Integrand size = 22, antiderivative size = 121 \[ \int (c x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {a^3 A (c x)^{1+m}}{c (1+m)}+\frac {a^2 (3 A b+a B) (c x)^{3+m}}{c^3 (3+m)}+\frac {3 a b (A b+a B) (c x)^{5+m}}{c^5 (5+m)}+\frac {b^2 (A b+3 a B) (c x)^{7+m}}{c^7 (7+m)}+\frac {b^3 B (c x)^{9+m}}{c^9 (9+m)} \] Output:
a^3*A*(c*x)^(1+m)/c/(1+m)+a^2*(3*A*b+B*a)*(c*x)^(3+m)/c^3/(3+m)+3*a*b*(A*b +B*a)*(c*x)^(5+m)/c^5/(5+m)+b^2*(A*b+3*B*a)*(c*x)^(7+m)/c^7/(7+m)+b^3*B*(c *x)^(9+m)/c^9/(9+m)
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.74 \[ \int (c x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=x (c x)^m \left (\frac {a^3 A}{1+m}+\frac {a^2 (3 A b+a B) x^2}{3+m}+\frac {3 a b (A b+a B) x^4}{5+m}+\frac {b^2 (A b+3 a B) x^6}{7+m}+\frac {b^3 B x^8}{9+m}\right ) \] Input:
Integrate[(c*x)^m*(a + b*x^2)^3*(A + B*x^2),x]
Output:
x*(c*x)^m*((a^3*A)/(1 + m) + (a^2*(3*A*b + a*B)*x^2)/(3 + m) + (3*a*b*(A*b + a*B)*x^4)/(5 + m) + (b^2*(A*b + 3*a*B)*x^6)/(7 + m) + (b^3*B*x^8)/(9 + m))
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {355, 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.
| \(\displaystyle \int \left (a+b x^2\right )^3 \left (A+B x^2\right ) (c x)^m \, dx\) |
| \(\Big \downarrow \) 355 |
| \(\displaystyle \int \left (a^3 A (c x)^m+\frac {a^2 (c x)^{m+2} (a B+3 A b)}{c^2}+\frac {b^2 (c x)^{m+6} (3 a B+A b)}{c^6}+\frac {3 a b (c x)^{m+4} (a B+A b)}{c^4}+\frac {b^3 B (c x)^{m+8}}{c^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 A (c x)^{m+1}}{c (m+1)}+\frac {a^2 (c x)^{m+3} (a B+3 A b)}{c^3 (m+3)}+\frac {b^2 (c x)^{m+7} (3 a B+A b)}{c^7 (m+7)}+\frac {3 a b (c x)^{m+5} (a B+A b)}{c^5 (m+5)}+\frac {b^3 B (c x)^{m+9}}{c^9 (m+9)}\) |
Input:
Int[(c*x)^m*(a + b*x^2)^3*(A + B*x^2),x]
Output:
(a^3*A*(c*x)^(1 + m))/(c*(1 + m)) + (a^2*(3*A*b + a*B)*(c*x)^(3 + m))/(c^3 *(3 + m)) + (3*a*b*(A*b + a*B)*(c*x)^(5 + m))/(c^5*(5 + m)) + (b^2*(A*b + 3*a*B)*(c*x)^(7 + m))/(c^7*(7 + m)) + (b^3*B*(c*x)^(9 + m))/(c^9*(9 + m))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & & IGtQ[q, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(474\) vs. \(2(121)=242\).
Time = 0.50 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.93
| method | result | size |
| gosper | \(\frac {x \left (B \,b^{3} m^{4} x^{8}+16 B \,b^{3} m^{3} x^{8}+A \,b^{3} m^{4} x^{6}+3 B a \,b^{2} m^{4} x^{6}+86 B \,b^{3} m^{2} x^{8}+18 A \,b^{3} m^{3} x^{6}+54 B a \,b^{2} m^{3} x^{6}+176 m \,x^{8} B \,b^{3}+3 A a \,b^{2} m^{4} x^{4}+104 A \,b^{3} m^{2} x^{6}+3 B \,a^{2} b \,m^{4} x^{4}+312 B a \,b^{2} m^{2} x^{6}+105 b^{3} B \,x^{8}+60 A a \,b^{2} m^{3} x^{4}+222 A \,b^{3} x^{6} m +60 B \,a^{2} b \,m^{3} x^{4}+666 B a \,b^{2} x^{6} m +3 A \,a^{2} b \,m^{4} x^{2}+390 A a \,b^{2} m^{2} x^{4}+135 A \,b^{3} x^{6}+B \,a^{3} m^{4} x^{2}+390 B \,a^{2} b \,m^{2} x^{4}+405 B a \,b^{2} x^{6}+66 A \,a^{2} b \,m^{3} x^{2}+900 a A \,b^{2} x^{4} m +22 B \,a^{3} m^{3} x^{2}+900 B \,a^{2} b \,x^{4} m +A \,a^{3} m^{4}+492 A \,a^{2} b \,m^{2} x^{2}+567 a A \,b^{2} x^{4}+164 B \,a^{3} m^{2} x^{2}+567 B \,a^{2} b \,x^{4}+24 A \,a^{3} m^{3}+1374 a^{2} A b \,x^{2} m +458 B \,a^{3} x^{2} m +206 A \,a^{3} m^{2}+945 a^{2} A b \,x^{2}+315 B \,a^{3} x^{2}+744 a^{3} A m +945 a^{3} A \right ) \left (c x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(475\) |
| risch | \(\frac {x \left (B \,b^{3} m^{4} x^{8}+16 B \,b^{3} m^{3} x^{8}+A \,b^{3} m^{4} x^{6}+3 B a \,b^{2} m^{4} x^{6}+86 B \,b^{3} m^{2} x^{8}+18 A \,b^{3} m^{3} x^{6}+54 B a \,b^{2} m^{3} x^{6}+176 m \,x^{8} B \,b^{3}+3 A a \,b^{2} m^{4} x^{4}+104 A \,b^{3} m^{2} x^{6}+3 B \,a^{2} b \,m^{4} x^{4}+312 B a \,b^{2} m^{2} x^{6}+105 b^{3} B \,x^{8}+60 A a \,b^{2} m^{3} x^{4}+222 A \,b^{3} x^{6} m +60 B \,a^{2} b \,m^{3} x^{4}+666 B a \,b^{2} x^{6} m +3 A \,a^{2} b \,m^{4} x^{2}+390 A a \,b^{2} m^{2} x^{4}+135 A \,b^{3} x^{6}+B \,a^{3} m^{4} x^{2}+390 B \,a^{2} b \,m^{2} x^{4}+405 B a \,b^{2} x^{6}+66 A \,a^{2} b \,m^{3} x^{2}+900 a A \,b^{2} x^{4} m +22 B \,a^{3} m^{3} x^{2}+900 B \,a^{2} b \,x^{4} m +A \,a^{3} m^{4}+492 A \,a^{2} b \,m^{2} x^{2}+567 a A \,b^{2} x^{4}+164 B \,a^{3} m^{2} x^{2}+567 B \,a^{2} b \,x^{4}+24 A \,a^{3} m^{3}+1374 a^{2} A b \,x^{2} m +458 B \,a^{3} x^{2} m +206 A \,a^{3} m^{2}+945 a^{2} A b \,x^{2}+315 B \,a^{3} x^{2}+744 a^{3} A m +945 a^{3} A \right ) \left (c x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(475\) |
| orering | \(\frac {x \left (B \,b^{3} m^{4} x^{8}+16 B \,b^{3} m^{3} x^{8}+A \,b^{3} m^{4} x^{6}+3 B a \,b^{2} m^{4} x^{6}+86 B \,b^{3} m^{2} x^{8}+18 A \,b^{3} m^{3} x^{6}+54 B a \,b^{2} m^{3} x^{6}+176 m \,x^{8} B \,b^{3}+3 A a \,b^{2} m^{4} x^{4}+104 A \,b^{3} m^{2} x^{6}+3 B \,a^{2} b \,m^{4} x^{4}+312 B a \,b^{2} m^{2} x^{6}+105 b^{3} B \,x^{8}+60 A a \,b^{2} m^{3} x^{4}+222 A \,b^{3} x^{6} m +60 B \,a^{2} b \,m^{3} x^{4}+666 B a \,b^{2} x^{6} m +3 A \,a^{2} b \,m^{4} x^{2}+390 A a \,b^{2} m^{2} x^{4}+135 A \,b^{3} x^{6}+B \,a^{3} m^{4} x^{2}+390 B \,a^{2} b \,m^{2} x^{4}+405 B a \,b^{2} x^{6}+66 A \,a^{2} b \,m^{3} x^{2}+900 a A \,b^{2} x^{4} m +22 B \,a^{3} m^{3} x^{2}+900 B \,a^{2} b \,x^{4} m +A \,a^{3} m^{4}+492 A \,a^{2} b \,m^{2} x^{2}+567 a A \,b^{2} x^{4}+164 B \,a^{3} m^{2} x^{2}+567 B \,a^{2} b \,x^{4}+24 A \,a^{3} m^{3}+1374 a^{2} A b \,x^{2} m +458 B \,a^{3} x^{2} m +206 A \,a^{3} m^{2}+945 a^{2} A b \,x^{2}+315 B \,a^{3} x^{2}+744 a^{3} A m +945 a^{3} A \right ) \left (c x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(475\) |
| parallelrisch | \(\frac {B \,x^{3} \left (c x \right )^{m} a^{3} m^{4}+22 B \,x^{3} \left (c x \right )^{m} a^{3} m^{3}+567 A \,x^{5} \left (c x \right )^{m} a \,b^{2}+A x \left (c x \right )^{m} a^{3} m^{4}+567 B \,x^{5} \left (c x \right )^{m} a^{2} b +164 B \,x^{3} \left (c x \right )^{m} a^{3} m^{2}+24 A x \left (c x \right )^{m} a^{3} m^{3}+458 B \,x^{3} \left (c x \right )^{m} a^{3} m +945 A \,x^{3} \left (c x \right )^{m} a^{2} b +206 A x \left (c x \right )^{m} a^{3} m^{2}+744 A x \left (c x \right )^{m} a^{3} m +60 A \,x^{5} \left (c x \right )^{m} a \,b^{2} m^{3}+3 B \,x^{7} \left (c x \right )^{m} a \,b^{2} m^{4}+54 B \,x^{7} \left (c x \right )^{m} a \,b^{2} m^{3}+3 A \,x^{5} \left (c x \right )^{m} a \,b^{2} m^{4}+666 B \,x^{7} \left (c x \right )^{m} a \,b^{2} m +60 B \,x^{5} \left (c x \right )^{m} a^{2} b \,m^{3}+390 A \,x^{5} \left (c x \right )^{m} a \,b^{2} m^{2}+3 A \,x^{3} \left (c x \right )^{m} a^{2} b \,m^{4}+390 B \,x^{5} \left (c x \right )^{m} a^{2} b \,m^{2}+900 A \,x^{5} \left (c x \right )^{m} a \,b^{2} m +66 A \,x^{3} \left (c x \right )^{m} a^{2} b \,m^{3}+900 B \,x^{5} \left (c x \right )^{m} a^{2} b m +492 A \,x^{3} \left (c x \right )^{m} a^{2} b \,m^{2}+1374 A \,x^{3} \left (c x \right )^{m} a^{2} b m +A \,x^{7} \left (c x \right )^{m} b^{3} m^{4}+86 B \,x^{9} \left (c x \right )^{m} b^{3} m^{2}+18 A \,x^{7} \left (c x \right )^{m} b^{3} m^{3}+176 B \,x^{9} \left (c x \right )^{m} b^{3} m +104 A \,x^{7} \left (c x \right )^{m} b^{3} m^{2}+222 A \,x^{7} \left (c x \right )^{m} b^{3} m +405 B \,x^{7} \left (c x \right )^{m} a \,b^{2}+B \,x^{9} \left (c x \right )^{m} b^{3} m^{4}+16 B \,x^{9} \left (c x \right )^{m} b^{3} m^{3}+105 B \,x^{9} \left (c x \right )^{m} b^{3}+135 A \,x^{7} \left (c x \right )^{m} b^{3}+315 B \,x^{3} \left (c x \right )^{m} a^{3}+945 A x \left (c x \right )^{m} a^{3}+312 B \,x^{7} \left (c x \right )^{m} a \,b^{2} m^{2}+3 B \,x^{5} \left (c x \right )^{m} a^{2} b \,m^{4}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(674\) |
Input:
int((c*x)^m*(b*x^2+a)^3*(B*x^2+A),x,method=_RETURNVERBOSE)
Output:
x*(B*b^3*m^4*x^8+16*B*b^3*m^3*x^8+A*b^3*m^4*x^6+3*B*a*b^2*m^4*x^6+86*B*b^3 *m^2*x^8+18*A*b^3*m^3*x^6+54*B*a*b^2*m^3*x^6+176*B*b^3*m*x^8+3*A*a*b^2*m^4 *x^4+104*A*b^3*m^2*x^6+3*B*a^2*b*m^4*x^4+312*B*a*b^2*m^2*x^6+105*B*b^3*x^8 +60*A*a*b^2*m^3*x^4+222*A*b^3*m*x^6+60*B*a^2*b*m^3*x^4+666*B*a*b^2*m*x^6+3 *A*a^2*b*m^4*x^2+390*A*a*b^2*m^2*x^4+135*A*b^3*x^6+B*a^3*m^4*x^2+390*B*a^2 *b*m^2*x^4+405*B*a*b^2*x^6+66*A*a^2*b*m^3*x^2+900*A*a*b^2*m*x^4+22*B*a^3*m ^3*x^2+900*B*a^2*b*m*x^4+A*a^3*m^4+492*A*a^2*b*m^2*x^2+567*A*a*b^2*x^4+164 *B*a^3*m^2*x^2+567*B*a^2*b*x^4+24*A*a^3*m^3+1374*A*a^2*b*m*x^2+458*B*a^3*m *x^2+206*A*a^3*m^2+945*A*a^2*b*x^2+315*B*a^3*x^2+744*A*a^3*m+945*A*a^3)*(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. 381 vs. \(2 (121) = 242\).
Time = 0.10 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.15 \[ \int (c x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {{\left ({\left (B b^{3} m^{4} + 16 \, B b^{3} m^{3} + 86 \, B b^{3} m^{2} + 176 \, B b^{3} m + 105 \, B b^{3}\right )} x^{9} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 405 \, B a b^{2} + 135 \, A b^{3} + 18 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 104 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 222 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{7} + 3 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 189 \, B a^{2} b + 189 \, A a b^{2} + 20 \, {\left (B a^{2} b + A a b^{2}\right )} m^{3} + 130 \, {\left (B a^{2} b + A a b^{2}\right )} m^{2} + 300 \, {\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{5} + {\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 315 \, B a^{3} + 945 \, A a^{2} b + 22 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 164 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 458 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{3} + {\left (A a^{3} m^{4} + 24 \, A a^{3} m^{3} + 206 \, A a^{3} m^{2} + 744 \, A a^{3} m + 945 \, A a^{3}\right )} x\right )} \left (c x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate((c*x)^m*(b*x^2+a)^3*(B*x^2+A),x, algorithm="fricas")
Output:
((B*b^3*m^4 + 16*B*b^3*m^3 + 86*B*b^3*m^2 + 176*B*b^3*m + 105*B*b^3)*x^9 + ((3*B*a*b^2 + A*b^3)*m^4 + 405*B*a*b^2 + 135*A*b^3 + 18*(3*B*a*b^2 + A*b^ 3)*m^3 + 104*(3*B*a*b^2 + A*b^3)*m^2 + 222*(3*B*a*b^2 + A*b^3)*m)*x^7 + 3* ((B*a^2*b + A*a*b^2)*m^4 + 189*B*a^2*b + 189*A*a*b^2 + 20*(B*a^2*b + A*a*b ^2)*m^3 + 130*(B*a^2*b + A*a*b^2)*m^2 + 300*(B*a^2*b + A*a*b^2)*m)*x^5 + ( (B*a^3 + 3*A*a^2*b)*m^4 + 315*B*a^3 + 945*A*a^2*b + 22*(B*a^3 + 3*A*a^2*b) *m^3 + 164*(B*a^3 + 3*A*a^2*b)*m^2 + 458*(B*a^3 + 3*A*a^2*b)*m)*x^3 + (A*a ^3*m^4 + 24*A*a^3*m^3 + 206*A*a^3*m^2 + 744*A*a^3*m + 945*A*a^3)*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. 2152 vs. \(2 (110) = 220\).
Time = 0.56 (sec) , antiderivative size = 2152, normalized size of antiderivative = 17.79 \[ \int (c x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((c*x)**m*(b*x**2+a)**3*(B*x**2+A),x)
Output:
Piecewise(((-A*a**3/(8*x**8) - A*a**2*b/(2*x**6) - 3*A*a*b**2/(4*x**4) - A *b**3/(2*x**2) - B*a**3/(6*x**6) - 3*B*a**2*b/(4*x**4) - 3*B*a*b**2/(2*x** 2) + B*b**3*log(x))/c**9, Eq(m, -9)), ((-A*a**3/(6*x**6) - 3*A*a**2*b/(4*x **4) - 3*A*a*b**2/(2*x**2) + A*b**3*log(x) - B*a**3/(4*x**4) - 3*B*a**2*b/ (2*x**2) + 3*B*a*b**2*log(x) + B*b**3*x**2/2)/c**7, Eq(m, -7)), ((-A*a**3/ (4*x**4) - 3*A*a**2*b/(2*x**2) + 3*A*a*b**2*log(x) + A*b**3*x**2/2 - B*a** 3/(2*x**2) + 3*B*a**2*b*log(x) + 3*B*a*b**2*x**2/2 + B*b**3*x**4/4)/c**5, Eq(m, -5)), ((-A*a**3/(2*x**2) + 3*A*a**2*b*log(x) + 3*A*a*b**2*x**2/2 + A *b**3*x**4/4 + B*a**3*log(x) + 3*B*a**2*b*x**2/2 + 3*B*a*b**2*x**4/4 + B*b **3*x**6/6)/c**3, Eq(m, -3)), ((A*a**3*log(x) + 3*A*a**2*b*x**2/2 + 3*A*a* b**2*x**4/4 + A*b**3*x**6/6 + B*a**3*x**2/2 + 3*B*a**2*b*x**4/4 + B*a*b**2 *x**6/2 + B*b**3*x**8/8)/c, Eq(m, -1)), (A*a**3*m**4*x*(c*x)**m/(m**5 + 25 *m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*A*a**3*m**3*x*(c*x)**m/(m **5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*A*a**3*m**2*x*(c *x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 744*A*a**3* m*x*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*A *a**3*x*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3 *A*a**2*b*m**4*x**3*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689* m + 945) + 66*A*a**2*b*m**3*x**3*(c*x)**m/(m**5 + 25*m**4 + 230*m**3 + 950 *m**2 + 1689*m + 945) + 492*A*a**2*b*m**2*x**3*(c*x)**m/(m**5 + 25*m**4...
Time = 0.05 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.34 \[ \int (c x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {B b^{3} c^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B a b^{2} c^{m} x^{7} x^{m}}{m + 7} + \frac {A b^{3} c^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a^{2} b c^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a b^{2} c^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{3} c^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a^{2} b c^{m} x^{3} x^{m}}{m + 3} + \frac {\left (c x\right )^{m + 1} A a^{3}}{c {\left (m + 1\right )}} \] Input:
integrate((c*x)^m*(b*x^2+a)^3*(B*x^2+A),x, algorithm="maxima")
Output:
B*b^3*c^m*x^9*x^m/(m + 9) + 3*B*a*b^2*c^m*x^7*x^m/(m + 7) + A*b^3*c^m*x^7* x^m/(m + 7) + 3*B*a^2*b*c^m*x^5*x^m/(m + 5) + 3*A*a*b^2*c^m*x^5*x^m/(m + 5 ) + B*a^3*c^m*x^3*x^m/(m + 3) + 3*A*a^2*b*c^m*x^3*x^m/(m + 3) + (c*x)^(m + 1)*A*a^3/(c*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (121) = 242\).
Time = 0.14 (sec) , antiderivative size = 673, normalized size of antiderivative = 5.56 \[ \int (c x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {\left (c x\right )^{m} B b^{3} m^{4} x^{9} + 16 \, \left (c x\right )^{m} B b^{3} m^{3} x^{9} + 3 \, \left (c x\right )^{m} B a b^{2} m^{4} x^{7} + \left (c x\right )^{m} A b^{3} m^{4} x^{7} + 86 \, \left (c x\right )^{m} B b^{3} m^{2} x^{9} + 54 \, \left (c x\right )^{m} B a b^{2} m^{3} x^{7} + 18 \, \left (c x\right )^{m} A b^{3} m^{3} x^{7} + 176 \, \left (c x\right )^{m} B b^{3} m x^{9} + 3 \, \left (c x\right )^{m} B a^{2} b m^{4} x^{5} + 3 \, \left (c x\right )^{m} A a b^{2} m^{4} x^{5} + 312 \, \left (c x\right )^{m} B a b^{2} m^{2} x^{7} + 104 \, \left (c x\right )^{m} A b^{3} m^{2} x^{7} + 105 \, \left (c x\right )^{m} B b^{3} x^{9} + 60 \, \left (c x\right )^{m} B a^{2} b m^{3} x^{5} + 60 \, \left (c x\right )^{m} A a b^{2} m^{3} x^{5} + 666 \, \left (c x\right )^{m} B a b^{2} m x^{7} + 222 \, \left (c x\right )^{m} A b^{3} m x^{7} + \left (c x\right )^{m} B a^{3} m^{4} x^{3} + 3 \, \left (c x\right )^{m} A a^{2} b m^{4} x^{3} + 390 \, \left (c x\right )^{m} B a^{2} b m^{2} x^{5} + 390 \, \left (c x\right )^{m} A a b^{2} m^{2} x^{5} + 405 \, \left (c x\right )^{m} B a b^{2} x^{7} + 135 \, \left (c x\right )^{m} A b^{3} x^{7} + 22 \, \left (c x\right )^{m} B a^{3} m^{3} x^{3} + 66 \, \left (c x\right )^{m} A a^{2} b m^{3} x^{3} + 900 \, \left (c x\right )^{m} B a^{2} b m x^{5} + 900 \, \left (c x\right )^{m} A a b^{2} m x^{5} + \left (c x\right )^{m} A a^{3} m^{4} x + 164 \, \left (c x\right )^{m} B a^{3} m^{2} x^{3} + 492 \, \left (c x\right )^{m} A a^{2} b m^{2} x^{3} + 567 \, \left (c x\right )^{m} B a^{2} b x^{5} + 567 \, \left (c x\right )^{m} A a b^{2} x^{5} + 24 \, \left (c x\right )^{m} A a^{3} m^{3} x + 458 \, \left (c x\right )^{m} B a^{3} m x^{3} + 1374 \, \left (c x\right )^{m} A a^{2} b m x^{3} + 206 \, \left (c x\right )^{m} A a^{3} m^{2} x + 315 \, \left (c x\right )^{m} B a^{3} x^{3} + 945 \, \left (c x\right )^{m} A a^{2} b x^{3} + 744 \, \left (c x\right )^{m} A a^{3} m x + 945 \, \left (c x\right )^{m} A a^{3} x}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \] Input:
integrate((c*x)^m*(b*x^2+a)^3*(B*x^2+A),x, algorithm="giac")
Output:
((c*x)^m*B*b^3*m^4*x^9 + 16*(c*x)^m*B*b^3*m^3*x^9 + 3*(c*x)^m*B*a*b^2*m^4* x^7 + (c*x)^m*A*b^3*m^4*x^7 + 86*(c*x)^m*B*b^3*m^2*x^9 + 54*(c*x)^m*B*a*b^ 2*m^3*x^7 + 18*(c*x)^m*A*b^3*m^3*x^7 + 176*(c*x)^m*B*b^3*m*x^9 + 3*(c*x)^m *B*a^2*b*m^4*x^5 + 3*(c*x)^m*A*a*b^2*m^4*x^5 + 312*(c*x)^m*B*a*b^2*m^2*x^7 + 104*(c*x)^m*A*b^3*m^2*x^7 + 105*(c*x)^m*B*b^3*x^9 + 60*(c*x)^m*B*a^2*b* m^3*x^5 + 60*(c*x)^m*A*a*b^2*m^3*x^5 + 666*(c*x)^m*B*a*b^2*m*x^7 + 222*(c* x)^m*A*b^3*m*x^7 + (c*x)^m*B*a^3*m^4*x^3 + 3*(c*x)^m*A*a^2*b*m^4*x^3 + 390 *(c*x)^m*B*a^2*b*m^2*x^5 + 390*(c*x)^m*A*a*b^2*m^2*x^5 + 405*(c*x)^m*B*a*b ^2*x^7 + 135*(c*x)^m*A*b^3*x^7 + 22*(c*x)^m*B*a^3*m^3*x^3 + 66*(c*x)^m*A*a ^2*b*m^3*x^3 + 900*(c*x)^m*B*a^2*b*m*x^5 + 900*(c*x)^m*A*a*b^2*m*x^5 + (c* x)^m*A*a^3*m^4*x + 164*(c*x)^m*B*a^3*m^2*x^3 + 492*(c*x)^m*A*a^2*b*m^2*x^3 + 567*(c*x)^m*B*a^2*b*x^5 + 567*(c*x)^m*A*a*b^2*x^5 + 24*(c*x)^m*A*a^3*m^ 3*x + 458*(c*x)^m*B*a^3*m*x^3 + 1374*(c*x)^m*A*a^2*b*m*x^3 + 206*(c*x)^m*A *a^3*m^2*x + 315*(c*x)^m*B*a^3*x^3 + 945*(c*x)^m*A*a^2*b*x^3 + 744*(c*x)^m *A*a^3*m*x + 945*(c*x)^m*A*a^3*x)/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689 *m + 945)
Time = 0.72 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.31 \[ \int (c x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx={\left (c\,x\right )}^m\,\left (\frac {A\,a^3\,x\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {B\,b^3\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a^2\,x^3\,\left (3\,A\,b+B\,a\right )\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {b^2\,x^7\,\left (A\,b+3\,B\,a\right )\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {3\,a\,b\,x^5\,\left (A\,b+B\,a\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}\right ) \] Input:
int((A + B*x^2)*(c*x)^m*(a + b*x^2)^3,x)
Output:
(c*x)^m*((A*a^3*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) + (B*b^3*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^2*x^3*( 3*A*b + B*a)*(458*m + 164*m^2 + 22*m^3 + m^4 + 315))/(1689*m + 950*m^2 + 2 30*m^3 + 25*m^4 + m^5 + 945) + (b^2*x^7*(A*b + 3*B*a)*(222*m + 104*m^2 + 1 8*m^3 + m^4 + 135))/(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945) + (3 *a*b*x^5*(A*b + B*a)*(300*m + 130*m^2 + 20*m^3 + m^4 + 189))/(1689*m + 950 *m^2 + 230*m^3 + 25*m^4 + m^5 + 945))
Time = 0.20 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.41 \[ \int (c x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {x^{m} c^{m} x \left (b^{4} m^{4} x^{8}+16 b^{4} m^{3} x^{8}+4 a \,b^{3} m^{4} x^{6}+86 b^{4} m^{2} x^{8}+72 a \,b^{3} m^{3} x^{6}+176 b^{4} m \,x^{8}+6 a^{2} b^{2} m^{4} x^{4}+416 a \,b^{3} m^{2} x^{6}+105 b^{4} x^{8}+120 a^{2} b^{2} m^{3} x^{4}+888 a \,b^{3} m \,x^{6}+4 a^{3} b \,m^{4} x^{2}+780 a^{2} b^{2} m^{2} x^{4}+540 a \,b^{3} x^{6}+88 a^{3} b \,m^{3} x^{2}+1800 a^{2} b^{2} m \,x^{4}+a^{4} m^{4}+656 a^{3} b \,m^{2} x^{2}+1134 a^{2} b^{2} x^{4}+24 a^{4} m^{3}+1832 a^{3} b m \,x^{2}+206 a^{4} m^{2}+1260 a^{3} b \,x^{2}+744 a^{4} m +945 a^{4}\right )}{m^{5}+25 m^{4}+230 m^{3}+950 m^{2}+1689 m +945} \] Input:
int((c*x)^m*(b*x^2+a)^3*(B*x^2+A),x)
Output:
(x**m*c**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)