3.351 \(\int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx\)
Optimal. Leaf size=135 \[ \frac{a^2 \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^5 (n+1)}-\frac{2 a \left (2 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{\left (6 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac{4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d (a+b x)^{n+5}}{b^5 (n+5)} \]
[Out]
(a^2*(b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b^2*c + 2*a^2*d)*(
a + b*x)^(2 + n))/(b^5*(2 + n)) + ((b^2*c + 6*a^2*d)*(a + b*x)^(3 + n))/(b^5*(3
+ n)) - (4*a*d*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d*(a + b*x)^(5 + n))/(b^5*(5
+ n))
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Rubi [A] time = 0.17588, antiderivative size = 135, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056
\[ \frac{a^2 \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^5 (n+1)}-\frac{2 a \left (2 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{\left (6 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac{4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d (a+b x)^{n+5}}{b^5 (n+5)} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*x)^n*(c + d*x^2),x]
[Out]
(a^2*(b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b^2*c + 2*a^2*d)*(
a + b*x)^(2 + n))/(b^5*(2 + n)) + ((b^2*c + 6*a^2*d)*(a + b*x)^(3 + n))/(b^5*(3
+ n)) - (4*a*d*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d*(a + b*x)^(5 + n))/(b^5*(5
+ n))
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Rubi in Sympy [A] time = 31.6645, size = 122, normalized size = 0.9 \[ \frac{a^{2} \left (a + b x\right )^{n + 1} \left (a^{2} d + b^{2} c\right )}{b^{5} \left (n + 1\right )} - \frac{4 a d \left (a + b x\right )^{n + 4}}{b^{5} \left (n + 4\right )} - \frac{2 a \left (a + b x\right )^{n + 2} \left (2 a^{2} d + b^{2} c\right )}{b^{5} \left (n + 2\right )} + \frac{d \left (a + b x\right )^{n + 5}}{b^{5} \left (n + 5\right )} + \frac{\left (a + b x\right )^{n + 3} \left (6 a^{2} d + b^{2} c\right )}{b^{5} \left (n + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)**n*(d*x**2+c),x)
[Out]
a**2*(a + b*x)**(n + 1)*(a**2*d + b**2*c)/(b**5*(n + 1)) - 4*a*d*(a + b*x)**(n +
4)/(b**5*(n + 4)) - 2*a*(a + b*x)**(n + 2)*(2*a**2*d + b**2*c)/(b**5*(n + 2)) +
d*(a + b*x)**(n + 5)/(b**5*(n + 5)) + (a + b*x)**(n + 3)*(6*a**2*d + b**2*c)/(b
**5*(n + 3))
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Mathematica [A] time = 0.135431, size = 158, normalized size = 1.17 \[ \frac{(a+b x)^{n+1} \left (24 a^4 d-24 a^3 b d (n+1) x+2 a^2 b^2 \left (c \left (n^2+9 n+20\right )+6 d \left (n^2+3 n+2\right ) x^2\right )-2 a b^3 (n+1) x \left (c \left (n^2+9 n+20\right )+2 d \left (n^2+5 n+6\right ) x^2\right )+b^4 \left (n^3+7 n^2+14 n+8\right ) x^2 \left (c (n+5)+d (n+3) x^2\right )\right )}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*x)^n*(c + d*x^2),x]
[Out]
((a + b*x)^(1 + n)*(24*a^4*d - 24*a^3*b*d*(1 + n)*x + b^4*(8 + 14*n + 7*n^2 + n^
3)*x^2*(c*(5 + n) + d*(3 + n)*x^2) + 2*a^2*b^2*(c*(20 + 9*n + n^2) + 6*d*(2 + 3*
n + n^2)*x^2) - 2*a*b^3*(1 + n)*x*(c*(20 + 9*n + n^2) + 2*d*(6 + 5*n + n^2)*x^2)
))/(b^5*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n))
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Maple [B] time = 0.01, size = 328, normalized size = 2.4 \[{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{4}d{n}^{4}{x}^{4}+10\,{b}^{4}d{n}^{3}{x}^{4}-4\,a{b}^{3}d{n}^{3}{x}^{3}+{b}^{4}c{n}^{4}{x}^{2}+35\,{b}^{4}d{n}^{2}{x}^{4}-24\,a{b}^{3}d{n}^{2}{x}^{3}+12\,{b}^{4}c{n}^{3}{x}^{2}+50\,{b}^{4}dn{x}^{4}+12\,{a}^{2}{b}^{2}d{n}^{2}{x}^{2}-2\,a{b}^{3}c{n}^{3}x-44\,a{b}^{3}dn{x}^{3}+49\,{b}^{4}c{n}^{2}{x}^{2}+24\,d{x}^{4}{b}^{4}+36\,{a}^{2}{b}^{2}dn{x}^{2}-20\,a{b}^{3}c{n}^{2}x-24\,da{x}^{3}{b}^{3}+78\,{b}^{4}cn{x}^{2}-24\,{a}^{3}bdnx+2\,{a}^{2}{b}^{2}c{n}^{2}+24\,{a}^{2}{b}^{2}d{x}^{2}-58\,a{b}^{3}cnx+40\,{b}^{4}c{x}^{2}-24\,{a}^{3}bdx+18\,{a}^{2}{b}^{2}cn-40\,a{b}^{3}cx+24\,{a}^{4}d+40\,{a}^{2}{b}^{2}c \right ) }{{b}^{5} \left ({n}^{5}+15\,{n}^{4}+85\,{n}^{3}+225\,{n}^{2}+274\,n+120 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)^n*(d*x^2+c),x)
[Out]
(b*x+a)^(1+n)*(b^4*d*n^4*x^4+10*b^4*d*n^3*x^4-4*a*b^3*d*n^3*x^3+b^4*c*n^4*x^2+35
*b^4*d*n^2*x^4-24*a*b^3*d*n^2*x^3+12*b^4*c*n^3*x^2+50*b^4*d*n*x^4+12*a^2*b^2*d*n
^2*x^2-2*a*b^3*c*n^3*x-44*a*b^3*d*n*x^3+49*b^4*c*n^2*x^2+24*b^4*d*x^4+36*a^2*b^2
*d*n*x^2-20*a*b^3*c*n^2*x-24*a*b^3*d*x^3+78*b^4*c*n*x^2-24*a^3*b*d*n*x+2*a^2*b^2
*c*n^2+24*a^2*b^2*d*x^2-58*a*b^3*c*n*x+40*b^4*c*x^2-24*a^3*b*d*x+18*a^2*b^2*c*n-
40*a*b^3*c*x+24*a^4*d+40*a^2*b^2*c)/b^5/(n^5+15*n^4+85*n^3+225*n^2+274*n+120)
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Maxima [A] time = 0.710032, size = 284, normalized size = 2.1 \[ \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} c}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac{{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} +{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \,{\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )}{\left (b x + a\right )}^{n} d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + 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^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^
5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 +
12*(n^2 + n)*a^3*b^2*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*d/((n^5 + 15*n^4
+ 85*n^3 + 225*n^2 + 274*n + 120)*b^5)
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Fricas [A] time = 0.281342, size = 497, normalized size = 3.68 \[ \frac{{\left (2 \, a^{3} b^{2} c n^{2} + 18 \, a^{3} b^{2} c n + 40 \, a^{3} b^{2} c + 24 \, a^{5} d +{\left (b^{5} d n^{4} + 10 \, b^{5} d n^{3} + 35 \, b^{5} d n^{2} + 50 \, b^{5} d n + 24 \, b^{5} d\right )} x^{5} +{\left (a b^{4} d n^{4} + 6 \, a b^{4} d n^{3} + 11 \, a b^{4} d n^{2} + 6 \, a b^{4} d n\right )} x^{4} +{\left (b^{5} c n^{4} + 40 \, b^{5} c + 4 \,{\left (3 \, b^{5} c - a^{2} b^{3} d\right )} n^{3} +{\left (49 \, b^{5} c - 12 \, a^{2} b^{3} d\right )} n^{2} + 2 \,{\left (39 \, b^{5} c - 4 \, a^{2} b^{3} d\right )} n\right )} x^{3} +{\left (a b^{4} c n^{4} + 10 \, a b^{4} c n^{3} +{\left (29 \, a b^{4} c + 12 \, a^{3} b^{2} d\right )} n^{2} + 4 \,{\left (5 \, a b^{4} c + 3 \, a^{3} b^{2} d\right )} n\right )} x^{2} - 2 \,{\left (a^{2} b^{3} c n^{3} + 9 \, a^{2} b^{3} c n^{2} + 4 \,{\left (5 \, a^{2} b^{3} c + 3 \, a^{4} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(b*x + a)^n*x^2,x, algorithm="fricas")
[Out]
(2*a^3*b^2*c*n^2 + 18*a^3*b^2*c*n + 40*a^3*b^2*c + 24*a^5*d + (b^5*d*n^4 + 10*b^
5*d*n^3 + 35*b^5*d*n^2 + 50*b^5*d*n + 24*b^5*d)*x^5 + (a*b^4*d*n^4 + 6*a*b^4*d*n
^3 + 11*a*b^4*d*n^2 + 6*a*b^4*d*n)*x^4 + (b^5*c*n^4 + 40*b^5*c + 4*(3*b^5*c - a^
2*b^3*d)*n^3 + (49*b^5*c - 12*a^2*b^3*d)*n^2 + 2*(39*b^5*c - 4*a^2*b^3*d)*n)*x^3
+ (a*b^4*c*n^4 + 10*a*b^4*c*n^3 + (29*a*b^4*c + 12*a^3*b^2*d)*n^2 + 4*(5*a*b^4*
c + 3*a^3*b^2*d)*n)*x^2 - 2*(a^2*b^3*c*n^3 + 9*a^2*b^3*c*n^2 + 4*(5*a^2*b^3*c +
3*a^4*b*d)*n)*x)*(b*x + a)^n/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*b^5*n^2 +
274*b^5*n + 120*b^5)
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Sympy [A] time = 12.8097, size = 4078, normalized size = 30.21 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x+a)**n*(d*x**2+c),x)
[Out]
Piecewise((a**n*(c*x**3/3 + d*x**5/5), Eq(b, 0)), (12*a**4*d*log(a/b + x)/(12*a*
*4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) +
25*a**4*d/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 +
12*b**9*x**4) + 48*a**3*b*d*x*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a
**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 88*a**3*b*d*x/(12*a**4*b**5 + 4
8*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - a**2*b**2*c
/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x
**4) + 72*a**2*b**2*d*x**2*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2
*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 108*a**2*b**2*d*x**2/(12*a**4*b**5
+ 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 4*a*b**
3*c*x/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b
**9*x**4) + 48*a*b**3*d*x**3*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a*
*2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) + 48*a*b**3*d*x**3/(12*a**4*b**5 +
48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4) - 6*b**4*c*
x**2/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*b**7*x**2 + 48*a*b**8*x**3 + 12*b*
*9*x**4) + 12*b**4*d*x**4*log(a/b + x)/(12*a**4*b**5 + 48*a**3*b**6*x + 72*a**2*
b**7*x**2 + 48*a*b**8*x**3 + 12*b**9*x**4), Eq(n, -5)), (-12*a**5*d*log(a/b + x)
/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) - 4*a**5*d/(3*
a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) - 36*a**4*b*d*x*lo
g(a/b + x)/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) - 36
*a**3*b**2*d*x**2*log(a/b + x)/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7*x**2 +
3*a*b**8*x**3) + 18*a**3*b**2*d*x**2/(3*a**4*b**5 + 9*a**3*b**6*x + 9*a**2*b**7
*x**2 + 3*a*b**8*x**3) - 12*a**2*b**3*d*x**3*log(a/b + x)/(3*a**4*b**5 + 9*a**3*
b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) + 18*a**2*b**3*d*x**3/(3*a**4*b**5 +
9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) + 3*a*b**4*d*x**4/(3*a**4*b**5
+ 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3) + b**5*c*x**3/(3*a**4*b**5
+ 9*a**3*b**6*x + 9*a**2*b**7*x**2 + 3*a*b**8*x**3), Eq(n, -4)), (12*a**4*d*log(
a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 6*a**4*d/(2*a**2*b**5 + 4*a*
b**6*x + 2*b**7*x**2) + 24*a**3*b*d*x*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2
*b**7*x**2) + 2*a**2*b**2*c*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2
) + a**2*b**2*c/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 12*a**2*b**2*d*x**2*l
og(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 12*a**2*b**2*d*x**2/(2*a*
*2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 4*a*b**3*c*x*log(a/b + x)/(2*a**2*b**5 + 4
*a*b**6*x + 2*b**7*x**2) - 4*a*b**3*d*x**3/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x*
*2) + 2*b**4*c*x**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 2*b*
*4*c*x**2/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + b**4*d*x**4/(2*a**2*b**5 +
4*a*b**6*x + 2*b**7*x**2), Eq(n, -3)), (-12*a**4*d*log(a/b + x)/(3*a*b**5 + 3*b*
*6*x) - 12*a**4*d/(3*a*b**5 + 3*b**6*x) - 12*a**3*b*d*x*log(a/b + x)/(3*a*b**5 +
3*b**6*x) - 6*a**2*b**2*c*log(a/b + x)/(3*a*b**5 + 3*b**6*x) - 6*a**2*b**2*c/(3
*a*b**5 + 3*b**6*x) + 6*a**2*b**2*d*x**2/(3*a*b**5 + 3*b**6*x) - 6*a*b**3*c*x*lo
g(a/b + x)/(3*a*b**5 + 3*b**6*x) - 2*a*b**3*d*x**3/(3*a*b**5 + 3*b**6*x) + 3*b**
4*c*x**2/(3*a*b**5 + 3*b**6*x) + b**4*d*x**4/(3*a*b**5 + 3*b**6*x), Eq(n, -2)),
(a**4*d*log(a/b + x)/b**5 - a**3*d*x/b**4 + a**2*c*log(a/b + x)/b**3 + a**2*d*x*
*2/(2*b**3) - a*c*x/b**2 - a*d*x**3/(3*b**2) + c*x**2/(2*b) + d*x**4/(4*b), Eq(n
, -1)), (24*a**5*d*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b
**5*n**2 + 274*b**5*n + 120*b**5) - 24*a**4*b*d*n*x*(a + b*x)**n/(b**5*n**5 + 15
*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 2*a**3*b**2
*c*n**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 +
274*b**5*n + 120*b**5) + 18*a**3*b**2*c*n*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4
+ 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 40*a**3*b**2*c*(a + b
*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 1
20*b**5) + 12*a**3*b**2*d*n**2*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*
b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 12*a**3*b**2*d*n*x**2*(a +
b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n +
120*b**5) - 2*a**2*b**3*c*n**3*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**
5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 18*a**2*b**3*c*n**2*x*(a + b*x
)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120
*b**5) - 40*a**2*b**3*c*n*x*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**
3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 4*a**2*b**3*d*n**3*x**3*(a + b*x)**
n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b*
*5) - 12*a**2*b**3*d*n**2*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*
n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) - 8*a**2*b**3*d*n*x**3*(a + b*x)**
n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b*
*5) + a*b**4*c*n**4*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 +
225*b**5*n**2 + 274*b**5*n + 120*b**5) + 10*a*b**4*c*n**3*x**2*(a + b*x)**n/(b*
*5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) +
29*a*b**4*c*n**2*x**2*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 2
25*b**5*n**2 + 274*b**5*n + 120*b**5) + 20*a*b**4*c*n*x**2*(a + b*x)**n/(b**5*n*
*5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + a*b*
*4*d*n**4*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*
n**2 + 274*b**5*n + 120*b**5) + 6*a*b**4*d*n**3*x**4*(a + b*x)**n/(b**5*n**5 + 1
5*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 11*a*b**4*
d*n**2*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**
2 + 274*b**5*n + 120*b**5) + 6*a*b**4*d*n*x**4*(a + b*x)**n/(b**5*n**5 + 15*b**5
*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + b**5*c*n**4*x**3
*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**
5*n + 120*b**5) + 12*b**5*c*n**3*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 8
5*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 49*b**5*c*n**2*x**3*(a +
b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n +
120*b**5) + 78*b**5*c*n*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n*
*3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 40*b**5*c*x**3*(a + b*x)**n/(b**5*
n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + b*
*5*d*n**4*x**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*
n**2 + 274*b**5*n + 120*b**5) + 10*b**5*d*n**3*x**5*(a + b*x)**n/(b**5*n**5 + 15
*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 35*b**5*d*n
**2*x**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 +
274*b**5*n + 120*b**5) + 50*b**5*d*n*x**5*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**
4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 120*b**5) + 24*b**5*d*x**5*(a +
b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n +
120*b**5), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.296885, size = 926, normalized size = 6.86 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*(b*x + a)^n*x^2,x, algorithm="giac")
[Out]
(b^5*d*n^4*x^5*e^(n*ln(b*x + a)) + a*b^4*d*n^4*x^4*e^(n*ln(b*x + a)) + 10*b^5*d*
n^3*x^5*e^(n*ln(b*x + a)) + b^5*c*n^4*x^3*e^(n*ln(b*x + a)) + 6*a*b^4*d*n^3*x^4*
e^(n*ln(b*x + a)) + 35*b^5*d*n^2*x^5*e^(n*ln(b*x + a)) + a*b^4*c*n^4*x^2*e^(n*ln
(b*x + a)) + 12*b^5*c*n^3*x^3*e^(n*ln(b*x + a)) - 4*a^2*b^3*d*n^3*x^3*e^(n*ln(b*
x + a)) + 11*a*b^4*d*n^2*x^4*e^(n*ln(b*x + a)) + 50*b^5*d*n*x^5*e^(n*ln(b*x + a)
) + 10*a*b^4*c*n^3*x^2*e^(n*ln(b*x + a)) + 49*b^5*c*n^2*x^3*e^(n*ln(b*x + a)) -
12*a^2*b^3*d*n^2*x^3*e^(n*ln(b*x + a)) + 6*a*b^4*d*n*x^4*e^(n*ln(b*x + a)) + 24*
b^5*d*x^5*e^(n*ln(b*x + a)) - 2*a^2*b^3*c*n^3*x*e^(n*ln(b*x + a)) + 29*a*b^4*c*n
^2*x^2*e^(n*ln(b*x + a)) + 12*a^3*b^2*d*n^2*x^2*e^(n*ln(b*x + a)) + 78*b^5*c*n*x
^3*e^(n*ln(b*x + a)) - 8*a^2*b^3*d*n*x^3*e^(n*ln(b*x + a)) - 18*a^2*b^3*c*n^2*x*
e^(n*ln(b*x + a)) + 20*a*b^4*c*n*x^2*e^(n*ln(b*x + a)) + 12*a^3*b^2*d*n*x^2*e^(n
*ln(b*x + a)) + 40*b^5*c*x^3*e^(n*ln(b*x + a)) + 2*a^3*b^2*c*n^2*e^(n*ln(b*x + a
)) - 40*a^2*b^3*c*n*x*e^(n*ln(b*x + a)) - 24*a^4*b*d*n*x*e^(n*ln(b*x + a)) + 18*
a^3*b^2*c*n*e^(n*ln(b*x + a)) + 40*a^3*b^2*c*e^(n*ln(b*x + a)) + 24*a^5*d*e^(n*l
n(b*x + a)))/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*b^5*n^2 + 274*b^5*n + 120*
b^5)