Integrand size = 30, antiderivative size = 106 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {a A (c x)^{1+m}}{c (1+m)}+\frac {(A b+a B) (c x)^{3+m}}{c^3 (3+m)}+\frac {(b B+a C) (c x)^{5+m}}{c^5 (5+m)}+\frac {(b C+a D) (c x)^{7+m}}{c^7 (7+m)}+\frac {b D (c x)^{9+m}}{c^9 (9+m)} \] Output:
a*A*(c*x)^(1+m)/c/(1+m)+(A*b+B*a)*(c*x)^(3+m)/c^3/(3+m)+(B*b+C*a)*(c*x)^(5 +m)/c^5/(5+m)+(C*b+D*a)*(c*x)^(7+m)/c^7/(7+m)+b*D*(c*x)^(9+m)/c^9/(9+m)
Time = 0.51 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x^2+C x^4+D x^6\right ) \, dx=x (c x)^m \left (\frac {a A}{1+m}+\frac {(A b+a B) x^2}{3+m}+\frac {(b B+a C) x^4}{5+m}+\frac {(b C+a D) x^6}{7+m}+\frac {b D x^8}{9+m}\right ) \] Input:
Integrate[(c*x)^m*(a + b*x^2)*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
x*(c*x)^m*((a*A)/(1 + m) + ((A*b + a*B)*x^2)/(3 + m) + ((b*B + a*C)*x^4)/( 5 + m) + ((b*C + a*D)*x^6)/(7 + m) + (b*D*x^8)/(9 + m))
Time = 0.33 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2333, 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.
| \(\displaystyle \int \left (a+b x^2\right ) (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \int \left (\frac {(c x)^{m+2} (a B+A b)}{c^2}+a A (c x)^m+\frac {(c x)^{m+4} (a C+b B)}{c^4}+\frac {(c x)^{m+6} (a D+b C)}{c^6}+\frac {b D (c x)^{m+8}}{c^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(c x)^{m+3} (a B+A b)}{c^3 (m+3)}+\frac {a A (c x)^{m+1}}{c (m+1)}+\frac {(c x)^{m+5} (a C+b B)}{c^5 (m+5)}+\frac {(c x)^{m+7} (a D+b C)}{c^7 (m+7)}+\frac {b D (c x)^{m+9}}{c^9 (m+9)}\) |
Input:
Int[(c*x)^m*(a + b*x^2)*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
(a*A*(c*x)^(1 + m))/(c*(1 + m)) + ((A*b + a*B)*(c*x)^(3 + m))/(c^3*(3 + m) ) + ((b*B + a*C)*(c*x)^(5 + m))/(c^5*(5 + m)) + ((b*C + a*D)*(c*x)^(7 + m) )/(c^7*(7 + m)) + (b*D*(c*x)^(9 + m))/(c^9*(9 + m))
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(106)=212\).
Time = 0.12 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.50
| method | result | size |
| gosper | \(\frac {x \left (D b \,m^{4} x^{8}+16 D b \,m^{3} x^{8}+C b \,m^{4} x^{6}+D a \,m^{4} x^{6}+86 D b \,m^{2} x^{8}+18 C b \,m^{3} x^{6}+18 D a \,m^{3} x^{6}+176 D b m \,x^{8}+B b \,m^{4} x^{4}+C a \,m^{4} x^{4}+104 C b \,m^{2} x^{6}+104 D a \,m^{2} x^{6}+105 D b \,x^{8}+20 B b \,m^{3} x^{4}+20 C a \,m^{3} x^{4}+222 C b m \,x^{6}+222 D a m \,x^{6}+A b \,m^{4} x^{2}+B a \,m^{4} x^{2}+130 B b \,m^{2} x^{4}+130 C a \,m^{2} x^{4}+135 C b \,x^{6}+135 D a \,x^{6}+22 A b \,m^{3} x^{2}+22 B a \,m^{3} x^{2}+300 B b m \,x^{4}+300 C a m \,x^{4}+A a \,m^{4}+164 A b \,m^{2} x^{2}+164 B a \,m^{2} x^{2}+189 b B \,x^{4}+189 C a \,x^{4}+24 A a \,m^{3}+458 A b m \,x^{2}+458 B a m \,x^{2}+206 A a \,m^{2}+315 A b \,x^{2}+315 B a \,x^{2}+744 A a m +945 A a \right ) \left (c x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(371\) |
| orering | \(\frac {x \left (D b \,m^{4} x^{8}+16 D b \,m^{3} x^{8}+C b \,m^{4} x^{6}+D a \,m^{4} x^{6}+86 D b \,m^{2} x^{8}+18 C b \,m^{3} x^{6}+18 D a \,m^{3} x^{6}+176 D b m \,x^{8}+B b \,m^{4} x^{4}+C a \,m^{4} x^{4}+104 C b \,m^{2} x^{6}+104 D a \,m^{2} x^{6}+105 D b \,x^{8}+20 B b \,m^{3} x^{4}+20 C a \,m^{3} x^{4}+222 C b m \,x^{6}+222 D a m \,x^{6}+A b \,m^{4} x^{2}+B a \,m^{4} x^{2}+130 B b \,m^{2} x^{4}+130 C a \,m^{2} x^{4}+135 C b \,x^{6}+135 D a \,x^{6}+22 A b \,m^{3} x^{2}+22 B a \,m^{3} x^{2}+300 B b m \,x^{4}+300 C a m \,x^{4}+A a \,m^{4}+164 A b \,m^{2} x^{2}+164 B a \,m^{2} x^{2}+189 b B \,x^{4}+189 C a \,x^{4}+24 A a \,m^{3}+458 A b m \,x^{2}+458 B a m \,x^{2}+206 A a \,m^{2}+315 A b \,x^{2}+315 B a \,x^{2}+744 A a m +945 A a \right ) \left (c x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(371\) |
| parallelrisch | \(\frac {105 D x^{9} \left (c x \right )^{m} b +135 C \,x^{7} \left (c x \right )^{m} b +135 D x^{7} \left (c x \right )^{m} a +189 B \,x^{5} \left (c x \right )^{m} b +189 C \,x^{5} \left (c x \right )^{m} a +315 A \,x^{3} \left (c x \right )^{m} b +315 B \,x^{3} \left (c x \right )^{m} a +945 A x \left (c x \right )^{m} a +164 A \,x^{3} \left (c x \right )^{m} b \,m^{2}+A x \left (c x \right )^{m} a \,m^{4}+164 B \,x^{3} \left (c x \right )^{m} a \,m^{2}+458 A \,x^{3} \left (c x \right )^{m} b m +24 A x \left (c x \right )^{m} a \,m^{3}+D x^{9} \left (c x \right )^{m} b \,m^{4}+B \,x^{5} \left (c x \right )^{m} b \,m^{4}+104 C \,x^{7} \left (c x \right )^{m} b \,m^{2}+C \,x^{5} \left (c x \right )^{m} a \,m^{4}+104 D x^{7} \left (c x \right )^{m} a \,m^{2}+20 B \,x^{5} \left (c x \right )^{m} b \,m^{3}+222 C \,x^{7} \left (c x \right )^{m} b m +20 C \,x^{5} \left (c x \right )^{m} a \,m^{3}+16 D x^{9} \left (c x \right )^{m} b \,m^{3}+C \,x^{7} \left (c x \right )^{m} b \,m^{4}+86 D x^{9} \left (c x \right )^{m} b \,m^{2}+D x^{7} \left (c x \right )^{m} a \,m^{4}+18 C \,x^{7} \left (c x \right )^{m} b \,m^{3}+176 D x^{9} \left (c x \right )^{m} b m +18 D x^{7} \left (c x \right )^{m} a \,m^{3}+222 D x^{7} \left (c x \right )^{m} a m +A \,x^{3} \left (c x \right )^{m} b \,m^{4}+130 B \,x^{5} \left (c x \right )^{m} b \,m^{2}+B \,x^{3} \left (c x \right )^{m} a \,m^{4}+130 C \,x^{5} \left (c x \right )^{m} a \,m^{2}+22 A \,x^{3} \left (c x \right )^{m} b \,m^{3}+300 B \,x^{5} \left (c x \right )^{m} b m +22 B \,x^{3} \left (c x \right )^{m} a \,m^{3}+300 C \,x^{5} \left (c x \right )^{m} a m +458 B \,x^{3} \left (c x \right )^{m} a m +206 A x \left (c x \right )^{m} a \,m^{2}+744 A x \left (c x \right )^{m} a m}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(570\) |
Input:
int((c*x)^m*(b*x^2+a)*(D*x^6+C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
x*(D*b*m^4*x^8+16*D*b*m^3*x^8+C*b*m^4*x^6+D*a*m^4*x^6+86*D*b*m^2*x^8+18*C* b*m^3*x^6+18*D*a*m^3*x^6+176*D*b*m*x^8+B*b*m^4*x^4+C*a*m^4*x^4+104*C*b*m^2 *x^6+104*D*a*m^2*x^6+105*D*b*x^8+20*B*b*m^3*x^4+20*C*a*m^3*x^4+222*C*b*m*x ^6+222*D*a*m*x^6+A*b*m^4*x^2+B*a*m^4*x^2+130*B*b*m^2*x^4+130*C*a*m^2*x^4+1 35*C*b*x^6+135*D*a*x^6+22*A*b*m^3*x^2+22*B*a*m^3*x^2+300*B*b*m*x^4+300*C*a *m*x^4+A*a*m^4+164*A*b*m^2*x^2+164*B*a*m^2*x^2+189*B*b*x^4+189*C*a*x^4+24* A*a*m^3+458*A*b*m*x^2+458*B*a*m*x^2+206*A*a*m^2+315*A*b*x^2+315*B*a*x^2+74 4*A*a*m+945*A*a)*(c*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. 272 vs. \(2 (106) = 212\).
Time = 0.08 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.57 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {{\left ({\left (D b m^{4} + 16 \, D b m^{3} + 86 \, D b m^{2} + 176 \, D b m + 105 \, D b\right )} x^{9} + {\left ({\left (D a + C b\right )} m^{4} + 18 \, {\left (D a + C b\right )} m^{3} + 104 \, {\left (D a + C b\right )} m^{2} + 135 \, D a + 135 \, C b + 222 \, {\left (D a + C b\right )} m\right )} x^{7} + {\left ({\left (C a + B b\right )} m^{4} + 20 \, {\left (C a + B b\right )} m^{3} + 130 \, {\left (C a + B b\right )} m^{2} + 189 \, C a + 189 \, B b + 300 \, {\left (C a + B b\right )} m\right )} x^{5} + {\left ({\left (B a + A b\right )} m^{4} + 22 \, {\left (B a + A b\right )} m^{3} + 164 \, {\left (B a + A b\right )} m^{2} + 315 \, B a + 315 \, A b + 458 \, {\left (B a + A b\right )} m\right )} x^{3} + {\left (A a m^{4} + 24 \, A a m^{3} + 206 \, A a m^{2} + 744 \, A a m + 945 \, A a\right )} x\right )} \left (c x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate((c*x)^m*(b*x^2+a)*(D*x^6+C*x^4+B*x^2+A),x, algorithm="fricas")
Output:
((D*b*m^4 + 16*D*b*m^3 + 86*D*b*m^2 + 176*D*b*m + 105*D*b)*x^9 + ((D*a + C *b)*m^4 + 18*(D*a + C*b)*m^3 + 104*(D*a + C*b)*m^2 + 135*D*a + 135*C*b + 2 22*(D*a + C*b)*m)*x^7 + ((C*a + B*b)*m^4 + 20*(C*a + B*b)*m^3 + 130*(C*a + B*b)*m^2 + 189*C*a + 189*B*b + 300*(C*a + B*b)*m)*x^5 + ((B*a + A*b)*m^4 + 22*(B*a + A*b)*m^3 + 164*(B*a + A*b)*m^2 + 315*B*a + 315*A*b + 458*(B*a + A*b)*m)*x^3 + (A*a*m^4 + 24*A*a*m^3 + 206*A*a*m^2 + 744*A*a*m + 945*A*a) *x)*(c*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. 1911 vs. \(2 (92) = 184\).
Time = 0.61 (sec) , antiderivative size = 1911, normalized size of antiderivative = 18.03 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x^2+C x^4+D x^6\right ) \, dx=\text {Too large to display} \] Input:
integrate((c*x)**m*(b*x**2+a)*(D*x**6+C*x**4+B*x**2+A),x)
Output:
Piecewise(((-A*a/(8*x**8) - A*b/(6*x**6) - B*a/(6*x**6) - B*b/(4*x**4) - C *a/(4*x**4) - C*b/(2*x**2) - D*a/(2*x**2) + D*b*log(x))/c**9, Eq(m, -9)), ((-A*a/(6*x**6) - A*b/(4*x**4) - B*a/(4*x**4) - B*b/(2*x**2) - C*a/(2*x**2 ) + C*b*log(x) + D*a*log(x) + D*b*x**2/2)/c**7, Eq(m, -7)), ((-A*a/(4*x**4 ) - A*b/(2*x**2) - B*a/(2*x**2) + B*b*log(x) + C*a*log(x) + C*b*x**2/2 + D *a*x**2/2 + D*b*x**4/4)/c**5, Eq(m, -5)), ((-A*a/(2*x**2) + A*b*log(x) + B *a*log(x) + B*b*x**2/2 + C*a*x**2/2 + C*b*x**4/4 + D*a*x**4/4 + D*b*x**6/6 )/c**3, Eq(m, -3)), ((A*a*log(x) + A*b*x**2/2 + B*a*x**2/2 + B*b*x**4/4 + C*a*x**4/4 + C*b*x**6/6 + D*a*x**6/6 + D*b*x**8/8)/c, Eq(m, -1)), (A*a*m** 4*x*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*A* a*m**3*x*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*A*a*m**2*x*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 9 45) + 744*A*a*m*x*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*A*a*x*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*b*m**4*x**3*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1 689*m + 945) + 22*A*b*m**3*x**3*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950* m**2 + 1689*m + 945) + 164*A*b*m**2*x**3*(c*x)**m/(m**5 + 25*m**4 + 230*m* *3 + 950*m**2 + 1689*m + 945) + 458*A*b*m*x**3*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 315*A*b*x**3*(c*x)**m/(m**5 + 25*m** 4 + 230*m**3 + 950*m**2 + 1689*m + 945) + B*a*m**4*x**3*(c*x)**m/(m**5 ...
Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.30 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {D b c^{m} x^{9} x^{m}}{m + 9} + \frac {D a c^{m} x^{7} x^{m}}{m + 7} + \frac {C b c^{m} x^{7} x^{m}}{m + 7} + \frac {C a c^{m} x^{5} x^{m}}{m + 5} + \frac {B b c^{m} x^{5} x^{m}}{m + 5} + \frac {B a c^{m} x^{3} x^{m}}{m + 3} + \frac {A b c^{m} x^{3} x^{m}}{m + 3} + \frac {\left (c x\right )^{m + 1} A a}{c {\left (m + 1\right )}} \] Input:
integrate((c*x)^m*(b*x^2+a)*(D*x^6+C*x^4+B*x^2+A),x, algorithm="maxima")
Output:
D*b*c^m*x^9*x^m/(m + 9) + D*a*c^m*x^7*x^m/(m + 7) + C*b*c^m*x^7*x^m/(m + 7 ) + C*a*c^m*x^5*x^m/(m + 5) + B*b*c^m*x^5*x^m/(m + 5) + B*a*c^m*x^3*x^m/(m + 3) + A*b*c^m*x^3*x^m/(m + 3) + (c*x)^(m + 1)*A*a/(c*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (106) = 212\).
Time = 0.13 (sec) , antiderivative size = 569, normalized size of antiderivative = 5.37 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {\left (c x\right )^{m} D b m^{4} x^{9} + 16 \, \left (c x\right )^{m} D b m^{3} x^{9} + \left (c x\right )^{m} D a m^{4} x^{7} + \left (c x\right )^{m} C b m^{4} x^{7} + 86 \, \left (c x\right )^{m} D b m^{2} x^{9} + 18 \, \left (c x\right )^{m} D a m^{3} x^{7} + 18 \, \left (c x\right )^{m} C b m^{3} x^{7} + 176 \, \left (c x\right )^{m} D b m x^{9} + \left (c x\right )^{m} C a m^{4} x^{5} + \left (c x\right )^{m} B b m^{4} x^{5} + 104 \, \left (c x\right )^{m} D a m^{2} x^{7} + 104 \, \left (c x\right )^{m} C b m^{2} x^{7} + 105 \, \left (c x\right )^{m} D b x^{9} + 20 \, \left (c x\right )^{m} C a m^{3} x^{5} + 20 \, \left (c x\right )^{m} B b m^{3} x^{5} + 222 \, \left (c x\right )^{m} D a m x^{7} + 222 \, \left (c x\right )^{m} C b m x^{7} + \left (c x\right )^{m} B a m^{4} x^{3} + \left (c x\right )^{m} A b m^{4} x^{3} + 130 \, \left (c x\right )^{m} C a m^{2} x^{5} + 130 \, \left (c x\right )^{m} B b m^{2} x^{5} + 135 \, \left (c x\right )^{m} D a x^{7} + 135 \, \left (c x\right )^{m} C b x^{7} + 22 \, \left (c x\right )^{m} B a m^{3} x^{3} + 22 \, \left (c x\right )^{m} A b m^{3} x^{3} + 300 \, \left (c x\right )^{m} C a m x^{5} + 300 \, \left (c x\right )^{m} B b m x^{5} + \left (c x\right )^{m} A a m^{4} x + 164 \, \left (c x\right )^{m} B a m^{2} x^{3} + 164 \, \left (c x\right )^{m} A b m^{2} x^{3} + 189 \, \left (c x\right )^{m} C a x^{5} + 189 \, \left (c x\right )^{m} B b x^{5} + 24 \, \left (c x\right )^{m} A a m^{3} x + 458 \, \left (c x\right )^{m} B a m x^{3} + 458 \, \left (c x\right )^{m} A b m x^{3} + 206 \, \left (c x\right )^{m} A a m^{2} x + 315 \, \left (c x\right )^{m} B a x^{3} + 315 \, \left (c x\right )^{m} A b x^{3} + 744 \, \left (c x\right )^{m} A a m x + 945 \, \left (c x\right )^{m} A a x}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate((c*x)^m*(b*x^2+a)*(D*x^6+C*x^4+B*x^2+A),x, algorithm="giac")
Output:
((c*x)^m*D*b*m^4*x^9 + 16*(c*x)^m*D*b*m^3*x^9 + (c*x)^m*D*a*m^4*x^7 + (c*x )^m*C*b*m^4*x^7 + 86*(c*x)^m*D*b*m^2*x^9 + 18*(c*x)^m*D*a*m^3*x^7 + 18*(c* x)^m*C*b*m^3*x^7 + 176*(c*x)^m*D*b*m*x^9 + (c*x)^m*C*a*m^4*x^5 + (c*x)^m*B *b*m^4*x^5 + 104*(c*x)^m*D*a*m^2*x^7 + 104*(c*x)^m*C*b*m^2*x^7 + 105*(c*x) ^m*D*b*x^9 + 20*(c*x)^m*C*a*m^3*x^5 + 20*(c*x)^m*B*b*m^3*x^5 + 222*(c*x)^m *D*a*m*x^7 + 222*(c*x)^m*C*b*m*x^7 + (c*x)^m*B*a*m^4*x^3 + (c*x)^m*A*b*m^4 *x^3 + 130*(c*x)^m*C*a*m^2*x^5 + 130*(c*x)^m*B*b*m^2*x^5 + 135*(c*x)^m*D*a *x^7 + 135*(c*x)^m*C*b*x^7 + 22*(c*x)^m*B*a*m^3*x^3 + 22*(c*x)^m*A*b*m^3*x ^3 + 300*(c*x)^m*C*a*m*x^5 + 300*(c*x)^m*B*b*m*x^5 + (c*x)^m*A*a*m^4*x + 1 64*(c*x)^m*B*a*m^2*x^3 + 164*(c*x)^m*A*b*m^2*x^3 + 189*(c*x)^m*C*a*x^5 + 1 89*(c*x)^m*B*b*x^5 + 24*(c*x)^m*A*a*m^3*x + 458*(c*x)^m*B*a*m*x^3 + 458*(c *x)^m*A*b*m*x^3 + 206*(c*x)^m*A*a*m^2*x + 315*(c*x)^m*B*a*x^3 + 315*(c*x)^ m*A*b*x^3 + 744*(c*x)^m*A*a*m*x + 945*(c*x)^m*A*a*x)/(m^5 + 25*m^4 + 230*m ^3 + 950*m^2 + 1689*m + 945)
Timed out. \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x^2+C x^4+D x^6\right ) \, dx=\int {\left (c\,x\right )}^m\,\left (b\,x^2+a\right )\,\left (A+B\,x^2+C\,x^4+x^6\,D\right ) \,d x \] Input:
int((c*x)^m*(a + b*x^2)*(A + B*x^2 + C*x^4 + x^6*D),x)
Output:
int((c*x)^m*(a + b*x^2)*(A + B*x^2 + C*x^4 + x^6*D), x)
Time = 0.15 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.19 \[ \int (c x)^m \left (a+b x^2\right ) \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {x^{m} c^{m} x \left (b d \,m^{4} x^{8}+16 b d \,m^{3} x^{8}+a d \,m^{4} x^{6}+b c \,m^{4} x^{6}+86 b d \,m^{2} x^{8}+18 a d \,m^{3} x^{6}+18 b c \,m^{3} x^{6}+176 b d m \,x^{8}+a c \,m^{4} x^{4}+104 a d \,m^{2} x^{6}+b^{2} m^{4} x^{4}+104 b c \,m^{2} x^{6}+105 b d \,x^{8}+20 a c \,m^{3} x^{4}+222 a d m \,x^{6}+20 b^{2} m^{3} x^{4}+222 b c m \,x^{6}+2 a b \,m^{4} x^{2}+130 a c \,m^{2} x^{4}+135 a d \,x^{6}+130 b^{2} m^{2} x^{4}+135 b c \,x^{6}+44 a b \,m^{3} x^{2}+300 a c m \,x^{4}+300 b^{2} m \,x^{4}+a^{2} m^{4}+328 a b \,m^{2} x^{2}+189 a c \,x^{4}+189 b^{2} x^{4}+24 a^{2} m^{3}+916 a b m \,x^{2}+206 a^{2} m^{2}+630 a b \,x^{2}+744 a^{2} m +945 a^{2}\right )}{m^{5}+25 m^{4}+230 m^{3}+950 m^{2}+1689 m +945} \] Input:
int((c*x)^m*(b*x^2+a)*(D*x^6+C*x^4+B*x^2+A),x)
Output:
(x**m*c**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 + a*c*m**4*x**4 + 20*a*c*m**3*x**4 + 130*a*c*m**2*x**4 + 300*a*c*m*x**4 + 189*a*c*x**4 + a*d*m**4*x**6 + 18*a*d*m**3*x**6 + 104* a*d*m**2*x**6 + 222*a*d*m*x**6 + 135*a*d*x**6 + 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 + b*c*m**4 *x**6 + 18*b*c*m**3*x**6 + 104*b*c*m**2*x**6 + 222*b*c*m*x**6 + 135*b*c*x* *6 + b*d*m**4*x**8 + 16*b*d*m**3*x**8 + 86*b*d*m**2*x**8 + 176*b*d*m*x**8 + 105*b*d*x**8))/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)