\(\int x (c+d x)^n (a+b x^2)^3 \, dx\) [1774]

Optimal result

Integrand size = 18, antiderivative size = 282 \[ \int x (c+d x)^n \left (a+b x^2\right )^3 \, dx=-\frac {c \left (b c^2+a d^2\right )^3 (c+d x)^{1+n}}{d^8 (1+n)}+\frac {\left (b c^2+a d^2\right )^2 \left (7 b c^2+a d^2\right ) (c+d x)^{2+n}}{d^8 (2+n)}-\frac {3 b c \left (b c^2+a d^2\right ) \left (7 b c^2+3 a d^2\right ) (c+d x)^{3+n}}{d^8 (3+n)}+\frac {b \left (35 b^2 c^4+30 a b c^2 d^2+3 a^2 d^4\right ) (c+d x)^{4+n}}{d^8 (4+n)}-\frac {5 b^2 c \left (7 b c^2+3 a d^2\right ) (c+d x)^{5+n}}{d^8 (5+n)}+\frac {3 b^2 \left (7 b c^2+a d^2\right ) (c+d x)^{6+n}}{d^8 (6+n)}-\frac {7 b^3 c (c+d x)^{7+n}}{d^8 (7+n)}+\frac {b^3 (c+d x)^{8+n}}{d^8 (8+n)} \] Output:

-c*(a*d^2+b*c^2)^3*(d*x+c)^(1+n)/d^8/(1+n)+(a*d^2+b*c^2)^2*(a*d^2+7*b*c^2) 
*(d*x+c)^(2+n)/d^8/(2+n)-3*b*c*(a*d^2+b*c^2)*(3*a*d^2+7*b*c^2)*(d*x+c)^(3+ 
n)/d^8/(3+n)+b*(3*a^2*d^4+30*a*b*c^2*d^2+35*b^2*c^4)*(d*x+c)^(4+n)/d^8/(4+ 
n)-5*b^2*c*(3*a*d^2+7*b*c^2)*(d*x+c)^(5+n)/d^8/(5+n)+3*b^2*(a*d^2+7*b*c^2) 
*(d*x+c)^(6+n)/d^8/(6+n)-7*b^3*c*(d*x+c)^(7+n)/d^8/(7+n)+b^3*(d*x+c)^(8+n) 
/d^8/(8+n)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(773\) vs. \(2(282)=564\).

Time = 1.23 (sec) , antiderivative size = 773, normalized size of antiderivative = 2.74 \[ \int x (c+d x)^n \left (a+b x^2\right )^3 \, dx=\frac {(c+d x)^{1+n} \left (d^6 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n) (7+n) (c+d x) \left (a+b x^2\right )^3-c (8+n) \left (d^6 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n) \left (a+b x^2\right )^3+6 \left (b c^2+a d^2\right ) (6+n) \left (d^4 (1+n) (2+n) (3+n) (4+n) \left (a+b x^2\right )^2+4 \left (b c^2+a d^2\right ) (4+n) \left (a d^2 \left (6+5 n+n^2\right )+b \left (2 c^2-2 c d (1+n) x+d^2 \left (2+3 n+n^2\right ) x^2\right )\right )-4 b c (1+n) (c+d x) \left (a d^2 \left (12+7 n+n^2\right )+b \left (2 c^2-2 c d (2+n) x+d^2 \left (6+5 n+n^2\right ) x^2\right )\right )\right )-6 b c (1+n) (c+d x) \left (d^4 (2+n) (3+n) (4+n) (5+n) \left (a+b x^2\right )^2+4 \left (b c^2+a d^2\right ) (5+n) \left (a d^2 \left (12+7 n+n^2\right )+b \left (2 c^2-2 c d (2+n) x+d^2 \left (6+5 n+n^2\right ) x^2\right )\right )-4 b c (2+n) (c+d x) \left (a d^2 \left (20+9 n+n^2\right )+b \left (2 c^2-2 c d (3+n) x+d^2 \left (12+7 n+n^2\right ) x^2\right )\right )\right )\right )+6 (1+n) (c+d x) \left (\left (b c^2+a d^2\right ) (7+n) \left (d^4 (2+n) (3+n) (4+n) (5+n) \left (a+b x^2\right )^2+4 \left (b c^2+a d^2\right ) (5+n) \left (a d^2 \left (12+7 n+n^2\right )+b \left (2 c^2-2 c d (2+n) x+d^2 \left (6+5 n+n^2\right ) x^2\right )\right )-4 b c (2+n) (c+d x) \left (a d^2 \left (20+9 n+n^2\right )+b \left (2 c^2-2 c d (3+n) x+d^2 \left (12+7 n+n^2\right ) x^2\right )\right )\right )-b c (2+n) (c+d x) \left (d^4 (3+n) (4+n) (5+n) (6+n) \left (a+b x^2\right )^2+4 \left (b c^2+a d^2\right ) (6+n) \left (a d^2 \left (20+9 n+n^2\right )+b \left (2 c^2-2 c d (3+n) x+d^2 \left (12+7 n+n^2\right ) x^2\right )\right )-4 b c (3+n) (c+d x) \left (a d^2 \left (30+11 n+n^2\right )+b \left (2 c^2-2 c d (4+n) x+d^2 \left (20+9 n+n^2\right ) x^2\right )\right )\right )\right )\right )}{d^8 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n) (7+n) (8+n)} \] Input:

Integrate[x*(c + d*x)^n*(a + b*x^2)^3,x]
 

Output:

((c + d*x)^(1 + n)*(d^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 
 + n)*(c + d*x)*(a + b*x^2)^3 - c*(8 + n)*(d^6*(1 + n)*(2 + n)*(3 + n)*(4 
+ n)*(5 + n)*(6 + n)*(a + b*x^2)^3 + 6*(b*c^2 + a*d^2)*(6 + n)*(d^4*(1 + n 
)*(2 + n)*(3 + n)*(4 + n)*(a + b*x^2)^2 + 4*(b*c^2 + a*d^2)*(4 + n)*(a*d^2 
*(6 + 5*n + n^2) + b*(2*c^2 - 2*c*d*(1 + n)*x + d^2*(2 + 3*n + n^2)*x^2)) 
- 4*b*c*(1 + n)*(c + d*x)*(a*d^2*(12 + 7*n + n^2) + b*(2*c^2 - 2*c*d*(2 + 
n)*x + d^2*(6 + 5*n + n^2)*x^2))) - 6*b*c*(1 + n)*(c + d*x)*(d^4*(2 + n)*( 
3 + n)*(4 + n)*(5 + n)*(a + b*x^2)^2 + 4*(b*c^2 + a*d^2)*(5 + n)*(a*d^2*(1 
2 + 7*n + n^2) + b*(2*c^2 - 2*c*d*(2 + n)*x + d^2*(6 + 5*n + n^2)*x^2)) - 
4*b*c*(2 + n)*(c + d*x)*(a*d^2*(20 + 9*n + n^2) + b*(2*c^2 - 2*c*d*(3 + n) 
*x + d^2*(12 + 7*n + n^2)*x^2)))) + 6*(1 + n)*(c + d*x)*((b*c^2 + a*d^2)*( 
7 + n)*(d^4*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(a + b*x^2)^2 + 4*(b*c^2 + a*d 
^2)*(5 + n)*(a*d^2*(12 + 7*n + n^2) + b*(2*c^2 - 2*c*d*(2 + n)*x + d^2*(6 
+ 5*n + n^2)*x^2)) - 4*b*c*(2 + n)*(c + d*x)*(a*d^2*(20 + 9*n + n^2) + b*( 
2*c^2 - 2*c*d*(3 + n)*x + d^2*(12 + 7*n + n^2)*x^2))) - b*c*(2 + n)*(c + d 
*x)*(d^4*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(a + b*x^2)^2 + 4*(b*c^2 + a*d^2) 
*(6 + n)*(a*d^2*(20 + 9*n + n^2) + b*(2*c^2 - 2*c*d*(3 + n)*x + d^2*(12 + 
7*n + n^2)*x^2)) - 4*b*c*(3 + n)*(c + d*x)*(a*d^2*(30 + 11*n + n^2) + b*(2 
*c^2 - 2*c*d*(4 + n)*x + d^2*(20 + 9*n + n^2)*x^2))))))/(d^8*(1 + n)*(2 + 
n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 + n))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {522, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \left (a+b x^2\right )^3 (c+d x)^n \, dx\)

\(\Big \downarrow \) 522

\(\displaystyle \int \left (\frac {b \left (3 a^2 d^4+30 a b c^2 d^2+35 b^2 c^4\right ) (c+d x)^{n+3}}{d^7}-\frac {5 b^2 c \left (3 a d^2+7 b c^2\right ) (c+d x)^{n+4}}{d^7}+\frac {3 b^2 \left (a d^2+7 b c^2\right ) (c+d x)^{n+5}}{d^7}-\frac {c \left (a d^2+b c^2\right )^3 (c+d x)^n}{d^7}+\frac {\left (a d^2+b c^2\right )^2 \left (a d^2+7 b c^2\right ) (c+d x)^{n+1}}{d^7}+\frac {3 b c \left (-3 a d^2-7 b c^2\right ) \left (a d^2+b c^2\right ) (c+d x)^{n+2}}{d^7}-\frac {7 b^3 c (c+d x)^{n+6}}{d^7}+\frac {b^3 (c+d x)^{n+7}}{d^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (3 a^2 d^4+30 a b c^2 d^2+35 b^2 c^4\right ) (c+d x)^{n+4}}{d^8 (n+4)}-\frac {5 b^2 c \left (3 a d^2+7 b c^2\right ) (c+d x)^{n+5}}{d^8 (n+5)}+\frac {3 b^2 \left (a d^2+7 b c^2\right ) (c+d x)^{n+6}}{d^8 (n+6)}-\frac {c \left (a d^2+b c^2\right )^3 (c+d x)^{n+1}}{d^8 (n+1)}+\frac {\left (a d^2+b c^2\right )^2 \left (a d^2+7 b c^2\right ) (c+d x)^{n+2}}{d^8 (n+2)}-\frac {3 b c \left (a d^2+b c^2\right ) \left (3 a d^2+7 b c^2\right ) (c+d x)^{n+3}}{d^8 (n+3)}-\frac {7 b^3 c (c+d x)^{n+7}}{d^8 (n+7)}+\frac {b^3 (c+d x)^{n+8}}{d^8 (n+8)}\)

Input:

Int[x*(c + d*x)^n*(a + b*x^2)^3,x]
 

Output:

-((c*(b*c^2 + a*d^2)^3*(c + d*x)^(1 + n))/(d^8*(1 + n))) + ((b*c^2 + a*d^2 
)^2*(7*b*c^2 + a*d^2)*(c + d*x)^(2 + n))/(d^8*(2 + n)) - (3*b*c*(b*c^2 + a 
*d^2)*(7*b*c^2 + 3*a*d^2)*(c + d*x)^(3 + n))/(d^8*(3 + n)) + (b*(35*b^2*c^ 
4 + 30*a*b*c^2*d^2 + 3*a^2*d^4)*(c + d*x)^(4 + n))/(d^8*(4 + n)) - (5*b^2* 
c*(7*b*c^2 + 3*a*d^2)*(c + d*x)^(5 + n))/(d^8*(5 + n)) + (3*b^2*(7*b*c^2 + 
 a*d^2)*(c + d*x)^(6 + n))/(d^8*(6 + n)) - (7*b^3*c*(c + d*x)^(7 + n))/(d^ 
8*(7 + n)) + (b^3*(c + d*x)^(8 + n))/(d^8*(8 + n))
 

Defintions of rubi rules used

rule 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1209\) vs. \(2(282)=564\).

Time = 0.40 (sec) , antiderivative size = 1210, normalized size of antiderivative = 4.29

method	result	size
norman	\(\text {Expression too large to display}\)	\(1210\)
gosper	\(\text {Expression too large to display}\)	\(1639\)
orering	\(\text {Expression too large to display}\)	\(1642\)
risch	\(\text {Expression too large to display}\)	\(1955\)
parallelrisch	\(\text {Expression too large to display}\)	\(2917\)

Input:

int(x*(d*x+c)^n*(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

b^3/(8+n)*x^8*exp(n*ln(d*x+c))+(a^3*d^6*n^6+33*a^3*d^6*n^5-9*a^2*b*c^2*d^4 
*n^5+445*a^3*d^6*n^4-234*a^2*b*c^2*d^4*n^4+3135*a^3*d^6*n^3-2259*a^2*b*c^2 
*d^4*n^3-180*a*b^2*c^4*d^2*n^3+12154*a^3*d^6*n^2-9594*a^2*b*c^2*d^4*n^2-27 
00*a*b^2*c^4*d^2*n^2+24552*a^3*d^6*n-15120*a^2*b*c^2*d^4*n-10080*a*b^2*c^4 
*d^2*n-2520*b^3*c^6*n+20160*a^3*d^6)/d^6/(n^7+35*n^6+511*n^5+4025*n^4+1842 
4*n^3+48860*n^2+69264*n+40320)*x^2*exp(n*ln(d*x+c))+(3*a*d^2*n^2+45*a*d^2* 
n-7*b*c^2*n+168*a*d^2)*b^2/d^2/(n^3+21*n^2+146*n+336)*x^6*exp(n*ln(d*x+c)) 
+1/d^7*n*c*(a^3*d^6*n^6+33*a^3*d^6*n^5+445*a^3*d^6*n^4+18*a^2*b*c^2*d^4*n^ 
4+3135*a^3*d^6*n^3+468*a^2*b*c^2*d^4*n^3+12154*a^3*d^6*n^2+4518*a^2*b*c^2* 
d^4*n^2+360*a*b^2*c^4*d^2*n^2+24552*a^3*d^6*n+19188*a^2*b*c^2*d^4*n+5400*a 
*b^2*c^4*d^2*n+20160*a^3*d^6+30240*a^2*b*c^2*d^4+20160*a*b^2*c^4*d^2+5040* 
b^3*c^6)/(n^8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3+118124*n^2+10958 
4*n+40320)*x*exp(n*ln(d*x+c))+n*c*b^3/d/(n^2+15*n+56)*x^7*exp(n*ln(d*x+c)) 
-c^2*(a^3*d^6*n^6+33*a^3*d^6*n^5+445*a^3*d^6*n^4+18*a^2*b*c^2*d^4*n^4+3135 
*a^3*d^6*n^3+468*a^2*b*c^2*d^4*n^3+12154*a^3*d^6*n^2+4518*a^2*b*c^2*d^4*n^ 
2+360*a*b^2*c^4*d^2*n^2+24552*a^3*d^6*n+19188*a^2*b*c^2*d^4*n+5400*a*b^2*c 
^4*d^2*n+20160*a^3*d^6+30240*a^2*b*c^2*d^4+20160*a*b^2*c^4*d^2+5040*b^3*c^ 
6)/d^8/(n^8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3+118124*n^2+109584* 
n+40320)*exp(n*ln(d*x+c))+3*(a^2*d^4*n^4+26*a^2*d^4*n^3-5*a*b*c^2*d^2*n^3+ 
251*a^2*d^4*n^2-75*a*b*c^2*d^2*n^2+1066*a^2*d^4*n-280*a*b*c^2*d^2*n-70*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1684 vs. \(2 (282) = 564\).

Time = 0.10 (sec) , antiderivative size = 1684, normalized size of antiderivative = 5.97 \[ \int x (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(x*(d*x+c)^n*(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

-(a^3*c^2*d^6*n^6 + 33*a^3*c^2*d^6*n^5 + 5040*b^3*c^8 + 20160*a*b^2*c^6*d^ 
2 + 30240*a^2*b*c^4*d^4 + 20160*a^3*c^2*d^6 - (b^3*d^8*n^7 + 28*b^3*d^8*n^ 
6 + 322*b^3*d^8*n^5 + 1960*b^3*d^8*n^4 + 6769*b^3*d^8*n^3 + 13132*b^3*d^8* 
n^2 + 13068*b^3*d^8*n + 5040*b^3*d^8)*x^8 - (b^3*c*d^7*n^7 + 21*b^3*c*d^7* 
n^6 + 175*b^3*c*d^7*n^5 + 735*b^3*c*d^7*n^4 + 1624*b^3*c*d^7*n^3 + 1764*b^ 
3*c*d^7*n^2 + 720*b^3*c*d^7*n)*x^7 - (3*a*b^2*d^8*n^7 + 20160*a*b^2*d^8 - 
(7*b^3*c^2*d^6 - 90*a*b^2*d^8)*n^6 - 3*(35*b^3*c^2*d^6 - 366*a*b^2*d^8)*n^ 
5 - 5*(119*b^3*c^2*d^6 - 1404*a*b^2*d^8)*n^4 - 9*(175*b^3*c^2*d^6 - 2803*a 
*b^2*d^8)*n^3 - 2*(959*b^3*c^2*d^6 - 25245*a*b^2*d^8)*n^2 - 24*(35*b^3*c^2 
*d^6 - 2143*a*b^2*d^8)*n)*x^6 - 3*(a*b^2*c*d^7*n^7 + 25*a*b^2*c*d^7*n^6 + 
(14*b^3*c^3*d^5 + 241*a*b^2*c*d^7)*n^5 + 5*(28*b^3*c^3*d^5 + 227*a*b^2*c*d 
^7)*n^4 + 2*(245*b^3*c^3*d^5 + 1367*a*b^2*c*d^7)*n^3 + 20*(35*b^3*c^3*d^5 
+ 158*a*b^2*c*d^7)*n^2 + 336*(b^3*c^3*d^5 + 4*a*b^2*c*d^7)*n)*x^5 + (18*a^ 
2*b*c^4*d^4 + 445*a^3*c^2*d^6)*n^4 - 3*(a^2*b*d^8*n^7 + 10080*a^2*b*d^8 - 
(5*a*b^2*c^2*d^6 - 32*a^2*b*d^8)*n^6 - (105*a*b^2*c^2*d^6 - 418*a^2*b*d^8) 
*n^5 - (70*b^3*c^4*d^4 + 785*a*b^2*c^2*d^6 - 2864*a^2*b*d^8)*n^4 - (420*b^ 
3*c^4*d^4 + 2535*a*b^2*c^2*d^6 - 10993*a^2*b*d^8)*n^3 - 2*(385*b^3*c^4*d^4 
 + 1765*a*b^2*c^2*d^6 - 11656*a^2*b*d^8)*n^2 - 12*(35*b^3*c^4*d^4 + 140*a* 
b^2*c^2*d^6 - 2073*a^2*b*d^8)*n)*x^4 + 3*(156*a^2*b*c^4*d^4 + 1045*a^3*c^2 
*d^6)*n^3 - 3*(a^2*b*c*d^7*n^7 + 29*a^2*b*c*d^7*n^6 + (20*a*b^2*c^3*d^5...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24687 vs. \(2 (265) = 530\).

Time = 5.99 (sec) , antiderivative size = 24687, normalized size of antiderivative = 87.54 \[ \int x (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(x*(d*x+c)**n*(b*x**2+a)**3,x)
 

Output:

Piecewise((c**n*(a**3*x**2/2 + 3*a**2*b*x**4/4 + a*b**2*x**6/2 + b**3*x**8 
/8), Eq(d, 0)), (-10*a**3*c*d**6/(420*c**7*d**8 + 2940*c**6*d**9*x + 8820* 
c**5*d**10*x**2 + 14700*c**4*d**11*x**3 + 14700*c**3*d**12*x**4 + 8820*c** 
2*d**13*x**5 + 2940*c*d**14*x**6 + 420*d**15*x**7) - 70*a**3*d**7*x/(420*c 
**7*d**8 + 2940*c**6*d**9*x + 8820*c**5*d**10*x**2 + 14700*c**4*d**11*x**3 
 + 14700*c**3*d**12*x**4 + 8820*c**2*d**13*x**5 + 2940*c*d**14*x**6 + 420* 
d**15*x**7) - 9*a**2*b*c**3*d**4/(420*c**7*d**8 + 2940*c**6*d**9*x + 8820* 
c**5*d**10*x**2 + 14700*c**4*d**11*x**3 + 14700*c**3*d**12*x**4 + 8820*c** 
2*d**13*x**5 + 2940*c*d**14*x**6 + 420*d**15*x**7) - 63*a**2*b*c**2*d**5*x 
/(420*c**7*d**8 + 2940*c**6*d**9*x + 8820*c**5*d**10*x**2 + 14700*c**4*d** 
11*x**3 + 14700*c**3*d**12*x**4 + 8820*c**2*d**13*x**5 + 2940*c*d**14*x**6 
 + 420*d**15*x**7) - 189*a**2*b*c*d**6*x**2/(420*c**7*d**8 + 2940*c**6*d** 
9*x + 8820*c**5*d**10*x**2 + 14700*c**4*d**11*x**3 + 14700*c**3*d**12*x**4 
 + 8820*c**2*d**13*x**5 + 2940*c*d**14*x**6 + 420*d**15*x**7) - 315*a**2*b 
*d**7*x**3/(420*c**7*d**8 + 2940*c**6*d**9*x + 8820*c**5*d**10*x**2 + 1470 
0*c**4*d**11*x**3 + 14700*c**3*d**12*x**4 + 8820*c**2*d**13*x**5 + 2940*c* 
d**14*x**6 + 420*d**15*x**7) - 30*a*b**2*c**5*d**2/(420*c**7*d**8 + 2940*c 
**6*d**9*x + 8820*c**5*d**10*x**2 + 14700*c**4*d**11*x**3 + 14700*c**3*d** 
12*x**4 + 8820*c**2*d**13*x**5 + 2940*c*d**14*x**6 + 420*d**15*x**7) - 210 
*a*b**2*c**4*d**3*x/(420*c**7*d**8 + 2940*c**6*d**9*x + 8820*c**5*d**10...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (282) = 564\).

Time = 0.05 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.22 \[ \int x (c+d x)^n \left (a+b x^2\right )^3 \, dx=\frac {{\left (d^{2} {\left (n + 1\right )} x^{2} + c d n x - c^{2}\right )} {\left (d x + c\right )}^{n} a^{3}}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {3 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c d^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} c^{2} d^{2} x^{2} + 6 \, c^{3} d n x - 6 \, c^{4}\right )} {\left (d x + c\right )}^{n} a^{2} b}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{4}} + \frac {3 \, {\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} d^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} c d^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c^{2} d^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{3} d^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} c^{4} d^{2} x^{2} + 120 \, c^{5} d n x - 120 \, c^{6}\right )} {\left (d x + c\right )}^{n} a b^{2}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} d^{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 )} d^{8} x^{8} + {\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} c d^{7} x^{7} - 7 \, {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} c^{2} d^{6} x^{6} + 42 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} c^{3} d^{5} x^{5} - 210 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c^{4} d^{4} x^{4} + 840 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{5} d^{3} x^{3} - 2520 \, {\left (n^{2} + n\right )} c^{6} d^{2} x^{2} + 5040 \, c^{7} d n x - 5040 \, c^{8}\right )} {\left (d x + c\right )}^{n} b^{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 )} d^{8}} \] Input:

integrate(x*(d*x+c)^n*(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*a^3/((n^2 + 3*n + 2)*d^2) + 
3*((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n 
^2 + n)*c^2*d^2*x^2 + 6*c^3*d*n*x - 6*c^4)*(d*x + c)^n*a^2*b/((n^4 + 10*n^ 
3 + 35*n^2 + 50*n + 24)*d^4) + 3*((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n 
 + 120)*d^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*c*d^5*x^5 - 5*(n 
^4 + 6*n^3 + 11*n^2 + 6*n)*c^2*d^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*c^3*d^3*x^ 
3 - 60*(n^2 + n)*c^4*d^2*x^2 + 120*c^5*d*n*x - 120*c^6)*(d*x + c)^n*a*b^2/ 
((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*d^6) + ((n^7 
 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*d^ 
8*x^8 + (n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*c 
*d^7*x^7 - 7*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*c^2*d^6*x 
^6 + 42*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*c^3*d^5*x^5 - 210*(n^4 + 6 
*n^3 + 11*n^2 + 6*n)*c^4*d^4*x^4 + 840*(n^3 + 3*n^2 + 2*n)*c^5*d^3*x^3 - 2 
520*(n^2 + n)*c^6*d^2*x^2 + 5040*c^7*d*n*x - 5040*c^8)*(d*x + c)^n*b^3/((n 
^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 10 
9584*n + 40320)*d^8)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2851 vs. \(2 (282) = 564\).

Time = 0.14 (sec) , antiderivative size = 2851, normalized size of antiderivative = 10.11 \[ \int x (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(x*(d*x+c)^n*(b*x^2+a)^3,x, algorithm="giac")
 

Output:

((d*x + c)^n*b^3*d^8*n^7*x^8 + (d*x + c)^n*b^3*c*d^7*n^7*x^7 + 28*(d*x + c 
)^n*b^3*d^8*n^6*x^8 + 3*(d*x + c)^n*a*b^2*d^8*n^7*x^6 + 21*(d*x + c)^n*b^3 
*c*d^7*n^6*x^7 + 322*(d*x + c)^n*b^3*d^8*n^5*x^8 + 3*(d*x + c)^n*a*b^2*c*d 
^7*n^7*x^5 - 7*(d*x + c)^n*b^3*c^2*d^6*n^6*x^6 + 90*(d*x + c)^n*a*b^2*d^8* 
n^6*x^6 + 175*(d*x + c)^n*b^3*c*d^7*n^5*x^7 + 1960*(d*x + c)^n*b^3*d^8*n^4 
*x^8 + 3*(d*x + c)^n*a^2*b*d^8*n^7*x^4 + 75*(d*x + c)^n*a*b^2*c*d^7*n^6*x^ 
5 - 105*(d*x + c)^n*b^3*c^2*d^6*n^5*x^6 + 1098*(d*x + c)^n*a*b^2*d^8*n^5*x 
^6 + 735*(d*x + c)^n*b^3*c*d^7*n^4*x^7 + 6769*(d*x + c)^n*b^3*d^8*n^3*x^8 
+ 3*(d*x + c)^n*a^2*b*c*d^7*n^7*x^3 - 15*(d*x + c)^n*a*b^2*c^2*d^6*n^6*x^4 
 + 96*(d*x + c)^n*a^2*b*d^8*n^6*x^4 + 42*(d*x + c)^n*b^3*c^3*d^5*n^5*x^5 + 
 723*(d*x + c)^n*a*b^2*c*d^7*n^5*x^5 - 595*(d*x + c)^n*b^3*c^2*d^6*n^4*x^6 
 + 7020*(d*x + c)^n*a*b^2*d^8*n^4*x^6 + 1624*(d*x + c)^n*b^3*c*d^7*n^3*x^7 
 + 13132*(d*x + c)^n*b^3*d^8*n^2*x^8 + (d*x + c)^n*a^3*d^8*n^7*x^2 + 87*(d 
*x + c)^n*a^2*b*c*d^7*n^6*x^3 - 315*(d*x + c)^n*a*b^2*c^2*d^6*n^5*x^4 + 12 
54*(d*x + c)^n*a^2*b*d^8*n^5*x^4 + 420*(d*x + c)^n*b^3*c^3*d^5*n^4*x^5 + 3 
405*(d*x + c)^n*a*b^2*c*d^7*n^4*x^5 - 1575*(d*x + c)^n*b^3*c^2*d^6*n^3*x^6 
 + 25227*(d*x + c)^n*a*b^2*d^8*n^3*x^6 + 1764*(d*x + c)^n*b^3*c*d^7*n^2*x^ 
7 + 13068*(d*x + c)^n*b^3*d^8*n*x^8 + (d*x + c)^n*a^3*c*d^7*n^7*x - 9*(d*x 
 + c)^n*a^2*b*c^2*d^6*n^6*x^2 + 34*(d*x + c)^n*a^3*d^8*n^6*x^2 + 60*(d*x + 
 c)^n*a*b^2*c^3*d^5*n^5*x^3 + 993*(d*x + c)^n*a^2*b*c*d^7*n^5*x^3 - 210...
 

Mupad [B] (verification not implemented)

Time = 11.19 (sec) , antiderivative size = 1459, normalized size of antiderivative = 5.17 \[ \int x (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

int(x*(a + b*x^2)^3*(c + d*x)^n,x)
 

Output:

(b^3*x^8*(c + d*x)^n*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 
+ 28*n^6 + n^7 + 5040))/(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4 
536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320) - (c^2*(c + d*x)^n*(20160*a^3*d^ 
6 + 5040*b^3*c^6 + 24552*a^3*d^6*n + 12154*a^3*d^6*n^2 + 3135*a^3*d^6*n^3 
+ 445*a^3*d^6*n^4 + 33*a^3*d^6*n^5 + a^3*d^6*n^6 + 20160*a*b^2*c^4*d^2 + 3 
0240*a^2*b*c^2*d^4 + 5400*a*b^2*c^4*d^2*n + 19188*a^2*b*c^2*d^4*n + 360*a* 
b^2*c^4*d^2*n^2 + 4518*a^2*b*c^2*d^4*n^2 + 468*a^2*b*c^2*d^4*n^3 + 18*a^2* 
b*c^2*d^4*n^4))/(d^8*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536 
*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) - (x^2*(n + 1)*(c + d*x)^n*(2520*b 
^3*c^6*n - 24552*a^3*d^6*n - 20160*a^3*d^6 - 12154*a^3*d^6*n^2 - 3135*a^3* 
d^6*n^3 - 445*a^3*d^6*n^4 - 33*a^3*d^6*n^5 - a^3*d^6*n^6 + 10080*a*b^2*c^4 
*d^2*n + 15120*a^2*b*c^2*d^4*n + 2700*a*b^2*c^4*d^2*n^2 + 9594*a^2*b*c^2*d 
^4*n^2 + 180*a*b^2*c^4*d^2*n^3 + 2259*a^2*b*c^2*d^4*n^3 + 234*a^2*b*c^2*d^ 
4*n^4 + 9*a^2*b*c^2*d^4*n^5))/(d^6*(109584*n + 118124*n^2 + 67284*n^3 + 22 
449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (b^2*x^6*(c + d*x) 
^n*(168*a*d^2 + 3*a*d^2*n^2 + 45*a*d^2*n - 7*b*c^2*n)*(274*n + 225*n^2 + 8 
5*n^3 + 15*n^4 + n^5 + 120))/(d^2*(109584*n + 118124*n^2 + 67284*n^3 + 224 
49*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (3*b*x^4*(c + d*x)^ 
n*(11*n + 6*n^2 + n^3 + 6)*(1680*a^2*d^4 + 1066*a^2*d^4*n - 70*b^2*c^4*n + 
 251*a^2*d^4*n^2 + 26*a^2*d^4*n^3 + a^2*d^4*n^4 - 280*a*b*c^2*d^2*n - 7...
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1950, normalized size of antiderivative = 6.91 \[ \int x (c+d x)^n \left (a+b x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int(x*(d*x+c)^n*(b*x^2+a)^3,x)
 

Output:

((c + d*x)**n*( - a**3*c**2*d**6*n**6 - 33*a**3*c**2*d**6*n**5 - 445*a**3* 
c**2*d**6*n**4 - 3135*a**3*c**2*d**6*n**3 - 12154*a**3*c**2*d**6*n**2 - 24 
552*a**3*c**2*d**6*n - 20160*a**3*c**2*d**6 + a**3*c*d**7*n**7*x + 33*a**3 
*c*d**7*n**6*x + 445*a**3*c*d**7*n**5*x + 3135*a**3*c*d**7*n**4*x + 12154* 
a**3*c*d**7*n**3*x + 24552*a**3*c*d**7*n**2*x + 20160*a**3*c*d**7*n*x + a* 
*3*d**8*n**7*x**2 + 34*a**3*d**8*n**6*x**2 + 478*a**3*d**8*n**5*x**2 + 358 
0*a**3*d**8*n**4*x**2 + 15289*a**3*d**8*n**3*x**2 + 36706*a**3*d**8*n**2*x 
**2 + 44712*a**3*d**8*n*x**2 + 20160*a**3*d**8*x**2 - 18*a**2*b*c**4*d**4* 
n**4 - 468*a**2*b*c**4*d**4*n**3 - 4518*a**2*b*c**4*d**4*n**2 - 19188*a**2 
*b*c**4*d**4*n - 30240*a**2*b*c**4*d**4 + 18*a**2*b*c**3*d**5*n**5*x + 468 
*a**2*b*c**3*d**5*n**4*x + 4518*a**2*b*c**3*d**5*n**3*x + 19188*a**2*b*c** 
3*d**5*n**2*x + 30240*a**2*b*c**3*d**5*n*x - 9*a**2*b*c**2*d**6*n**6*x**2 
- 243*a**2*b*c**2*d**6*n**5*x**2 - 2493*a**2*b*c**2*d**6*n**4*x**2 - 11853 
*a**2*b*c**2*d**6*n**3*x**2 - 24714*a**2*b*c**2*d**6*n**2*x**2 - 15120*a** 
2*b*c**2*d**6*n*x**2 + 3*a**2*b*c*d**7*n**7*x**3 + 87*a**2*b*c*d**7*n**6*x 
**3 + 993*a**2*b*c*d**7*n**5*x**3 + 5613*a**2*b*c*d**7*n**4*x**3 + 16140*a 
**2*b*c*d**7*n**3*x**3 + 21516*a**2*b*c*d**7*n**2*x**3 + 10080*a**2*b*c*d* 
*7*n*x**3 + 3*a**2*b*d**8*n**7*x**4 + 96*a**2*b*d**8*n**6*x**4 + 1254*a**2 
*b*d**8*n**5*x**4 + 8592*a**2*b*d**8*n**4*x**4 + 32979*a**2*b*d**8*n**3*x* 
*4 + 69936*a**2*b*d**8*n**2*x**4 + 74628*a**2*b*d**8*n*x**4 + 30240*a**...