3.152 \(\int x (a+b x)^n \left (c+d x^3\right ) \, dx\)
Optimal. Leaf size=126 \[ -\frac{a \left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^5 (n+1)}+\frac{\left (b^3 c-4 a^3 d\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{6 a^2 d (a+b x)^{n+3}}{b^5 (n+3)}-\frac{4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d (a+b x)^{n+5}}{b^5 (n+5)} \]
[Out]
-((a*(b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^5*(1 + n))) + ((b^3*c - 4*a^3*d)*(a +
b*x)^(2 + n))/(b^5*(2 + n)) + (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 = 0.147343, antiderivative size = 126, 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
\[ -\frac{a \left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^5 (n+1)}+\frac{\left (b^3 c-4 a^3 d\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac{6 a^2 d (a+b x)^{n+3}}{b^5 (n+3)}-\frac{4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac{d (a+b x)^{n+5}}{b^5 (n+5)} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x)^n*(c + d*x^3),x]
[Out]
-((a*(b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^5*(1 + n))) + ((b^3*c - 4*a^3*d)*(a +
b*x)^(2 + n))/(b^5*(2 + n)) + (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 = 31.4633, size = 112, normalized size = 0.89 \[ \frac{6 a^{2} d \left (a + b x\right )^{n + 3}}{b^{5} \left (n + 3\right )} - \frac{4 a d \left (a + b x\right )^{n + 4}}{b^{5} \left (n + 4\right )} + \frac{a \left (a + b x\right )^{n + 1} \left (a^{3} d - b^{3} c\right )}{b^{5} \left (n + 1\right )} + \frac{d \left (a + b x\right )^{n + 5}}{b^{5} \left (n + 5\right )} - \frac{\left (a + b x\right )^{n + 2} \left (4 a^{3} d - b^{3} c\right )}{b^{5} \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**n*(d*x**3+c),x)
[Out]
6*a**2*d*(a + b*x)**(n + 3)/(b**5*(n + 3)) - 4*a*d*(a + b*x)**(n + 4)/(b**5*(n +
4)) + a*(a + b*x)**(n + 1)*(a**3*d - b**3*c)/(b**5*(n + 1)) + d*(a + b*x)**(n +
5)/(b**5*(n + 5)) - (a + b*x)**(n + 2)*(4*a**3*d - b**3*c)/(b**5*(n + 2))
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Mathematica [A] time = 0.119842, size = 142, normalized size = 1.13 \[ \frac{(a+b x)^{n+1} \left (24 a^4 d-24 a^3 b d (n+1) x+12 a^2 b^2 d \left (n^2+3 n+2\right ) x^2-a b^3 (n+3) \left (c \left (n^2+9 n+20\right )+4 d \left (n^2+3 n+2\right ) x^3\right )+b^4 \left (n^3+8 n^2+19 n+12\right ) x \left (c (n+5)+d (n+2) x^3\right )\right )}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5)} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x)^n*(c + d*x^3),x]
[Out]
((a + b*x)^(1 + n)*(24*a^4*d - 24*a^3*b*d*(1 + n)*x + 12*a^2*b^2*d*(2 + 3*n + n^
2)*x^2 + b^4*(12 + 19*n + 8*n^2 + n^3)*x*(c*(5 + n) + d*(2 + n)*x^3) - a*b^3*(3
+ n)*(c*(20 + 9*n + n^2) + 4*d*(2 + 3*n + n^2)*x^3)))/(b^5*(1 + n)*(2 + n)*(3 +
n)*(4 + n)*(5 + n))
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Maple [B] time = 0.008, size = 283, normalized size = 2.3 \[{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{4}d{n}^{4}{x}^{4}+10\,{b}^{4}d{n}^{3}{x}^{4}-4\,a{b}^{3}d{n}^{3}{x}^{3}+35\,{b}^{4}d{n}^{2}{x}^{4}-24\,a{b}^{3}d{n}^{2}{x}^{3}+{b}^{4}c{n}^{4}x+50\,{b}^{4}dn{x}^{4}+12\,{a}^{2}{b}^{2}d{n}^{2}{x}^{2}-44\,a{b}^{3}dn{x}^{3}+13\,{b}^{4}c{n}^{3}x+24\,d{x}^{4}{b}^{4}+36\,{a}^{2}{b}^{2}dn{x}^{2}-a{b}^{3}c{n}^{3}-24\,ad{x}^{3}{b}^{3}+59\,{b}^{4}c{n}^{2}x-24\,{a}^{3}bdnx+24\,d{a}^{2}{x}^{2}{b}^{2}-12\,a{b}^{3}c{n}^{2}+107\,{b}^{4}cnx-24\,{a}^{3}bdx-47\,a{b}^{3}cn+60\,{b}^{4}cx+24\,{a}^{4}d-60\,a{b}^{3}c \right ) }{{b}^{5} \left ({n}^{5}+15\,{n}^{4}+85\,{n}^{3}+225\,{n}^{2}+274\,n+120 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^n*(d*x^3+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+35*b^4*d*n^2*x^4
-24*a*b^3*d*n^2*x^3+b^4*c*n^4*x+50*b^4*d*n*x^4+12*a^2*b^2*d*n^2*x^2-44*a*b^3*d*n
*x^3+13*b^4*c*n^3*x+24*b^4*d*x^4+36*a^2*b^2*d*n*x^2-a*b^3*c*n^3-24*a*b^3*d*x^3+5
9*b^4*c*n^2*x-24*a^3*b*d*n*x+24*a^2*b^2*d*x^2-12*a*b^3*c*n^2+107*b^4*c*n*x-24*a^
3*b*d*x-47*a*b^3*c*n+60*b^4*c*x+24*a^4*d-60*a*b^3*c)/b^5/(n^5+15*n^4+85*n^3+225*
n^2+274*n+120)
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Maxima [A] time = 0.712591, size = 248, normalized size = 1.97 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac{{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} +{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \,{\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )}{\left (b x + a\right )}^{n} d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)*(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/((n^2 + 3*n + 2)*b^2) + ((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 = 0.28876, size = 470, normalized size = 3.73 \[ -\frac{{\left (a^{2} b^{3} c n^{3} + 12 \, a^{2} b^{3} c n^{2} + 47 \, a^{2} b^{3} c n + 60 \, a^{2} b^{3} c - 24 \, a^{5} d -{\left (b^{5} d n^{4} + 10 \, b^{5} d n^{3} + 35 \, b^{5} d n^{2} + 50 \, b^{5} d n + 24 \, b^{5} d\right )} x^{5} -{\left (a b^{4} d n^{4} + 6 \, a b^{4} d n^{3} + 11 \, a b^{4} d n^{2} + 6 \, a b^{4} d n\right )} x^{4} + 4 \,{\left (a^{2} b^{3} d n^{3} + 3 \, a^{2} b^{3} d n^{2} + 2 \, a^{2} b^{3} d n\right )} x^{3} -{\left (b^{5} c n^{4} + 13 \, b^{5} c n^{3} + 60 \, b^{5} c +{\left (59 \, b^{5} c + 12 \, a^{3} b^{2} d\right )} n^{2} +{\left (107 \, b^{5} c + 12 \, a^{3} b^{2} d\right )} n\right )} x^{2} -{\left (a b^{4} c n^{4} + 12 \, a b^{4} c n^{3} + 47 \, a b^{4} c n^{2} + 12 \,{\left (5 \, a b^{4} c - 2 \, a^{4} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)*(b*x + a)^n*x,x, algorithm="fricas")
[Out]
-(a^2*b^3*c*n^3 + 12*a^2*b^3*c*n^2 + 47*a^2*b^3*c*n + 60*a^2*b^3*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 + 4*(a^2*b^3*d*n^3 + 3*
a^2*b^3*d*n^2 + 2*a^2*b^3*d*n)*x^3 - (b^5*c*n^4 + 13*b^5*c*n^3 + 60*b^5*c + (59*
b^5*c + 12*a^3*b^2*d)*n^2 + (107*b^5*c + 12*a^3*b^2*d)*n)*x^2 - (a*b^4*c*n^4 + 1
2*a*b^4*c*n^3 + 47*a*b^4*c*n^2 + 12*(5*a*b^4*c - 2*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 = 15.3252, size = 3699, normalized size = 29.36 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)**n*(d*x**3+c),x)
[Out]
Piecewise((a**n*(c*x**2/2 + 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) +
7*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 + 1
2*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) + 16*a**3*b*d*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) + 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) - a*b**3*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) + 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) - 24
*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) - 4*b**4*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) + 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) - 18*b**4*
d*x**4/(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)), (-24*a**4*d*log(a/b + x)/(6*a**3*b**5 + 18*a**2*b**6*x +
18*a*b**7*x**2 + 6*b**8*x**3) - 20*a**4*d/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*
b**7*x**2 + 6*b**8*x**3) - 72*a**3*b*d*x*log(a/b + x)/(6*a**3*b**5 + 18*a**2*b**
6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - 36*a**3*b*d*x/(6*a**3*b**5 + 18*a**2*b**6*
x + 18*a*b**7*x**2 + 6*b**8*x**3) - 72*a**2*b**2*d*x**2*log(a/b + x)/(6*a**3*b**
5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - a*b**3*c/(6*a**3*b**5 + 18*
a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - 24*a*b**3*d*x**3*log(a/b + x)/(6*a
**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) + 24*a*b**3*d*x**3/(6*
a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - 3*b**4*c*x/(6*a**3*
b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) + 6*b**4*d*x**4/(6*a**3*b*
*5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*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) + 18*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) + 24*a**3*b*d*x/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) + 12*a**2*
b**2*d*x**2*log(a/b + x)/(2*a**2*b**5 + 4*a*b**6*x + 2*b**7*x**2) - a*b**3*c/(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*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*d*x**2/(3*a*b**5 + 3*b**6*x) + 3*a*
b**3*c*log(a/b + x)/(3*a*b**5 + 3*b**6*x) + 3*a*b**3*c/(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*log(a/b + x)/(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*d*x**2/(2*b**3) - a*c*log(a/b + x)/b**2 - a*d*x**3/(3*b**
2) + c*x/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) + 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) - a**2*b**3*c*n**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) - 12*a**2*b**3*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) - 47*a**2*b**3*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) - 60*a**2*b**3*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 + 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 + 22
5*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 + 22
5*b**5*n**2 + 274*b**5*n + 120*b**5) + a*b**4*c*n**4*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*b**
4*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) + 47*a*b**4*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) + 60*a*b**4*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) + 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 + 15*b**5*n**4 + 85*b**5*n**3 + 225*b**5*n**2 + 274*b**5*n + 1
20*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**2*(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) + 13*b**5*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) + 59*b
**5*c*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) + 107*b**5*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) + 60*b**5*c*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) + 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 + 1
20*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 = 0.270256, size = 857, normalized size = 6.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)*(b*x + a)^n*x,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)) + 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)) + b^5*c*n^4*x^2*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)) + a*b^4*c*n^4*x*e^(n*ln(b*x + a)) + 13*b^5*c*n^3*x^2*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)) + 2
4*b^5*d*x^5*e^(n*ln(b*x + a)) + 12*a*b^4*c*n^3*x*e^(n*ln(b*x + a)) + 59*b^5*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)) - 8*a^2*b^3*d*n
*x^3*e^(n*ln(b*x + a)) - a^2*b^3*c*n^3*e^(n*ln(b*x + a)) + 47*a*b^4*c*n^2*x*e^(n
*ln(b*x + a)) + 107*b^5*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)) - 12*a^2*b^3*c*n^2*e^(n*ln(b*x + a)) + 60*a*b^4*c*n*x*e^(n*ln(b*x + a))
- 24*a^4*b*d*n*x*e^(n*ln(b*x + a)) + 60*b^5*c*x^2*e^(n*ln(b*x + a)) - 47*a^2*b^
3*c*n*e^(n*ln(b*x + a)) - 60*a^2*b^3*c*e^(n*ln(b*x + a)) + 24*a^5*d*e^(n*ln(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)