*b^2*d^4)*n^3 + 3*(b^3*c*d^3 + 7*a*b^2*d^4)*n^2 + 2*(b^3*c*d^3 +
21*a*b^2*d^4)*n)*x^3 - 3*(a^2*b*c^2*d^2 - 3*a^3*c*d^3)*n^2 + 3*(12*a^2*b*d^4 + (
a*b^2*c*d^3 + a^2*b*d^4)*n^3 - (b^3*c^2*d^2 - 5*a*b^2*c*d^3 - 8*a^2*b*d^4)*n^2 -
(b^3*c^2*d^2 - 4*a*b^2*c*d^3 - 19*a^2*b*d^4)*n)*x^2 + (6*a*b^2*c^3*d - 21*a^2*b
*c^2*d^2 + 26*a^3*c*d^3)*n + (24*a^3*d^4 + (3*a^2*b*c*d^3 + a^3*d^4)*n^3 - 3*(2*
a*b^2*c^2*d^2 - 7*a^2*b*c*d^3 - 3*a^3*d^4)*n^2 + 2*(3*b^3*c^3*d - 12*a*b^2*c^2*d
^2 + 18*a^2*b*c*d^3 + 13*a^3*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 = 6.3583, size = 4004, normalized size = 36.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(d*x+c)**n,x)

[Out]

Piecewise((c**n*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), Eq(d, 0)
), (-2*a**3*c*d**3/(6*c**4*d**4 + 18*c**3*d**5*x + 18*c**2*d**6*x**2 + 6*c*d**7*
x**3) - 3*a**2*b*c**2*d**2/(6*c**4*d**4 + 18*c**3*d**5*x + 18*c**2*d**6*x**2 + 6
*c*d**7*x**3) - 9*a**2*b*c*d**3*x/(6*c**4*d**4 + 18*c**3*d**5*x + 18*c**2*d**6*x
**2 + 6*c*d**7*x**3) + 6*a*b**2*d**4*x**3/(6*c**4*d**4 + 18*c**3*d**5*x + 18*c**
2*d**6*x**2 + 6*c*d**7*x**3) + 6*b**3*c**4*log(c/d + x)/(6*c**4*d**4 + 18*c**3*d
**5*x + 18*c**2*d**6*x**2 + 6*c*d**7*x**3) + 5*b**3*c**4/(6*c**4*d**4 + 18*c**3*
d**5*x + 18*c**2*d**6*x**2 + 6*c*d**7*x**3) + 18*b**3*c**3*d*x*log(c/d + x)/(6*c
**4*d**4 + 18*c**3*d**5*x + 18*c**2*d**6*x**2 + 6*c*d**7*x**3) + 9*b**3*c**3*d*x
/(6*c**4*d**4 + 18*c**3*d**5*x + 18*c**2*d**6*x**2 + 6*c*d**7*x**3) + 18*b**3*c*
*2*d**2*x**2*log(c/d + x)/(6*c**4*d**4 + 18*c**3*d**5*x + 18*c**2*d**6*x**2 + 6*
c*d**7*x**3) + 6*b**3*c*d**3*x**3*log(c/d + x)/(6*c**4*d**4 + 18*c**3*d**5*x + 1
8*c**2*d**6*x**2 + 6*c*d**7*x**3) - 6*b**3*c*d**3*x**3/(6*c**4*d**4 + 18*c**3*d*
*5*x + 18*c**2*d**6*x**2 + 6*c*d**7*x**3), Eq(n, -4)), (-a**3*d**3/(2*c**2*d**4
+ 4*c*d**5*x + 2*d**6*x**2) - 3*a**2*b*c*d**2/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6
*x**2) - 6*a**2*b*d**3*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 6*a*b**2*c**
2*d*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 9*a*b**2*c**2*d/(2*c
**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 12*a*b**2*c*d**2*x*log(c/d + x)/(2*c**2*d
**4 + 4*c*d**5*x + 2*d**6*x**2) + 12*a*b**2*c*d**2*x/(2*c**2*d**4 + 4*c*d**5*x +
2*d**6*x**2) + 6*a*b**2*d**3*x**2*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d*
*6*x**2) - 6*b**3*c**3*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 9
*b**3*c**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*b**3*c**2*d*x*log(c/d +
x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*b**3*c**2*d*x/(2*c**2*d**4 + 4
*c*d**5*x + 2*d**6*x**2) - 6*b**3*c*d**2*x**2*log(c/d + x)/(2*c**2*d**4 + 4*c*d*
*5*x + 2*d**6*x**2) + 2*b**3*d**3*x**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2),
Eq(n, -3)), (-2*a**3*d**3/(2*c*d**4 + 2*d**5*x) + 6*a**2*b*c*d**2*log(c/d + x)/
(2*c*d**4 + 2*d**5*x) + 6*a**2*b*c*d**2/(2*c*d**4 + 2*d**5*x) + 6*a**2*b*d**3*x*
log(c/d + x)/(2*c*d**4 + 2*d**5*x) - 12*a*b**2*c**2*d*log(c/d + x)/(2*c*d**4 + 2
*d**5*x) - 12*a*b**2*c**2*d/(2*c*d**4 + 2*d**5*x) - 12*a*b**2*c*d**2*x*log(c/d +
x)/(2*c*d**4 + 2*d**5*x) + 6*a*b**2*d**3*x**2/(2*c*d**4 + 2*d**5*x) + 6*b**3*c*
*3*log(c/d + x)/(2*c*d**4 + 2*d**5*x) + 6*b**3*c**3/(2*c*d**4 + 2*d**5*x) + 6*b*
*3*c**2*d*x*log(c/d + x)/(2*c*d**4 + 2*d**5*x) - 3*b**3*c*d**2*x**2/(2*c*d**4 +
2*d**5*x) + b**3*d**3*x**3/(2*c*d**4 + 2*d**5*x), Eq(n, -2)), (a**3*log(c/d + x)
/d - 3*a**2*b*c*log(c/d + x)/d**2 + 3*a**2*b*x/d + 3*a*b**2*c**2*log(c/d + x)/d*
*3 - 3*a*b**2*c*x/d**2 + 3*a*b**2*x**2/(2*d) - b**3*c**3*log(c/d + x)/d**4 + b**
3*c**2*x/d**3 - b**3*c*x**2/(2*d**2) + b**3*x**3/(3*d), Eq(n, -1)), (a**3*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**3*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**3*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**3*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**3*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**3*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**3*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**3*d**4*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 3
5*d**4*n**2 + 50*d**4*n + 24*d**4) - 3*a**2*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) - 21*a**2*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) -
36*a**2*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) + 3*a**2*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) + 21*a**2*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) + 36*a**2*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) + 3*a**2*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) + 24*a**2*b*d**4*n**2*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) + 57*a**2*b*d**4*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) + 36*a**2
*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) + 6*a*b**2*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) + 24*a*b**2*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) - 6*a*b**2*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) - 24*a*b**2*
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) + 3*a*b**2*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) + 15*a*b**2*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) + 12*a*b**2*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) + 3*a*b**2*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) + 21*a*b**2*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) + 42*a*b**2*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) + 24*a*
b**2*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*b**3*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*b**3*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*b**3*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*b**3*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) + b**3*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*b**3*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*b**3*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) + b**3*
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*b**3*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*b**3*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*b**3*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 = 0.248451, size = 1233, normalized size = 11.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*(d*x + c)^n,x, algorithm="giac")

[Out]

(b^3*d^4*n^3*x^4*e^(n*ln(d*x + c)) + b^3*c*d^3*n^3*x^3*e^(n*ln(d*x + c)) + 3*a*b
^2*d^4*n^3*x^3*e^(n*ln(d*x + c)) + 6*b^3*d^4*n^2*x^4*e^(n*ln(d*x + c)) + 3*a*b^2
*c*d^3*n^3*x^2*e^(n*ln(d*x + c)) + 3*a^2*b*d^4*n^3*x^2*e^(n*ln(d*x + c)) + 3*b^3
*c*d^3*n^2*x^3*e^(n*ln(d*x + c)) + 21*a*b^2*d^4*n^2*x^3*e^(n*ln(d*x + c)) + 11*b
^3*d^4*n*x^4*e^(n*ln(d*x + c)) + 3*a^2*b*c*d^3*n^3*x*e^(n*ln(d*x + c)) + a^3*d^4
*n^3*x*e^(n*ln(d*x + c)) - 3*b^3*c^2*d^2*n^2*x^2*e^(n*ln(d*x + c)) + 15*a*b^2*c*
d^3*n^2*x^2*e^(n*ln(d*x + c)) + 24*a^2*b*d^4*n^2*x^2*e^(n*ln(d*x + c)) + 2*b^3*c
*d^3*n*x^3*e^(n*ln(d*x + c)) + 42*a*b^2*d^4*n*x^3*e^(n*ln(d*x + c)) + 6*b^3*d^4*
x^4*e^(n*ln(d*x + c)) + a^3*c*d^3*n^3*e^(n*ln(d*x + c)) - 6*a*b^2*c^2*d^2*n^2*x*
e^(n*ln(d*x + c)) + 21*a^2*b*c*d^3*n^2*x*e^(n*ln(d*x + c)) + 9*a^3*d^4*n^2*x*e^(
n*ln(d*x + c)) - 3*b^3*c^2*d^2*n*x^2*e^(n*ln(d*x + c)) + 12*a*b^2*c*d^3*n*x^2*e^
(n*ln(d*x + c)) + 57*a^2*b*d^4*n*x^2*e^(n*ln(d*x + c)) + 24*a*b^2*d^4*x^3*e^(n*l
n(d*x + c)) - 3*a^2*b*c^2*d^2*n^2*e^(n*ln(d*x + c)) + 9*a^3*c*d^3*n^2*e^(n*ln(d*
x + c)) + 6*b^3*c^3*d*n*x*e^(n*ln(d*x + c)) - 24*a*b^2*c^2*d^2*n*x*e^(n*ln(d*x +
c)) + 36*a^2*b*c*d^3*n*x*e^(n*ln(d*x + c)) + 26*a^3*d^4*n*x*e^(n*ln(d*x + c)) +
36*a^2*b*d^4*x^2*e^(n*ln(d*x + c)) + 6*a*b^2*c^3*d*n*e^(n*ln(d*x + c)) - 21*a^2
*b*c^2*d^2*n*e^(n*ln(d*x + c)) + 26*a^3*c*d^3*n*e^(n*ln(d*x + c)) + 24*a^3*d^4*x
*e^(n*ln(d*x + c)) - 6*b^3*c^4*e^(n*ln(d*x + c)) + 24*a*b^2*c^3*d*e^(n*ln(d*x +
c)) - 36*a^2*b*c^2*d^2*e^(n*ln(d*x + c)) + 24*a^3*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)