\(\int x^2 (c+d x)^n (a+b x^2)^3 \, dx\) [203]

Optimal result

Integrand size = 20, antiderivative size = 343 \[ \int x^2 (c+d x)^n \left (a+b x^2\right )^3 \, dx=\frac {c^2 \left (b c^2+a d^2\right )^3 (c+d x)^{1+n}}{d^9 (1+n)}-\frac {2 c \left (b c^2+a d^2\right )^2 \left (4 b c^2+a d^2\right ) (c+d x)^{2+n}}{d^9 (2+n)}+\frac {\left (b c^2+a d^2\right ) \left (28 b^2 c^4+17 a b c^2 d^2+a^2 d^4\right ) (c+d x)^{3+n}}{d^9 (3+n)}-\frac {4 b c \left (14 b^2 c^4+15 a b c^2 d^2+3 a^2 d^4\right ) (c+d x)^{4+n}}{d^9 (4+n)}+\frac {b \left (70 b^2 c^4+45 a b c^2 d^2+3 a^2 d^4\right ) (c+d x)^{5+n}}{d^9 (5+n)}-\frac {2 b^2 c \left (28 b c^2+9 a d^2\right ) (c+d x)^{6+n}}{d^9 (6+n)}+\frac {b^2 \left (28 b c^2+3 a d^2\right ) (c+d x)^{7+n}}{d^9 (7+n)}-\frac {8 b^3 c (c+d x)^{8+n}}{d^9 (8+n)}+\frac {b^3 (c+d x)^{9+n}}{d^9 (9+n)} \] Output:

c^2*(a*d^2+b*c^2)^3*(d*x+c)^(1+n)/d^9/(1+n)-2*c*(a*d^2+b*c^2)^2*(a*d^2+4*b 
*c^2)*(d*x+c)^(2+n)/d^9/(2+n)+(a*d^2+b*c^2)*(a^2*d^4+17*a*b*c^2*d^2+28*b^2 
*c^4)*(d*x+c)^(3+n)/d^9/(3+n)-4*b*c*(3*a^2*d^4+15*a*b*c^2*d^2+14*b^2*c^4)* 
(d*x+c)^(4+n)/d^9/(4+n)+b*(3*a^2*d^4+45*a*b*c^2*d^2+70*b^2*c^4)*(d*x+c)^(5 
+n)/d^9/(5+n)-2*b^2*c*(9*a*d^2+28*b*c^2)*(d*x+c)^(6+n)/d^9/(6+n)+b^2*(3*a* 
d^2+28*b*c^2)*(d*x+c)^(7+n)/d^9/(7+n)-8*b^3*c*(d*x+c)^(8+n)/d^9/(8+n)+b^3* 
(d*x+c)^(9+n)/d^9/(9+n)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.88 \[ \int x^2 (c+d x)^n \left (a+b x^2\right )^3 \, dx=\frac {(c+d x)^{1+n} \left (\frac {c^2 \left (b c^2+a d^2\right )^3}{1+n}-\frac {2 c \left (b c^2+a d^2\right )^2 \left (4 b c^2+a d^2\right ) (c+d x)}{2+n}+\frac {\left (b c^2+a d^2\right ) \left (28 b^2 c^4+17 a b c^2 d^2+a^2 d^4\right ) (c+d x)^2}{3+n}-\frac {4 b c \left (14 b^2 c^4+15 a b c^2 d^2+3 a^2 d^4\right ) (c+d x)^3}{4+n}+\frac {b \left (70 b^2 c^4+45 a b c^2 d^2+3 a^2 d^4\right ) (c+d x)^4}{5+n}-\frac {2 b^2 c \left (28 b c^2+9 a d^2\right ) (c+d x)^5}{6+n}+\frac {b^2 \left (28 b c^2+3 a d^2\right ) (c+d x)^6}{7+n}-\frac {8 b^3 c (c+d x)^7}{8+n}+\frac {b^3 (c+d x)^8}{9+n}\right )}{d^9} \] Input:

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

Output:

((c + d*x)^(1 + n)*((c^2*(b*c^2 + a*d^2)^3)/(1 + n) - (2*c*(b*c^2 + a*d^2) 
^2*(4*b*c^2 + a*d^2)*(c + d*x))/(2 + n) + ((b*c^2 + a*d^2)*(28*b^2*c^4 + 1 
7*a*b*c^2*d^2 + a^2*d^4)*(c + d*x)^2)/(3 + n) - (4*b*c*(14*b^2*c^4 + 15*a* 
b*c^2*d^2 + 3*a^2*d^4)*(c + d*x)^3)/(4 + n) + (b*(70*b^2*c^4 + 45*a*b*c^2* 
d^2 + 3*a^2*d^4)*(c + d*x)^4)/(5 + n) - (2*b^2*c*(28*b*c^2 + 9*a*d^2)*(c + 
 d*x)^5)/(6 + n) + (b^2*(28*b*c^2 + 3*a*d^2)*(c + d*x)^6)/(7 + n) - (8*b^3 
*c*(c + d*x)^7)/(8 + n) + (b^3*(c + d*x)^8)/(9 + n)))/d^9
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

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

\(\Big \downarrow \) 522

\(\displaystyle \int \left (\frac {\left (a d^2+b c^2\right ) \left (a^2 d^4+17 a b c^2 d^2+28 b^2 c^4\right ) (c+d x)^{n+2}}{d^8}-\frac {4 b c \left (3 a^2 d^4+15 a b c^2 d^2+14 b^2 c^4\right ) (c+d x)^{n+3}}{d^8}+\frac {b \left (3 a^2 d^4+45 a b c^2 d^2+70 b^2 c^4\right ) (c+d x)^{n+4}}{d^8}-\frac {2 b^2 c \left (9 a d^2+28 b c^2\right ) (c+d x)^{n+5}}{d^8}+\frac {b^2 \left (3 a d^2+28 b c^2\right ) (c+d x)^{n+6}}{d^8}+\frac {c^2 \left (a d^2+b c^2\right )^3 (c+d x)^n}{d^8}-\frac {2 c \left (a d^2+b c^2\right )^2 \left (a d^2+4 b c^2\right ) (c+d x)^{n+1}}{d^8}-\frac {8 b^3 c (c+d x)^{n+7}}{d^8}+\frac {b^3 (c+d x)^{n+8}}{d^8}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (a d^2+b c^2\right ) \left (a^2 d^4+17 a b c^2 d^2+28 b^2 c^4\right ) (c+d x)^{n+3}}{d^9 (n+3)}-\frac {4 b c \left (3 a^2 d^4+15 a b c^2 d^2+14 b^2 c^4\right ) (c+d x)^{n+4}}{d^9 (n+4)}+\frac {b \left (3 a^2 d^4+45 a b c^2 d^2+70 b^2 c^4\right ) (c+d x)^{n+5}}{d^9 (n+5)}-\frac {2 b^2 c \left (9 a d^2+28 b c^2\right ) (c+d x)^{n+6}}{d^9 (n+6)}+\frac {b^2 \left (3 a d^2+28 b c^2\right ) (c+d x)^{n+7}}{d^9 (n+7)}+\frac {c^2 \left (a d^2+b c^2\right )^3 (c+d x)^{n+1}}{d^9 (n+1)}-\frac {2 c \left (a d^2+b c^2\right )^2 \left (a d^2+4 b c^2\right ) (c+d x)^{n+2}}{d^9 (n+2)}-\frac {8 b^3 c (c+d x)^{n+8}}{d^9 (n+8)}+\frac {b^3 (c+d x)^{n+9}}{d^9 (n+9)}\)

Input:

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

Output:

(c^2*(b*c^2 + a*d^2)^3*(c + d*x)^(1 + n))/(d^9*(1 + n)) - (2*c*(b*c^2 + a* 
d^2)^2*(4*b*c^2 + a*d^2)*(c + d*x)^(2 + n))/(d^9*(2 + n)) + ((b*c^2 + a*d^ 
2)*(28*b^2*c^4 + 17*a*b*c^2*d^2 + a^2*d^4)*(c + d*x)^(3 + n))/(d^9*(3 + n) 
) - (4*b*c*(14*b^2*c^4 + 15*a*b*c^2*d^2 + 3*a^2*d^4)*(c + d*x)^(4 + n))/(d 
^9*(4 + n)) + (b*(70*b^2*c^4 + 45*a*b*c^2*d^2 + 3*a^2*d^4)*(c + d*x)^(5 + 
n))/(d^9*(5 + n)) - (2*b^2*c*(28*b*c^2 + 9*a*d^2)*(c + d*x)^(6 + n))/(d^9* 
(6 + n)) + (b^2*(28*b*c^2 + 3*a*d^2)*(c + d*x)^(7 + n))/(d^9*(7 + n)) - (8 
*b^3*c*(c + d*x)^(8 + n))/(d^9*(8 + n)) + (b^3*(c + d*x)^(9 + n))/(d^9*(9 
+ 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. \(2231\) vs. \(2(343)=686\).

Time = 0.41 (sec) , antiderivative size = 2232, normalized size of antiderivative = 6.51

method	result	size
gosper	\(\text {Expression too large to display}\)	\(2232\)
orering	\(\text {Expression too large to display}\)	\(2235\)
risch	\(\text {Expression too large to display}\)	\(2558\)
parallelrisch	\(\text {Expression too large to display}\)	\(3685\)

Input:

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

Output:

1/d^9*(d*x+c)^(1+n)/(n^9+45*n^8+870*n^7+9450*n^6+63273*n^5+269325*n^4+7236 
80*n^3+1172700*n^2+1026576*n+362880)*(b^3*d^8*n^8*x^8+36*b^3*d^8*n^7*x^8+3 
*a*b^2*d^8*n^8*x^6-8*b^3*c*d^7*n^7*x^7+546*b^3*d^8*n^6*x^8+114*a*b^2*d^8*n 
^7*x^6-224*b^3*c*d^7*n^6*x^7+4536*b^3*d^8*n^5*x^8+3*a^2*b*d^8*n^8*x^4-18*a 
*b^2*c*d^7*n^7*x^5+1812*a*b^2*d^8*n^6*x^6+56*b^3*c^2*d^6*n^6*x^6-2576*b^3* 
c*d^7*n^5*x^7+22449*b^3*d^8*n^4*x^8+120*a^2*b*d^8*n^7*x^4-576*a*b^2*c*d^7* 
n^6*x^5+15666*a*b^2*d^8*n^5*x^6+1176*b^3*c^2*d^6*n^5*x^6-15680*b^3*c*d^7*n 
^4*x^7+67284*b^3*d^8*n^3*x^8+a^3*d^8*n^8*x^2-12*a^2*b*c*d^7*n^7*x^3+2010*a 
^2*b*d^8*n^6*x^4+90*a*b^2*c^2*d^6*n^6*x^4-7416*a*b^2*c*d^7*n^5*x^5+80157*a 
*b^2*d^8*n^4*x^6-336*b^3*c^3*d^5*n^5*x^5+9800*b^3*c^2*d^6*n^4*x^6-54152*b^ 
3*c*d^7*n^3*x^7+118124*b^3*d^8*n^2*x^8+42*a^3*d^8*n^7*x^2-432*a^2*b*c*d^7* 
n^6*x^3+18300*a^2*b*d^8*n^5*x^4+2430*a*b^2*c^2*d^6*n^5*x^4-49500*a*b^2*c*d 
^7*n^4*x^5+246876*a*b^2*d^8*n^3*x^6-5040*b^3*c^3*d^5*n^4*x^5+41160*b^3*c^2 
*d^6*n^3*x^6-105056*b^3*c*d^7*n^2*x^7+109584*b^3*d^8*n*x^8-2*a^3*c*d^7*n^7 
*x+744*a^3*d^8*n^6*x^2+36*a^2*b*c^2*d^6*n^6*x^2-6312*a^2*b*c*d^7*n^5*x^3+9 
8319*a^2*b*d^8*n^4*x^4-360*a*b^2*c^3*d^5*n^5*x^3+24930*a*b^2*c^2*d^6*n^4*x 
^4-183942*a*b^2*c*d^7*n^3*x^5+442908*a*b^2*d^8*n^2*x^6+1680*b^3*c^4*d^4*n^ 
4*x^4-28560*b^3*c^3*d^5*n^3*x^5+90944*b^3*c^2*d^6*n^2*x^6-104544*b^3*c*d^7 
*n*x^7+40320*b^3*d^8*x^8-80*a^3*c*d^7*n^6*x+7218*a^3*d^8*n^5*x^2+1188*a^2* 
b*c^2*d^6*n^5*x^2-47952*a^2*b*c*d^7*n^4*x^3+316380*a^2*b*d^8*n^3*x^4-82...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2166 vs. \(2 (343) = 686\).

Time = 0.12 (sec) , antiderivative size = 2166, normalized size of antiderivative = 6.31 \[ \int x^2 (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(2*a^3*c^3*d^6*n^6 + 78*a^3*c^3*d^6*n^5 + 40320*b^3*c^9 + 155520*a*b^2*c^7 
*d^2 + 217728*a^2*b*c^5*d^4 + 120960*a^3*c^3*d^6 + (b^3*d^9*n^8 + 36*b^3*d 
^9*n^7 + 546*b^3*d^9*n^6 + 4536*b^3*d^9*n^5 + 22449*b^3*d^9*n^4 + 67284*b^ 
3*d^9*n^3 + 118124*b^3*d^9*n^2 + 109584*b^3*d^9*n + 40320*b^3*d^9)*x^9 + ( 
b^3*c*d^8*n^8 + 28*b^3*c*d^8*n^7 + 322*b^3*c*d^8*n^6 + 1960*b^3*c*d^8*n^5 
+ 6769*b^3*c*d^8*n^4 + 13132*b^3*c*d^8*n^3 + 13068*b^3*c*d^8*n^2 + 5040*b^ 
3*c*d^8*n)*x^8 + (3*a*b^2*d^9*n^8 + 155520*a*b^2*d^9 - 2*(4*b^3*c^2*d^7 - 
57*a*b^2*d^9)*n^7 - 12*(14*b^3*c^2*d^7 - 151*a*b^2*d^9)*n^6 - 14*(100*b^3* 
c^2*d^7 - 1119*a*b^2*d^9)*n^5 - 21*(280*b^3*c^2*d^7 - 3817*a*b^2*d^9)*n^4 
- 28*(464*b^3*c^2*d^7 - 8817*a*b^2*d^9)*n^3 - 36*(392*b^3*c^2*d^7 - 12303* 
a*b^2*d^9)*n^2 - 144*(40*b^3*c^2*d^7 - 2901*a*b^2*d^9)*n)*x^7 + (3*a*b^2*c 
*d^8*n^8 + 96*a*b^2*c*d^8*n^7 + 4*(14*b^3*c^3*d^6 + 309*a*b^2*c*d^8)*n^6 + 
 30*(28*b^3*c^3*d^6 + 275*a*b^2*c*d^8)*n^5 + (4760*b^3*c^3*d^6 + 30657*a*b 
^2*c*d^8)*n^4 + 6*(2100*b^3*c^3*d^6 + 10489*a*b^2*c*d^8)*n^3 + 8*(1918*b^3 
*c^3*d^6 + 8163*a*b^2*c*d^8)*n^2 + 960*(7*b^3*c^3*d^6 + 27*a*b^2*c*d^8)*n) 
*x^6 + 3*(a^2*b*d^9*n^8 + 72576*a^2*b*d^9 - 2*(3*a*b^2*c^2*d^7 - 20*a^2*b* 
d^9)*n^7 - 2*(81*a*b^2*c^2*d^7 - 335*a^2*b*d^9)*n^6 - 2*(56*b^3*c^4*d^5 + 
831*a*b^2*c^2*d^7 - 3050*a^2*b*d^9)*n^5 - (1120*b^3*c^4*d^5 + 8190*a*b^2*c 
^2*d^7 - 32773*a^2*b*d^9)*n^4 - 4*(980*b^3*c^4*d^5 + 5091*a*b^2*c^2*d^7 - 
26365*a^2*b*d^9)*n^3 - 4*(1400*b^3*c^4*d^5 + 6012*a*b^2*c^2*d^7 - 49095...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35984 vs. \(2 (328) = 656\).

Time = 8.82 (sec) , antiderivative size = 35984, normalized size of antiderivative = 104.91 \[ \int x^2 (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((c**n*(a**3*x**3/3 + 3*a**2*b*x**5/5 + 3*a*b**2*x**7/7 + b**3*x* 
*9/9), Eq(d, 0)), (-5*a**3*c**2*d**6/(840*c**8*d**9 + 6720*c**7*d**10*x + 
23520*c**6*d**11*x**2 + 47040*c**5*d**12*x**3 + 58800*c**4*d**13*x**4 + 47 
040*c**3*d**14*x**5 + 23520*c**2*d**15*x**6 + 6720*c*d**16*x**7 + 840*d**1 
7*x**8) - 40*a**3*c*d**7*x/(840*c**8*d**9 + 6720*c**7*d**10*x + 23520*c**6 
*d**11*x**2 + 47040*c**5*d**12*x**3 + 58800*c**4*d**13*x**4 + 47040*c**3*d 
**14*x**5 + 23520*c**2*d**15*x**6 + 6720*c*d**16*x**7 + 840*d**17*x**8) - 
140*a**3*d**8*x**2/(840*c**8*d**9 + 6720*c**7*d**10*x + 23520*c**6*d**11*x 
**2 + 47040*c**5*d**12*x**3 + 58800*c**4*d**13*x**4 + 47040*c**3*d**14*x** 
5 + 23520*c**2*d**15*x**6 + 6720*c*d**16*x**7 + 840*d**17*x**8) - 9*a**2*b 
*c**4*d**4/(840*c**8*d**9 + 6720*c**7*d**10*x + 23520*c**6*d**11*x**2 + 47 
040*c**5*d**12*x**3 + 58800*c**4*d**13*x**4 + 47040*c**3*d**14*x**5 + 2352 
0*c**2*d**15*x**6 + 6720*c*d**16*x**7 + 840*d**17*x**8) - 72*a**2*b*c**3*d 
**5*x/(840*c**8*d**9 + 6720*c**7*d**10*x + 23520*c**6*d**11*x**2 + 47040*c 
**5*d**12*x**3 + 58800*c**4*d**13*x**4 + 47040*c**3*d**14*x**5 + 23520*c** 
2*d**15*x**6 + 6720*c*d**16*x**7 + 840*d**17*x**8) - 252*a**2*b*c**2*d**6* 
x**2/(840*c**8*d**9 + 6720*c**7*d**10*x + 23520*c**6*d**11*x**2 + 47040*c* 
*5*d**12*x**3 + 58800*c**4*d**13*x**4 + 47040*c**3*d**14*x**5 + 23520*c**2 
*d**15*x**6 + 6720*c*d**16*x**7 + 840*d**17*x**8) - 504*a**2*b*c*d**7*x**3 
/(840*c**8*d**9 + 6720*c**7*d**10*x + 23520*c**6*d**11*x**2 + 47040*c**...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 795 vs. \(2 (343) = 686\).

Time = 0.05 (sec) , antiderivative size = 795, normalized size of antiderivative = 2.32 \[ \int x^2 (c+d x)^n \left (a+b x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

((n^2 + 3*n + 2)*d^3*x^3 + (n^2 + n)*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x 
 + c)^n*a^3/((n^3 + 6*n^2 + 11*n + 6)*d^3) + 3*((n^4 + 10*n^3 + 35*n^2 + 5 
0*n + 24)*d^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*c*d^4*x^4 - 4*(n^3 + 3*n^ 
2 + 2*n)*c^2*d^3*x^3 + 12*(n^2 + n)*c^3*d^2*x^2 - 24*c^4*d*n*x + 24*c^5)*( 
d*x + c)^n*a^2*b/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*d^5) + 3 
*((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*d^7*x^7 + ( 
n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*c*d^6*x^6 - 6*(n^5 + 10 
*n^4 + 35*n^3 + 50*n^2 + 24*n)*c^2*d^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6* 
n)*c^3*d^4*x^4 - 120*(n^3 + 3*n^2 + 2*n)*c^4*d^3*x^3 + 360*(n^2 + n)*c^5*d 
^2*x^2 - 720*c^6*d*n*x + 720*c^7)*(d*x + c)^n*a*b^2/((n^7 + 28*n^6 + 322*n 
^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*d^7) + ((n^8 + 36*n 
^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 109584*n + 
40320)*d^9*x^9 + (n^8 + 28*n^7 + 322*n^6 + 1960*n^5 + 6769*n^4 + 13132*n^3 
 + 13068*n^2 + 5040*n)*c*d^8*x^8 - 8*(n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1 
624*n^3 + 1764*n^2 + 720*n)*c^2*d^7*x^7 + 56*(n^6 + 15*n^5 + 85*n^4 + 225* 
n^3 + 274*n^2 + 120*n)*c^3*d^6*x^6 - 336*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 
 24*n)*c^4*d^5*x^5 + 1680*(n^4 + 6*n^3 + 11*n^2 + 6*n)*c^5*d^4*x^4 - 6720* 
(n^3 + 3*n^2 + 2*n)*c^6*d^3*x^3 + 20160*(n^2 + n)*c^7*d^2*x^2 - 40320*c^8* 
d*n*x + 40320*c^9)*(d*x + c)^n*b^3/((n^9 + 45*n^8 + 870*n^7 + 9450*n^6 + 6 
3273*n^5 + 269325*n^4 + 723680*n^3 + 1172700*n^2 + 1026576*n + 362880)*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3713 vs. \(2 (343) = 686\).

Time = 0.16 (sec) , antiderivative size = 3713, normalized size of antiderivative = 10.83 \[ \int x^2 (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

((d*x + c)^n*b^3*d^9*n^8*x^9 + (d*x + c)^n*b^3*c*d^8*n^8*x^8 + 36*(d*x + c 
)^n*b^3*d^9*n^7*x^9 + 3*(d*x + c)^n*a*b^2*d^9*n^8*x^7 + 28*(d*x + c)^n*b^3 
*c*d^8*n^7*x^8 + 546*(d*x + c)^n*b^3*d^9*n^6*x^9 + 3*(d*x + c)^n*a*b^2*c*d 
^8*n^8*x^6 - 8*(d*x + c)^n*b^3*c^2*d^7*n^7*x^7 + 114*(d*x + c)^n*a*b^2*d^9 
*n^7*x^7 + 322*(d*x + c)^n*b^3*c*d^8*n^6*x^8 + 4536*(d*x + c)^n*b^3*d^9*n^ 
5*x^9 + 3*(d*x + c)^n*a^2*b*d^9*n^8*x^5 + 96*(d*x + c)^n*a*b^2*c*d^8*n^7*x 
^6 - 168*(d*x + c)^n*b^3*c^2*d^7*n^6*x^7 + 1812*(d*x + c)^n*a*b^2*d^9*n^6* 
x^7 + 1960*(d*x + c)^n*b^3*c*d^8*n^5*x^8 + 22449*(d*x + c)^n*b^3*d^9*n^4*x 
^9 + 3*(d*x + c)^n*a^2*b*c*d^8*n^8*x^4 - 18*(d*x + c)^n*a*b^2*c^2*d^7*n^7* 
x^5 + 120*(d*x + c)^n*a^2*b*d^9*n^7*x^5 + 56*(d*x + c)^n*b^3*c^3*d^6*n^6*x 
^6 + 1236*(d*x + c)^n*a*b^2*c*d^8*n^6*x^6 - 1400*(d*x + c)^n*b^3*c^2*d^7*n 
^5*x^7 + 15666*(d*x + c)^n*a*b^2*d^9*n^5*x^7 + 6769*(d*x + c)^n*b^3*c*d^8* 
n^4*x^8 + 67284*(d*x + c)^n*b^3*d^9*n^3*x^9 + (d*x + c)^n*a^3*d^9*n^8*x^3 
+ 108*(d*x + c)^n*a^2*b*c*d^8*n^7*x^4 - 486*(d*x + c)^n*a*b^2*c^2*d^7*n^6* 
x^5 + 2010*(d*x + c)^n*a^2*b*d^9*n^6*x^5 + 840*(d*x + c)^n*b^3*c^3*d^6*n^5 
*x^6 + 8250*(d*x + c)^n*a*b^2*c*d^8*n^5*x^6 - 5880*(d*x + c)^n*b^3*c^2*d^7 
*n^4*x^7 + 80157*(d*x + c)^n*a*b^2*d^9*n^4*x^7 + 13132*(d*x + c)^n*b^3*c*d 
^8*n^3*x^8 + 118124*(d*x + c)^n*b^3*d^9*n^2*x^9 + (d*x + c)^n*a^3*c*d^8*n^ 
8*x^2 - 12*(d*x + c)^n*a^2*b*c^2*d^7*n^7*x^3 + 42*(d*x + c)^n*a^3*d^9*n^7* 
x^3 + 90*(d*x + c)^n*a*b^2*c^3*d^6*n^6*x^4 + 1578*(d*x + c)^n*a^2*b*c*d...
 

Mupad [B] (verification not implemented)

Time = 9.78 (sec) , antiderivative size = 1796, normalized size of antiderivative = 5.24 \[ \int x^2 (c+d x)^n \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(b^3*x^9*(c + d*x)^n*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536 
*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320))/(1026576*n + 1172700*n^2 + 723680* 
n^3 + 269325*n^4 + 63273*n^5 + 9450*n^6 + 870*n^7 + 45*n^8 + n^9 + 362880) 
 + (2*c^3*(c + d*x)^n*(60480*a^3*d^6 + 20160*b^3*c^6 + 60216*a^3*d^6*n + 2 
4574*a^3*d^6*n^2 + 5265*a^3*d^6*n^3 + 625*a^3*d^6*n^4 + 39*a^3*d^6*n^5 + a 
^3*d^6*n^6 + 77760*a*b^2*c^4*d^2 + 108864*a^2*b*c^2*d^4 + 18360*a*b^2*c^4* 
d^2*n + 59400*a^2*b*c^2*d^4*n + 1080*a*b^2*c^4*d^2*n^2 + 12060*a^2*b*c^2*d 
^4*n^2 + 1080*a^2*b*c^2*d^4*n^3 + 36*a^2*b*c^2*d^4*n^4))/(d^9*(1026576*n + 
 1172700*n^2 + 723680*n^3 + 269325*n^4 + 63273*n^5 + 9450*n^6 + 870*n^7 + 
45*n^8 + n^9 + 362880)) - (x^3*(c + d*x)^n*(3*n + n^2 + 2)*(6720*b^3*c^6*n 
 - 60216*a^3*d^6*n - 60480*a^3*d^6 - 24574*a^3*d^6*n^2 - 5265*a^3*d^6*n^3 
- 625*a^3*d^6*n^4 - 39*a^3*d^6*n^5 - a^3*d^6*n^6 + 25920*a*b^2*c^4*d^2*n + 
 36288*a^2*b*c^2*d^4*n + 6120*a*b^2*c^4*d^2*n^2 + 19800*a^2*b*c^2*d^4*n^2 
+ 360*a*b^2*c^4*d^2*n^3 + 4020*a^2*b*c^2*d^4*n^3 + 360*a^2*b*c^2*d^4*n^4 + 
 12*a^2*b*c^2*d^4*n^5))/(d^6*(1026576*n + 1172700*n^2 + 723680*n^3 + 26932 
5*n^4 + 63273*n^5 + 9450*n^6 + 870*n^7 + 45*n^8 + n^9 + 362880)) + (3*b*x^ 
5*(c + d*x)^n*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)*(3024*a^2*d^4 + 1650*a^2 
*d^4*n - 112*b^2*c^4*n + 335*a^2*d^4*n^2 + 30*a^2*d^4*n^3 + a^2*d^4*n^4 - 
432*a*b*c^2*d^2*n - 102*a*b*c^2*d^2*n^2 - 6*a*b*c^2*d^2*n^3))/(d^4*(102657 
6*n + 1172700*n^2 + 723680*n^3 + 269325*n^4 + 63273*n^5 + 9450*n^6 + 87...
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 2557, normalized size of antiderivative = 7.45 \[ \int x^2 (c+d x)^n \left (a+b x^2\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

((c + d*x)**n*(2*a**3*c**3*d**6*n**6 + 78*a**3*c**3*d**6*n**5 + 1250*a**3* 
c**3*d**6*n**4 + 10530*a**3*c**3*d**6*n**3 + 49148*a**3*c**3*d**6*n**2 + 1 
20432*a**3*c**3*d**6*n + 120960*a**3*c**3*d**6 - 2*a**3*c**2*d**7*n**7*x - 
 78*a**3*c**2*d**7*n**6*x - 1250*a**3*c**2*d**7*n**5*x - 10530*a**3*c**2*d 
**7*n**4*x - 49148*a**3*c**2*d**7*n**3*x - 120432*a**3*c**2*d**7*n**2*x - 
120960*a**3*c**2*d**7*n*x + a**3*c*d**8*n**8*x**2 + 40*a**3*c*d**8*n**7*x* 
*2 + 664*a**3*c*d**8*n**6*x**2 + 5890*a**3*c*d**8*n**5*x**2 + 29839*a**3*c 
*d**8*n**4*x**2 + 84790*a**3*c*d**8*n**3*x**2 + 120696*a**3*c*d**8*n**2*x* 
*2 + 60480*a**3*c*d**8*n*x**2 + a**3*d**9*n**8*x**3 + 42*a**3*d**9*n**7*x* 
*3 + 744*a**3*d**9*n**6*x**3 + 7218*a**3*d**9*n**5*x**3 + 41619*a**3*d**9* 
n**4*x**3 + 144468*a**3*d**9*n**3*x**3 + 290276*a**3*d**9*n**2*x**3 + 3018 
72*a**3*d**9*n*x**3 + 120960*a**3*d**9*x**3 + 72*a**2*b*c**5*d**4*n**4 + 2 
160*a**2*b*c**5*d**4*n**3 + 24120*a**2*b*c**5*d**4*n**2 + 118800*a**2*b*c* 
*5*d**4*n + 217728*a**2*b*c**5*d**4 - 72*a**2*b*c**4*d**5*n**5*x - 2160*a* 
*2*b*c**4*d**5*n**4*x - 24120*a**2*b*c**4*d**5*n**3*x - 118800*a**2*b*c**4 
*d**5*n**2*x - 217728*a**2*b*c**4*d**5*n*x + 36*a**2*b*c**3*d**6*n**6*x**2 
 + 1116*a**2*b*c**3*d**6*n**5*x**2 + 13140*a**2*b*c**3*d**6*n**4*x**2 + 71 
460*a**2*b*c**3*d**6*n**3*x**2 + 168264*a**2*b*c**3*d**6*n**2*x**2 + 10886 
4*a**2*b*c**3*d**6*n*x**2 - 12*a**2*b*c**2*d**7*n**7*x**3 - 396*a**2*b*c** 
2*d**7*n**6*x**3 - 5124*a**2*b*c**2*d**7*n**5*x**3 - 32580*a**2*b*c**2*...