3.360 \(\int x (a+b x)^n (c+d x^2)^3 \, dx\)
Optimal. Leaf size=282 \[ \frac{d \left (30 a^2 b^2 c d+35 a^4 d^2+3 b^4 c^2\right ) (a+b x)^{n+4}}{b^8 (n+4)}-\frac{5 a d^2 \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac{3 d^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+6}}{b^8 (n+6)}-\frac{a \left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^8 (n+1)}+\frac{\left (a^2 d+b^2 c\right )^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac{3 a d \left (a^2 d+b^2 c\right ) \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^8 (n+3)}-\frac{7 a d^3 (a+b x)^{n+7}}{b^8 (n+7)}+\frac{d^3 (a+b x)^{n+8}}{b^8 (n+8)} \]
[Out]
-((a*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^2*c + a^2*d)^2*(b^2*c + 7*a^2*d)*(a + b*x)^(2 +
n))/(b^8*(2 + n)) - (3*a*d*(b^2*c + a^2*d)*(3*b^2*c + 7*a^2*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) + (d*(3*b^4*c
^2 + 30*a^2*b^2*c*d + 35*a^4*d^2)*(a + b*x)^(4 + n))/(b^8*(4 + n)) - (5*a*d^2*(3*b^2*c + 7*a^2*d)*(a + b*x)^(5
+ n))/(b^8*(5 + n)) + (3*d^2*(b^2*c + 7*a^2*d)*(a + b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^3*(a + b*x)^(7 + n))
/(b^8*(7 + n)) + (d^3*(a + b*x)^(8 + n))/(b^8*(8 + n))
________________________________________________________________________________________
Rubi [A] time = 0.167943, antiderivative size = 282, 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, Rules used =
{772} \[ \frac{d \left (30 a^2 b^2 c d+35 a^4 d^2+3 b^4 c^2\right ) (a+b x)^{n+4}}{b^8 (n+4)}-\frac{5 a d^2 \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac{3 d^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+6}}{b^8 (n+6)}-\frac{a \left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^8 (n+1)}+\frac{\left (a^2 d+b^2 c\right )^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac{3 a d \left (a^2 d+b^2 c\right ) \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^8 (n+3)}-\frac{7 a d^3 (a+b x)^{n+7}}{b^8 (n+7)}+\frac{d^3 (a+b x)^{n+8}}{b^8 (n+8)} \]
Antiderivative was successfully verified.
[In]
Int[x*(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
-((a*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^2*c + a^2*d)^2*(b^2*c + 7*a^2*d)*(a + b*x)^(2 +
n))/(b^8*(2 + n)) - (3*a*d*(b^2*c + a^2*d)*(3*b^2*c + 7*a^2*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) + (d*(3*b^4*c
^2 + 30*a^2*b^2*c*d + 35*a^4*d^2)*(a + b*x)^(4 + n))/(b^8*(4 + n)) - (5*a*d^2*(3*b^2*c + 7*a^2*d)*(a + b*x)^(5
+ n))/(b^8*(5 + n)) + (3*d^2*(b^2*c + 7*a^2*d)*(a + b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^3*(a + b*x)^(7 + n))
/(b^8*(7 + n)) + (d^3*(a + b*x)^(8 + n))/(b^8*(8 + n))
Rule 772
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]
Rubi steps
\begin{align*} \int x (a+b x)^n \left (c+d x^2\right )^3 \, dx &=\int \left (-\frac{a \left (b^2 c+a^2 d\right )^3 (a+b x)^n}{b^7}+\frac{\left (b^2 c+a^2 d\right )^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{1+n}}{b^7}+\frac{3 a d \left (-3 b^2 c-7 a^2 d\right ) \left (b^2 c+a^2 d\right ) (a+b x)^{2+n}}{b^7}+\frac{d \left (3 b^4 c^2+30 a^2 b^2 c d+35 a^4 d^2\right ) (a+b x)^{3+n}}{b^7}-\frac{5 a d^2 \left (3 b^2 c+7 a^2 d\right ) (a+b x)^{4+n}}{b^7}+\frac{3 d^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{5+n}}{b^7}-\frac{7 a d^3 (a+b x)^{6+n}}{b^7}+\frac{d^3 (a+b x)^{7+n}}{b^7}\right ) \, dx\\ &=-\frac{a \left (b^2 c+a^2 d\right )^3 (a+b x)^{1+n}}{b^8 (1+n)}+\frac{\left (b^2 c+a^2 d\right )^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{2+n}}{b^8 (2+n)}-\frac{3 a d \left (b^2 c+a^2 d\right ) \left (3 b^2 c+7 a^2 d\right ) (a+b x)^{3+n}}{b^8 (3+n)}+\frac{d \left (3 b^4 c^2+30 a^2 b^2 c d+35 a^4 d^2\right ) (a+b x)^{4+n}}{b^8 (4+n)}-\frac{5 a d^2 \left (3 b^2 c+7 a^2 d\right ) (a+b x)^{5+n}}{b^8 (5+n)}+\frac{3 d^2 \left (b^2 c+7 a^2 d\right ) (a+b x)^{6+n}}{b^8 (6+n)}-\frac{7 a d^3 (a+b x)^{7+n}}{b^8 (7+n)}+\frac{d^3 (a+b x)^{8+n}}{b^8 (8+n)}\\ \end{align*}
Mathematica [B] time = 1.65199, size = 709, normalized size = 2.51 \[ \frac{(a+b x)^{n+1} \left (-a (n+8) \left (6 (n+6) \left (a^2 d+b^2 c\right ) \left (4 (n+4) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+1) x+b^2 (n+2) \left (c (n+3)+d (n+1) x^2\right )\right )-4 a d (n+1) (a+b x) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )+b^4 (n+1) (n+2) (n+3) (n+4) \left (c+d x^2\right )^2\right )-6 a d (n+1) (a+b x) \left (4 (n+5) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )-4 a d (n+2) (a+b x) \left (2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left (c (n+5)+d (n+3) x^2\right )\right )+b^4 (n+2) (n+3) (n+4) (n+5) \left (c+d x^2\right )^2\right )+b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) \left (c+d x^2\right )^3\right )+6 (n+1) (a+b x) \left ((n+7) \left (a^2 d+b^2 c\right ) \left (4 (n+5) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+2) x+b^2 (n+3) \left (c (n+4)+d (n+2) x^2\right )\right )-4 a d (n+2) (a+b x) \left (2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left (c (n+5)+d (n+3) x^2\right )\right )+b^4 (n+2) (n+3) (n+4) (n+5) \left (c+d x^2\right )^2\right )-a d (n+2) (a+b x) \left (4 (n+6) \left (a^2 d+b^2 c\right ) \left (2 a^2 d-2 a b d (n+3) x+b^2 (n+4) \left (c (n+5)+d (n+3) x^2\right )\right )-4 a d (n+3) (a+b x) \left (2 a^2 d-2 a b d (n+4) x+b^2 (n+5) \left (c (n+6)+d (n+4) x^2\right )\right )+b^4 (n+3) (n+4) (n+5) (n+6) \left (c+d x^2\right )^2\right )\right )+b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (a+b x) \left (c+d x^2\right )^3\right )}{b^8 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8)} \]
Antiderivative was successfully verified.
[In]
Integrate[x*(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
((a + b*x)^(1 + n)*(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(a + b*x)*(c + d*x^2)^3 - a*(8
+ n)*(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(c + d*x^2)^3 + 6*(b^2*c + a^2*d)*(6 + n)*(b^4*(1 +
n)*(2 + n)*(3 + n)*(4 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(4 + n)*(2*a^2*d - 2*a*b*d*(1 + n)*x + b^2*(2 +
n)*(c*(3 + n) + d*(1 + n)*x^2)) - 4*a*d*(1 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n
) + d*(2 + n)*x^2))) - 6*a*d*(1 + n)*(a + b*x)*(b^4*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(c + d*x^2)^2 + 4*(b^2*c +
a^2*d)*(5 + n)*(2*a^2*d - 2*a*b*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2)) - 4*a*d*(2 + n)*(a + b
*x)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*(4 + n)*(c*(5 + n) + d*(3 + n)*x^2)))) + 6*(1 + n)*(a + b*x)*((b^2*c +
a^2*d)*(7 + n)*(b^4*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(c + d*x^2)^2 + 4*(b^2*c + a^2*d)*(5 + n)*(2*a^2*d - 2*a*b
*d*(2 + n)*x + b^2*(3 + n)*(c*(4 + n) + d*(2 + n)*x^2)) - 4*a*d*(2 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(3 + n)*x
+ b^2*(4 + n)*(c*(5 + n) + d*(3 + n)*x^2))) - a*d*(2 + n)*(a + b*x)*(b^4*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(c +
d*x^2)^2 + 4*(b^2*c + a^2*d)*(6 + n)*(2*a^2*d - 2*a*b*d*(3 + n)*x + b^2*(4 + n)*(c*(5 + n) + d*(3 + n)*x^2))
- 4*a*d*(3 + n)*(a + b*x)*(2*a^2*d - 2*a*b*d*(4 + n)*x + b^2*(5 + n)*(c*(6 + n) + d*(4 + n)*x^2))))))/(b^8*(1
+ n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n))
________________________________________________________________________________________
Maple [B] time = 0.059, size = 1639, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(b*x+a)^n*(d*x^2+c)^3,x)
[Out]
-(b*x+a)^(1+n)*(-b^7*d^3*n^7*x^7-28*b^7*d^3*n^6*x^7+7*a*b^6*d^3*n^6*x^6-3*b^7*c*d^2*n^7*x^5-322*b^7*d^3*n^5*x^
7+147*a*b^6*d^3*n^5*x^6-90*b^7*c*d^2*n^6*x^5-1960*b^7*d^3*n^4*x^7-42*a^2*b^5*d^3*n^5*x^5+15*a*b^6*c*d^2*n^6*x^
4+1225*a*b^6*d^3*n^4*x^6-3*b^7*c^2*d*n^7*x^3-1098*b^7*c*d^2*n^5*x^5-6769*b^7*d^3*n^3*x^7-630*a^2*b^5*d^3*n^4*x
^5+375*a*b^6*c*d^2*n^5*x^4+5145*a*b^6*d^3*n^3*x^6-96*b^7*c^2*d*n^6*x^3-7020*b^7*c*d^2*n^4*x^5-13132*b^7*d^3*n^
2*x^7+210*a^3*b^4*d^3*n^4*x^4-60*a^2*b^5*c*d^2*n^5*x^3-3570*a^2*b^5*d^3*n^3*x^5+9*a*b^6*c^2*d*n^6*x^2+3615*a*b
^6*c*d^2*n^4*x^4+11368*a*b^6*d^3*n^2*x^6-b^7*c^3*n^7*x-1254*b^7*c^2*d*n^5*x^3-25227*b^7*c*d^2*n^3*x^5-13068*b^
7*d^3*n*x^7+2100*a^3*b^4*d^3*n^3*x^4-1260*a^2*b^5*c*d^2*n^4*x^3-9450*a^2*b^5*d^3*n^2*x^5+261*a*b^6*c^2*d*n^5*x
^2+17025*a*b^6*c*d^2*n^3*x^4+12348*a*b^6*d^3*n*x^6-34*b^7*c^3*n^6*x-8592*b^7*c^2*d*n^4*x^3-50490*b^7*c*d^2*n^2
*x^5-5040*b^7*d^3*x^7-840*a^4*b^3*d^3*n^3*x^3+180*a^3*b^4*c*d^2*n^4*x^2+7350*a^3*b^4*d^3*n^2*x^4-18*a^2*b^5*c^
2*d*n^5*x-9420*a^2*b^5*c*d^2*n^3*x^3-11508*a^2*b^5*d^3*n*x^5+a*b^6*c^3*n^6+2979*a*b^6*c^2*d*n^4*x^2+41010*a*b^
6*c*d^2*n^2*x^4+5040*a*b^6*d^3*x^6-478*b^7*c^3*n^5*x-32979*b^7*c^2*d*n^3*x^3-51432*b^7*c*d^2*n*x^5-5040*a^4*b^
3*d^3*n^2*x^3+3240*a^3*b^4*c*d^2*n^3*x^2+10500*a^3*b^4*d^3*n*x^4-486*a^2*b^5*c^2*d*n^4*x-30420*a^2*b^5*c*d^2*n
^2*x^3-5040*a^2*b^5*d^3*x^5+33*a*b^6*c^3*n^5+16839*a*b^6*c^2*d*n^3*x^2+47400*a*b^6*c*d^2*n*x^4-3580*b^7*c^3*n^
4*x-69936*b^7*c^2*d*n^2*x^3-20160*b^7*c*d^2*x^5+2520*a^5*b^2*d^3*n^2*x^2-360*a^4*b^3*c*d^2*n^3*x-9240*a^4*b^3*
d^3*n*x^3+18*a^3*b^4*c^2*d*n^4+18540*a^3*b^4*c*d^2*n^2*x^2+5040*a^3*b^4*d^3*x^4-4986*a^2*b^5*c^2*d*n^3*x-42360
*a^2*b^5*c*d^2*n*x^3+445*a*b^6*c^3*n^4+48420*a*b^6*c^2*d*n^2*x^2+20160*a*b^6*c*d^2*x^4-15289*b^7*c^3*n^3*x-746
28*b^7*c^2*d*n*x^3+7560*a^5*b^2*d^3*n*x^2-5760*a^4*b^3*c*d^2*n^2*x-5040*a^4*b^3*d^3*x^3+468*a^3*b^4*c^2*d*n^3+
35640*a^3*b^4*c*d^2*n*x^2-23706*a^2*b^5*c^2*d*n^2*x-20160*a^2*b^5*c*d^2*x^3+3135*a*b^6*c^3*n^3+64548*a*b^6*c^2
*d*n*x^2-36706*b^7*c^3*n^2*x-30240*b^7*c^2*d*x^3-5040*a^6*b*d^3*n*x+360*a^5*b^2*c*d^2*n^2+5040*a^5*b^2*d^3*x^2
-25560*a^4*b^3*c*d^2*n*x+4518*a^3*b^4*c^2*d*n^2+20160*a^3*b^4*c*d^2*x^2-49428*a^2*b^5*c^2*d*n*x+12154*a*b^6*c^
3*n^2+30240*a*b^6*c^2*d*x^2-44712*b^7*c^3*n*x-5040*a^6*b*d^3*x+5400*a^5*b^2*c*d^2*n-20160*a^4*b^3*c*d^2*x+1918
8*a^3*b^4*c^2*d*n-30240*a^2*b^5*c^2*d*x+24552*a*b^6*c^3*n-20160*b^7*c^3*x+5040*a^7*d^3+20160*a^5*b^2*c*d^2+302
40*a^3*b^4*c^2*d+20160*a*b^6*c^3)/b^8/(n^8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3+118124*n^2+109584*n+403
20)
________________________________________________________________________________________
Maxima [B] time = 1.08426, size = 844, normalized size = 2.99 \begin{align*} \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n} c^{3}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac{3 \,{\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^{2} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac{3 \,{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} +{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \,{\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )}{\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} + \frac{{\left ({\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{8} x^{8} +{\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a b^{7} x^{7} - 7 \,{\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{2} b^{6} x^{6} + 42 \,{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{3} b^{5} x^{5} - 210 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{4} b^{4} x^{4} + 840 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{5} b^{3} x^{3} - 2520 \,{\left (n^{2} + n\right )} a^{6} b^{2} x^{2} + 5040 \, a^{7} b n x - 5040 \, a^{8}\right )}{\left (b x + a\right )}^{n} d^{3}}{{\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x^2+c)^3,x, algorithm="maxima")
[Out]
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^3/((n^2 + 3*n + 2)*b^2) + 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*c^2*d/((n^4 + 10*
n^3 + 35*n^2 + 50*n + 24)*b^4) + 3*((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*n^4 +
35*n^3 + 50*n^2 + 24*n)*a*b^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^
3*x^3 - 60*(n^2 + n)*a^4*b^2*x^2 + 120*a^5*b*n*x - 120*a^6)*(b*x + a)^n*c*d^2/((n^6 + 21*n^5 + 175*n^4 + 735*n
^3 + 1624*n^2 + 1764*n + 720)*b^6) + ((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 50
40)*b^8*x^8 + (n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a*b^7*x^7 - 7*(n^6 + 15*n^5 + 8
5*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^2*b^6*x^6 + 42*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^3*b^5*x^5 - 210*
(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^4*b^4*x^4 + 840*(n^3 + 3*n^2 + 2*n)*a^5*b^3*x^3 - 2520*(n^2 + n)*a^6*b^2*x^2 +
5040*a^7*b*n*x - 5040*a^8)*(b*x + a)^n*d^3/((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 11812
4*n^2 + 109584*n + 40320)*b^8)
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Fricas [B] time = 2.24505, size = 3702, normalized size = 13.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x^2+c)^3,x, algorithm="fricas")
[Out]
-(a^2*b^6*c^3*n^6 + 33*a^2*b^6*c^3*n^5 + 20160*a^2*b^6*c^3 + 30240*a^4*b^4*c^2*d + 20160*a^6*b^2*c*d^2 + 5040*
a^8*d^3 - (b^8*d^3*n^7 + 28*b^8*d^3*n^6 + 322*b^8*d^3*n^5 + 1960*b^8*d^3*n^4 + 6769*b^8*d^3*n^3 + 13132*b^8*d^
3*n^2 + 13068*b^8*d^3*n + 5040*b^8*d^3)*x^8 - (a*b^7*d^3*n^7 + 21*a*b^7*d^3*n^6 + 175*a*b^7*d^3*n^5 + 735*a*b^
7*d^3*n^4 + 1624*a*b^7*d^3*n^3 + 1764*a*b^7*d^3*n^2 + 720*a*b^7*d^3*n)*x^7 - (3*b^8*c*d^2*n^7 + 20160*b^8*c*d^
2 + (90*b^8*c*d^2 - 7*a^2*b^6*d^3)*n^6 + 3*(366*b^8*c*d^2 - 35*a^2*b^6*d^3)*n^5 + 5*(1404*b^8*c*d^2 - 119*a^2*
b^6*d^3)*n^4 + 9*(2803*b^8*c*d^2 - 175*a^2*b^6*d^3)*n^3 + 2*(25245*b^8*c*d^2 - 959*a^2*b^6*d^3)*n^2 + 24*(2143
*b^8*c*d^2 - 35*a^2*b^6*d^3)*n)*x^6 - 3*(a*b^7*c*d^2*n^7 + 25*a*b^7*c*d^2*n^6 + (241*a*b^7*c*d^2 + 14*a^3*b^5*
d^3)*n^5 + 5*(227*a*b^7*c*d^2 + 28*a^3*b^5*d^3)*n^4 + 2*(1367*a*b^7*c*d^2 + 245*a^3*b^5*d^3)*n^3 + 20*(158*a*b
^7*c*d^2 + 35*a^3*b^5*d^3)*n^2 + 336*(4*a*b^7*c*d^2 + a^3*b^5*d^3)*n)*x^5 + (445*a^2*b^6*c^3 + 18*a^4*b^4*c^2*
d)*n^4 - 3*(b^8*c^2*d*n^7 + 10080*b^8*c^2*d + (32*b^8*c^2*d - 5*a^2*b^6*c*d^2)*n^6 + (418*b^8*c^2*d - 105*a^2*
b^6*c*d^2)*n^5 + (2864*b^8*c^2*d - 785*a^2*b^6*c*d^2 - 70*a^4*b^4*d^3)*n^4 + (10993*b^8*c^2*d - 2535*a^2*b^6*c
*d^2 - 420*a^4*b^4*d^3)*n^3 + 2*(11656*b^8*c^2*d - 1765*a^2*b^6*c*d^2 - 385*a^4*b^4*d^3)*n^2 + 12*(2073*b^8*c^
2*d - 140*a^2*b^6*c*d^2 - 35*a^4*b^4*d^3)*n)*x^4 + 3*(1045*a^2*b^6*c^3 + 156*a^4*b^4*c^2*d)*n^3 - 3*(a*b^7*c^2
*d*n^7 + 29*a*b^7*c^2*d*n^6 + (331*a*b^7*c^2*d + 20*a^3*b^5*c*d^2)*n^5 + (1871*a*b^7*c^2*d + 360*a^3*b^5*c*d^2
)*n^4 + 20*(269*a*b^7*c^2*d + 103*a^3*b^5*c*d^2 + 14*a^5*b^3*d^3)*n^3 + 4*(1793*a*b^7*c^2*d + 990*a^3*b^5*c*d^
2 + 210*a^5*b^3*d^3)*n^2 + 560*(6*a*b^7*c^2*d + 4*a^3*b^5*c*d^2 + a^5*b^3*d^3)*n)*x^3 + 2*(6077*a^2*b^6*c^3 +
2259*a^4*b^4*c^2*d + 180*a^6*b^2*c*d^2)*n^2 - (b^8*c^3*n^7 + 20160*b^8*c^3 + (34*b^8*c^3 - 9*a^2*b^6*c^2*d)*n^
6 + (478*b^8*c^3 - 243*a^2*b^6*c^2*d)*n^5 + (3580*b^8*c^3 - 2493*a^2*b^6*c^2*d - 180*a^4*b^4*c*d^2)*n^4 + (152
89*b^8*c^3 - 11853*a^2*b^6*c^2*d - 2880*a^4*b^4*c*d^2)*n^3 + 2*(18353*b^8*c^3 - 12357*a^2*b^6*c^2*d - 6390*a^4
*b^4*c*d^2 - 1260*a^6*b^2*d^3)*n^2 + 72*(621*b^8*c^3 - 210*a^2*b^6*c^2*d - 140*a^4*b^4*c*d^2 - 35*a^6*b^2*d^3)
*n)*x^2 + 36*(682*a^2*b^6*c^3 + 533*a^4*b^4*c^2*d + 150*a^6*b^2*c*d^2)*n - (a*b^7*c^3*n^7 + 33*a*b^7*c^3*n^6 +
(445*a*b^7*c^3 + 18*a^3*b^5*c^2*d)*n^5 + 3*(1045*a*b^7*c^3 + 156*a^3*b^5*c^2*d)*n^4 + 2*(6077*a*b^7*c^3 + 225
9*a^3*b^5*c^2*d + 180*a^5*b^3*c*d^2)*n^3 + 36*(682*a*b^7*c^3 + 533*a^3*b^5*c^2*d + 150*a^5*b^3*c*d^2)*n^2 + 50
40*(4*a*b^7*c^3 + 6*a^3*b^5*c^2*d + 4*a^5*b^3*c*d^2 + a^7*b*d^3)*n)*x)*(b*x + a)^n/(b^8*n^8 + 36*b^8*n^7 + 546
*b^8*n^6 + 4536*b^8*n^5 + 22449*b^8*n^4 + 67284*b^8*n^3 + 118124*b^8*n^2 + 109584*b^8*n + 40320*b^8)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)**n*(d*x**2+c)**3,x)
[Out]
Timed out
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Giac [B] time = 1.26606, size = 3849, normalized size = 13.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(b*x+a)^n*(d*x^2+c)^3,x, algorithm="giac")
[Out]
((b*x + a)^n*b^8*d^3*n^7*x^8 + (b*x + a)^n*a*b^7*d^3*n^7*x^7 + 28*(b*x + a)^n*b^8*d^3*n^6*x^8 + 3*(b*x + a)^n*
b^8*c*d^2*n^7*x^6 + 21*(b*x + a)^n*a*b^7*d^3*n^6*x^7 + 322*(b*x + a)^n*b^8*d^3*n^5*x^8 + 3*(b*x + a)^n*a*b^7*c
*d^2*n^7*x^5 + 90*(b*x + a)^n*b^8*c*d^2*n^6*x^6 - 7*(b*x + a)^n*a^2*b^6*d^3*n^6*x^6 + 175*(b*x + a)^n*a*b^7*d^
3*n^5*x^7 + 1960*(b*x + a)^n*b^8*d^3*n^4*x^8 + 3*(b*x + a)^n*b^8*c^2*d*n^7*x^4 + 75*(b*x + a)^n*a*b^7*c*d^2*n^
6*x^5 + 1098*(b*x + a)^n*b^8*c*d^2*n^5*x^6 - 105*(b*x + a)^n*a^2*b^6*d^3*n^5*x^6 + 735*(b*x + a)^n*a*b^7*d^3*n
^4*x^7 + 6769*(b*x + a)^n*b^8*d^3*n^3*x^8 + 3*(b*x + a)^n*a*b^7*c^2*d*n^7*x^3 + 96*(b*x + a)^n*b^8*c^2*d*n^6*x
^4 - 15*(b*x + a)^n*a^2*b^6*c*d^2*n^6*x^4 + 723*(b*x + a)^n*a*b^7*c*d^2*n^5*x^5 + 42*(b*x + a)^n*a^3*b^5*d^3*n
^5*x^5 + 7020*(b*x + a)^n*b^8*c*d^2*n^4*x^6 - 595*(b*x + a)^n*a^2*b^6*d^3*n^4*x^6 + 1624*(b*x + a)^n*a*b^7*d^3
*n^3*x^7 + 13132*(b*x + a)^n*b^8*d^3*n^2*x^8 + (b*x + a)^n*b^8*c^3*n^7*x^2 + 87*(b*x + a)^n*a*b^7*c^2*d*n^6*x^
3 + 1254*(b*x + a)^n*b^8*c^2*d*n^5*x^4 - 315*(b*x + a)^n*a^2*b^6*c*d^2*n^5*x^4 + 3405*(b*x + a)^n*a*b^7*c*d^2*
n^4*x^5 + 420*(b*x + a)^n*a^3*b^5*d^3*n^4*x^5 + 25227*(b*x + a)^n*b^8*c*d^2*n^3*x^6 - 1575*(b*x + a)^n*a^2*b^6
*d^3*n^3*x^6 + 1764*(b*x + a)^n*a*b^7*d^3*n^2*x^7 + 13068*(b*x + a)^n*b^8*d^3*n*x^8 + (b*x + a)^n*a*b^7*c^3*n^
7*x + 34*(b*x + a)^n*b^8*c^3*n^6*x^2 - 9*(b*x + a)^n*a^2*b^6*c^2*d*n^6*x^2 + 993*(b*x + a)^n*a*b^7*c^2*d*n^5*x
^3 + 60*(b*x + a)^n*a^3*b^5*c*d^2*n^5*x^3 + 8592*(b*x + a)^n*b^8*c^2*d*n^4*x^4 - 2355*(b*x + a)^n*a^2*b^6*c*d^
2*n^4*x^4 - 210*(b*x + a)^n*a^4*b^4*d^3*n^4*x^4 + 8202*(b*x + a)^n*a*b^7*c*d^2*n^3*x^5 + 1470*(b*x + a)^n*a^3*
b^5*d^3*n^3*x^5 + 50490*(b*x + a)^n*b^8*c*d^2*n^2*x^6 - 1918*(b*x + a)^n*a^2*b^6*d^3*n^2*x^6 + 720*(b*x + a)^n
*a*b^7*d^3*n*x^7 + 5040*(b*x + a)^n*b^8*d^3*x^8 + 33*(b*x + a)^n*a*b^7*c^3*n^6*x + 478*(b*x + a)^n*b^8*c^3*n^5
*x^2 - 243*(b*x + a)^n*a^2*b^6*c^2*d*n^5*x^2 + 5613*(b*x + a)^n*a*b^7*c^2*d*n^4*x^3 + 1080*(b*x + a)^n*a^3*b^5
*c*d^2*n^4*x^3 + 32979*(b*x + a)^n*b^8*c^2*d*n^3*x^4 - 7605*(b*x + a)^n*a^2*b^6*c*d^2*n^3*x^4 - 1260*(b*x + a)
^n*a^4*b^4*d^3*n^3*x^4 + 9480*(b*x + a)^n*a*b^7*c*d^2*n^2*x^5 + 2100*(b*x + a)^n*a^3*b^5*d^3*n^2*x^5 + 51432*(
b*x + a)^n*b^8*c*d^2*n*x^6 - 840*(b*x + a)^n*a^2*b^6*d^3*n*x^6 - (b*x + a)^n*a^2*b^6*c^3*n^6 + 445*(b*x + a)^n
*a*b^7*c^3*n^5*x + 18*(b*x + a)^n*a^3*b^5*c^2*d*n^5*x + 3580*(b*x + a)^n*b^8*c^3*n^4*x^2 - 2493*(b*x + a)^n*a^
2*b^6*c^2*d*n^4*x^2 - 180*(b*x + a)^n*a^4*b^4*c*d^2*n^4*x^2 + 16140*(b*x + a)^n*a*b^7*c^2*d*n^3*x^3 + 6180*(b*
x + a)^n*a^3*b^5*c*d^2*n^3*x^3 + 840*(b*x + a)^n*a^5*b^3*d^3*n^3*x^3 + 69936*(b*x + a)^n*b^8*c^2*d*n^2*x^4 - 1
0590*(b*x + a)^n*a^2*b^6*c*d^2*n^2*x^4 - 2310*(b*x + a)^n*a^4*b^4*d^3*n^2*x^4 + 4032*(b*x + a)^n*a*b^7*c*d^2*n
*x^5 + 1008*(b*x + a)^n*a^3*b^5*d^3*n*x^5 + 20160*(b*x + a)^n*b^8*c*d^2*x^6 - 33*(b*x + a)^n*a^2*b^6*c^3*n^5 +
3135*(b*x + a)^n*a*b^7*c^3*n^4*x + 468*(b*x + a)^n*a^3*b^5*c^2*d*n^4*x + 15289*(b*x + a)^n*b^8*c^3*n^3*x^2 -
11853*(b*x + a)^n*a^2*b^6*c^2*d*n^3*x^2 - 2880*(b*x + a)^n*a^4*b^4*c*d^2*n^3*x^2 + 21516*(b*x + a)^n*a*b^7*c^2
*d*n^2*x^3 + 11880*(b*x + a)^n*a^3*b^5*c*d^2*n^2*x^3 + 2520*(b*x + a)^n*a^5*b^3*d^3*n^2*x^3 + 74628*(b*x + a)^
n*b^8*c^2*d*n*x^4 - 5040*(b*x + a)^n*a^2*b^6*c*d^2*n*x^4 - 1260*(b*x + a)^n*a^4*b^4*d^3*n*x^4 - 445*(b*x + a)^
n*a^2*b^6*c^3*n^4 - 18*(b*x + a)^n*a^4*b^4*c^2*d*n^4 + 12154*(b*x + a)^n*a*b^7*c^3*n^3*x + 4518*(b*x + a)^n*a^
3*b^5*c^2*d*n^3*x + 360*(b*x + a)^n*a^5*b^3*c*d^2*n^3*x + 36706*(b*x + a)^n*b^8*c^3*n^2*x^2 - 24714*(b*x + a)^
n*a^2*b^6*c^2*d*n^2*x^2 - 12780*(b*x + a)^n*a^4*b^4*c*d^2*n^2*x^2 - 2520*(b*x + a)^n*a^6*b^2*d^3*n^2*x^2 + 100
80*(b*x + a)^n*a*b^7*c^2*d*n*x^3 + 6720*(b*x + a)^n*a^3*b^5*c*d^2*n*x^3 + 1680*(b*x + a)^n*a^5*b^3*d^3*n*x^3 +
30240*(b*x + a)^n*b^8*c^2*d*x^4 - 3135*(b*x + a)^n*a^2*b^6*c^3*n^3 - 468*(b*x + a)^n*a^4*b^4*c^2*d*n^3 + 2455
2*(b*x + a)^n*a*b^7*c^3*n^2*x + 19188*(b*x + a)^n*a^3*b^5*c^2*d*n^2*x + 5400*(b*x + a)^n*a^5*b^3*c*d^2*n^2*x +
44712*(b*x + a)^n*b^8*c^3*n*x^2 - 15120*(b*x + a)^n*a^2*b^6*c^2*d*n*x^2 - 10080*(b*x + a)^n*a^4*b^4*c*d^2*n*x
^2 - 2520*(b*x + a)^n*a^6*b^2*d^3*n*x^2 - 12154*(b*x + a)^n*a^2*b^6*c^3*n^2 - 4518*(b*x + a)^n*a^4*b^4*c^2*d*n
^2 - 360*(b*x + a)^n*a^6*b^2*c*d^2*n^2 + 20160*(b*x + a)^n*a*b^7*c^3*n*x + 30240*(b*x + a)^n*a^3*b^5*c^2*d*n*x
+ 20160*(b*x + a)^n*a^5*b^3*c*d^2*n*x + 5040*(b*x + a)^n*a^7*b*d^3*n*x + 20160*(b*x + a)^n*b^8*c^3*x^2 - 2455
2*(b*x + a)^n*a^2*b^6*c^3*n - 19188*(b*x + a)^n*a^4*b^4*c^2*d*n - 5400*(b*x + a)^n*a^6*b^2*c*d^2*n - 20160*(b*
x + a)^n*a^2*b^6*c^3 - 30240*(b*x + a)^n*a^4*b^4*c^2*d - 20160*(b*x + a)^n*a^6*b^2*c*d^2 - 5040*(b*x + a)^n*a^
8*d^3)/(b^8*n^8 + 36*b^8*n^7 + 546*b^8*n^6 + 4536*b^8*n^5 + 22449*b^8*n^4 + 67284*b^8*n^3 + 118124*b^8*n^2 + 1
09584*b^8*n + 40320*b^8)