Integrand size = 26, antiderivative size = 104 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {a^4 (d x)^{1+m}}{d (1+m)}+\frac {4 a^3 b (d x)^{3+m}}{d^3 (3+m)}+\frac {6 a^2 b^2 (d x)^{5+m}}{d^5 (5+m)}+\frac {4 a b^3 (d x)^{7+m}}{d^7 (7+m)}+\frac {b^4 (d x)^{9+m}}{d^9 (9+m)} \] Output:
a^4*(d*x)^(1+m)/d/(1+m)+4*a^3*b*(d*x)^(3+m)/d^3/(3+m)+6*a^2*b^2*(d*x)^(5+m )/d^5/(5+m)+4*a*b^3*(d*x)^(7+m)/d^7/(7+m)+b^4*(d*x)^(9+m)/d^9/(9+m)
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=x (d x)^m \left (\frac {a^4}{1+m}+\frac {4 a^3 b x^2}{3+m}+\frac {6 a^2 b^2 x^4}{5+m}+\frac {4 a b^3 x^6}{7+m}+\frac {b^4 x^8}{9+m}\right ) \] Input:
Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
x*(d*x)^m*(a^4/(1 + m) + (4*a^3*b*x^2)/(3 + m) + (6*a^2*b^2*x^4)/(5 + m) + (4*a*b^3*x^6)/(7 + m) + (b^4*x^8)/(9 + m))
Time = 0.45 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1380, 27, 244, 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.
| \(\displaystyle \int \left (a^2+2 a b x^2+b^2 x^4\right )^2 (d x)^m \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle \frac {\int b^4 (d x)^m \left (b x^2+a\right )^4dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \left (a+b x^2\right )^4 (d x)^mdx\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \int \left (a^4 (d x)^m+\frac {4 a^3 b (d x)^{m+2}}{d^2}+\frac {6 a^2 b^2 (d x)^{m+4}}{d^4}+\frac {4 a b^3 (d x)^{m+6}}{d^6}+\frac {b^4 (d x)^{m+8}}{d^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^4 (d x)^{m+1}}{d (m+1)}+\frac {4 a^3 b (d x)^{m+3}}{d^3 (m+3)}+\frac {6 a^2 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac {4 a b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac {b^4 (d x)^{m+9}}{d^9 (m+9)}\) |
Input:
Int[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
(a^4*(d*x)^(1 + m))/(d*(1 + m)) + (4*a^3*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (6*a^2*b^2*(d*x)^(5 + m))/(d^5*(5 + m)) + (4*a*b^3*(d*x)^(7 + m))/(d^7*(7 + m)) + (b^4*(d*x)^(9 + m))/(d^9*(9 + m))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(291\) vs. \(2(104)=208\).
Time = 0.18 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.81
| method | result | size |
| gosper | \(\frac {\left (d x \right )^{m} \left (b^{4} m^{4} x^{8}+16 b^{4} m^{3} x^{8}+4 a \,b^{3} m^{4} x^{6}+86 b^{4} m^{2} x^{8}+72 a \,b^{3} m^{3} x^{6}+176 m \,x^{8} b^{4}+6 a^{2} b^{2} m^{4} x^{4}+416 a \,b^{3} m^{2} x^{6}+105 b^{4} x^{8}+120 a^{2} b^{2} m^{3} x^{4}+888 m \,x^{6} b^{3} a +4 a^{3} b \,m^{4} x^{2}+780 a^{2} b^{2} m^{2} x^{4}+540 a \,b^{3} x^{6}+88 a^{3} b \,m^{3} x^{2}+1800 m \,x^{4} a^{2} b^{2}+a^{4} m^{4}+656 a^{3} b \,m^{2} x^{2}+1134 a^{2} b^{2} x^{4}+24 a^{4} m^{3}+1832 m \,x^{2} a^{3} b +206 a^{4} m^{2}+1260 a^{3} b \,x^{2}+744 m \,a^{4}+945 a^{4}\right ) x}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(292\) |
| risch | \(\frac {\left (d x \right )^{m} \left (b^{4} m^{4} x^{8}+16 b^{4} m^{3} x^{8}+4 a \,b^{3} m^{4} x^{6}+86 b^{4} m^{2} x^{8}+72 a \,b^{3} m^{3} x^{6}+176 m \,x^{8} b^{4}+6 a^{2} b^{2} m^{4} x^{4}+416 a \,b^{3} m^{2} x^{6}+105 b^{4} x^{8}+120 a^{2} b^{2} m^{3} x^{4}+888 m \,x^{6} b^{3} a +4 a^{3} b \,m^{4} x^{2}+780 a^{2} b^{2} m^{2} x^{4}+540 a \,b^{3} x^{6}+88 a^{3} b \,m^{3} x^{2}+1800 m \,x^{4} a^{2} b^{2}+a^{4} m^{4}+656 a^{3} b \,m^{2} x^{2}+1134 a^{2} b^{2} x^{4}+24 a^{4} m^{3}+1832 m \,x^{2} a^{3} b +206 a^{4} m^{2}+1260 a^{3} b \,x^{2}+744 m \,a^{4}+945 a^{4}\right ) x}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(292\) |
| orering | \(\frac {\left (b^{4} m^{4} x^{8}+16 b^{4} m^{3} x^{8}+4 a \,b^{3} m^{4} x^{6}+86 b^{4} m^{2} x^{8}+72 a \,b^{3} m^{3} x^{6}+176 m \,x^{8} b^{4}+6 a^{2} b^{2} m^{4} x^{4}+416 a \,b^{3} m^{2} x^{6}+105 b^{4} x^{8}+120 a^{2} b^{2} m^{3} x^{4}+888 m \,x^{6} b^{3} a +4 a^{3} b \,m^{4} x^{2}+780 a^{2} b^{2} m^{2} x^{4}+540 a \,b^{3} x^{6}+88 a^{3} b \,m^{3} x^{2}+1800 m \,x^{4} a^{2} b^{2}+a^{4} m^{4}+656 a^{3} b \,m^{2} x^{2}+1134 a^{2} b^{2} x^{4}+24 a^{4} m^{3}+1832 m \,x^{2} a^{3} b +206 a^{4} m^{2}+1260 a^{3} b \,x^{2}+744 m \,a^{4}+945 a^{4}\right ) x \left (d x \right )^{m} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right ) \left (b \,x^{2}+a \right )^{4}}\) | \(321\) |
| parallelrisch | \(\frac {x \left (d x \right )^{m} a^{4} m^{4}+24 x \left (d x \right )^{m} a^{4} m^{3}+1260 x^{3} \left (d x \right )^{m} a^{3} b +206 x \left (d x \right )^{m} a^{4} m^{2}+744 x \left (d x \right )^{m} a^{4} m +540 x^{7} \left (d x \right )^{m} a \,b^{3}+x^{9} \left (d x \right )^{m} b^{4} m^{4}+16 x^{9} \left (d x \right )^{m} b^{4} m^{3}+86 x^{9} \left (d x \right )^{m} b^{4} m^{2}+1134 x^{5} \left (d x \right )^{m} a^{2} b^{2}+176 x^{9} \left (d x \right )^{m} b^{4} m +4 x^{7} \left (d x \right )^{m} a \,b^{3} m^{4}+72 x^{7} \left (d x \right )^{m} a \,b^{3} m^{3}+416 x^{7} \left (d x \right )^{m} a \,b^{3} m^{2}+6 x^{5} \left (d x \right )^{m} a^{2} b^{2} m^{4}+888 x^{7} \left (d x \right )^{m} a \,b^{3} m +120 x^{5} \left (d x \right )^{m} a^{2} b^{2} m^{3}+780 x^{5} \left (d x \right )^{m} a^{2} b^{2} m^{2}+4 x^{3} \left (d x \right )^{m} a^{3} b \,m^{4}+1800 x^{5} \left (d x \right )^{m} a^{2} b^{2} m +88 x^{3} \left (d x \right )^{m} a^{3} b \,m^{3}+656 x^{3} \left (d x \right )^{m} a^{3} b \,m^{2}+1832 x^{3} \left (d x \right )^{m} a^{3} b m +105 x^{9} \left (d x \right )^{m} b^{4}+945 x \left (d x \right )^{m} a^{4}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(416\) |
Input:
int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
(d*x)^m*(b^4*m^4*x^8+16*b^4*m^3*x^8+4*a*b^3*m^4*x^6+86*b^4*m^2*x^8+72*a*b^ 3*m^3*x^6+176*b^4*m*x^8+6*a^2*b^2*m^4*x^4+416*a*b^3*m^2*x^6+105*b^4*x^8+12 0*a^2*b^2*m^3*x^4+888*a*b^3*m*x^6+4*a^3*b*m^4*x^2+780*a^2*b^2*m^2*x^4+540* a*b^3*x^6+88*a^3*b*m^3*x^2+1800*a^2*b^2*m*x^4+a^4*m^4+656*a^3*b*m^2*x^2+11 34*a^2*b^2*x^4+24*a^4*m^3+1832*a^3*b*m*x^2+206*a^4*m^2+1260*a^3*b*x^2+744* a^4*m+945*a^4)*x/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (104) = 208\).
Time = 0.09 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.43 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {{\left ({\left (b^{4} m^{4} + 16 \, b^{4} m^{3} + 86 \, b^{4} m^{2} + 176 \, b^{4} m + 105 \, b^{4}\right )} x^{9} + 4 \, {\left (a b^{3} m^{4} + 18 \, a b^{3} m^{3} + 104 \, a b^{3} m^{2} + 222 \, a b^{3} m + 135 \, a b^{3}\right )} x^{7} + 6 \, {\left (a^{2} b^{2} m^{4} + 20 \, a^{2} b^{2} m^{3} + 130 \, a^{2} b^{2} m^{2} + 300 \, a^{2} b^{2} m + 189 \, a^{2} b^{2}\right )} x^{5} + 4 \, {\left (a^{3} b m^{4} + 22 \, a^{3} b m^{3} + 164 \, a^{3} b m^{2} + 458 \, a^{3} b m + 315 \, a^{3} b\right )} x^{3} + {\left (a^{4} m^{4} + 24 \, a^{4} m^{3} + 206 \, a^{4} m^{2} + 744 \, a^{4} m + 945 \, a^{4}\right )} x\right )} \left (d x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
Output:
((b^4*m^4 + 16*b^4*m^3 + 86*b^4*m^2 + 176*b^4*m + 105*b^4)*x^9 + 4*(a*b^3* m^4 + 18*a*b^3*m^3 + 104*a*b^3*m^2 + 222*a*b^3*m + 135*a*b^3)*x^7 + 6*(a^2 *b^2*m^4 + 20*a^2*b^2*m^3 + 130*a^2*b^2*m^2 + 300*a^2*b^2*m + 189*a^2*b^2) *x^5 + 4*(a^3*b*m^4 + 22*a^3*b*m^3 + 164*a^3*b*m^2 + 458*a^3*b*m + 315*a^3 *b)*x^3 + (a^4*m^4 + 24*a^4*m^3 + 206*a^4*m^2 + 744*a^4*m + 945*a^4)*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. 1278 vs. \(2 (94) = 188\).
Time = 0.49 (sec) , antiderivative size = 1278, normalized size of antiderivative = 12.29 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**2,x)
Output:
Piecewise(((-a**4/(8*x**8) - 2*a**3*b/(3*x**6) - 3*a**2*b**2/(2*x**4) - 2* a*b**3/x**2 + b**4*log(x))/d**9, Eq(m, -9)), ((-a**4/(6*x**6) - a**3*b/x** 4 - 3*a**2*b**2/x**2 + 4*a*b**3*log(x) + b**4*x**2/2)/d**7, Eq(m, -7)), (( -a**4/(4*x**4) - 2*a**3*b/x**2 + 6*a**2*b**2*log(x) + 2*a*b**3*x**2 + b**4 *x**4/4)/d**5, Eq(m, -5)), ((-a**4/(2*x**2) + 4*a**3*b*log(x) + 3*a**2*b** 2*x**2 + a*b**3*x**4 + b**4*x**6/6)/d**3, Eq(m, -3)), ((a**4*log(x) + 2*a* *3*b*x**2 + 3*a**2*b**2*x**4/2 + 2*a*b**3*x**6/3 + b**4*x**8/8)/d, Eq(m, - 1)), (a**4*m**4*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*a**4*m**3*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1 689*m + 945) + 206*a**4*m**2*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m **2 + 1689*m + 945) + 744*a**4*m*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 9 50*m**2 + 1689*m + 945) + 945*a**4*x*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 4*a**3*b*m**4*x**3*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 88*a**3*b*m**3*x**3*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 656*a**3*b*m**2*x**3*(d*x )**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1832*a**3*b*m *x**3*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 126 0*a**3*b*x**3*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94 5) + 6*a**2*b**2*m**4*x**5*(d*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 120*a**2*b**2*m**3*x**5*(d*x)**m/(m**5 + 25*m**4 + 23...
Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {b^{4} d^{m} x^{9} x^{m}}{m + 9} + \frac {4 \, a b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, a^{2} b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {4 \, a^{3} b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{4}}{d {\left (m + 1\right )}} \] Input:
integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
Output:
b^4*d^m*x^9*x^m/(m + 9) + 4*a*b^3*d^m*x^7*x^m/(m + 7) + 6*a^2*b^2*d^m*x^5* x^m/(m + 5) + 4*a^3*b*d^m*x^3*x^m/(m + 3) + (d*x)^(m + 1)*a^4/(d*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (104) = 208\).
Time = 0.11 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.99 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {\left (d x\right )^{m} b^{4} m^{4} x^{9} + 16 \, \left (d x\right )^{m} b^{4} m^{3} x^{9} + 4 \, \left (d x\right )^{m} a b^{3} m^{4} x^{7} + 86 \, \left (d x\right )^{m} b^{4} m^{2} x^{9} + 72 \, \left (d x\right )^{m} a b^{3} m^{3} x^{7} + 176 \, \left (d x\right )^{m} b^{4} m x^{9} + 6 \, \left (d x\right )^{m} a^{2} b^{2} m^{4} x^{5} + 416 \, \left (d x\right )^{m} a b^{3} m^{2} x^{7} + 105 \, \left (d x\right )^{m} b^{4} x^{9} + 120 \, \left (d x\right )^{m} a^{2} b^{2} m^{3} x^{5} + 888 \, \left (d x\right )^{m} a b^{3} m x^{7} + 4 \, \left (d x\right )^{m} a^{3} b m^{4} x^{3} + 780 \, \left (d x\right )^{m} a^{2} b^{2} m^{2} x^{5} + 540 \, \left (d x\right )^{m} a b^{3} x^{7} + 88 \, \left (d x\right )^{m} a^{3} b m^{3} x^{3} + 1800 \, \left (d x\right )^{m} a^{2} b^{2} m x^{5} + \left (d x\right )^{m} a^{4} m^{4} x + 656 \, \left (d x\right )^{m} a^{3} b m^{2} x^{3} + 1134 \, \left (d x\right )^{m} a^{2} b^{2} x^{5} + 24 \, \left (d x\right )^{m} a^{4} m^{3} x + 1832 \, \left (d x\right )^{m} a^{3} b m x^{3} + 206 \, \left (d x\right )^{m} a^{4} m^{2} x + 1260 \, \left (d x\right )^{m} a^{3} b x^{3} + 744 \, \left (d x\right )^{m} a^{4} m x + 945 \, \left (d x\right )^{m} a^{4} x}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
Output:
((d*x)^m*b^4*m^4*x^9 + 16*(d*x)^m*b^4*m^3*x^9 + 4*(d*x)^m*a*b^3*m^4*x^7 + 86*(d*x)^m*b^4*m^2*x^9 + 72*(d*x)^m*a*b^3*m^3*x^7 + 176*(d*x)^m*b^4*m*x^9 + 6*(d*x)^m*a^2*b^2*m^4*x^5 + 416*(d*x)^m*a*b^3*m^2*x^7 + 105*(d*x)^m*b^4* x^9 + 120*(d*x)^m*a^2*b^2*m^3*x^5 + 888*(d*x)^m*a*b^3*m*x^7 + 4*(d*x)^m*a^ 3*b*m^4*x^3 + 780*(d*x)^m*a^2*b^2*m^2*x^5 + 540*(d*x)^m*a*b^3*x^7 + 88*(d* x)^m*a^3*b*m^3*x^3 + 1800*(d*x)^m*a^2*b^2*m*x^5 + (d*x)^m*a^4*m^4*x + 656* (d*x)^m*a^3*b*m^2*x^3 + 1134*(d*x)^m*a^2*b^2*x^5 + 24*(d*x)^m*a^4*m^3*x + 1832*(d*x)^m*a^3*b*m*x^3 + 206*(d*x)^m*a^4*m^2*x + 1260*(d*x)^m*a^3*b*x^3 + 744*(d*x)^m*a^4*m*x + 945*(d*x)^m*a^4*x)/(m^5 + 25*m^4 + 230*m^3 + 950*m ^2 + 1689*m + 945)
Time = 18.42 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.53 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx={\left (d\,x\right )}^m\,\left (\frac {b^4\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a^4\,x\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {4\,a\,b^3\,x^7\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {4\,a^3\,b\,x^3\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {6\,a^2\,b^2\,x^5\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}\right ) \] Input:
int((d*x)^m*(a^2 + b^2*x^4 + 2*a*b*x^2)^2,x)
Output:
(d*x)^m*((b^4*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) + (a^4*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) + (4*a*b^3*x^7*( 222*m + 104*m^2 + 18*m^3 + m^4 + 135))/(1689*m + 950*m^2 + 230*m^3 + 25*m^ 4 + m^5 + 945) + (4*a^3*b*x^3*(458*m + 164*m^2 + 22*m^3 + m^4 + 315))/(168 9*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (6*a^2*b^2*x^5*(300*m + 13 0*m^2 + 20*m^3 + m^4 + 189))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945))
Time = 0.16 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.81 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx=\frac {x^{m} d^{m} x \left (b^{4} m^{4} x^{8}+16 b^{4} m^{3} x^{8}+4 a \,b^{3} m^{4} x^{6}+86 b^{4} m^{2} x^{8}+72 a \,b^{3} m^{3} x^{6}+176 b^{4} m \,x^{8}+6 a^{2} b^{2} m^{4} x^{4}+416 a \,b^{3} m^{2} x^{6}+105 b^{4} x^{8}+120 a^{2} b^{2} m^{3} x^{4}+888 a \,b^{3} m \,x^{6}+4 a^{3} b \,m^{4} x^{2}+780 a^{2} b^{2} m^{2} x^{4}+540 a \,b^{3} x^{6}+88 a^{3} b \,m^{3} x^{2}+1800 a^{2} b^{2} m \,x^{4}+a^{4} m^{4}+656 a^{3} b \,m^{2} x^{2}+1134 a^{2} b^{2} x^{4}+24 a^{4} m^{3}+1832 a^{3} b m \,x^{2}+206 a^{4} m^{2}+1260 a^{3} b \,x^{2}+744 a^{4} m +945 a^{4}\right )}{m^{5}+25 m^{4}+230 m^{3}+950 m^{2}+1689 m +945} \] Input:
int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^2,x)
Output:
(x**m*d**m*x*(a**4*m**4 + 24*a**4*m**3 + 206*a**4*m**2 + 744*a**4*m + 945* a**4 + 4*a**3*b*m**4*x**2 + 88*a**3*b*m**3*x**2 + 656*a**3*b*m**2*x**2 + 1 832*a**3*b*m*x**2 + 1260*a**3*b*x**2 + 6*a**2*b**2*m**4*x**4 + 120*a**2*b* *2*m**3*x**4 + 780*a**2*b**2*m**2*x**4 + 1800*a**2*b**2*m*x**4 + 1134*a**2 *b**2*x**4 + 4*a*b**3*m**4*x**6 + 72*a*b**3*m**3*x**6 + 416*a*b**3*m**2*x* *6 + 888*a*b**3*m*x**6 + 540*a*b**3*x**6 + b**4*m**4*x**8 + 16*b**4*m**3*x **8 + 86*b**4*m**2*x**8 + 176*b**4*m*x**8 + 105*b**4*x**8))/(m**5 + 25*m** 4 + 230*m**3 + 950*m**2 + 1689*m + 945)