3.28 \(\int (c+d x)^n (A+B x+C x^2+D x^3) \, dx\)

Optimal. Leaf size=126 \[ \frac{(c+d x)^{n+1} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4 (n+1)}-\frac{(c+d x)^{n+2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4 (n+2)}+\frac{(C d-3 c D) (c+d x)^{n+3}}{d^4 (n+3)}+\frac{D (c+d x)^{n+4}}{d^4 (n+4)} \]

[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 = 0.0718049, 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, Rules used = {1850} \[ \frac{(c+d x)^{n+1} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^4 (n+1)}-\frac{(c+d x)^{n+2} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4 (n+2)}+\frac{(C d-3 c D) (c+d x)^{n+3}}{d^4 (n+3)}+\frac{D (c+d x)^{n+4}}{d^4 (n+4)} \]

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)
)

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (\frac{\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^n}{d^3}+\frac{\left (-2 c C d+B d^2+3 c^2 D\right ) (c+d x)^{1+n}}{d^3}+\frac{(C d-3 c D) (c+d x)^{2+n}}{d^3}+\frac{D (c+d x)^{3+n}}{d^3}\right ) \, dx\\ &=\frac{\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{1+n}}{d^4 (1+n)}-\frac{\left (2 c C d-B d^2-3 c^2 D\right ) (c+d x)^{2+n}}{d^4 (2+n)}+\frac{(C d-3 c D) (c+d x)^{3+n}}{d^4 (3+n)}+\frac{D (c+d x)^{4+n}}{d^4 (4+n)}\\ \end{align*}

Mathematica [A]  time = 0.105272, size = 108, normalized size = 0.86 \[ \frac{(c+d x)^{n+1} \left (\frac{A d^3-B c d^2+c^2 C d+c^3 (-D)}{n+1}+\frac{(c+d x) \left (B d^2+3 c^2 D-2 c C d\right )}{n+2}+\frac{(c+d x)^2 (C d-3 c D)}{n+3}+\frac{D (c+d x)^3}{n+4}\right )}{d^4} \]

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)*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)/(1 + n) + ((-2*c*C*d + B*d^2 + 3*c^2*D)*(c + d*x))/(2
+ n) + ((C*d - 3*c*D)*(c + d*x)^2)/(3 + n) + (D*(c + d*x)^3)/(4 + n)))/d^4

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Maple [B]  time = 0.005, size = 308, normalized size = 2.4 \begin{align*}{\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 ) }} \end{align*}

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(-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((d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 4.69687, size = 3750, normalized size = 29.76 \begin{align*} \text{result too large to display} \end{align*}

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*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) - 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) - 3*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) + 2*C*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*D*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*D*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*D*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*D*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*D*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*D*c*d**3*x**3*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*D*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*d**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - B*c*d**2/(
2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 2*B*d**3*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 2*C*c**2*d*log
(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 3*C*c**2*d/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 4*C
*c*d**2*x*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 4*C*c*d**2*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d
**6*x**2) + 2*C*d**3*x**2*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 6*D*c**3*log(c/d + x)/(2*c**
2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 9*D*c**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*D*c**2*d*x*log(c/d
+ x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*D*c**2*d*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 6*D*
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*D*d**3*x**3/(2*c**2*d**4 + 4*c*d**5*x +
2*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) + 2
4*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 + 10*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))

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Giac [B]  time = 2.57679, size = 983, normalized size = 7.8 \begin{align*} \text{result too large to display} \end{align*}

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, algorithm="giac")

[Out]

((d*x + c)^n*D*d^4*n^3*x^4 + (d*x + c)^n*D*c*d^3*n^3*x^3 + (d*x + c)^n*C*d^4*n^3*x^3 + 6*(d*x + c)^n*D*d^4*n^2
*x^4 + (d*x + c)^n*C*c*d^3*n^3*x^2 + (d*x + c)^n*B*d^4*n^3*x^2 + 3*(d*x + c)^n*D*c*d^3*n^2*x^3 + 7*(d*x + c)^n
*C*d^4*n^2*x^3 + 11*(d*x + c)^n*D*d^4*n*x^4 + (d*x + c)^n*B*c*d^3*n^3*x + (d*x + c)^n*A*d^4*n^3*x - 3*(d*x + c
)^n*D*c^2*d^2*n^2*x^2 + 5*(d*x + c)^n*C*c*d^3*n^2*x^2 + 8*(d*x + c)^n*B*d^4*n^2*x^2 + 2*(d*x + c)^n*D*c*d^3*n*
x^3 + 14*(d*x + c)^n*C*d^4*n*x^3 + 6*(d*x + c)^n*D*d^4*x^4 + (d*x + c)^n*A*c*d^3*n^3 - 2*(d*x + c)^n*C*c^2*d^2
*n^2*x + 7*(d*x + c)^n*B*c*d^3*n^2*x + 9*(d*x + c)^n*A*d^4*n^2*x - 3*(d*x + c)^n*D*c^2*d^2*n*x^2 + 4*(d*x + c)
^n*C*c*d^3*n*x^2 + 19*(d*x + c)^n*B*d^4*n*x^2 + 8*(d*x + c)^n*C*d^4*x^3 - (d*x + c)^n*B*c^2*d^2*n^2 + 9*(d*x +
c)^n*A*c*d^3*n^2 + 6*(d*x + c)^n*D*c^3*d*n*x - 8*(d*x + c)^n*C*c^2*d^2*n*x + 12*(d*x + c)^n*B*c*d^3*n*x + 26*
(d*x + c)^n*A*d^4*n*x + 12*(d*x + c)^n*B*d^4*x^2 + 2*(d*x + c)^n*C*c^3*d*n - 7*(d*x + c)^n*B*c^2*d^2*n + 26*(d
*x + c)^n*A*c*d^3*n + 24*(d*x + c)^n*A*d^4*x - 6*(d*x + c)^n*D*c^4 + 8*(d*x + c)^n*C*c^3*d - 12*(d*x + c)^n*B*
c^2*d^2 + 24*(d*x + c)^n*A*c*d^3)/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n + 24*d^4)