3.918 \(\int x^2 (a+b x)^n (c+d x) \, dx\)

Optimal. Leaf size=104 \[ \frac{a^2 (b c-a d) (a+b x)^{n+1}}{b^4 (n+1)}-\frac{a (2 b c-3 a d) (a+b x)^{n+2}}{b^4 (n+2)}+\frac{(b c-3 a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]

[Out]

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

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Rubi [A]  time = 0.052563, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{a^2 (b c-a d) (a+b x)^{n+1}}{b^4 (n+1)}-\frac{a (2 b c-3 a d) (a+b x)^{n+2}}{b^4 (n+2)}+\frac{(b c-3 a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^n*(c + d*x),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0696848, size = 87, normalized size = 0.84 \[ \frac{(a+b x)^{n+1} \left (\frac{a^2 (b c-a d)}{n+1}+\frac{(a+b x)^2 (b c-3 a d)}{n+3}+\frac{a (a+b x) (3 a d-2 b c)}{n+2}+\frac{d (a+b x)^3}{n+4}\right )}{b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^n*(c + d*x),x]

[Out]

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

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Maple [B]  time = 0.006, size = 222, normalized size = 2.1 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}d{n}^{3}{x}^{3}-{b}^{3}c{n}^{3}{x}^{2}-6\,{b}^{3}d{n}^{2}{x}^{3}+3\,a{b}^{2}d{n}^{2}{x}^{2}-7\,{b}^{3}c{n}^{2}{x}^{2}-11\,{b}^{3}dn{x}^{3}+2\,a{b}^{2}c{n}^{2}x+9\,a{b}^{2}dn{x}^{2}-14\,{b}^{3}cn{x}^{2}-6\,{b}^{3}d{x}^{3}-6\,{a}^{2}bdnx+10\,a{b}^{2}cnx+6\,a{b}^{2}d{x}^{2}-8\,{b}^{3}c{x}^{2}-2\,{a}^{2}bcn-6\,{a}^{2}bdx+8\,a{b}^{2}cx+6\,{a}^{3}d-8\,{a}^{2}bc \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(x^2*(b*x+a)^n*(d*x+c),x)

[Out]

-(b*x+a)^(1+n)*(-b^3*d*n^3*x^3-b^3*c*n^3*x^2-6*b^3*d*n^2*x^3+3*a*b^2*d*n^2*x^2-7*b^3*c*n^2*x^2-11*b^3*d*n*x^3+
2*a*b^2*c*n^2*x+9*a*b^2*d*n*x^2-14*b^3*c*n*x^2-6*b^3*d*x^3-6*a^2*b*d*n*x+10*a*b^2*c*n*x+6*a*b^2*d*x^2-8*b^3*c*
x^2-2*a^2*b*c*n-6*a^2*b*d*x+8*a*b^2*c*x+6*a^3*d-8*a^2*b*c)/b^4/(n^4+10*n^3+35*n^2+50*n+24)

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Maxima [A]  time = 1.08589, size = 232, normalized size = 2.23 \begin{align*} \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} c}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )}{\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x+a)^n*(d*x+c),x, algorithm="maxima")

[Out]

((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c/((n^3 + 6*n^2 + 11*n + 6)*
b^3) + ((n^3 + 6*n^2 + 11*n + 6)*b^4*x^4 + (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n
*x - 6*a^4)*(b*x + a)^n*d/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4)

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Fricas [B]  time = 1.57925, size = 524, normalized size = 5.04 \begin{align*} \frac{{\left (2 \, a^{3} b c n + 8 \, a^{3} b c - 6 \, a^{4} d +{\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} +{\left (8 \, b^{4} c +{\left (b^{4} c + a b^{3} d\right )} n^{3} +{\left (7 \, b^{4} c + 3 \, a b^{3} d\right )} n^{2} + 2 \,{\left (7 \, b^{4} c + a b^{3} d\right )} n\right )} x^{3} +{\left (a b^{3} c n^{3} +{\left (5 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} n^{2} +{\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} n\right )} x^{2} - 2 \,{\left (a^{2} b^{2} c n^{2} +{\left (4 \, a^{2} b^{2} c - 3 \, a^{3} b d\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(x^2*(b*x+a)^n*(d*x+c),x, algorithm="fricas")

[Out]

(2*a^3*b*c*n + 8*a^3*b*c - 6*a^4*d + (b^4*d*n^3 + 6*b^4*d*n^2 + 11*b^4*d*n + 6*b^4*d)*x^4 + (8*b^4*c + (b^4*c
+ a*b^3*d)*n^3 + (7*b^4*c + 3*a*b^3*d)*n^2 + 2*(7*b^4*c + a*b^3*d)*n)*x^3 + (a*b^3*c*n^3 + (5*a*b^3*c - 3*a^2*
b^2*d)*n^2 + (4*a*b^3*c - 3*a^2*b^2*d)*n)*x^2 - 2*(a^2*b^2*c*n^2 + (4*a^2*b^2*c - 3*a^3*b*d)*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 = 4.77823, size = 2399, normalized size = 23.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x+a)**n*(d*x+c),x)

[Out]

Piecewise((a**n*(c*x**3/3 + d*x**4/4), Eq(b, 0)), (6*a**4*d*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/(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*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*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) + 18*a**2*b**2*d*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) + 6*a*b**3*d*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*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) + 2*b**4*c*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*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3*d/(2*a**2*b**4 + 4*a*b**5*x +
2*b**6*x**2) + 2*a**2*b*c*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 3*a**2*b*c/(2*a**2*b**4 + 4
*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d*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*x
/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 4*a*b**2*c*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2)
+ 4*a*b**2*c*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*a*b**2*d*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**
5*x + 2*b**6*x**2) + 2*b**3*c*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*x**3/(2*a*
*2*b**4 + 4*a*b**5*x + 2*b**6*x**2), Eq(n, -3)), (6*a**3*d*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3*d/(2*a*
b**4 + 2*b**5*x) - 4*a**2*b*c*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 4*a**2*b*c/(2*a*b**4 + 2*b**5*x) + 6*a**2*b
*d*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 4*a*b**2*c*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*d*x**2/(2
*a*b**4 + 2*b**5*x) + 2*b**3*c*x**2/(2*a*b**4 + 2*b**5*x) + b**3*d*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a
**3*d*log(a/b + x)/b**4 + a**2*c*log(a/b + x)/b**3 + a**2*d*x/b**3 - a*c*x/b**2 - a*d*x**2/(2*b**2) + c*x**2/(
2*b) + d*x**3/(3*b), Eq(n, -1)), (-6*a**4*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) + 2*a**3*b*c*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) + 8*a**
3*b*c*(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*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) - 2*a**2*b**2*c*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) - 8*a**2*b**2*c*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*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*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*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) + 5*a*b**3*c*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) + 4*a*b**3*c*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*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*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*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*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) + 7*b**4*c*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) + 14*b**4*c*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) + 8*b**4*c*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*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*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*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*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))

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Giac [B]  time = 2.59953, size = 582, normalized size = 5.6 \begin{align*} \frac{{\left (b x + a\right )}^{n} b^{4} d n^{3} x^{4} +{\left (b x + a\right )}^{n} b^{4} c n^{3} x^{3} +{\left (b x + a\right )}^{n} a b^{3} d n^{3} x^{3} + 6 \,{\left (b x + a\right )}^{n} b^{4} d n^{2} x^{4} +{\left (b x + a\right )}^{n} a b^{3} c n^{3} x^{2} + 7 \,{\left (b x + a\right )}^{n} b^{4} c n^{2} x^{3} + 3 \,{\left (b x + a\right )}^{n} a b^{3} d n^{2} x^{3} + 11 \,{\left (b x + a\right )}^{n} b^{4} d n x^{4} + 5 \,{\left (b x + a\right )}^{n} a b^{3} c n^{2} x^{2} - 3 \,{\left (b x + a\right )}^{n} a^{2} b^{2} d n^{2} x^{2} + 14 \,{\left (b x + a\right )}^{n} b^{4} c n x^{3} + 2 \,{\left (b x + a\right )}^{n} a b^{3} d n x^{3} + 6 \,{\left (b x + a\right )}^{n} b^{4} d x^{4} - 2 \,{\left (b x + a\right )}^{n} a^{2} b^{2} c n^{2} x + 4 \,{\left (b x + a\right )}^{n} a b^{3} c n x^{2} - 3 \,{\left (b x + a\right )}^{n} a^{2} b^{2} d n x^{2} + 8 \,{\left (b x + a\right )}^{n} b^{4} c x^{3} - 8 \,{\left (b x + a\right )}^{n} a^{2} b^{2} c n x + 6 \,{\left (b x + a\right )}^{n} a^{3} b d n x + 2 \,{\left (b x + a\right )}^{n} a^{3} b c n + 8 \,{\left (b x + a\right )}^{n} a^{3} b c - 6 \,{\left (b x + a\right )}^{n} a^{4} d}{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(x^2*(b*x+a)^n*(d*x+c),x, algorithm="giac")

[Out]

((b*x + a)^n*b^4*d*n^3*x^4 + (b*x + a)^n*b^4*c*n^3*x^3 + (b*x + a)^n*a*b^3*d*n^3*x^3 + 6*(b*x + a)^n*b^4*d*n^2
*x^4 + (b*x + a)^n*a*b^3*c*n^3*x^2 + 7*(b*x + a)^n*b^4*c*n^2*x^3 + 3*(b*x + a)^n*a*b^3*d*n^2*x^3 + 11*(b*x + a
)^n*b^4*d*n*x^4 + 5*(b*x + a)^n*a*b^3*c*n^2*x^2 - 3*(b*x + a)^n*a^2*b^2*d*n^2*x^2 + 14*(b*x + a)^n*b^4*c*n*x^3
+ 2*(b*x + a)^n*a*b^3*d*n*x^3 + 6*(b*x + a)^n*b^4*d*x^4 - 2*(b*x + a)^n*a^2*b^2*c*n^2*x + 4*(b*x + a)^n*a*b^3
*c*n*x^2 - 3*(b*x + a)^n*a^2*b^2*d*n*x^2 + 8*(b*x + a)^n*b^4*c*x^3 - 8*(b*x + a)^n*a^2*b^2*c*n*x + 6*(b*x + a)
^n*a^3*b*d*n*x + 2*(b*x + a)^n*a^3*b*c*n + 8*(b*x + a)^n*a^3*b*c - 6*(b*x + a)^n*a^4*d)/(b^4*n^4 + 10*b^4*n^3
+ 35*b^4*n^2 + 50*b^4*n + 24*b^4)