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 = 0.0566002, 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, Rules used = {43} \[ -\frac{3 b^2 (b c-a d) (c+d x)^{n+3}}{d^4 (n+3)}-\frac{(b c-a d)^3 (c+d x)^{n+1}}{d^4 (n+1)}+\frac{3 b (b c-a d)^2 (c+d x)^{n+2}}{d^4 (n+2)}+\frac{b^3 (c+d x)^{n+4}}{d^4 (n+4)} \]

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))

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)^3 (c+d x)^n \, dx &=\int \left (\frac{(-b c+a d)^3 (c+d x)^n}{d^3}+\frac{3 b (b c-a d)^2 (c+d x)^{1+n}}{d^3}-\frac{3 b^2 (b c-a d) (c+d x)^{2+n}}{d^3}+\frac{b^3 (c+d x)^{3+n}}{d^3}\right ) \, dx\\ &=-\frac{(b c-a d)^3 (c+d x)^{1+n}}{d^4 (1+n)}+\frac{3 b (b c-a d)^2 (c+d x)^{2+n}}{d^4 (2+n)}-\frac{3 b^2 (b c-a d) (c+d x)^{3+n}}{d^4 (3+n)}+\frac{b^3 (c+d x)^{4+n}}{d^4 (4+n)}\\ \end{align*}

Mathematica [A]  time = 0.0745138, size = 95, normalized size = 0.86 \[ \frac{(c+d x)^{n+1} \left (-\frac{3 b^2 (c+d x)^2 (b c-a d)}{n+3}+\frac{3 b (c+d x) (b c-a d)^2}{n+2}-\frac{(b c-a d)^3}{n+1}+\frac{b^3 (c+d x)^3}{n+4}\right )}{d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(c + d*x)^n,x]

[Out]

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

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Maple [B]  time = 0.008, size = 386, normalized size = 3.5 \begin{align*}{\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 ) }} \end{align*}

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(-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)^3*(d*x+c)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.76685, size = 1011, normalized size = 9.11 \begin{align*} \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}} \end{align*}

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 = 4.44555, size = 4056, normalized size = 36.54 \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)**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**2*d**3/(6*c**5*
d**4 + 18*c**4*d**5*x + 18*c**3*d**6*x**2 + 6*c**2*d**7*x**3) + 9*a**2*b*c*d**4*x**2/(6*c**5*d**4 + 18*c**4*d*
*5*x + 18*c**3*d**6*x**2 + 6*c**2*d**7*x**3) + 3*a**2*b*d**5*x**3/(6*c**5*d**4 + 18*c**4*d**5*x + 18*c**3*d**6
*x**2 + 6*c**2*d**7*x**3) + 6*a*b**2*c*d**4*x**3/(6*c**5*d**4 + 18*c**4*d**5*x + 18*c**3*d**6*x**2 + 6*c**2*d*
*7*x**3) + 6*b**3*c**5*log(c/d + x)/(6*c**5*d**4 + 18*c**4*d**5*x + 18*c**3*d**6*x**2 + 6*c**2*d**7*x**3) + 2*
b**3*c**5/(6*c**5*d**4 + 18*c**4*d**5*x + 18*c**3*d**6*x**2 + 6*c**2*d**7*x**3) + 18*b**3*c**4*d*x*log(c/d + x
)/(6*c**5*d**4 + 18*c**4*d**5*x + 18*c**3*d**6*x**2 + 6*c**2*d**7*x**3) + 18*b**3*c**3*d**2*x**2*log(c/d + x)/
(6*c**5*d**4 + 18*c**4*d**5*x + 18*c**3*d**6*x**2 + 6*c**2*d**7*x**3) - 9*b**3*c**3*d**2*x**2/(6*c**5*d**4 + 1
8*c**4*d**5*x + 18*c**3*d**6*x**2 + 6*c**2*d**7*x**3) + 6*b**3*c**2*d**3*x**3*log(c/d + x)/(6*c**5*d**4 + 18*c
**4*d**5*x + 18*c**3*d**6*x**2 + 6*c**2*d**7*x**3) - 9*b**3*c**2*d**3*x**3/(6*c**5*d**4 + 18*c**4*d**5*x + 18*
c**3*d**6*x**2 + 6*c**2*d**7*x**3), Eq(n, -4)), (-a**3*c*d**3/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**2) +
3*a**2*b*d**4*x**2/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**2) + 6*a*b**2*c**3*d*log(c/d + x)/(2*c**3*d**4 +
4*c**2*d**5*x + 2*c*d**6*x**2) + 3*a*b**2*c**3*d/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**2) + 12*a*b**2*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) + 6*a*b**2*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) - 6*a*b**2*c*d**3*x**2/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**2)
- 6*b**3*c**4*log(c/d + x)/(2*c**3*d**4 + 4*c**2*d**5*x + 2*c*d**6*x**2) - 3*b**3*c**4/(2*c**3*d**4 + 4*c**2*d
**5*x + 2*c*d**6*x**2) - 12*b**3*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*b**3*
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*b**3*c**2*d**2*x**2/(2*c**3*d**4
+ 4*c**2*d**5*x + 2*c*d**6*x**2) + 2*b**3*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**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*lo
g(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 + 35*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 + 10*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 [B]  time = 1.0581, size = 1125, normalized size = 10.14 \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)^3*(d*x+c)^n,x, algorithm="giac")

[Out]

((d*x + c)^n*b^3*d^4*n^3*x^4 + (d*x + c)^n*b^3*c*d^3*n^3*x^3 + 3*(d*x + c)^n*a*b^2*d^4*n^3*x^3 + 6*(d*x + c)^n
*b^3*d^4*n^2*x^4 + 3*(d*x + c)^n*a*b^2*c*d^3*n^3*x^2 + 3*(d*x + c)^n*a^2*b*d^4*n^3*x^2 + 3*(d*x + c)^n*b^3*c*d
^3*n^2*x^3 + 21*(d*x + c)^n*a*b^2*d^4*n^2*x^3 + 11*(d*x + c)^n*b^3*d^4*n*x^4 + 3*(d*x + c)^n*a^2*b*c*d^3*n^3*x
+ (d*x + c)^n*a^3*d^4*n^3*x - 3*(d*x + c)^n*b^3*c^2*d^2*n^2*x^2 + 15*(d*x + c)^n*a*b^2*c*d^3*n^2*x^2 + 24*(d*
x + c)^n*a^2*b*d^4*n^2*x^2 + 2*(d*x + c)^n*b^3*c*d^3*n*x^3 + 42*(d*x + c)^n*a*b^2*d^4*n*x^3 + 6*(d*x + c)^n*b^
3*d^4*x^4 + (d*x + c)^n*a^3*c*d^3*n^3 - 6*(d*x + c)^n*a*b^2*c^2*d^2*n^2*x + 21*(d*x + c)^n*a^2*b*c*d^3*n^2*x +
9*(d*x + c)^n*a^3*d^4*n^2*x - 3*(d*x + c)^n*b^3*c^2*d^2*n*x^2 + 12*(d*x + c)^n*a*b^2*c*d^3*n*x^2 + 57*(d*x +
c)^n*a^2*b*d^4*n*x^2 + 24*(d*x + c)^n*a*b^2*d^4*x^3 - 3*(d*x + c)^n*a^2*b*c^2*d^2*n^2 + 9*(d*x + c)^n*a^3*c*d^
3*n^2 + 6*(d*x + c)^n*b^3*c^3*d*n*x - 24*(d*x + c)^n*a*b^2*c^2*d^2*n*x + 36*(d*x + c)^n*a^2*b*c*d^3*n*x + 26*(
d*x + c)^n*a^3*d^4*n*x + 36*(d*x + c)^n*a^2*b*d^4*x^2 + 6*(d*x + c)^n*a*b^2*c^3*d*n - 21*(d*x + c)^n*a^2*b*c^2
*d^2*n + 26*(d*x + c)^n*a^3*c*d^3*n + 24*(d*x + c)^n*a^3*d^4*x - 6*(d*x + c)^n*b^3*c^4 + 24*(d*x + c)^n*a*b^2*
c^3*d - 36*(d*x + c)^n*a^2*b*c^2*d^2 + 24*(d*x + c)^n*a^3*c*d^3)/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n
+ 24*d^4)