3.325 \(\int x^m (a+b x^2)^2 (c+d x^2)^3 \, dx\)
Optimal. Leaf size=151 \[ \frac{c x^{m+5} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )}{m+5}+\frac{d x^{m+7} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )}{m+7}+\frac{a^2 c^3 x^{m+1}}{m+1}+\frac{a c^2 x^{m+3} (3 a d+2 b c)}{m+3}+\frac{b d^2 x^{m+9} (2 a d+3 b c)}{m+9}+\frac{b^2 d^3 x^{m+11}}{m+11} \]
[Out]
(a^2*c^3*x^(1 + m))/(1 + m) + (a*c^2*(2*b*c + 3*a*d)*x^(3 + m))/(3 + m) + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)
*x^(5 + m))/(5 + m) + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(7 + m))/(7 + m) + (b*d^2*(3*b*c + 2*a*d)*x^(9 +
m))/(9 + m) + (b^2*d^3*x^(11 + m))/(11 + m)
________________________________________________________________________________________
Rubi [A] time = 0.0885689, antiderivative size = 151, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used =
{448} \[ \frac{c x^{m+5} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )}{m+5}+\frac{d x^{m+7} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )}{m+7}+\frac{a^2 c^3 x^{m+1}}{m+1}+\frac{a c^2 x^{m+3} (3 a d+2 b c)}{m+3}+\frac{b d^2 x^{m+9} (2 a d+3 b c)}{m+9}+\frac{b^2 d^3 x^{m+11}}{m+11} \]
Antiderivative was successfully verified.
[In]
Int[x^m*(a + b*x^2)^2*(c + d*x^2)^3,x]
[Out]
(a^2*c^3*x^(1 + m))/(1 + m) + (a*c^2*(2*b*c + 3*a*d)*x^(3 + m))/(3 + m) + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)
*x^(5 + m))/(5 + m) + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(7 + m))/(7 + m) + (b*d^2*(3*b*c + 2*a*d)*x^(9 +
m))/(9 + m) + (b^2*d^3*x^(11 + m))/(11 + m)
Rule 448
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]
Rubi steps
\begin{align*} \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^m+a c^2 (2 b c+3 a d) x^{2+m}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{4+m}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{6+m}+b d^2 (3 b c+2 a d) x^{8+m}+b^2 d^3 x^{10+m}\right ) \, dx\\ &=\frac{a^2 c^3 x^{1+m}}{1+m}+\frac{a c^2 (2 b c+3 a d) x^{3+m}}{3+m}+\frac{c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{5+m}}{5+m}+\frac{d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{7+m}}{7+m}+\frac{b d^2 (3 b c+2 a d) x^{9+m}}{9+m}+\frac{b^2 d^3 x^{11+m}}{11+m}\\ \end{align*}
Mathematica [A] time = 0.103946, size = 141, normalized size = 0.93 \[ x^m \left (\frac{d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )}{m+7}+\frac{c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )}{m+5}+\frac{a^2 c^3 x}{m+1}+\frac{a c^2 x^3 (3 a d+2 b c)}{m+3}+\frac{b d^2 x^9 (2 a d+3 b c)}{m+9}+\frac{b^2 d^3 x^{11}}{m+11}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^m*(a + b*x^2)^2*(c + d*x^2)^3,x]
[Out]
x^m*((a^2*c^3*x)/(1 + m) + (a*c^2*(2*b*c + 3*a*d)*x^3)/(3 + m) + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^5)/(5
+ m) + (d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7)/(7 + m) + (b*d^2*(3*b*c + 2*a*d)*x^9)/(9 + m) + (b^2*d^3*x^11
)/(11 + m))
________________________________________________________________________________________
Maple [B] time = 0.01, size = 976, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(b*x^2+a)^2*(d*x^2+c)^3,x)
[Out]
x^(1+m)*(b^2*d^3*m^5*x^10+25*b^2*d^3*m^4*x^10+2*a*b*d^3*m^5*x^8+3*b^2*c*d^2*m^5*x^8+230*b^2*d^3*m^3*x^10+54*a*
b*d^3*m^4*x^8+81*b^2*c*d^2*m^4*x^8+950*b^2*d^3*m^2*x^10+a^2*d^3*m^5*x^6+6*a*b*c*d^2*m^5*x^6+524*a*b*d^3*m^3*x^
8+3*b^2*c^2*d*m^5*x^6+786*b^2*c*d^2*m^3*x^8+1689*b^2*d^3*m*x^10+29*a^2*d^3*m^4*x^6+174*a*b*c*d^2*m^4*x^6+2244*
a*b*d^3*m^2*x^8+87*b^2*c^2*d*m^4*x^6+3366*b^2*c*d^2*m^2*x^8+945*b^2*d^3*x^10+3*a^2*c*d^2*m^5*x^4+302*a^2*d^3*m
^3*x^6+6*a*b*c^2*d*m^5*x^4+1812*a*b*c*d^2*m^3*x^6+4082*a*b*d^3*m*x^8+b^2*c^3*m^5*x^4+906*b^2*c^2*d*m^3*x^6+612
3*b^2*c*d^2*m*x^8+93*a^2*c*d^2*m^4*x^4+1366*a^2*d^3*m^2*x^6+186*a*b*c^2*d*m^4*x^4+8196*a*b*c*d^2*m^2*x^6+2310*
a*b*d^3*x^8+31*b^2*c^3*m^4*x^4+4098*b^2*c^2*d*m^2*x^6+3465*b^2*c*d^2*x^8+3*a^2*c^2*d*m^5*x^2+1050*a^2*c*d^2*m^
3*x^4+2577*a^2*d^3*m*x^6+2*a*b*c^3*m^5*x^2+2100*a*b*c^2*d*m^3*x^4+15462*a*b*c*d^2*m*x^6+350*b^2*c^3*m^3*x^4+77
31*b^2*c^2*d*m*x^6+99*a^2*c^2*d*m^4*x^2+5190*a^2*c*d^2*m^2*x^4+1485*a^2*d^3*x^6+66*a*b*c^3*m^4*x^2+10380*a*b*c
^2*d*m^2*x^4+8910*a*b*c*d^2*x^6+1730*b^2*c^3*m^2*x^4+4455*b^2*c^2*d*x^6+a^2*c^3*m^5+1218*a^2*c^2*d*m^3*x^2+104
67*a^2*c*d^2*m*x^4+812*a*b*c^3*m^3*x^2+20934*a*b*c^2*d*m*x^4+3489*b^2*c^3*m*x^4+35*a^2*c^3*m^4+6786*a^2*c^2*d*
m^2*x^2+6237*a^2*c*d^2*x^4+4524*a*b*c^3*m^2*x^2+12474*a*b*c^2*d*x^4+2079*b^2*c^3*x^4+470*a^2*c^3*m^3+16059*a^2
*c^2*d*m*x^2+10706*a*b*c^3*m*x^2+3010*a^2*c^3*m^2+10395*a^2*c^2*d*x^2+6930*a*b*c^3*x^2+9129*a^2*c^3*m+10395*a^
2*c^3)/(11+m)/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.01973, size = 1740, normalized size = 11.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="fricas")
[Out]
((b^2*d^3*m^5 + 25*b^2*d^3*m^4 + 230*b^2*d^3*m^3 + 950*b^2*d^3*m^2 + 1689*b^2*d^3*m + 945*b^2*d^3)*x^11 + ((3*
b^2*c*d^2 + 2*a*b*d^3)*m^5 + 3465*b^2*c*d^2 + 2310*a*b*d^3 + 27*(3*b^2*c*d^2 + 2*a*b*d^3)*m^4 + 262*(3*b^2*c*d
^2 + 2*a*b*d^3)*m^3 + 1122*(3*b^2*c*d^2 + 2*a*b*d^3)*m^2 + 2041*(3*b^2*c*d^2 + 2*a*b*d^3)*m)*x^9 + ((3*b^2*c^2
*d + 6*a*b*c*d^2 + a^2*d^3)*m^5 + 4455*b^2*c^2*d + 8910*a*b*c*d^2 + 1485*a^2*d^3 + 29*(3*b^2*c^2*d + 6*a*b*c*d
^2 + a^2*d^3)*m^4 + 302*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*m^3 + 1366*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)
*m^2 + 2577*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*m)*x^7 + ((b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*m^5 + 2079*b
^2*c^3 + 12474*a*b*c^2*d + 6237*a^2*c*d^2 + 31*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*m^4 + 350*(b^2*c^3 + 6*a*
b*c^2*d + 3*a^2*c*d^2)*m^3 + 1730*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*m^2 + 3489*(b^2*c^3 + 6*a*b*c^2*d + 3*
a^2*c*d^2)*m)*x^5 + ((2*a*b*c^3 + 3*a^2*c^2*d)*m^5 + 6930*a*b*c^3 + 10395*a^2*c^2*d + 33*(2*a*b*c^3 + 3*a^2*c^
2*d)*m^4 + 406*(2*a*b*c^3 + 3*a^2*c^2*d)*m^3 + 2262*(2*a*b*c^3 + 3*a^2*c^2*d)*m^2 + 5353*(2*a*b*c^3 + 3*a^2*c^
2*d)*m)*x^3 + (a^2*c^3*m^5 + 35*a^2*c^3*m^4 + 470*a^2*c^3*m^3 + 3010*a^2*c^3*m^2 + 9129*a^2*c^3*m + 10395*a^2*
c^3)*x)*x^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)
________________________________________________________________________________________
Sympy [A] time = 5.01584, size = 4345, normalized size = 28.77 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m*(b*x**2+a)**2*(d*x**2+c)**3,x)
[Out]
Piecewise((-a**2*c**3/(10*x**10) - 3*a**2*c**2*d/(8*x**8) - a**2*c*d**2/(2*x**6) - a**2*d**3/(4*x**4) - a*b*c*
*3/(4*x**8) - a*b*c**2*d/x**6 - 3*a*b*c*d**2/(2*x**4) - a*b*d**3/x**2 - b**2*c**3/(6*x**6) - 3*b**2*c**2*d/(4*
x**4) - 3*b**2*c*d**2/(2*x**2) + b**2*d**3*log(x), Eq(m, -11)), (-a**2*c**3/(8*x**8) - a**2*c**2*d/(2*x**6) -
3*a**2*c*d**2/(4*x**4) - a**2*d**3/(2*x**2) - a*b*c**3/(3*x**6) - 3*a*b*c**2*d/(2*x**4) - 3*a*b*c*d**2/x**2 +
2*a*b*d**3*log(x) - b**2*c**3/(4*x**4) - 3*b**2*c**2*d/(2*x**2) + 3*b**2*c*d**2*log(x) + b**2*d**3*x**2/2, Eq(
m, -9)), (-a**2*c**3/(6*x**6) - 3*a**2*c**2*d/(4*x**4) - 3*a**2*c*d**2/(2*x**2) + a**2*d**3*log(x) - a*b*c**3/
(2*x**4) - 3*a*b*c**2*d/x**2 + 6*a*b*c*d**2*log(x) + a*b*d**3*x**2 - b**2*c**3/(2*x**2) + 3*b**2*c**2*d*log(x)
+ 3*b**2*c*d**2*x**2/2 + b**2*d**3*x**4/4, Eq(m, -7)), (-a**2*c**3/(4*x**4) - 3*a**2*c**2*d/(2*x**2) + 3*a**2
*c*d**2*log(x) + a**2*d**3*x**2/2 - a*b*c**3/x**2 + 6*a*b*c**2*d*log(x) + 3*a*b*c*d**2*x**2 + a*b*d**3*x**4/2
+ b**2*c**3*log(x) + 3*b**2*c**2*d*x**2/2 + 3*b**2*c*d**2*x**4/4 + b**2*d**3*x**6/6, Eq(m, -5)), (-a**2*c**3/(
2*x**2) + 3*a**2*c**2*d*log(x) + 3*a**2*c*d**2*x**2/2 + a**2*d**3*x**4/4 + 2*a*b*c**3*log(x) + 3*a*b*c**2*d*x*
*2 + 3*a*b*c*d**2*x**4/2 + a*b*d**3*x**6/3 + b**2*c**3*x**2/2 + 3*b**2*c**2*d*x**4/4 + b**2*c*d**2*x**6/2 + b*
*2*d**3*x**8/8, Eq(m, -3)), (a**2*c**3*log(x) + 3*a**2*c**2*d*x**2/2 + 3*a**2*c*d**2*x**4/4 + a**2*d**3*x**6/6
+ a*b*c**3*x**2 + 3*a*b*c**2*d*x**4/2 + a*b*c*d**2*x**6 + a*b*d**3*x**8/4 + b**2*c**3*x**4/4 + b**2*c**2*d*x*
*6/2 + 3*b**2*c*d**2*x**8/8 + b**2*d**3*x**10/10, Eq(m, -1)), (a**2*c**3*m**5*x*x**m/(m**6 + 36*m**5 + 505*m**
4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 35*a**2*c**3*m**4*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**
3 + 12139*m**2 + 19524*m + 10395) + 470*a**2*c**3*m**3*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m
**2 + 19524*m + 10395) + 3010*a**2*c**3*m**2*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1952
4*m + 10395) + 9129*a**2*c**3*m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 10395*a**2*c**3*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*a**2*c**2*
d*m**5*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 99*a**2*c**2*d*m**4*
x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1218*a**2*c**2*d*m**3*x**3*
x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6786*a**2*c**2*d*m**2*x**3*x**m/
(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 16059*a**2*c**2*d*m*x**3*x**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10395*a**2*c**2*d*x**3*x**m/(m**6 + 36*m**5
+ 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*a**2*c*d**2*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*m*
*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 93*a**2*c*d**2*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 34
80*m**3 + 12139*m**2 + 19524*m + 10395) + 1050*a**2*c*d**2*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m*
*3 + 12139*m**2 + 19524*m + 10395) + 5190*a**2*c*d**2*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 +
12139*m**2 + 19524*m + 10395) + 10467*a**2*c*d**2*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m
**2 + 19524*m + 10395) + 6237*a**2*c*d**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1952
4*m + 10395) + a**2*d**3*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 29*a**2*d**3*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 302*a*
*2*d**3*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1366*a**2*d**3
*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2577*a**2*d**3*m*x**7
*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1485*a**2*d**3*x**7*x**m/(m**6
+ 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*a*b*c**3*m**5*x**3*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 66*a*b*c**3*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4
+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 812*a*b*c**3*m**3*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m*
*3 + 12139*m**2 + 19524*m + 10395) + 4524*a*b*c**3*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 121
39*m**2 + 19524*m + 10395) + 10706*a*b*c**3*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 +
19524*m + 10395) + 6930*a*b*c**3*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103
95) + 6*a*b*c**2*d*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 186
*a*b*c**2*d*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2100*a*b*c
**2*d*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10380*a*b*c**2*d
*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 20934*a*b*c**2*d*m*x*
*5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 12474*a*b*c**2*d*x**5*x**m/(m
**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6*a*b*c*d**2*m**5*x**7*x**m/(m**6 + 36*
m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 174*a*b*c*d**2*m**4*x**7*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1812*a*b*c*d**2*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m*
*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 8196*a*b*c*d**2*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3
480*m**3 + 12139*m**2 + 19524*m + 10395) + 15462*a*b*c*d**2*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 8910*a*b*c*d**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m*
*2 + 19524*m + 10395) + 2*a*b*d**3*m**5*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*
m + 10395) + 54*a*b*d**3*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 524*a*b*d**3*m**3*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2244*a
*b*d**3*m**2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4082*a*b*d**3*
m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2310*a*b*d**3*x**9*x**m/(
m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + b**2*c**3*m**5*x**5*x**m/(m**6 + 36*m*
*5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 31*b**2*c**3*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*
m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 350*b**2*c**3*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3
480*m**3 + 12139*m**2 + 19524*m + 10395) + 1730*b**2*c**3*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**
3 + 12139*m**2 + 19524*m + 10395) + 3489*b**2*c**3*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*
m**2 + 19524*m + 10395) + 2079*b**2*c**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524
*m + 10395) + 3*b**2*c**2*d*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103
95) + 87*b**2*c**2*d*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 9
06*b**2*c**2*d*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4098*b*
*2*c**2*d*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 7731*b**2*c*
*2*d*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4455*b**2*c**2*d*x**
7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*b**2*c*d**2*m**5*x**9*x**m/(
m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 81*b**2*c*d**2*m**4*x**9*x**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 786*b**2*c*d**2*m**3*x**9*x**m/(m**6 + 36*m**
5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3366*b**2*c*d**2*m**2*x**9*x**m/(m**6 + 36*m**5 + 5
05*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6123*b**2*c*d**2*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4
+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3465*b**2*c*d**2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**
3 + 12139*m**2 + 19524*m + 10395) + b**2*d**3*m**5*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m
**2 + 19524*m + 10395) + 25*b**2*d**3*m**4*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19
524*m + 10395) + 230*b**2*d**3*m**3*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +
10395) + 950*b**2*d**3*m**2*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 1689*b**2*d**3*m*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 945*b*
*2*d**3*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395), True))
________________________________________________________________________________________
Giac [B] time = 1.2044, size = 1609, normalized size = 10.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="giac")
[Out]
(b^2*d^3*m^5*x^11*x^m + 25*b^2*d^3*m^4*x^11*x^m + 3*b^2*c*d^2*m^5*x^9*x^m + 2*a*b*d^3*m^5*x^9*x^m + 230*b^2*d^
3*m^3*x^11*x^m + 81*b^2*c*d^2*m^4*x^9*x^m + 54*a*b*d^3*m^4*x^9*x^m + 950*b^2*d^3*m^2*x^11*x^m + 3*b^2*c^2*d*m^
5*x^7*x^m + 6*a*b*c*d^2*m^5*x^7*x^m + a^2*d^3*m^5*x^7*x^m + 786*b^2*c*d^2*m^3*x^9*x^m + 524*a*b*d^3*m^3*x^9*x^
m + 1689*b^2*d^3*m*x^11*x^m + 87*b^2*c^2*d*m^4*x^7*x^m + 174*a*b*c*d^2*m^4*x^7*x^m + 29*a^2*d^3*m^4*x^7*x^m +
3366*b^2*c*d^2*m^2*x^9*x^m + 2244*a*b*d^3*m^2*x^9*x^m + 945*b^2*d^3*x^11*x^m + b^2*c^3*m^5*x^5*x^m + 6*a*b*c^2
*d*m^5*x^5*x^m + 3*a^2*c*d^2*m^5*x^5*x^m + 906*b^2*c^2*d*m^3*x^7*x^m + 1812*a*b*c*d^2*m^3*x^7*x^m + 302*a^2*d^
3*m^3*x^7*x^m + 6123*b^2*c*d^2*m*x^9*x^m + 4082*a*b*d^3*m*x^9*x^m + 31*b^2*c^3*m^4*x^5*x^m + 186*a*b*c^2*d*m^4
*x^5*x^m + 93*a^2*c*d^2*m^4*x^5*x^m + 4098*b^2*c^2*d*m^2*x^7*x^m + 8196*a*b*c*d^2*m^2*x^7*x^m + 1366*a^2*d^3*m
^2*x^7*x^m + 3465*b^2*c*d^2*x^9*x^m + 2310*a*b*d^3*x^9*x^m + 2*a*b*c^3*m^5*x^3*x^m + 3*a^2*c^2*d*m^5*x^3*x^m +
350*b^2*c^3*m^3*x^5*x^m + 2100*a*b*c^2*d*m^3*x^5*x^m + 1050*a^2*c*d^2*m^3*x^5*x^m + 7731*b^2*c^2*d*m*x^7*x^m
+ 15462*a*b*c*d^2*m*x^7*x^m + 2577*a^2*d^3*m*x^7*x^m + 66*a*b*c^3*m^4*x^3*x^m + 99*a^2*c^2*d*m^4*x^3*x^m + 173
0*b^2*c^3*m^2*x^5*x^m + 10380*a*b*c^2*d*m^2*x^5*x^m + 5190*a^2*c*d^2*m^2*x^5*x^m + 4455*b^2*c^2*d*x^7*x^m + 89
10*a*b*c*d^2*x^7*x^m + 1485*a^2*d^3*x^7*x^m + a^2*c^3*m^5*x*x^m + 812*a*b*c^3*m^3*x^3*x^m + 1218*a^2*c^2*d*m^3
*x^3*x^m + 3489*b^2*c^3*m*x^5*x^m + 20934*a*b*c^2*d*m*x^5*x^m + 10467*a^2*c*d^2*m*x^5*x^m + 35*a^2*c^3*m^4*x*x
^m + 4524*a*b*c^3*m^2*x^3*x^m + 6786*a^2*c^2*d*m^2*x^3*x^m + 2079*b^2*c^3*x^5*x^m + 12474*a*b*c^2*d*x^5*x^m +
6237*a^2*c*d^2*x^5*x^m + 470*a^2*c^3*m^3*x*x^m + 10706*a*b*c^3*m*x^3*x^m + 16059*a^2*c^2*d*m*x^3*x^m + 3010*a^
2*c^3*m^2*x*x^m + 6930*a*b*c^3*x^3*x^m + 10395*a^2*c^2*d*x^3*x^m + 9129*a^2*c^3*m*x*x^m + 10395*a^2*c^3*x*x^m)
/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)