3.912 \(\int x (a+b x)^n (c+d x)^2 \, dx\)
Optimal. Leaf size=114 \[ -\frac{a (b c-a d)^2 (a+b x)^{n+1}}{b^4 (n+1)}+\frac{(b c-3 a d) (b c-a d) (a+b x)^{n+2}}{b^4 (n+2)}+\frac{d (2 b c-3 a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d^2 (a+b x)^{n+4}}{b^4 (n+4)} \]
[Out]
-((a*(b*c - a*d)^2*(a + b*x)^(1 + n))/(b^4*(1 + n))) + ((b*c - 3*a*d)*(b*c - a*d
)*(a + b*x)^(2 + n))/(b^4*(2 + n)) + (d*(2*b*c - 3*a*d)*(a + b*x)^(3 + n))/(b^4*
(3 + n)) + (d^2*(a + b*x)^(4 + n))/(b^4*(4 + n))
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Rubi [A] time = 0.141097, antiderivative size = 114, 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
\[ -\frac{a (b c-a d)^2 (a+b x)^{n+1}}{b^4 (n+1)}+\frac{(b c-3 a d) (b c-a d) (a+b x)^{n+2}}{b^4 (n+2)}+\frac{d (2 b c-3 a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d^2 (a+b x)^{n+4}}{b^4 (n+4)} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x)^n*(c + d*x)^2,x]
[Out]
-((a*(b*c - a*d)^2*(a + b*x)^(1 + n))/(b^4*(1 + n))) + ((b*c - 3*a*d)*(b*c - a*d
)*(a + b*x)^(2 + n))/(b^4*(2 + n)) + (d*(2*b*c - 3*a*d)*(a + b*x)^(3 + n))/(b^4*
(3 + n)) + (d^2*(a + b*x)^(4 + n))/(b^4*(4 + n))
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Rubi in Sympy [A] time = 27.9194, size = 100, normalized size = 0.88 \[ - \frac{a \left (a + b x\right )^{n + 1} \left (a d - b c\right )^{2}}{b^{4} \left (n + 1\right )} + \frac{d^{2} \left (a + b x\right )^{n + 4}}{b^{4} \left (n + 4\right )} - \frac{d \left (a + b x\right )^{n + 3} \left (3 a d - 2 b c\right )}{b^{4} \left (n + 3\right )} + \frac{\left (a + b x\right )^{n + 2} \left (a d - b c\right ) \left (3 a d - b c\right )}{b^{4} \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**n*(d*x+c)**2,x)
[Out]
-a*(a + b*x)**(n + 1)*(a*d - b*c)**2/(b**4*(n + 1)) + d**2*(a + b*x)**(n + 4)/(b
**4*(n + 4)) - d*(a + b*x)**(n + 3)*(3*a*d - 2*b*c)/(b**4*(n + 3)) + (a + b*x)**
(n + 2)*(a*d - b*c)*(3*a*d - b*c)/(b**4*(n + 2))
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Mathematica [A] time = 0.159342, size = 160, normalized size = 1.4 \[ \frac{(a+b x)^{n+1} \left (-6 a^3 d^2+2 a^2 b d (2 c (n+4)+3 d (n+1) x)-a b^2 \left (c^2 \left (n^2+7 n+12\right )+4 c d \left (n^2+5 n+4\right ) x+3 d^2 \left (n^2+3 n+2\right ) x^2\right )+b^3 (n+1) x \left (c^2 \left (n^2+7 n+12\right )+2 c d \left (n^2+6 n+8\right ) x+d^2 \left (n^2+5 n+6\right ) x^2\right )\right )}{b^4 (n+1) (n+2) (n+3) (n+4)} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x)^n*(c + d*x)^2,x]
[Out]
((a + b*x)^(1 + n)*(-6*a^3*d^2 + 2*a^2*b*d*(2*c*(4 + n) + 3*d*(1 + n)*x) - a*b^2
*(c^2*(12 + 7*n + n^2) + 4*c*d*(4 + 5*n + n^2)*x + 3*d^2*(2 + 3*n + n^2)*x^2) +
b^3*(1 + n)*x*(c^2*(12 + 7*n + n^2) + 2*c*d*(8 + 6*n + n^2)*x + d^2*(6 + 5*n + n
^2)*x^2)))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))
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Maple [B] time = 0.013, size = 324, normalized size = 2.8 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}{d}^{2}{n}^{3}{x}^{3}-2\,{b}^{3}cd{n}^{3}{x}^{2}-6\,{b}^{3}{d}^{2}{n}^{2}{x}^{3}+3\,a{b}^{2}{d}^{2}{n}^{2}{x}^{2}-{b}^{3}{c}^{2}{n}^{3}x-14\,{b}^{3}cd{n}^{2}{x}^{2}-11\,{b}^{3}{d}^{2}n{x}^{3}+4\,a{b}^{2}cd{n}^{2}x+9\,a{b}^{2}{d}^{2}n{x}^{2}-8\,{b}^{3}{c}^{2}{n}^{2}x-28\,{b}^{3}cdn{x}^{2}-6\,{d}^{2}{x}^{3}{b}^{3}-6\,{a}^{2}b{d}^{2}nx+a{b}^{2}{c}^{2}{n}^{2}+20\,a{b}^{2}cdnx+6\,a{b}^{2}{d}^{2}{x}^{2}-19\,{b}^{3}{c}^{2}nx-16\,{b}^{3}cd{x}^{2}-4\,{a}^{2}bcdn-6\,{a}^{2}b{d}^{2}x+7\,a{b}^{2}{c}^{2}n+16\,a{b}^{2}cdx-12\,{b}^{3}{c}^{2}x+6\,{a}^{3}{d}^{2}-16\,{a}^{2}bcd+12\,a{b}^{2}{c}^{2} \right ) }{{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^n*(d*x+c)^2,x)
[Out]
-(b*x+a)^(1+n)*(-b^3*d^2*n^3*x^3-2*b^3*c*d*n^3*x^2-6*b^3*d^2*n^2*x^3+3*a*b^2*d^2
*n^2*x^2-b^3*c^2*n^3*x-14*b^3*c*d*n^2*x^2-11*b^3*d^2*n*x^3+4*a*b^2*c*d*n^2*x+9*a
*b^2*d^2*n*x^2-8*b^3*c^2*n^2*x-28*b^3*c*d*n*x^2-6*b^3*d^2*x^3-6*a^2*b*d^2*n*x+a*
b^2*c^2*n^2+20*a*b^2*c*d*n*x+6*a*b^2*d^2*x^2-19*b^3*c^2*n*x-16*b^3*c*d*x^2-4*a^2
*b*c*d*n-6*a^2*b*d^2*x+7*a*b^2*c^2*n+16*a*b^2*c*d*x-12*b^3*c^2*x+6*a^3*d^2-16*a^
2*b*c*d+12*a*b^2*c^2)/b^4/(n^4+10*n^3+35*n^2+50*n+24)
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Maxima [A] time = 1.38233, size = 298, normalized size = 2.61 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n} c^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac{2 \,{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n} c d}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )}{\left (b x + a\right )}^{n} d^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*(b*x + a)^n*x,x, algorithm="maxima")
[Out]
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2/((n^2 + 3*n + 2)*b^2) + 2*((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*
d/((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^2/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4)
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Fricas [A] time = 0.22553, size = 531, normalized size = 4.66 \[ -\frac{{\left (a^{2} b^{2} c^{2} n^{2} + 12 \, a^{2} b^{2} c^{2} - 16 \, a^{3} b c d + 6 \, a^{4} d^{2} -{\left (b^{4} d^{2} n^{3} + 6 \, b^{4} d^{2} n^{2} + 11 \, b^{4} d^{2} n + 6 \, b^{4} d^{2}\right )} x^{4} -{\left (16 \, b^{4} c d +{\left (2 \, b^{4} c d + a b^{3} d^{2}\right )} n^{3} +{\left (14 \, b^{4} c d + 3 \, a b^{3} d^{2}\right )} n^{2} + 2 \,{\left (14 \, b^{4} c d + a b^{3} d^{2}\right )} n\right )} x^{3} -{\left (12 \, b^{4} c^{2} +{\left (b^{4} c^{2} + 2 \, a b^{3} c d\right )} n^{3} +{\left (8 \, b^{4} c^{2} + 10 \, a b^{3} c d - 3 \, a^{2} b^{2} d^{2}\right )} n^{2} +{\left (19 \, b^{4} c^{2} + 8 \, a b^{3} c d - 3 \, a^{2} b^{2} d^{2}\right )} n\right )} x^{2} +{\left (7 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d\right )} n -{\left (a b^{3} c^{2} n^{3} +{\left (7 \, a b^{3} c^{2} - 4 \, a^{2} b^{2} c d\right )} n^{2} + 2 \,{\left (6 \, a b^{3} c^{2} - 8 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*(b*x + a)^n*x,x, algorithm="fricas")
[Out]
-(a^2*b^2*c^2*n^2 + 12*a^2*b^2*c^2 - 16*a^3*b*c*d + 6*a^4*d^2 - (b^4*d^2*n^3 + 6
*b^4*d^2*n^2 + 11*b^4*d^2*n + 6*b^4*d^2)*x^4 - (16*b^4*c*d + (2*b^4*c*d + a*b^3*
d^2)*n^3 + (14*b^4*c*d + 3*a*b^3*d^2)*n^2 + 2*(14*b^4*c*d + a*b^3*d^2)*n)*x^3 -
(12*b^4*c^2 + (b^4*c^2 + 2*a*b^3*c*d)*n^3 + (8*b^4*c^2 + 10*a*b^3*c*d - 3*a^2*b^
2*d^2)*n^2 + (19*b^4*c^2 + 8*a*b^3*c*d - 3*a^2*b^2*d^2)*n)*x^2 + (7*a^2*b^2*c^2
- 4*a^3*b*c*d)*n - (a*b^3*c^2*n^3 + (7*a*b^3*c^2 - 4*a^2*b^2*c*d)*n^2 + 2*(6*a*b
^3*c^2 - 8*a^2*b^2*c*d + 3*a^3*b*d^2)*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 = 9.81202, size = 3402, normalized size = 29.84 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)**n*(d*x+c)**2,x)
[Out]
Piecewise((a**n*(c**2*x**2/2 + 2*c*d*x**3/3 + d**2*x**4/4), Eq(b, 0)), (6*a**5*d
**2*log(a/b + x)/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7
*x**3) + 2*a**5*d**2/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*
b**7*x**3) + 18*a**4*b*d**2*x*log(a/b + x)/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a*
*3*b**6*x**2 + 6*a**2*b**7*x**3) + 18*a**3*b**2*d**2*x**2*log(a/b + x)/(6*a**5*b
**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3) - 9*a**3*b**2*d**2*
x**2/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3) + 6*a
**2*b**3*d**2*x**3*log(a/b + x)/(6*a**5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**
2 + 6*a**2*b**7*x**3) - 9*a**2*b**3*d**2*x**3/(6*a**5*b**4 + 18*a**4*b**5*x + 18
*a**3*b**6*x**2 + 6*a**2*b**7*x**3) + 3*a*b**4*c**2*x**2/(6*a**5*b**4 + 18*a**4*
b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3) + 4*a*b**4*c*d*x**3/(6*a**5*b**4
+ 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3) + b**5*c**2*x**3/(6*a**
5*b**4 + 18*a**4*b**5*x + 18*a**3*b**6*x**2 + 6*a**2*b**7*x**3), Eq(n, -4)), (-6
*a**4*d**2*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) - 3*a**4*d
**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 4*a**3*b*c*d*log(a/b + x)/(2
*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + 2*a**3*b*c*d/(2*a**3*b**4 + 4*a**2
*b**5*x + 2*a*b**6*x**2) - 12*a**3*b*d**2*x*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b
**5*x + 2*a*b**6*x**2) + 8*a**2*b**2*c*d*x*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b*
*5*x + 2*a*b**6*x**2) - 6*a**2*b**2*d**2*x**2*log(a/b + x)/(2*a**3*b**4 + 4*a**2
*b**5*x + 2*a*b**6*x**2) + 6*a**2*b**2*d**2*x**2/(2*a**3*b**4 + 4*a**2*b**5*x +
2*a*b**6*x**2) + 4*a*b**3*c*d*x**2*log(a/b + x)/(2*a**3*b**4 + 4*a**2*b**5*x + 2
*a*b**6*x**2) - 4*a*b**3*c*d*x**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2)
+ 2*a*b**3*d**2*x**3/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2) + b**4*c**2*x
**2/(2*a**3*b**4 + 4*a**2*b**5*x + 2*a*b**6*x**2), Eq(n, -3)), (6*a**3*d**2*log(
a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3*d**2/(2*a*b**4 + 2*b**5*x) - 8*a**2*b*c*
d*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 8*a**2*b*c*d/(2*a*b**4 + 2*b**5*x) + 6*a*
*2*b*d**2*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 2*a*b**2*c**2*log(a/b + x)/(2*a
*b**4 + 2*b**5*x) + 2*a*b**2*c**2/(2*a*b**4 + 2*b**5*x) - 8*a*b**2*c*d*x*log(a/b
+ x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*d**2*x**2/(2*a*b**4 + 2*b**5*x) + 2*b**3*
c**2*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 4*b**3*c*d*x**2/(2*a*b**4 + 2*b**5*x
) + b**3*d**2*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*d**2*log(a/b + x)/b
**4 + 2*a**2*c*d*log(a/b + x)/b**3 + a**2*d**2*x/b**3 - a*c**2*log(a/b + x)/b**2
- 2*a*c*d*x/b**2 - a*d**2*x**2/(2*b**2) + c**2*x/b + c*d*x**2/b + d**2*x**3/(3*
b), Eq(n, -1)), (-6*a**4*d**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**3*b*c*d*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) + 16*a**3*b*c*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) + 6*a**3*b*d**2*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) - a**2*b*
*2*c**2*n**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) - 7*a**2*b**2*c**2*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) - 12*a**2*b**2*c**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**2*b**2*c*d*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) - 16*a**2*b**
2*c*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) - 3*a**2*b**2*d**2*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**2*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**2*n**3*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) + 7*a*
b**3*c**2*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) + 12*a*b**3*c**2*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*b**3*c*d*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) + 10*a*b**3*c*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) + 8
*a*b**3*c*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*d**2*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**2*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**2*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**2*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) + 8*b**4*c**2*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) + 19*b**4*c**2*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) + 12*b**4*c**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) + 2*
b**4*c*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) + 14*b**4*c*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) + 28*b**4*c*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) + 16*b**4*c*d*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*
*2*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**2*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**2*n*x**4*(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*b**4*d**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), True))
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.237082, size = 976, normalized size = 8.56 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*(b*x + a)^n*x,x, algorithm="giac")
[Out]
(b^4*d^2*n^3*x^4*e^(n*ln(b*x + a)) + 2*b^4*c*d*n^3*x^3*e^(n*ln(b*x + a)) + a*b^3
*d^2*n^3*x^3*e^(n*ln(b*x + a)) + 6*b^4*d^2*n^2*x^4*e^(n*ln(b*x + a)) + b^4*c^2*n
^3*x^2*e^(n*ln(b*x + a)) + 2*a*b^3*c*d*n^3*x^2*e^(n*ln(b*x + a)) + 14*b^4*c*d*n^
2*x^3*e^(n*ln(b*x + a)) + 3*a*b^3*d^2*n^2*x^3*e^(n*ln(b*x + a)) + 11*b^4*d^2*n*x
^4*e^(n*ln(b*x + a)) + a*b^3*c^2*n^3*x*e^(n*ln(b*x + a)) + 8*b^4*c^2*n^2*x^2*e^(
n*ln(b*x + a)) + 10*a*b^3*c*d*n^2*x^2*e^(n*ln(b*x + a)) - 3*a^2*b^2*d^2*n^2*x^2*
e^(n*ln(b*x + a)) + 28*b^4*c*d*n*x^3*e^(n*ln(b*x + a)) + 2*a*b^3*d^2*n*x^3*e^(n*
ln(b*x + a)) + 6*b^4*d^2*x^4*e^(n*ln(b*x + a)) + 7*a*b^3*c^2*n^2*x*e^(n*ln(b*x +
a)) - 4*a^2*b^2*c*d*n^2*x*e^(n*ln(b*x + a)) + 19*b^4*c^2*n*x^2*e^(n*ln(b*x + a)
) + 8*a*b^3*c*d*n*x^2*e^(n*ln(b*x + a)) - 3*a^2*b^2*d^2*n*x^2*e^(n*ln(b*x + a))
+ 16*b^4*c*d*x^3*e^(n*ln(b*x + a)) - a^2*b^2*c^2*n^2*e^(n*ln(b*x + a)) + 12*a*b^
3*c^2*n*x*e^(n*ln(b*x + a)) - 16*a^2*b^2*c*d*n*x*e^(n*ln(b*x + a)) + 6*a^3*b*d^2
*n*x*e^(n*ln(b*x + a)) + 12*b^4*c^2*x^2*e^(n*ln(b*x + a)) - 7*a^2*b^2*c^2*n*e^(n
*ln(b*x + a)) + 4*a^3*b*c*d*n*e^(n*ln(b*x + a)) - 12*a^2*b^2*c^2*e^(n*ln(b*x + a
)) + 16*a^3*b*c*d*e^(n*ln(b*x + a)) - 6*a^4*d^2*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)