3.932 \(\int (a+b x)^n (c+d x)^3 \, dx\)
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 = 0.0497695, 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, Rules used =
{43} \[ \frac{3 d^2 (b c-a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac{(b c-a d)^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 d (b c-a d)^2 (a+b x)^{n+2}}{b^4 (n+2)}+\frac{d^3 (a+b x)^{n+4}}{b^4 (n+4)} \]
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))
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int (a+b x)^n (c+d x)^3 \, dx &=\int \left (\frac{(b c-a d)^3 (a+b x)^n}{b^3}+\frac{3 d (b c-a d)^2 (a+b x)^{1+n}}{b^3}+\frac{3 d^2 (b c-a d) (a+b x)^{2+n}}{b^3}+\frac{d^3 (a+b x)^{3+n}}{b^3}\right ) \, dx\\ &=\frac{(b c-a d)^3 (a+b x)^{1+n}}{b^4 (1+n)}+\frac{3 d (b c-a d)^2 (a+b x)^{2+n}}{b^4 (2+n)}+\frac{3 d^2 (b c-a d) (a+b x)^{3+n}}{b^4 (3+n)}+\frac{d^3 (a+b x)^{4+n}}{b^4 (4+n)}\\ \end{align*}
Mathematica [A] time = 0.0661798, size = 94, normalized size = 0.85 \[ \frac{(a+b x)^{n+1} \left (\frac{3 d^2 (a+b x)^2 (b c-a d)}{n+3}+\frac{3 d (a+b x) (b c-a d)^2}{n+2}+\frac{(b c-a d)^3}{n+1}+\frac{d^3 (a+b x)^3}{n+4}\right )}{b^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^n*(c + d*x)^3,x]
[Out]
((a + b*x)^(1 + n)*((b*c - a*d)^3/(1 + n) + (3*d*(b*c - a*d)^2*(a + b*x))/(2 + n) + (3*d^2*(b*c - a*d)*(a + b*
x)^2)/(3 + n) + (d^3*(a + b*x)^3)/(4 + n)))/b^4
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Maple [B] time = 0.007, size = 389, normalized size = 3.5 \begin{align*} -{\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\,{d}^{3}{x}^{3}{b}^{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 ) }} \end{align*}
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-2
1*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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x+c)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.67852, size = 1011, normalized size = 9.19 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x+c)^3,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 = 6.09511, size = 4004, normalized size = 36.4 \begin{align*} \text{result too large to display} \end{align*}
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**4*d**3*log(a/b + x)/(
6*a**4*b**4 + 18*a**3*b**5*x + 18*a**2*b**6*x**2 + 6*a*b**7*x**3) + 5*a**4*d**3/(6*a**4*b**4 + 18*a**3*b**5*x
+ 18*a**2*b**6*x**2 + 6*a*b**7*x**3) + 18*a**3*b*d**3*x*log(a/b + x)/(6*a**4*b**4 + 18*a**3*b**5*x + 18*a**2*b
**6*x**2 + 6*a*b**7*x**3) + 9*a**3*b*d**3*x/(6*a**4*b**4 + 18*a**3*b**5*x + 18*a**2*b**6*x**2 + 6*a*b**7*x**3)
- 3*a**2*b**2*c**2*d/(6*a**4*b**4 + 18*a**3*b**5*x + 18*a**2*b**6*x**2 + 6*a*b**7*x**3) + 18*a**2*b**2*d**3*x
**2*log(a/b + x)/(6*a**4*b**4 + 18*a**3*b**5*x + 18*a**2*b**6*x**2 + 6*a*b**7*x**3) - 2*a*b**3*c**3/(6*a**4*b*
*4 + 18*a**3*b**5*x + 18*a**2*b**6*x**2 + 6*a*b**7*x**3) - 9*a*b**3*c**2*d*x/(6*a**4*b**4 + 18*a**3*b**5*x + 1
8*a**2*b**6*x**2 + 6*a*b**7*x**3) + 6*a*b**3*d**3*x**3*log(a/b + x)/(6*a**4*b**4 + 18*a**3*b**5*x + 18*a**2*b*
*6*x**2 + 6*a*b**7*x**3) - 6*a*b**3*d**3*x**3/(6*a**4*b**4 + 18*a**3*b**5*x + 18*a**2*b**6*x**2 + 6*a*b**7*x**
3) + 6*b**4*c*d**2*x**3/(6*a**4*b**4 + 18*a**3*b**5*x + 18*a**2*b**6*x**2 + 6*a*b**7*x**3), Eq(n, -4)), (-6*a*
*3*d**3*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3*d**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6
*x**2) + 6*a**2*b*c*d**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 9*a**2*b*c*d**2/(2*a**2*b**4
+ 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d**3*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2
*b*d**3*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 3*a*b**2*c**2*d/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2)
+ 12*a*b**2*c*d**2*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 12*a*b**2*c*d**2*x/(2*a**2*b**4 +
4*a*b**5*x + 2*b**6*x**2) - 6*a*b**2*d**3*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - b**3*c
**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*b**3*c**2*d*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 6*b*
*3*c*d**2*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*d**3*x**3/(2*a**2*b**4 + 4*a*b**
5*x + 2*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 + 2
4*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 [B] time = 2.39108, size = 1125, normalized size = 10.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x+c)^3,x, algorithm="giac")
[Out]
((b*x + a)^n*b^4*d^3*n^3*x^4 + 3*(b*x + a)^n*b^4*c*d^2*n^3*x^3 + (b*x + a)^n*a*b^3*d^3*n^3*x^3 + 6*(b*x + a)^n
*b^4*d^3*n^2*x^4 + 3*(b*x + a)^n*b^4*c^2*d*n^3*x^2 + 3*(b*x + a)^n*a*b^3*c*d^2*n^3*x^2 + 21*(b*x + a)^n*b^4*c*
d^2*n^2*x^3 + 3*(b*x + a)^n*a*b^3*d^3*n^2*x^3 + 11*(b*x + a)^n*b^4*d^3*n*x^4 + (b*x + a)^n*b^4*c^3*n^3*x + 3*(
b*x + a)^n*a*b^3*c^2*d*n^3*x + 24*(b*x + a)^n*b^4*c^2*d*n^2*x^2 + 15*(b*x + a)^n*a*b^3*c*d^2*n^2*x^2 - 3*(b*x
+ a)^n*a^2*b^2*d^3*n^2*x^2 + 42*(b*x + a)^n*b^4*c*d^2*n*x^3 + 2*(b*x + a)^n*a*b^3*d^3*n*x^3 + 6*(b*x + a)^n*b^
4*d^3*x^4 + (b*x + a)^n*a*b^3*c^3*n^3 + 9*(b*x + a)^n*b^4*c^3*n^2*x + 21*(b*x + a)^n*a*b^3*c^2*d*n^2*x - 6*(b*
x + a)^n*a^2*b^2*c*d^2*n^2*x + 57*(b*x + a)^n*b^4*c^2*d*n*x^2 + 12*(b*x + a)^n*a*b^3*c*d^2*n*x^2 - 3*(b*x + a)
^n*a^2*b^2*d^3*n*x^2 + 24*(b*x + a)^n*b^4*c*d^2*x^3 + 9*(b*x + a)^n*a*b^3*c^3*n^2 - 3*(b*x + a)^n*a^2*b^2*c^2*
d*n^2 + 26*(b*x + a)^n*b^4*c^3*n*x + 36*(b*x + a)^n*a*b^3*c^2*d*n*x - 24*(b*x + a)^n*a^2*b^2*c*d^2*n*x + 6*(b*
x + a)^n*a^3*b*d^3*n*x + 36*(b*x + a)^n*b^4*c^2*d*x^2 + 26*(b*x + a)^n*a*b^3*c^3*n - 21*(b*x + a)^n*a^2*b^2*c^
2*d*n + 6*(b*x + a)^n*a^3*b*c*d^2*n + 24*(b*x + a)^n*b^4*c^3*x + 24*(b*x + a)^n*a*b^3*c^3 - 36*(b*x + a)^n*a^2
*b^2*c^2*d + 24*(b*x + a)^n*a^3*b*c*d^2 - 6*(b*x + a)^n*a^4*d^3)/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n
+ 24*b^4)