Integrand size = 20, antiderivative size = 101 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx=\frac {a^2 (d x)^{1+m}}{d (1+m)}+\frac {2 a b (d x)^{3+m}}{d^3 (3+m)}+\frac {\left (b^2+2 a c\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {2 b c (d x)^{7+m}}{d^7 (7+m)}+\frac {c^2 (d x)^{9+m}}{d^9 (9+m)} \] Output:
a^2*(d*x)^(1+m)/d/(1+m)+2*a*b*(d*x)^(3+m)/d^3/(3+m)+(2*a*c+b^2)*(d*x)^(5+m )/d^5/(5+m)+2*b*c*(d*x)^(7+m)/d^7/(7+m)+c^2*(d*x)^(9+m)/d^9/(9+m)
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx=x (d x)^m \left (\frac {a^2}{1+m}+\frac {2 a b x^2}{3+m}+\frac {\left (b^2+2 a c\right ) x^4}{5+m}+\frac {2 b c x^6}{7+m}+\frac {c^2 x^8}{9+m}\right ) \] Input:
Integrate[(d*x)^m*(a + b*x^2 + c*x^4)^2,x]
Output:
x*(d*x)^m*(a^2/(1 + m) + (2*a*b*x^2)/(3 + m) + ((b^2 + 2*a*c)*x^4)/(5 + m) + (2*b*c*x^6)/(7 + m) + (c^2*x^8)/(9 + m))
Time = 0.43 (sec) , antiderivative size = 101, 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 = {1433, 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.
| \(\displaystyle \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx\) |
| \(\Big \downarrow \) 1433 |
| \(\displaystyle \int \left (a^2 (d x)^m+\frac {\left (2 a c+b^2\right ) (d x)^{m+4}}{d^4}+\frac {2 a b (d x)^{m+2}}{d^2}+\frac {2 b c (d x)^{m+6}}{d^6}+\frac {c^2 (d x)^{m+8}}{d^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 (d x)^{m+1}}{d (m+1)}+\frac {\left (2 a c+b^2\right ) (d x)^{m+5}}{d^5 (m+5)}+\frac {2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac {2 b c (d x)^{m+7}}{d^7 (m+7)}+\frac {c^2 (d x)^{m+9}}{d^9 (m+9)}\) |
Input:
Int[(d*x)^m*(a + b*x^2 + c*x^4)^2,x]
Output:
(a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (( b^2 + 2*a*c)*(d*x)^(5 + m))/(d^5*(5 + m)) + (2*b*c*(d*x)^(7 + m))/(d^7*(7 + m)) + (c^2*(d*x)^(9 + m))/(d^9*(9 + m))
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || !IntegerQ[(m + 1)/2])
Leaf count of result is larger than twice the leaf count of optimal. \(300\) vs. \(2(101)=202\).
Time = 0.14 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.98
| method | result | size |
| gosper | \(\frac {x \left (c^{2} m^{4} x^{8}+16 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+86 c^{2} m^{2} x^{8}+36 b c \,m^{3} x^{6}+176 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+208 b c \,m^{2} x^{6}+105 c^{2} x^{8}+40 a c \,m^{3} x^{4}+20 b^{2} m^{3} x^{4}+444 m \,x^{6} b c +2 a b \,m^{4} x^{2}+260 a c \,m^{2} x^{4}+130 b^{2} m^{2} x^{4}+270 b c \,x^{6}+44 a b \,m^{3} x^{2}+600 a c \,x^{4} m +300 b^{2} x^{4} m +a^{2} m^{4}+328 a b \,m^{2} x^{2}+378 a c \,x^{4}+189 b^{2} x^{4}+24 a^{2} m^{3}+916 a b \,x^{2} m +206 a^{2} m^{2}+630 a b \,x^{2}+744 a^{2} m +945 a^{2}\right ) \left (d x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(301\) |
| risch | \(\frac {x \left (c^{2} m^{4} x^{8}+16 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+86 c^{2} m^{2} x^{8}+36 b c \,m^{3} x^{6}+176 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+208 b c \,m^{2} x^{6}+105 c^{2} x^{8}+40 a c \,m^{3} x^{4}+20 b^{2} m^{3} x^{4}+444 m \,x^{6} b c +2 a b \,m^{4} x^{2}+260 a c \,m^{2} x^{4}+130 b^{2} m^{2} x^{4}+270 b c \,x^{6}+44 a b \,m^{3} x^{2}+600 a c \,x^{4} m +300 b^{2} x^{4} m +a^{2} m^{4}+328 a b \,m^{2} x^{2}+378 a c \,x^{4}+189 b^{2} x^{4}+24 a^{2} m^{3}+916 a b \,x^{2} m +206 a^{2} m^{2}+630 a b \,x^{2}+744 a^{2} m +945 a^{2}\right ) \left (d x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(301\) |
| orering | \(\frac {x \left (c^{2} m^{4} x^{8}+16 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+86 c^{2} m^{2} x^{8}+36 b c \,m^{3} x^{6}+176 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+208 b c \,m^{2} x^{6}+105 c^{2} x^{8}+40 a c \,m^{3} x^{4}+20 b^{2} m^{3} x^{4}+444 m \,x^{6} b c +2 a b \,m^{4} x^{2}+260 a c \,m^{2} x^{4}+130 b^{2} m^{2} x^{4}+270 b c \,x^{6}+44 a b \,m^{3} x^{2}+600 a c \,x^{4} m +300 b^{2} x^{4} m +a^{2} m^{4}+328 a b \,m^{2} x^{2}+378 a c \,x^{4}+189 b^{2} x^{4}+24 a^{2} m^{3}+916 a b \,x^{2} m +206 a^{2} m^{2}+630 a b \,x^{2}+744 a^{2} m +945 a^{2}\right ) \left (d x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(301\) |
| parallelrisch | \(\frac {x^{9} \left (d x \right )^{m} c^{2} m^{4}+16 x^{9} \left (d x \right )^{m} c^{2} m^{3}+86 x^{9} \left (d x \right )^{m} c^{2} m^{2}+176 x^{9} \left (d x \right )^{m} c^{2} m +x^{5} \left (d x \right )^{m} b^{2} m^{4}+20 x^{5} \left (d x \right )^{m} b^{2} m^{3}+270 x^{7} \left (d x \right )^{m} b c +130 x^{5} \left (d x \right )^{m} b^{2} m^{2}+300 x^{5} \left (d x \right )^{m} b^{2} m +378 x^{5} \left (d x \right )^{m} a c +x \left (d x \right )^{m} a^{2} m^{4}+2 x^{7} \left (d x \right )^{m} b c \,m^{4}+36 x^{7} \left (d x \right )^{m} b c \,m^{3}+208 x^{7} \left (d x \right )^{m} b c \,m^{2}+2 x^{5} \left (d x \right )^{m} a c \,m^{4}+444 x^{7} \left (d x \right )^{m} b c m +40 x^{5} \left (d x \right )^{m} a c \,m^{3}+260 x^{5} \left (d x \right )^{m} a c \,m^{2}+2 x^{3} \left (d x \right )^{m} a b \,m^{4}+600 x^{5} \left (d x \right )^{m} a c m +44 x^{3} \left (d x \right )^{m} a b \,m^{3}+328 x^{3} \left (d x \right )^{m} a b \,m^{2}+916 x^{3} \left (d x \right )^{m} a b m +24 x \left (d x \right )^{m} a^{2} m^{3}+630 x^{3} \left (d x \right )^{m} a b +206 x \left (d x \right )^{m} a^{2} m^{2}+744 x \left (d x \right )^{m} a^{2} m +945 x \left (d x \right )^{m} a^{2}+189 x^{5} \left (d x \right )^{m} b^{2}+105 x^{9} \left (d x \right )^{m} c^{2}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(450\) |
Input:
int((d*x)^m*(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
x*(c^2*m^4*x^8+16*c^2*m^3*x^8+2*b*c*m^4*x^6+86*c^2*m^2*x^8+36*b*c*m^3*x^6+ 176*c^2*m*x^8+2*a*c*m^4*x^4+b^2*m^4*x^4+208*b*c*m^2*x^6+105*c^2*x^8+40*a*c *m^3*x^4+20*b^2*m^3*x^4+444*b*c*m*x^6+2*a*b*m^4*x^2+260*a*c*m^2*x^4+130*b^ 2*m^2*x^4+270*b*c*x^6+44*a*b*m^3*x^2+600*a*c*m*x^4+300*b^2*m*x^4+a^2*m^4+3 28*a*b*m^2*x^2+378*a*c*x^4+189*b^2*x^4+24*a^2*m^3+916*a*b*m*x^2+206*a^2*m^ 2+630*a*b*x^2+744*a^2*m+945*a^2)*(d*x)^m/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (101) = 202\).
Time = 0.07 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.39 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx=\frac {{\left ({\left (c^{2} m^{4} + 16 \, c^{2} m^{3} + 86 \, c^{2} m^{2} + 176 \, c^{2} m + 105 \, c^{2}\right )} x^{9} + 2 \, {\left (b c m^{4} + 18 \, b c m^{3} + 104 \, b c m^{2} + 222 \, b c m + 135 \, b c\right )} x^{7} + {\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 20 \, {\left (b^{2} + 2 \, a c\right )} m^{3} + 130 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + 189 \, b^{2} + 378 \, a c + 300 \, {\left (b^{2} + 2 \, a c\right )} m\right )} x^{5} + 2 \, {\left (a b m^{4} + 22 \, a b m^{3} + 164 \, a b m^{2} + 458 \, a b m + 315 \, a b\right )} x^{3} + {\left (a^{2} m^{4} + 24 \, a^{2} m^{3} + 206 \, a^{2} m^{2} + 744 \, a^{2} m + 945 \, a^{2}\right )} x\right )} \left (d x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate((d*x)^m*(c*x^4+b*x^2+a)^2,x, algorithm="fricas")
Output:
((c^2*m^4 + 16*c^2*m^3 + 86*c^2*m^2 + 176*c^2*m + 105*c^2)*x^9 + 2*(b*c*m^ 4 + 18*b*c*m^3 + 104*b*c*m^2 + 222*b*c*m + 135*b*c)*x^7 + ((b^2 + 2*a*c)*m ^4 + 20*(b^2 + 2*a*c)*m^3 + 130*(b^2 + 2*a*c)*m^2 + 189*b^2 + 378*a*c + 30 0*(b^2 + 2*a*c)*m)*x^5 + 2*(a*b*m^4 + 22*a*b*m^3 + 164*a*b*m^2 + 458*a*b*m + 315*a*b)*x^3 + (a^2*m^4 + 24*a^2*m^3 + 206*a^2*m^2 + 744*a^2*m + 945*a^ 2)*x)*(d*x)^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)
Leaf count of result is larger than twice the leaf count of optimal. 1435 vs. \(2 (90) = 180\).
Time = 0.51 (sec) , antiderivative size = 1435, normalized size of antiderivative = 14.21 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((d*x)**m*(c*x**4+b*x**2+a)**2,x)
Output:
Piecewise(((-a**2/(8*x**8) - a*b/(3*x**6) - a*c/(2*x**4) - b**2/(4*x**4) - b*c/x**2 + c**2*log(x))/d**9, Eq(m, -9)), ((-a**2/(6*x**6) - a*b/(2*x**4) - a*c/x**2 - b**2/(2*x**2) + 2*b*c*log(x) + c**2*x**2/2)/d**7, Eq(m, -7)) , ((-a**2/(4*x**4) - a*b/x**2 + 2*a*c*log(x) + b**2*log(x) + b*c*x**2 + c* *2*x**4/4)/d**5, Eq(m, -5)), ((-a**2/(2*x**2) + 2*a*b*log(x) + a*c*x**2 + b**2*x**2/2 + b*c*x**4/2 + c**2*x**6/6)/d**3, Eq(m, -3)), ((a**2*log(x) + a*b*x**2 + a*c*x**4/2 + b**2*x**4/4 + b*c*x**6/3 + c**2*x**8/8)/d, Eq(m, - 1)), (a**2*m**4*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*a**2*m**3*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1 689*m + 945) + 206*a**2*m**2*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m **2 + 1689*m + 945) + 744*a**2*m*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 9 50*m**2 + 1689*m + 945) + 945*a**2*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*a*b*m**4*x**3*(d*x)**m/(m**5 + 25*m**4 + 230 *m**3 + 950*m**2 + 1689*m + 945) + 44*a*b*m**3*x**3*(d*x)**m/(m**5 + 25*m* *4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 328*a*b*m**2*x**3*(d*x)**m/(m** 5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 916*a*b*m*x**3*(d*x)** m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 630*a*b*x**3*(d* x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*a*c*m**4*x **5*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 40*a* c*m**3*x**5*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 9...
Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx=\frac {c^{2} d^{m} x^{9} x^{m}}{m + 9} + \frac {2 \, b c d^{m} x^{7} x^{m}}{m + 7} + \frac {b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a c d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} \] Input:
integrate((d*x)^m*(c*x^4+b*x^2+a)^2,x, algorithm="maxima")
Output:
c^2*d^m*x^9*x^m/(m + 9) + 2*b*c*d^m*x^7*x^m/(m + 7) + b^2*d^m*x^5*x^m/(m + 5) + 2*a*c*d^m*x^5*x^m/(m + 5) + 2*a*b*d^m*x^3*x^m/(m + 3) + (d*x)^(m + 1 )*a^2/(d*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (101) = 202\).
Time = 0.11 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.45 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx=\frac {\left (d x\right )^{m} c^{2} m^{4} x^{9} + 16 \, \left (d x\right )^{m} c^{2} m^{3} x^{9} + 2 \, \left (d x\right )^{m} b c m^{4} x^{7} + 86 \, \left (d x\right )^{m} c^{2} m^{2} x^{9} + 36 \, \left (d x\right )^{m} b c m^{3} x^{7} + 176 \, \left (d x\right )^{m} c^{2} m x^{9} + \left (d x\right )^{m} b^{2} m^{4} x^{5} + 2 \, \left (d x\right )^{m} a c m^{4} x^{5} + 208 \, \left (d x\right )^{m} b c m^{2} x^{7} + 105 \, \left (d x\right )^{m} c^{2} x^{9} + 20 \, \left (d x\right )^{m} b^{2} m^{3} x^{5} + 40 \, \left (d x\right )^{m} a c m^{3} x^{5} + 444 \, \left (d x\right )^{m} b c m x^{7} + 2 \, \left (d x\right )^{m} a b m^{4} x^{3} + 130 \, \left (d x\right )^{m} b^{2} m^{2} x^{5} + 260 \, \left (d x\right )^{m} a c m^{2} x^{5} + 270 \, \left (d x\right )^{m} b c x^{7} + 44 \, \left (d x\right )^{m} a b m^{3} x^{3} + 300 \, \left (d x\right )^{m} b^{2} m x^{5} + 600 \, \left (d x\right )^{m} a c m x^{5} + \left (d x\right )^{m} a^{2} m^{4} x + 328 \, \left (d x\right )^{m} a b m^{2} x^{3} + 189 \, \left (d x\right )^{m} b^{2} x^{5} + 378 \, \left (d x\right )^{m} a c x^{5} + 24 \, \left (d x\right )^{m} a^{2} m^{3} x + 916 \, \left (d x\right )^{m} a b m x^{3} + 206 \, \left (d x\right )^{m} a^{2} m^{2} x + 630 \, \left (d x\right )^{m} a b x^{3} + 744 \, \left (d x\right )^{m} a^{2} m x + 945 \, \left (d x\right )^{m} a^{2} x}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate((d*x)^m*(c*x^4+b*x^2+a)^2,x, algorithm="giac")
Output:
((d*x)^m*c^2*m^4*x^9 + 16*(d*x)^m*c^2*m^3*x^9 + 2*(d*x)^m*b*c*m^4*x^7 + 86 *(d*x)^m*c^2*m^2*x^9 + 36*(d*x)^m*b*c*m^3*x^7 + 176*(d*x)^m*c^2*m*x^9 + (d *x)^m*b^2*m^4*x^5 + 2*(d*x)^m*a*c*m^4*x^5 + 208*(d*x)^m*b*c*m^2*x^7 + 105* (d*x)^m*c^2*x^9 + 20*(d*x)^m*b^2*m^3*x^5 + 40*(d*x)^m*a*c*m^3*x^5 + 444*(d *x)^m*b*c*m*x^7 + 2*(d*x)^m*a*b*m^4*x^3 + 130*(d*x)^m*b^2*m^2*x^5 + 260*(d *x)^m*a*c*m^2*x^5 + 270*(d*x)^m*b*c*x^7 + 44*(d*x)^m*a*b*m^3*x^3 + 300*(d* x)^m*b^2*m*x^5 + 600*(d*x)^m*a*c*m*x^5 + (d*x)^m*a^2*m^4*x + 328*(d*x)^m*a *b*m^2*x^3 + 189*(d*x)^m*b^2*x^5 + 378*(d*x)^m*a*c*x^5 + 24*(d*x)^m*a^2*m^ 3*x + 916*(d*x)^m*a*b*m*x^3 + 206*(d*x)^m*a^2*m^2*x + 630*(d*x)^m*a*b*x^3 + 744*(d*x)^m*a^2*m*x + 945*(d*x)^m*a^2*x)/(m^5 + 25*m^4 + 230*m^3 + 950*m ^2 + 1689*m + 945)
Time = 17.84 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.57 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx={\left (d\,x\right )}^m\,\left (\frac {c^2\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {x^5\,\left (b^2+2\,a\,c\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a^2\,x\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {2\,a\,b\,x^3\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {2\,b\,c\,x^7\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}\right ) \] Input:
int((d*x)^m*(a + b*x^2 + c*x^4)^2,x)
Output:
(d*x)^m*((c^2*x^9*(176*m + 86*m^2 + 16*m^3 + m^4 + 105))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (x^5*(2*a*c + b^2)*(300*m + 130*m^2 + 2 0*m^3 + m^4 + 189))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (a ^2*x*(744*m + 206*m^2 + 24*m^3 + m^4 + 945))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (2*a*b*x^3*(458*m + 164*m^2 + 22*m^3 + m^4 + 315))/ (1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (2*b*c*x^7*(222*m + 10 4*m^2 + 18*m^3 + m^4 + 135))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945))
Time = 0.18 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.98 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx=\frac {x^{m} d^{m} x \left (c^{2} m^{4} x^{8}+16 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+86 c^{2} m^{2} x^{8}+36 b c \,m^{3} x^{6}+176 c^{2} m \,x^{8}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+208 b c \,m^{2} x^{6}+105 c^{2} x^{8}+40 a c \,m^{3} x^{4}+20 b^{2} m^{3} x^{4}+444 b c m \,x^{6}+2 a b \,m^{4} x^{2}+260 a c \,m^{2} x^{4}+130 b^{2} m^{2} x^{4}+270 b c \,x^{6}+44 a b \,m^{3} x^{2}+600 a c m \,x^{4}+300 b^{2} m \,x^{4}+a^{2} m^{4}+328 a b \,m^{2} x^{2}+378 a c \,x^{4}+189 b^{2} x^{4}+24 a^{2} m^{3}+916 a b m \,x^{2}+206 a^{2} m^{2}+630 a b \,x^{2}+744 a^{2} m +945 a^{2}\right )}{m^{5}+25 m^{4}+230 m^{3}+950 m^{2}+1689 m +945} \] Input:
int((d*x)^m*(c*x^4+b*x^2+a)^2,x)
Output:
(x**m*d**m*x*(a**2*m**4 + 24*a**2*m**3 + 206*a**2*m**2 + 744*a**2*m + 945* a**2 + 2*a*b*m**4*x**2 + 44*a*b*m**3*x**2 + 328*a*b*m**2*x**2 + 916*a*b*m* x**2 + 630*a*b*x**2 + 2*a*c*m**4*x**4 + 40*a*c*m**3*x**4 + 260*a*c*m**2*x* *4 + 600*a*c*m*x**4 + 378*a*c*x**4 + b**2*m**4*x**4 + 20*b**2*m**3*x**4 + 130*b**2*m**2*x**4 + 300*b**2*m*x**4 + 189*b**2*x**4 + 2*b*c*m**4*x**6 + 3 6*b*c*m**3*x**6 + 208*b*c*m**2*x**6 + 444*b*c*m*x**6 + 270*b*c*x**6 + c**2 *m**4*x**8 + 16*c**2*m**3*x**8 + 86*c**2*m**2*x**8 + 176*c**2*m*x**8 + 105 *c**2*x**8))/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)