3.911 \(\int x^2 (a+b x)^n (c+d x)^2 \, dx\)
Optimal. Leaf size=157 \[ \frac{a^2 (b c-a d)^2 (a+b x)^{n+1}}{b^5 (n+1)}+\frac{\left (6 a^2 d^2-6 a b c d+b^2 c^2\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac{2 a (b c-2 a d) (b c-a d) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{2 d (b c-2 a d) (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d^2 (a+b x)^{n+5}}{b^5 (n+5)} \]
[Out]
(a^2*(b*c - a*d)^2*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b*c - 2*a*d)*(b*c -
a*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)) + ((b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2)*(a +
b*x)^(3 + n))/(b^5*(3 + n)) + (2*d*(b*c - 2*a*d)*(a + b*x)^(4 + n))/(b^5*(4 + n)
) + (d^2*(a + b*x)^(5 + n))/(b^5*(5 + n))
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Rubi [A] time = 0.205637, antiderivative size = 157, 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
\[ \frac{a^2 (b c-a d)^2 (a+b x)^{n+1}}{b^5 (n+1)}+\frac{\left (6 a^2 d^2-6 a b c d+b^2 c^2\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac{2 a (b c-2 a d) (b c-a d) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{2 d (b c-2 a d) (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d^2 (a+b x)^{n+5}}{b^5 (n+5)} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*x)^n*(c + d*x)^2,x]
[Out]
(a^2*(b*c - a*d)^2*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b*c - 2*a*d)*(b*c -
a*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)) + ((b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2)*(a +
b*x)^(3 + n))/(b^5*(3 + n)) + (2*d*(b*c - 2*a*d)*(a + b*x)^(4 + n))/(b^5*(4 + n)
) + (d^2*(a + b*x)^(5 + n))/(b^5*(5 + n))
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Rubi in Sympy [A] time = 42.7943, size = 144, normalized size = 0.92 \[ \frac{a^{2} \left (a + b x\right )^{n + 1} \left (a d - b c\right )^{2}}{b^{5} \left (n + 1\right )} - \frac{2 a \left (a + b x\right )^{n + 2} \left (a d - b c\right ) \left (2 a d - b c\right )}{b^{5} \left (n + 2\right )} + \frac{d^{2} \left (a + b x\right )^{n + 5}}{b^{5} \left (n + 5\right )} - \frac{2 d \left (a + b x\right )^{n + 4} \left (2 a d - b c\right )}{b^{5} \left (n + 4\right )} + \frac{\left (a + b x\right )^{n + 3} \left (6 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}\right )}{b^{5} \left (n + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)**n*(d*x+c)**2,x)
[Out]
a**2*(a + b*x)**(n + 1)*(a*d - b*c)**2/(b**5*(n + 1)) - 2*a*(a + b*x)**(n + 2)*(
a*d - b*c)*(2*a*d - b*c)/(b**5*(n + 2)) + d**2*(a + b*x)**(n + 5)/(b**5*(n + 5))
- 2*d*(a + b*x)**(n + 4)*(2*a*d - b*c)/(b**5*(n + 4)) + (a + b*x)**(n + 3)*(6*a
**2*d**2 - 6*a*b*c*d + b**2*c**2)/(b**5*(n + 3))
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Mathematica [A] time = 0.218701, size = 225, normalized size = 1.43 \[ \frac{(a+b x)^{n+1} \left (24 a^4 d^2-12 a^3 b d (c (n+5)+2 d (n+1) x)+2 a^2 b^2 \left (c^2 \left (n^2+9 n+20\right )+6 c d \left (n^2+6 n+5\right ) x+6 d^2 \left (n^2+3 n+2\right ) x^2\right )-2 a b^3 (n+1) x \left (c^2 \left (n^2+9 n+20\right )+3 c d \left (n^2+7 n+10\right ) x+2 d^2 \left (n^2+5 n+6\right ) x^2\right )+b^4 \left (n^2+3 n+2\right ) x^2 \left (c^2 \left (n^2+9 n+20\right )+2 c d \left (n^2+8 n+15\right ) x+d^2 \left (n^2+7 n+12\right ) x^2\right )\right )}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5)} \]
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^2 - 12*a^3*b*d*(c*(5 + n) + 2*d*(1 + n)*x) + 2*a^2*
b^2*(c^2*(20 + 9*n + n^2) + 6*c*d*(5 + 6*n + n^2)*x + 6*d^2*(2 + 3*n + n^2)*x^2)
- 2*a*b^3*(1 + n)*x*(c^2*(20 + 9*n + n^2) + 3*c*d*(10 + 7*n + n^2)*x + 2*d^2*(6
+ 5*n + n^2)*x^2) + b^4*(2 + 3*n + n^2)*x^2*(c^2*(20 + 9*n + n^2) + 2*c*d*(15 +
8*n + n^2)*x + d^2*(12 + 7*n + n^2)*x^2)))/(b^5*(1 + n)*(2 + n)*(3 + n)*(4 + n)
*(5 + n))
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Maple [B] time = 0.013, size = 547, normalized size = 3.5 \[{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{4}{d}^{2}{n}^{4}{x}^{4}+2\,{b}^{4}cd{n}^{4}{x}^{3}+10\,{b}^{4}{d}^{2}{n}^{3}{x}^{4}-4\,a{b}^{3}{d}^{2}{n}^{3}{x}^{3}+{b}^{4}{c}^{2}{n}^{4}{x}^{2}+22\,{b}^{4}cd{n}^{3}{x}^{3}+35\,{b}^{4}{d}^{2}{n}^{2}{x}^{4}-6\,a{b}^{3}cd{n}^{3}{x}^{2}-24\,a{b}^{3}{d}^{2}{n}^{2}{x}^{3}+12\,{b}^{4}{c}^{2}{n}^{3}{x}^{2}+82\,{b}^{4}cd{n}^{2}{x}^{3}+50\,{b}^{4}{d}^{2}n{x}^{4}+12\,{a}^{2}{b}^{2}{d}^{2}{n}^{2}{x}^{2}-2\,a{b}^{3}{c}^{2}{n}^{3}x-48\,a{b}^{3}cd{n}^{2}{x}^{2}-44\,a{b}^{3}{d}^{2}n{x}^{3}+49\,{b}^{4}{c}^{2}{n}^{2}{x}^{2}+122\,{b}^{4}cdn{x}^{3}+24\,{d}^{2}{x}^{4}{b}^{4}+12\,{a}^{2}{b}^{2}cd{n}^{2}x+36\,{a}^{2}{b}^{2}{d}^{2}n{x}^{2}-20\,a{b}^{3}{c}^{2}{n}^{2}x-102\,a{b}^{3}cdn{x}^{2}-24\,a{b}^{3}{d}^{2}{x}^{3}+78\,{b}^{4}{c}^{2}n{x}^{2}+60\,{b}^{4}cd{x}^{3}-24\,{a}^{3}b{d}^{2}nx+2\,{a}^{2}{b}^{2}{c}^{2}{n}^{2}+72\,{a}^{2}{b}^{2}cdnx+24\,{a}^{2}{b}^{2}{d}^{2}{x}^{2}-58\,a{b}^{3}{c}^{2}nx-60\,a{b}^{3}cd{x}^{2}+40\,{b}^{4}{c}^{2}{x}^{2}-12\,{a}^{3}bcdn-24\,{a}^{3}b{d}^{2}x+18\,{a}^{2}{b}^{2}{c}^{2}n+60\,{a}^{2}{b}^{2}cdx-40\,a{b}^{3}{c}^{2}x+24\,{a}^{4}{d}^{2}-60\,{a}^{3}bcd+40\,{a}^{2}{b}^{2}{c}^{2} \right ) }{{b}^{5} \left ({n}^{5}+15\,{n}^{4}+85\,{n}^{3}+225\,{n}^{2}+274\,n+120 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)^n*(d*x+c)^2,x)
[Out]
(b*x+a)^(1+n)*(b^4*d^2*n^4*x^4+2*b^4*c*d*n^4*x^3+10*b^4*d^2*n^3*x^4-4*a*b^3*d^2*
n^3*x^3+b^4*c^2*n^4*x^2+22*b^4*c*d*n^3*x^3+35*b^4*d^2*n^2*x^4-6*a*b^3*c*d*n^3*x^
2-24*a*b^3*d^2*n^2*x^3+12*b^4*c^2*n^3*x^2+82*b^4*c*d*n^2*x^3+50*b^4*d^2*n*x^4+12
*a^2*b^2*d^2*n^2*x^2-2*a*b^3*c^2*n^3*x-48*a*b^3*c*d*n^2*x^2-44*a*b^3*d^2*n*x^3+4
9*b^4*c^2*n^2*x^2+122*b^4*c*d*n*x^3+24*b^4*d^2*x^4+12*a^2*b^2*c*d*n^2*x+36*a^2*b
^2*d^2*n*x^2-20*a*b^3*c^2*n^2*x-102*a*b^3*c*d*n*x^2-24*a*b^3*d^2*x^3+78*b^4*c^2*
n*x^2+60*b^4*c*d*x^3-24*a^3*b*d^2*n*x+2*a^2*b^2*c^2*n^2+72*a^2*b^2*c*d*n*x+24*a^
2*b^2*d^2*x^2-58*a*b^3*c^2*n*x-60*a*b^3*c*d*x^2+40*b^4*c^2*x^2-12*a^3*b*c*d*n-24
*a^3*b*d^2*x+18*a^2*b^2*c^2*n+60*a^2*b^2*c*d*x-40*a*b^3*c^2*x+24*a^4*d^2-60*a^3*
b*c*d+40*a^2*b^2*c^2)/b^5/(n^5+15*n^4+85*n^3+225*n^2+274*n+120)
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Maxima [A] time = 1.38344, size = 429, normalized size = 2.73 \[ \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} c^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac{2 \,{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )}{\left (b x + a\right )}^{n} c d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac{{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} +{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \,{\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )}{\left (b x + a\right )}^{n} d^{2}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*(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^2/((n^3 + 6*n^2 + 11*n + 6)*b^3) + 2*((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*c*d/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4) + ((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^2/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5)
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Fricas [A] time = 0.232709, size = 788, normalized size = 5.02 \[ \frac{{\left (2 \, a^{3} b^{2} c^{2} n^{2} + 40 \, a^{3} b^{2} c^{2} - 60 \, a^{4} b c d + 24 \, a^{5} d^{2} +{\left (b^{5} d^{2} n^{4} + 10 \, b^{5} d^{2} n^{3} + 35 \, b^{5} d^{2} n^{2} + 50 \, b^{5} d^{2} n + 24 \, b^{5} d^{2}\right )} x^{5} +{\left (60 \, b^{5} c d +{\left (2 \, b^{5} c d + a b^{4} d^{2}\right )} n^{4} + 2 \,{\left (11 \, b^{5} c d + 3 \, a b^{4} d^{2}\right )} n^{3} +{\left (82 \, b^{5} c d + 11 \, a b^{4} d^{2}\right )} n^{2} + 2 \,{\left (61 \, b^{5} c d + 3 \, a b^{4} d^{2}\right )} n\right )} x^{4} +{\left (40 \, b^{5} c^{2} +{\left (b^{5} c^{2} + 2 \, a b^{4} c d\right )} n^{4} + 4 \,{\left (3 \, b^{5} c^{2} + 4 \, a b^{4} c d - a^{2} b^{3} d^{2}\right )} n^{3} +{\left (49 \, b^{5} c^{2} + 34 \, a b^{4} c d - 12 \, a^{2} b^{3} d^{2}\right )} n^{2} + 2 \,{\left (39 \, b^{5} c^{2} + 10 \, a b^{4} c d - 4 \, a^{2} b^{3} d^{2}\right )} n\right )} x^{3} +{\left (a b^{4} c^{2} n^{4} + 2 \,{\left (5 \, a b^{4} c^{2} - 3 \, a^{2} b^{3} c d\right )} n^{3} +{\left (29 \, a b^{4} c^{2} - 36 \, a^{2} b^{3} c d + 12 \, a^{3} b^{2} d^{2}\right )} n^{2} + 2 \,{\left (10 \, a b^{4} c^{2} - 15 \, a^{2} b^{3} c d + 6 \, a^{3} b^{2} d^{2}\right )} n\right )} x^{2} + 6 \,{\left (3 \, a^{3} b^{2} c^{2} - 2 \, a^{4} b c d\right )} n - 2 \,{\left (a^{2} b^{3} c^{2} n^{3} + 3 \,{\left (3 \, a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d\right )} n^{2} + 2 \,{\left (10 \, a^{2} b^{3} c^{2} - 15 \, a^{3} b^{2} c d + 6 \, a^{4} b d^{2}\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*(b*x + a)^n*x^2,x, algorithm="fricas")
[Out]
(2*a^3*b^2*c^2*n^2 + 40*a^3*b^2*c^2 - 60*a^4*b*c*d + 24*a^5*d^2 + (b^5*d^2*n^4 +
10*b^5*d^2*n^3 + 35*b^5*d^2*n^2 + 50*b^5*d^2*n + 24*b^5*d^2)*x^5 + (60*b^5*c*d
+ (2*b^5*c*d + a*b^4*d^2)*n^4 + 2*(11*b^5*c*d + 3*a*b^4*d^2)*n^3 + (82*b^5*c*d +
11*a*b^4*d^2)*n^2 + 2*(61*b^5*c*d + 3*a*b^4*d^2)*n)*x^4 + (40*b^5*c^2 + (b^5*c^
2 + 2*a*b^4*c*d)*n^4 + 4*(3*b^5*c^2 + 4*a*b^4*c*d - a^2*b^3*d^2)*n^3 + (49*b^5*c
^2 + 34*a*b^4*c*d - 12*a^2*b^3*d^2)*n^2 + 2*(39*b^5*c^2 + 10*a*b^4*c*d - 4*a^2*b
^3*d^2)*n)*x^3 + (a*b^4*c^2*n^4 + 2*(5*a*b^4*c^2 - 3*a^2*b^3*c*d)*n^3 + (29*a*b^
4*c^2 - 36*a^2*b^3*c*d + 12*a^3*b^2*d^2)*n^2 + 2*(10*a*b^4*c^2 - 15*a^2*b^3*c*d
+ 6*a^3*b^2*d^2)*n)*x^2 + 6*(3*a^3*b^2*c^2 - 2*a^4*b*c*d)*n - 2*(a^2*b^3*c^2*n^3
+ 3*(3*a^2*b^3*c^2 - 2*a^3*b^2*c*d)*n^2 + 2*(10*a^2*b^3*c^2 - 15*a^3*b^2*c*d +
6*a^4*b*d^2)*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 = 16.7152, size = 6215, normalized size = 39.59 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x+a)**n*(d*x+c)**2,x)
[Out]
Piecewise((a**n*(c**2*x**3/3 + c*d*x**4/2 + d**2*x**5/5), Eq(b, 0)), (12*a**6*d*
*2*log(a/b + x)/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**
8*x**3 + 12*a**2*b**9*x**4) + 3*a**6*d**2/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a*
*4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) + 48*a**5*b*d**2*x*log(a/b
+ x)/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 1
2*a**2*b**9*x**4) + 72*a**4*b**2*d**2*x**2*log(a/b + x)/(12*a**6*b**5 + 48*a**5*
b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) - 24*a**4*b*
*2*d**2*x**2/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x
**3 + 12*a**2*b**9*x**4) + 48*a**3*b**3*d**2*x**3*log(a/b + x)/(12*a**6*b**5 + 4
8*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) - 40*
a**3*b**3*d**2*x**3/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3
*b**8*x**3 + 12*a**2*b**9*x**4) + 12*a**2*b**4*d**2*x**4*log(a/b + x)/(12*a**6*b
**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4
) - 22*a**2*b**4*d**2*x**4/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 +
48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) + 4*a*b**5*c**2*x**3/(12*a**6*b**5 + 48*a
**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4) + 6*a*b*
*5*c*d*x**4/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**4*b**7*x**2 + 48*a**3*b**8*x*
*3 + 12*a**2*b**9*x**4) + b**6*c**2*x**4/(12*a**6*b**5 + 48*a**5*b**6*x + 72*a**
4*b**7*x**2 + 48*a**3*b**8*x**3 + 12*a**2*b**9*x**4), Eq(n, -5)), (-12*a**5*d**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) -
4*a**5*d**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) +
6*a**4*b*c*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) + 2*a**4*b*c*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**2*x*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*c*d*x*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) - 36*a**3*b**2*d**2*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**2*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)
+ 18*a**2*b**3*c*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) - 9*a**2*b**3*c*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**2*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**2*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) + 6*a*b**4*c*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)
- 9*a*b**4*c*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**2*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**2*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**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x
+ 2*b**7*x**2) + 6*a**4*d**2/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 12*a**3*
b*c*d*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 6*a**3*b*c*d/(2*a*
*2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 24*a**3*b*d**2*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**2*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/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2)
- 24*a**2*b**2*c*d*x*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 12*
a**2*b**2*d**2*x**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 12*a
**2*b**2*d**2*x**2/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 4*a*b**3*c**2*x*lo
g(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - 12*a*b**3*c*d*x**2*log(a/b
+ x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 12*a*b**3*c*d*x**2/(2*a**2*b**5
+ 4*a*b**6*x + 2*b**7*x**2) - 4*a*b**3*d**2*x**3/(2*a**2*b**5 + 4*a*b**6*x + 2*
b**7*x**2) + 2*b**4*c**2*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**2*x**2/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 4*b**4*c*d*x**
3/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + b**4*d**2*x**4/(2*a**2*b**5 + 4*a*b
**6*x + 2*b**7*x**2), Eq(n, -3)), (-12*a**4*d**2*log(a/b + x)/(3*a*b**5 + 3*b**6
*x) - 12*a**4*d**2/(3*a*b**5 + 3*b**6*x) + 18*a**3*b*c*d*log(a/b + x)/(3*a*b**5
+ 3*b**6*x) + 18*a**3*b*c*d/(3*a*b**5 + 3*b**6*x) - 12*a**3*b*d**2*x*log(a/b + x
)/(3*a*b**5 + 3*b**6*x) - 6*a**2*b**2*c**2*log(a/b + x)/(3*a*b**5 + 3*b**6*x) -
6*a**2*b**2*c**2/(3*a*b**5 + 3*b**6*x) + 18*a**2*b**2*c*d*x*log(a/b + x)/(3*a*b*
*5 + 3*b**6*x) + 6*a**2*b**2*d**2*x**2/(3*a*b**5 + 3*b**6*x) - 6*a*b**3*c**2*x*l
og(a/b + x)/(3*a*b**5 + 3*b**6*x) - 9*a*b**3*c*d*x**2/(3*a*b**5 + 3*b**6*x) - 2*
a*b**3*d**2*x**3/(3*a*b**5 + 3*b**6*x) + 3*b**4*c**2*x**2/(3*a*b**5 + 3*b**6*x)
+ 3*b**4*c*d*x**3/(3*a*b**5 + 3*b**6*x) + b**4*d**2*x**4/(3*a*b**5 + 3*b**6*x),
Eq(n, -2)), (a**4*d**2*log(a/b + x)/b**5 - 2*a**3*c*d*log(a/b + x)/b**4 - a**3*d
**2*x/b**4 + a**2*c**2*log(a/b + x)/b**3 + 2*a**2*c*d*x/b**3 + a**2*d**2*x**2/(2
*b**3) - a*c**2*x/b**2 - a*c*d*x**2/b**2 - a*d**2*x**3/(3*b**2) + c**2*x**2/(2*b
) + 2*c*d*x**3/(3*b) + d**2*x**4/(4*b), Eq(n, -1)), (24*a**5*d**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**4*b*c*d*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) - 60*a**4*b*c*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**
2*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 + 27
4*b**5*n + 120*b**5) + 2*a**3*b**2*c**2*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**2*
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**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*c*d*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) + 60*a**3*b**2*c*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) + 12*a**3*b**2*d**2*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**2*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**2*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**2*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**2
*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) - 6*a**2*b**3*c*d*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) - 36*a**2*b**3*c*
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) - 30*a**2*b**3*c*d*n*x**2*(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) - 4*a**2*b**
3*d**2*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**2*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**2*n*x**3*(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) + a*b**4*c**2*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) + 1
0*a*b**4*c**2*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**2*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)
+ 20*a*b**4*c**2*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*b**4*c*d*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)
+ 16*a*b**4*c*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) + 34*a*b**4*c*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
) + 20*a*b**4*c*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*d**2*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**2*n**3*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) + 11*a*b**4*d**2*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**2*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**2*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) + 1
2*b**5*c**2*n**3*x**3*(a + b*x)**n/(b**5*n**5 + 15*b**5*n**4 + 85*b**5*n**3 + 22
5*b**5*n**2 + 274*b**5*n + 120*b**5) + 49*b**5*c**2*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) + 7
8*b**5*c**2*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**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) + 2*b**5*c*
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) + 22*b**5*c*d*n**3*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) + 82*b**5*c*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) + 122*b**5*c*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) + 60*b**5*c*d*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*d**2*n**4*x**5*(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) + 10*b**5*d**2*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**2*n**2*x**5*(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) + 50*b**5*d**2*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**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), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.223485, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*(b*x + a)^n*x^2,x, algorithm="giac")
[Out]
Done