Integrand size = 22, antiderivative size = 101 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {a^2 (d x)^{3+m}}{d^3 (3+m)}+\frac {2 a b (d x)^{5+m}}{d^5 (5+m)}+\frac {\left (b^2+2 a c\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {2 b c (d x)^{9+m}}{d^9 (9+m)}+\frac {c^2 (d x)^{11+m}}{d^{11} (11+m)} \] Output:
a^2*(d*x)^(3+m)/d^3/(3+m)+2*a*b*(d*x)^(5+m)/d^5/(5+m)+(2*a*c+b^2)*(d*x)^(7 +m)/d^7/(7+m)+2*b*c*(d*x)^(9+m)/d^9/(9+m)+c^2*(d*x)^(11+m)/d^11/(11+m)
Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.44 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {x^3 (d x)^m \left (a^2 \left (3465+1888 m+374 m^2+32 m^3+m^4\right )+2 a \left (297+159 m+23 m^2+m^3\right ) x^2 \left (b (7+m)+c (5+m) x^2\right )+\left (15+8 m+m^2\right ) x^4 \left (b^2 \left (99+20 m+m^2\right )+2 b c \left (77+18 m+m^2\right ) x^2+c^2 \left (63+16 m+m^2\right ) x^4\right )\right )}{(3+m) (5+m) (7+m) (9+m) (11+m)} \] Input:
Integrate[(d*x)^m*(a*x + b*x^3 + c*x^5)^2,x]
Output:
(x^3*(d*x)^m*(a^2*(3465 + 1888*m + 374*m^2 + 32*m^3 + m^4) + 2*a*(297 + 15 9*m + 23*m^2 + m^3)*x^2*(b*(7 + m) + c*(5 + m)*x^2) + (15 + 8*m + m^2)*x^4 *(b^2*(99 + 20*m + m^2) + 2*b*c*(77 + 18*m + m^2)*x^2 + c^2*(63 + 16*m + m ^2)*x^4)))/((3 + m)*(5 + m)*(7 + m)*(9 + m)*(11 + m))
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {9, 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.
| \(\displaystyle \int (d x)^m \left (a x+b x^3+c x^5\right )^2 \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\int (d x)^{m+2} \left (c x^4+b x^2+a\right )^2dx}{d^2}\) |
| \(\Big \downarrow \) 1433 |
| \(\displaystyle \frac {\int \left (a^2 (d x)^{m+2}+\frac {2 a b (d x)^{m+4}}{d^2}+\frac {\left (b^2+2 a c\right ) (d x)^{m+6}}{d^4}+\frac {2 b c (d x)^{m+8}}{d^6}+\frac {c^2 (d x)^{m+10}}{d^8}\right )dx}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^2 (d x)^{m+3}}{d (m+3)}+\frac {\left (2 a c+b^2\right ) (d x)^{m+7}}{d^5 (m+7)}+\frac {2 a b (d x)^{m+5}}{d^3 (m+5)}+\frac {2 b c (d x)^{m+9}}{d^7 (m+9)}+\frac {c^2 (d x)^{m+11}}{d^9 (m+11)}}{d^2}\) |
Input:
Int[(d*x)^m*(a*x + b*x^3 + c*x^5)^2,x]
Output:
((a^2*(d*x)^(3 + m))/(d*(3 + m)) + (2*a*b*(d*x)^(5 + m))/(d^3*(5 + m)) + ( (b^2 + 2*a*c)*(d*x)^(7 + m))/(d^5*(7 + m)) + (2*b*c*(d*x)^(9 + m))/(d^7*(9 + m)) + (c^2*(d*x)^(11 + m))/(d^9*(11 + m)))/d^2
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
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. \(302\) vs. \(2(101)=202\).
Time = 0.17 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.00
| method | result | size |
| gosper | \(\frac {\left (d x \right )^{m} \left (c^{2} m^{4} x^{8}+24 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+206 c^{2} m^{2} x^{8}+52 b c \,m^{3} x^{6}+744 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+472 b c \,m^{2} x^{6}+945 x^{8} c^{2}+56 a c \,m^{3} x^{4}+28 b^{2} m^{3} x^{4}+1772 m \,x^{6} b c +2 a b \,m^{4} x^{2}+548 a c \,m^{2} x^{4}+274 b^{2} m^{2} x^{4}+2310 b c \,x^{6}+60 a b \,m^{3} x^{2}+2184 a c m \,x^{4}+1092 b^{2} m \,x^{4}+a^{2} m^{4}+640 a b \,m^{2} x^{2}+2970 x^{4} a c +1485 b^{2} x^{4}+32 a^{2} m^{3}+2820 a b m \,x^{2}+374 a^{2} m^{2}+4158 a b \,x^{2}+1888 a^{2} m +3465 a^{2}\right ) x^{3}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (m +5\right ) \left (3+m \right )}\) | \(303\) |
| risch | \(\frac {\left (d x \right )^{m} \left (c^{2} m^{4} x^{8}+24 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+206 c^{2} m^{2} x^{8}+52 b c \,m^{3} x^{6}+744 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+472 b c \,m^{2} x^{6}+945 x^{8} c^{2}+56 a c \,m^{3} x^{4}+28 b^{2} m^{3} x^{4}+1772 m \,x^{6} b c +2 a b \,m^{4} x^{2}+548 a c \,m^{2} x^{4}+274 b^{2} m^{2} x^{4}+2310 b c \,x^{6}+60 a b \,m^{3} x^{2}+2184 a c m \,x^{4}+1092 b^{2} m \,x^{4}+a^{2} m^{4}+640 a b \,m^{2} x^{2}+2970 x^{4} a c +1485 b^{2} x^{4}+32 a^{2} m^{3}+2820 a b m \,x^{2}+374 a^{2} m^{2}+4158 a b \,x^{2}+1888 a^{2} m +3465 a^{2}\right ) x^{3}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (m +5\right ) \left (3+m \right )}\) | \(303\) |
| orering | \(\frac {\left (c^{2} m^{4} x^{8}+24 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+206 c^{2} m^{2} x^{8}+52 b c \,m^{3} x^{6}+744 m \,x^{8} c^{2}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+472 b c \,m^{2} x^{6}+945 x^{8} c^{2}+56 a c \,m^{3} x^{4}+28 b^{2} m^{3} x^{4}+1772 m \,x^{6} b c +2 a b \,m^{4} x^{2}+548 a c \,m^{2} x^{4}+274 b^{2} m^{2} x^{4}+2310 b c \,x^{6}+60 a b \,m^{3} x^{2}+2184 a c m \,x^{4}+1092 b^{2} m \,x^{4}+a^{2} m^{4}+640 a b \,m^{2} x^{2}+2970 x^{4} a c +1485 b^{2} x^{4}+32 a^{2} m^{3}+2820 a b m \,x^{2}+374 a^{2} m^{2}+4158 a b \,x^{2}+1888 a^{2} m +3465 a^{2}\right ) x \left (d x \right )^{m} \left (c \,x^{5}+b \,x^{3}+x a \right )^{2}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (m +5\right ) \left (3+m \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}\) | \(331\) |
| parallelrisch | \(\frac {2 x^{7} \left (d x \right )^{m} a c \,m^{4}+1772 x^{9} \left (d x \right )^{m} b c m +56 x^{7} \left (d x \right )^{m} a c \,m^{3}+548 x^{7} \left (d x \right )^{m} a c \,m^{2}+2 x^{5} \left (d x \right )^{m} a b \,m^{4}+2184 x^{7} \left (d x \right )^{m} a c m +60 x^{5} \left (d x \right )^{m} a b \,m^{3}+640 x^{5} \left (d x \right )^{m} a b \,m^{2}+2820 x^{5} \left (d x \right )^{m} a b m +206 x^{11} \left (d x \right )^{m} c^{2} m^{2}+744 x^{11} \left (d x \right )^{m} c^{2} m +x^{7} \left (d x \right )^{m} b^{2} m^{4}+28 x^{7} \left (d x \right )^{m} b^{2} m^{3}+2310 x^{9} \left (d x \right )^{m} b c +274 x^{7} \left (d x \right )^{m} b^{2} m^{2}+1092 x^{7} \left (d x \right )^{m} b^{2} m +2970 x^{7} \left (d x \right )^{m} a c +x^{3} \left (d x \right )^{m} a^{2} m^{4}+32 x^{3} \left (d x \right )^{m} a^{2} m^{3}+4158 x^{5} \left (d x \right )^{m} a b +374 x^{3} \left (d x \right )^{m} a^{2} m^{2}+1888 x^{3} \left (d x \right )^{m} a^{2} m +x^{11} \left (d x \right )^{m} c^{2} m^{4}+24 x^{11} \left (d x \right )^{m} c^{2} m^{3}+2 x^{9} \left (d x \right )^{m} b c \,m^{4}+52 x^{9} \left (d x \right )^{m} b c \,m^{3}+472 x^{9} \left (d x \right )^{m} b c \,m^{2}+1485 x^{7} \left (d x \right )^{m} b^{2}+945 x^{11} \left (d x \right )^{m} c^{2}+3465 x^{3} \left (d x \right )^{m} a^{2}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (m +5\right ) \left (3+m \right )}\) | \(460\) |
Input:
int((d*x)^m*(c*x^5+b*x^3+a*x)^2,x,method=_RETURNVERBOSE)
Output:
(d*x)^m*(c^2*m^4*x^8+24*c^2*m^3*x^8+2*b*c*m^4*x^6+206*c^2*m^2*x^8+52*b*c*m ^3*x^6+744*c^2*m*x^8+2*a*c*m^4*x^4+b^2*m^4*x^4+472*b*c*m^2*x^6+945*c^2*x^8 +56*a*c*m^3*x^4+28*b^2*m^3*x^4+1772*b*c*m*x^6+2*a*b*m^4*x^2+548*a*c*m^2*x^ 4+274*b^2*m^2*x^4+2310*b*c*x^6+60*a*b*m^3*x^2+2184*a*c*m*x^4+1092*b^2*m*x^ 4+a^2*m^4+640*a*b*m^2*x^2+2970*a*c*x^4+1485*b^2*x^4+32*a^2*m^3+2820*a*b*m* x^2+374*a^2*m^2+4158*a*b*x^2+1888*a^2*m+3465*a^2)*x^3/(11+m)/(9+m)/(7+m)/( m+5)/(3+m)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (101) = 202\).
Time = 0.08 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.41 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {{\left ({\left (c^{2} m^{4} + 24 \, c^{2} m^{3} + 206 \, c^{2} m^{2} + 744 \, c^{2} m + 945 \, c^{2}\right )} x^{11} + 2 \, {\left (b c m^{4} + 26 \, b c m^{3} + 236 \, b c m^{2} + 886 \, b c m + 1155 \, b c\right )} x^{9} + {\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 28 \, {\left (b^{2} + 2 \, a c\right )} m^{3} + 274 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + 1485 \, b^{2} + 2970 \, a c + 1092 \, {\left (b^{2} + 2 \, a c\right )} m\right )} x^{7} + 2 \, {\left (a b m^{4} + 30 \, a b m^{3} + 320 \, a b m^{2} + 1410 \, a b m + 2079 \, a b\right )} x^{5} + {\left (a^{2} m^{4} + 32 \, a^{2} m^{3} + 374 \, a^{2} m^{2} + 1888 \, a^{2} m + 3465 \, a^{2}\right )} x^{3}\right )} \left (d x\right )^{m}}{m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395} \] Input:
integrate((d*x)^m*(c*x^5+b*x^3+a*x)^2,x, algorithm="fricas")
Output:
((c^2*m^4 + 24*c^2*m^3 + 206*c^2*m^2 + 744*c^2*m + 945*c^2)*x^11 + 2*(b*c* m^4 + 26*b*c*m^3 + 236*b*c*m^2 + 886*b*c*m + 1155*b*c)*x^9 + ((b^2 + 2*a*c )*m^4 + 28*(b^2 + 2*a*c)*m^3 + 274*(b^2 + 2*a*c)*m^2 + 1485*b^2 + 2970*a*c + 1092*(b^2 + 2*a*c)*m)*x^7 + 2*(a*b*m^4 + 30*a*b*m^3 + 320*a*b*m^2 + 141 0*a*b*m + 2079*a*b)*x^5 + (a^2*m^4 + 32*a^2*m^3 + 374*a^2*m^2 + 1888*a^2*m + 3465*a^2)*x^3)*(d*x)^m/(m^5 + 35*m^4 + 470*m^3 + 3010*m^2 + 9129*m + 10 395)
Leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (92) = 184\).
Time = 0.88 (sec) , antiderivative size = 1445, normalized size of antiderivative = 14.31 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((d*x)**m*(c*x**5+b*x**3+a*x)**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**11, Eq(m, -11)), ((-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**9, Eq(m, -9 )), ((-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**7, Eq(m, -7)), ((-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**5, Eq(m, -5)), ((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**3, Eq (m, -3)), (a**2*m**4*x**3*(d*x)**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 32*a**2*m**3*x**3*(d*x)**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 374*a**2*m**2*x**3*(d*x)**m/(m**5 + 35*m** 4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 1888*a**2*m*x**3*(d*x)**m/(m* *5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 3465*a**2*x**3*(d* x)**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 2*a*b*m** 4*x**5*(d*x)**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 60*a*b*m**3*x**5*(d*x)**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 640*a*b*m**2*x**5*(d*x)**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m **2 + 9129*m + 10395) + 2820*a*b*m*x**5*(d*x)**m/(m**5 + 35*m**4 + 470*m** 3 + 3010*m**2 + 9129*m + 10395) + 4158*a*b*x**5*(d*x)**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 2*a*c*m**4*x**7*(d*x)**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 56*a*c*m**3*x**7*(d...
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {2 \, b c d^{m} x^{9} x^{m}}{m + 9} + \frac {b^{2} d^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, a c d^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, a b d^{m} x^{5} x^{m}}{m + 5} + \frac {a^{2} d^{m} x^{3} x^{m}}{m + 3} \] Input:
integrate((d*x)^m*(c*x^5+b*x^3+a*x)^2,x, algorithm="maxima")
Output:
c^2*d^m*x^11*x^m/(m + 11) + 2*b*c*d^m*x^9*x^m/(m + 9) + b^2*d^m*x^7*x^m/(m + 7) + 2*a*c*d^m*x^7*x^m/(m + 7) + 2*a*b*d^m*x^5*x^m/(m + 5) + a^2*d^m*x^ 3*x^m/(m + 3)
Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (101) = 202\).
Time = 0.13 (sec) , antiderivative size = 459, normalized size of antiderivative = 4.54 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {\left (d x\right )^{m} c^{2} m^{4} x^{11} + 24 \, \left (d x\right )^{m} c^{2} m^{3} x^{11} + 2 \, \left (d x\right )^{m} b c m^{4} x^{9} + 206 \, \left (d x\right )^{m} c^{2} m^{2} x^{11} + 52 \, \left (d x\right )^{m} b c m^{3} x^{9} + 744 \, \left (d x\right )^{m} c^{2} m x^{11} + \left (d x\right )^{m} b^{2} m^{4} x^{7} + 2 \, \left (d x\right )^{m} a c m^{4} x^{7} + 472 \, \left (d x\right )^{m} b c m^{2} x^{9} + 945 \, \left (d x\right )^{m} c^{2} x^{11} + 28 \, \left (d x\right )^{m} b^{2} m^{3} x^{7} + 56 \, \left (d x\right )^{m} a c m^{3} x^{7} + 1772 \, \left (d x\right )^{m} b c m x^{9} + 2 \, \left (d x\right )^{m} a b m^{4} x^{5} + 274 \, \left (d x\right )^{m} b^{2} m^{2} x^{7} + 548 \, \left (d x\right )^{m} a c m^{2} x^{7} + 2310 \, \left (d x\right )^{m} b c x^{9} + 60 \, \left (d x\right )^{m} a b m^{3} x^{5} + 1092 \, \left (d x\right )^{m} b^{2} m x^{7} + 2184 \, \left (d x\right )^{m} a c m x^{7} + \left (d x\right )^{m} a^{2} m^{4} x^{3} + 640 \, \left (d x\right )^{m} a b m^{2} x^{5} + 1485 \, \left (d x\right )^{m} b^{2} x^{7} + 2970 \, \left (d x\right )^{m} a c x^{7} + 32 \, \left (d x\right )^{m} a^{2} m^{3} x^{3} + 2820 \, \left (d x\right )^{m} a b m x^{5} + 374 \, \left (d x\right )^{m} a^{2} m^{2} x^{3} + 4158 \, \left (d x\right )^{m} a b x^{5} + 1888 \, \left (d x\right )^{m} a^{2} m x^{3} + 3465 \, \left (d x\right )^{m} a^{2} x^{3}}{m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395} \] Input:
integrate((d*x)^m*(c*x^5+b*x^3+a*x)^2,x, algorithm="giac")
Output:
((d*x)^m*c^2*m^4*x^11 + 24*(d*x)^m*c^2*m^3*x^11 + 2*(d*x)^m*b*c*m^4*x^9 + 206*(d*x)^m*c^2*m^2*x^11 + 52*(d*x)^m*b*c*m^3*x^9 + 744*(d*x)^m*c^2*m*x^11 + (d*x)^m*b^2*m^4*x^7 + 2*(d*x)^m*a*c*m^4*x^7 + 472*(d*x)^m*b*c*m^2*x^9 + 945*(d*x)^m*c^2*x^11 + 28*(d*x)^m*b^2*m^3*x^7 + 56*(d*x)^m*a*c*m^3*x^7 + 1772*(d*x)^m*b*c*m*x^9 + 2*(d*x)^m*a*b*m^4*x^5 + 274*(d*x)^m*b^2*m^2*x^7 + 548*(d*x)^m*a*c*m^2*x^7 + 2310*(d*x)^m*b*c*x^9 + 60*(d*x)^m*a*b*m^3*x^5 + 1092*(d*x)^m*b^2*m*x^7 + 2184*(d*x)^m*a*c*m*x^7 + (d*x)^m*a^2*m^4*x^3 + 6 40*(d*x)^m*a*b*m^2*x^5 + 1485*(d*x)^m*b^2*x^7 + 2970*(d*x)^m*a*c*x^7 + 32* (d*x)^m*a^2*m^3*x^3 + 2820*(d*x)^m*a*b*m*x^5 + 374*(d*x)^m*a^2*m^2*x^3 + 4 158*(d*x)^m*a*b*x^5 + 1888*(d*x)^m*a^2*m*x^3 + 3465*(d*x)^m*a^2*x^3)/(m^5 + 35*m^4 + 470*m^3 + 3010*m^2 + 9129*m + 10395)
Time = 12.59 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.59 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^2 \, dx={\left (d\,x\right )}^m\,\left (\frac {a^2\,x^3\,\left (m^4+32\,m^3+374\,m^2+1888\,m+3465\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {c^2\,x^{11}\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {x^7\,\left (b^2+2\,a\,c\right )\,\left (m^4+28\,m^3+274\,m^2+1092\,m+1485\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {2\,a\,b\,x^5\,\left (m^4+30\,m^3+320\,m^2+1410\,m+2079\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {2\,b\,c\,x^9\,\left (m^4+26\,m^3+236\,m^2+886\,m+1155\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}\right ) \] Input:
int((d*x)^m*(a*x + b*x^3 + c*x^5)^2,x)
Output:
(d*x)^m*((a^2*x^3*(1888*m + 374*m^2 + 32*m^3 + m^4 + 3465))/(9129*m + 3010 *m^2 + 470*m^3 + 35*m^4 + m^5 + 10395) + (c^2*x^11*(744*m + 206*m^2 + 24*m ^3 + m^4 + 945))/(9129*m + 3010*m^2 + 470*m^3 + 35*m^4 + m^5 + 10395) + (x ^7*(2*a*c + b^2)*(1092*m + 274*m^2 + 28*m^3 + m^4 + 1485))/(9129*m + 3010* m^2 + 470*m^3 + 35*m^4 + m^5 + 10395) + (2*a*b*x^5*(1410*m + 320*m^2 + 30* m^3 + m^4 + 2079))/(9129*m + 3010*m^2 + 470*m^3 + 35*m^4 + m^5 + 10395) + (2*b*c*x^9*(886*m + 236*m^2 + 26*m^3 + m^4 + 1155))/(9129*m + 3010*m^2 + 4 70*m^3 + 35*m^4 + m^5 + 10395))
Time = 0.19 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.00 \[ \int (d x)^m \left (a x+b x^3+c x^5\right )^2 \, dx=\frac {x^{m} d^{m} x^{3} \left (c^{2} m^{4} x^{8}+24 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+206 c^{2} m^{2} x^{8}+52 b c \,m^{3} x^{6}+744 c^{2} m \,x^{8}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+472 b c \,m^{2} x^{6}+945 c^{2} x^{8}+56 a c \,m^{3} x^{4}+28 b^{2} m^{3} x^{4}+1772 b c m \,x^{6}+2 a b \,m^{4} x^{2}+548 a c \,m^{2} x^{4}+274 b^{2} m^{2} x^{4}+2310 b c \,x^{6}+60 a b \,m^{3} x^{2}+2184 a c m \,x^{4}+1092 b^{2} m \,x^{4}+a^{2} m^{4}+640 a b \,m^{2} x^{2}+2970 a c \,x^{4}+1485 b^{2} x^{4}+32 a^{2} m^{3}+2820 a b m \,x^{2}+374 a^{2} m^{2}+4158 a b \,x^{2}+1888 a^{2} m +3465 a^{2}\right )}{m^{5}+35 m^{4}+470 m^{3}+3010 m^{2}+9129 m +10395} \] Input:
int((d*x)^m*(c*x^5+b*x^3+a*x)^2,x)
Output:
(x**m*d**m*x**3*(a**2*m**4 + 32*a**2*m**3 + 374*a**2*m**2 + 1888*a**2*m + 3465*a**2 + 2*a*b*m**4*x**2 + 60*a*b*m**3*x**2 + 640*a*b*m**2*x**2 + 2820* a*b*m*x**2 + 4158*a*b*x**2 + 2*a*c*m**4*x**4 + 56*a*c*m**3*x**4 + 548*a*c* m**2*x**4 + 2184*a*c*m*x**4 + 2970*a*c*x**4 + b**2*m**4*x**4 + 28*b**2*m** 3*x**4 + 274*b**2*m**2*x**4 + 1092*b**2*m*x**4 + 1485*b**2*x**4 + 2*b*c*m* *4*x**6 + 52*b*c*m**3*x**6 + 472*b*c*m**2*x**6 + 1772*b*c*m*x**6 + 2310*b* c*x**6 + c**2*m**4*x**8 + 24*c**2*m**3*x**8 + 206*c**2*m**2*x**8 + 744*c** 2*m*x**8 + 945*c**2*x**8))/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395)