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

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))

_______________________________________________________________________________________

Rubi [A]  time = 0.140222, 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 \[ \frac{a^2 (b c-a d) (a+b x)^{n+1}}{b^4 (n+1)}-\frac{a (2 b c-3 a d) (a+b x)^{n+2}}{b^4 (n+2)}+\frac{(b c-3 a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]

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))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.2625, size = 92, normalized size = 0.88 \[ - \frac{a^{2} \left (a + b x\right )^{n + 1} \left (a d - b c\right )}{b^{4} \left (n + 1\right )} + \frac{a \left (a + b x\right )^{n + 2} \left (3 a d - 2 b c\right )}{b^{4} \left (n + 2\right )} + \frac{d \left (a + b x\right )^{n + 4}}{b^{4} \left (n + 4\right )} - \frac{\left (a + b x\right )^{n + 3} \left (3 a d - b c\right )}{b^{4} \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)**n*(d*x+c),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.139777, size = 110, normalized size = 1.06 \[ \frac{(a+b x)^{n+1} \left (-6 a^3 d+2 a^2 b (c (n+4)+3 d (n+1) x)-a b^2 (n+1) x (2 c (n+4)+3 d (n+2) x)+b^3 \left (n^2+3 n+2\right ) x^2 (c (n+4)+d (n+3) x)\right )}{b^4 (n+1) (n+2) (n+3) (n+4)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1 + n)*(-6*a^3*d + 2*a^2*b*(c*(4 + n) + 3*d*(1 + n)*x) - a*b^2*(1 +
n)*x*(2*c*(4 + n) + 3*d*(2 + n)*x) + b^3*(2 + 3*n + n^2)*x^2*(c*(4 + n) + d*(3 +
n)*x)))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))

_______________________________________________________________________________________

Maple [B]  time = 0.01, size = 222, normalized size = 2.1 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}d{n}^{3}{x}^{3}-{b}^{3}c{n}^{3}{x}^{2}-6\,{b}^{3}d{n}^{2}{x}^{3}+3\,a{b}^{2}d{n}^{2}{x}^{2}-7\,{b}^{3}c{n}^{2}{x}^{2}-11\,{b}^{3}dn{x}^{3}+2\,a{b}^{2}c{n}^{2}x+9\,a{b}^{2}dn{x}^{2}-14\,{b}^{3}cn{x}^{2}-6\,d{x}^{3}{b}^{3}-6\,{a}^{2}bdnx+10\,a{b}^{2}cnx+6\,a{b}^{2}d{x}^{2}-8\,{b}^{3}c{x}^{2}-2\,{a}^{2}bcn-6\,{a}^{2}bdx+8\,a{b}^{2}cx+6\,{a}^{3}d-8\,{a}^{2}bc \right ) }{{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \]

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)

_______________________________________________________________________________________

Maxima [A]  time = 1.36515, size = 232, normalized size = 2.23 \[ \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} c}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )}{\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Fricas [A]  time = 0.237457, size = 339, normalized size = 3.26 \[ \frac{{\left (2 \, a^{3} b c n + 8 \, a^{3} b c - 6 \, a^{4} d +{\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} +{\left (8 \, b^{4} c +{\left (b^{4} c + a b^{3} d\right )} n^{3} +{\left (7 \, b^{4} c + 3 \, a b^{3} d\right )} n^{2} + 2 \,{\left (7 \, b^{4} c + a b^{3} d\right )} n\right )} x^{3} +{\left (a b^{3} c n^{3} +{\left (5 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} n^{2} +{\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} n\right )} x^{2} - 2 \,{\left (a^{2} b^{2} c n^{2} +{\left (4 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Sympy [A]  time = 7.80271, size = 2402, normalized size = 23.1 \[ \text{result too large to display} \]

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) + 2*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) + 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) - 9*a**2*b**2*d*x**2/(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) - 9*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) - 3*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) + 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) + 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) - 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) + 6*a*b**2*d*x**2/(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*c*x**2/(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 + 1
0*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))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.300777, size = 641, normalized size = 6.16 \[ \frac{b^{4} d n^{3} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + b^{4} c n^{3} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{3} d n^{3} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 6 \, b^{4} d n^{2} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{3} c n^{3} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 7 \, b^{4} c n^{2} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 3 \, a b^{3} d n^{2} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 11 \, b^{4} d n x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 5 \, a b^{3} c n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 3 \, a^{2} b^{2} d n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 14 \, b^{4} c n x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, a b^{3} d n x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 6 \, b^{4} d x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 2 \, a^{2} b^{2} c n^{2} x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 4 \, a b^{3} c n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 3 \, a^{2} b^{2} d n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 8 \, b^{4} c x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 8 \, a^{2} b^{2} c n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 6 \, a^{3} b d n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, a^{3} b c n e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 8 \, a^{3} b c e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 6 \, a^{4} d e^{\left (n{\rm ln}\left (b x + a\right )\right )}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)*(b*x + a)^n*x^2,x, algorithm="giac")

[Out]

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