Integrand size = 32, antiderivative size = 155 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {a^2 A (c x)^{1+m}}{c (1+m)}+\frac {a (2 A b+a B) (c x)^{3+m}}{c^3 (3+m)}+\frac {\left (A b^2+a (2 b B+a C)\right ) (c x)^{5+m}}{c^5 (5+m)}+\frac {\left (b^2 B+2 a b C+a^2 D\right ) (c x)^{7+m}}{c^7 (7+m)}+\frac {b (b C+2 a D) (c x)^{9+m}}{c^9 (9+m)}+\frac {b^2 D (c x)^{11+m}}{c^{11} (11+m)} \] Output:
a^2*A*(c*x)^(1+m)/c/(1+m)+a*(2*A*b+B*a)*(c*x)^(3+m)/c^3/(3+m)+(A*b^2+a*(2* B*b+C*a))*(c*x)^(5+m)/c^5/(5+m)+(B*b^2+2*C*a*b+D*a^2)*(c*x)^(7+m)/c^7/(7+m )+b*(C*b+2*D*a)*(c*x)^(9+m)/c^9/(9+m)+b^2*D*(c*x)^(11+m)/c^11/(11+m)
Time = 0.96 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.75 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=x (c x)^m \left (\frac {a^2 A}{1+m}+\frac {a (2 A b+a B) x^2}{3+m}+\frac {\left (A b^2+a (2 b B+a C)\right ) x^4}{5+m}+\frac {\left (b^2 B+2 a b C+a^2 D\right ) x^6}{7+m}+\frac {b (b C+2 a D) x^8}{9+m}+\frac {b^2 D x^{10}}{11+m}\right ) \] Input:
Integrate[(c*x)^m*(a + b*x^2)^2*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
x*(c*x)^m*((a^2*A)/(1 + m) + (a*(2*A*b + a*B)*x^2)/(3 + m) + ((A*b^2 + a*( 2*b*B + a*C))*x^4)/(5 + m) + ((b^2*B + 2*a*b*C + a^2*D)*x^6)/(7 + m) + (b* (b*C + 2*a*D)*x^8)/(9 + m) + (b^2*D*x^10)/(11 + m))
Time = 0.42 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, 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.
| \(\displaystyle \int \left (a+b x^2\right )^2 (c x)^m \left (A+B x^2+C x^4+D x^6\right ) \, dx\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \int \left (a^2 A (c x)^m+\frac {(c x)^{m+6} \left (a^2 D+2 a b C+b^2 B\right )}{c^6}+\frac {(c x)^{m+4} \left (a (a C+2 b B)+A b^2\right )}{c^4}+\frac {a (c x)^{m+2} (a B+2 A b)}{c^2}+\frac {b (c x)^{m+8} (2 a D+b C)}{c^8}+\frac {b^2 D (c x)^{m+10}}{c^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 A (c x)^{m+1}}{c (m+1)}+\frac {(c x)^{m+7} \left (a^2 D+2 a b C+b^2 B\right )}{c^7 (m+7)}+\frac {(c x)^{m+5} \left (a (a C+2 b B)+A b^2\right )}{c^5 (m+5)}+\frac {a (c x)^{m+3} (a B+2 A b)}{c^3 (m+3)}+\frac {b (c x)^{m+9} (2 a D+b C)}{c^9 (m+9)}+\frac {b^2 D (c x)^{m+11}}{c^{11} (m+11)}\) |
Input:
Int[(c*x)^m*(a + b*x^2)^2*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
(a^2*A*(c*x)^(1 + m))/(c*(1 + m)) + (a*(2*A*b + a*B)*(c*x)^(3 + m))/(c^3*( 3 + m)) + ((A*b^2 + a*(2*b*B + a*C))*(c*x)^(5 + m))/(c^5*(5 + m)) + ((b^2* B + 2*a*b*C + a^2*D)*(c*x)^(7 + m))/(c^7*(7 + m)) + (b*(b*C + 2*a*D)*(c*x) ^(9 + m))/(c^9*(9 + m)) + (b^2*D*(c*x)^(11 + m))/(c^11*(11 + 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. \(792\) vs. \(2(155)=310\).
Time = 0.51 (sec) , antiderivative size = 793, normalized size of antiderivative = 5.12
| method | result | size |
| gosper | \(\frac {x \left (D b^{2} m^{5} x^{10}+25 D b^{2} m^{4} x^{10}+C \,b^{2} m^{5} x^{8}+2 D a b \,m^{5} x^{8}+230 D b^{2} m^{3} x^{10}+27 C \,b^{2} m^{4} x^{8}+54 D a b \,m^{4} x^{8}+950 D b^{2} m^{2} x^{10}+B \,b^{2} m^{5} x^{6}+2 C a b \,m^{5} x^{6}+262 C \,b^{2} m^{3} x^{8}+D a^{2} m^{5} x^{6}+524 D a b \,m^{3} x^{8}+1689 D b^{2} m \,x^{10}+29 B \,b^{2} m^{4} x^{6}+58 C a b \,m^{4} x^{6}+1122 C \,b^{2} m^{2} x^{8}+29 D a^{2} m^{4} x^{6}+2244 D a b \,m^{2} x^{8}+945 b^{2} x^{10} D+A \,b^{2} m^{5} x^{4}+2 B a b \,m^{5} x^{4}+302 B \,b^{2} m^{3} x^{6}+C \,a^{2} m^{5} x^{4}+604 C a b \,m^{3} x^{6}+2041 C \,b^{2} m \,x^{8}+302 D a^{2} m^{3} x^{6}+4082 D a b m \,x^{8}+31 A \,b^{2} m^{4} x^{4}+62 B a b \,m^{4} x^{4}+1366 B \,b^{2} m^{2} x^{6}+31 C \,a^{2} m^{4} x^{4}+2732 C a b \,m^{2} x^{6}+1155 C \,b^{2} x^{8}+1366 D a^{2} m^{2} x^{6}+2310 D a b \,x^{8}+2 A a b \,m^{5} x^{2}+350 A \,b^{2} m^{3} x^{4}+B \,a^{2} m^{5} x^{2}+700 B a b \,m^{3} x^{4}+2577 B \,b^{2} m \,x^{6}+350 C \,a^{2} m^{3} x^{4}+5154 C a b m \,x^{6}+2577 D a^{2} m \,x^{6}+66 A a b \,m^{4} x^{2}+1730 A \,b^{2} m^{2} x^{4}+33 B \,a^{2} m^{4} x^{2}+3460 B a b \,m^{2} x^{4}+1485 b^{2} B \,x^{6}+1730 C \,a^{2} m^{2} x^{4}+2970 b \,x^{6} C a +1485 D a^{2} x^{6}+A \,a^{2} m^{5}+812 A a b \,m^{3} x^{2}+3489 A \,b^{2} m \,x^{4}+406 B \,a^{2} m^{3} x^{2}+6978 B a b m \,x^{4}+3489 C \,a^{2} m \,x^{4}+35 A \,a^{2} m^{4}+4524 A a b \,m^{2} x^{2}+2079 A \,b^{2} x^{4}+2262 B \,a^{2} m^{2} x^{2}+4158 B a b \,x^{4}+2079 C \,a^{2} x^{4}+470 A \,a^{2} m^{3}+10706 A a b m \,x^{2}+5353 B \,a^{2} m \,x^{2}+3010 A \,a^{2} m^{2}+6930 a A b \,x^{2}+3465 B \,a^{2} x^{2}+9129 A \,a^{2} m +10395 a^{2} A \right ) \left (c x \right )^{m}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(793\) |
| orering | \(\frac {x \left (D b^{2} m^{5} x^{10}+25 D b^{2} m^{4} x^{10}+C \,b^{2} m^{5} x^{8}+2 D a b \,m^{5} x^{8}+230 D b^{2} m^{3} x^{10}+27 C \,b^{2} m^{4} x^{8}+54 D a b \,m^{4} x^{8}+950 D b^{2} m^{2} x^{10}+B \,b^{2} m^{5} x^{6}+2 C a b \,m^{5} x^{6}+262 C \,b^{2} m^{3} x^{8}+D a^{2} m^{5} x^{6}+524 D a b \,m^{3} x^{8}+1689 D b^{2} m \,x^{10}+29 B \,b^{2} m^{4} x^{6}+58 C a b \,m^{4} x^{6}+1122 C \,b^{2} m^{2} x^{8}+29 D a^{2} m^{4} x^{6}+2244 D a b \,m^{2} x^{8}+945 b^{2} x^{10} D+A \,b^{2} m^{5} x^{4}+2 B a b \,m^{5} x^{4}+302 B \,b^{2} m^{3} x^{6}+C \,a^{2} m^{5} x^{4}+604 C a b \,m^{3} x^{6}+2041 C \,b^{2} m \,x^{8}+302 D a^{2} m^{3} x^{6}+4082 D a b m \,x^{8}+31 A \,b^{2} m^{4} x^{4}+62 B a b \,m^{4} x^{4}+1366 B \,b^{2} m^{2} x^{6}+31 C \,a^{2} m^{4} x^{4}+2732 C a b \,m^{2} x^{6}+1155 C \,b^{2} x^{8}+1366 D a^{2} m^{2} x^{6}+2310 D a b \,x^{8}+2 A a b \,m^{5} x^{2}+350 A \,b^{2} m^{3} x^{4}+B \,a^{2} m^{5} x^{2}+700 B a b \,m^{3} x^{4}+2577 B \,b^{2} m \,x^{6}+350 C \,a^{2} m^{3} x^{4}+5154 C a b m \,x^{6}+2577 D a^{2} m \,x^{6}+66 A a b \,m^{4} x^{2}+1730 A \,b^{2} m^{2} x^{4}+33 B \,a^{2} m^{4} x^{2}+3460 B a b \,m^{2} x^{4}+1485 b^{2} B \,x^{6}+1730 C \,a^{2} m^{2} x^{4}+2970 b \,x^{6} C a +1485 D a^{2} x^{6}+A \,a^{2} m^{5}+812 A a b \,m^{3} x^{2}+3489 A \,b^{2} m \,x^{4}+406 B \,a^{2} m^{3} x^{2}+6978 B a b m \,x^{4}+3489 C \,a^{2} m \,x^{4}+35 A \,a^{2} m^{4}+4524 A a b \,m^{2} x^{2}+2079 A \,b^{2} x^{4}+2262 B \,a^{2} m^{2} x^{2}+4158 B a b \,x^{4}+2079 C \,a^{2} x^{4}+470 A \,a^{2} m^{3}+10706 A a b m \,x^{2}+5353 B \,a^{2} m \,x^{2}+3010 A \,a^{2} m^{2}+6930 a A b \,x^{2}+3465 B \,a^{2} x^{2}+9129 A \,a^{2} m +10395 a^{2} A \right ) \left (c x \right )^{m}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(793\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1153\) |
Input:
int((c*x)^m*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
x*(D*b^2*m^5*x^10+25*D*b^2*m^4*x^10+C*b^2*m^5*x^8+2*D*a*b*m^5*x^8+230*D*b^ 2*m^3*x^10+27*C*b^2*m^4*x^8+54*D*a*b*m^4*x^8+950*D*b^2*m^2*x^10+B*b^2*m^5* x^6+2*C*a*b*m^5*x^6+262*C*b^2*m^3*x^8+D*a^2*m^5*x^6+524*D*a*b*m^3*x^8+1689 *D*b^2*m*x^10+29*B*b^2*m^4*x^6+58*C*a*b*m^4*x^6+1122*C*b^2*m^2*x^8+29*D*a^ 2*m^4*x^6+2244*D*a*b*m^2*x^8+945*D*b^2*x^10+A*b^2*m^5*x^4+2*B*a*b*m^5*x^4+ 302*B*b^2*m^3*x^6+C*a^2*m^5*x^4+604*C*a*b*m^3*x^6+2041*C*b^2*m*x^8+302*D*a ^2*m^3*x^6+4082*D*a*b*m*x^8+31*A*b^2*m^4*x^4+62*B*a*b*m^4*x^4+1366*B*b^2*m ^2*x^6+31*C*a^2*m^4*x^4+2732*C*a*b*m^2*x^6+1155*C*b^2*x^8+1366*D*a^2*m^2*x ^6+2310*D*a*b*x^8+2*A*a*b*m^5*x^2+350*A*b^2*m^3*x^4+B*a^2*m^5*x^2+700*B*a* b*m^3*x^4+2577*B*b^2*m*x^6+350*C*a^2*m^3*x^4+5154*C*a*b*m*x^6+2577*D*a^2*m *x^6+66*A*a*b*m^4*x^2+1730*A*b^2*m^2*x^4+33*B*a^2*m^4*x^2+3460*B*a*b*m^2*x ^4+1485*B*b^2*x^6+1730*C*a^2*m^2*x^4+2970*C*a*b*x^6+1485*D*a^2*x^6+A*a^2*m ^5+812*A*a*b*m^3*x^2+3489*A*b^2*m*x^4+406*B*a^2*m^3*x^2+6978*B*a*b*m*x^4+3 489*C*a^2*m*x^4+35*A*a^2*m^4+4524*A*a*b*m^2*x^2+2079*A*b^2*x^4+2262*B*a^2* m^2*x^2+4158*B*a*b*x^4+2079*C*a^2*x^4+470*A*a^2*m^3+10706*A*a*b*m*x^2+5353 *B*a^2*m*x^2+3010*A*a^2*m^2+6930*A*a*b*x^2+3465*B*a^2*x^2+9129*A*a^2*m+103 95*A*a^2)*(c*x)^m/(11+m)/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)
Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (155) = 310\).
Time = 0.09 (sec) , antiderivative size = 575, normalized size of antiderivative = 3.71 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {{\left ({\left (D b^{2} m^{5} + 25 \, D b^{2} m^{4} + 230 \, D b^{2} m^{3} + 950 \, D b^{2} m^{2} + 1689 \, D b^{2} m + 945 \, D b^{2}\right )} x^{11} + {\left ({\left (2 \, D a b + C b^{2}\right )} m^{5} + 27 \, {\left (2 \, D a b + C b^{2}\right )} m^{4} + 262 \, {\left (2 \, D a b + C b^{2}\right )} m^{3} + 2310 \, D a b + 1155 \, C b^{2} + 1122 \, {\left (2 \, D a b + C b^{2}\right )} m^{2} + 2041 \, {\left (2 \, D a b + C b^{2}\right )} m\right )} x^{9} + {\left ({\left (D a^{2} + 2 \, C a b + B b^{2}\right )} m^{5} + 29 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} m^{4} + 302 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} m^{3} + 1485 \, D a^{2} + 2970 \, C a b + 1485 \, B b^{2} + 1366 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} m^{2} + 2577 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} m\right )} x^{7} + {\left ({\left (C a^{2} + 2 \, B a b + A b^{2}\right )} m^{5} + 31 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} m^{4} + 350 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} m^{3} + 2079 \, C a^{2} + 4158 \, B a b + 2079 \, A b^{2} + 1730 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} m^{2} + 3489 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} m\right )} x^{5} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{5} + 33 \, {\left (B a^{2} + 2 \, A a b\right )} m^{4} + 406 \, {\left (B a^{2} + 2 \, A a b\right )} m^{3} + 3465 \, B a^{2} + 6930 \, A a b + 2262 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 5353 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{3} + {\left (A a^{2} m^{5} + 35 \, A a^{2} m^{4} + 470 \, A a^{2} m^{3} + 3010 \, A a^{2} m^{2} + 9129 \, A a^{2} m + 10395 \, A a^{2}\right )} x\right )} \left (c x\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \] Input:
integrate((c*x)^m*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x, algorithm="fricas")
Output:
((D*b^2*m^5 + 25*D*b^2*m^4 + 230*D*b^2*m^3 + 950*D*b^2*m^2 + 1689*D*b^2*m + 945*D*b^2)*x^11 + ((2*D*a*b + C*b^2)*m^5 + 27*(2*D*a*b + C*b^2)*m^4 + 26 2*(2*D*a*b + C*b^2)*m^3 + 2310*D*a*b + 1155*C*b^2 + 1122*(2*D*a*b + C*b^2) *m^2 + 2041*(2*D*a*b + C*b^2)*m)*x^9 + ((D*a^2 + 2*C*a*b + B*b^2)*m^5 + 29 *(D*a^2 + 2*C*a*b + B*b^2)*m^4 + 302*(D*a^2 + 2*C*a*b + B*b^2)*m^3 + 1485* D*a^2 + 2970*C*a*b + 1485*B*b^2 + 1366*(D*a^2 + 2*C*a*b + B*b^2)*m^2 + 257 7*(D*a^2 + 2*C*a*b + B*b^2)*m)*x^7 + ((C*a^2 + 2*B*a*b + A*b^2)*m^5 + 31*( C*a^2 + 2*B*a*b + A*b^2)*m^4 + 350*(C*a^2 + 2*B*a*b + A*b^2)*m^3 + 2079*C* a^2 + 4158*B*a*b + 2079*A*b^2 + 1730*(C*a^2 + 2*B*a*b + A*b^2)*m^2 + 3489* (C*a^2 + 2*B*a*b + A*b^2)*m)*x^5 + ((B*a^2 + 2*A*a*b)*m^5 + 33*(B*a^2 + 2* A*a*b)*m^4 + 406*(B*a^2 + 2*A*a*b)*m^3 + 3465*B*a^2 + 6930*A*a*b + 2262*(B *a^2 + 2*A*a*b)*m^2 + 5353*(B*a^2 + 2*A*a*b)*m)*x^3 + (A*a^2*m^5 + 35*A*a^ 2*m^4 + 470*A*a^2*m^3 + 3010*A*a^2*m^2 + 9129*A*a^2*m + 10395*A*a^2)*x)*(c *x)^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)
Leaf count of result is larger than twice the leaf count of optimal. 4068 vs. \(2 (141) = 282\).
Time = 0.87 (sec) , antiderivative size = 4068, normalized size of antiderivative = 26.25 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\text {Too large to display} \] Input:
integrate((c*x)**m*(b*x**2+a)**2*(D*x**6+C*x**4+B*x**2+A),x)
Output:
Piecewise(((-A*a**2/(10*x**10) - A*a*b/(4*x**8) - A*b**2/(6*x**6) - B*a**2 /(8*x**8) - B*a*b/(3*x**6) - B*b**2/(4*x**4) - C*a**2/(6*x**6) - C*a*b/(2* x**4) - C*b**2/(2*x**2) - D*a**2/(4*x**4) - D*a*b/x**2 + D*b**2*log(x))/c* *11, Eq(m, -11)), ((-A*a**2/(8*x**8) - A*a*b/(3*x**6) - A*b**2/(4*x**4) - B*a**2/(6*x**6) - B*a*b/(2*x**4) - B*b**2/(2*x**2) - C*a**2/(4*x**4) - C*a *b/x**2 + C*b**2*log(x) - D*a**2/(2*x**2) + 2*D*a*b*log(x) + D*b**2*x**2/2 )/c**9, Eq(m, -9)), ((-A*a**2/(6*x**6) - A*a*b/(2*x**4) - A*b**2/(2*x**2) - B*a**2/(4*x**4) - B*a*b/x**2 + B*b**2*log(x) - C*a**2/(2*x**2) + 2*C*a*b *log(x) + C*b**2*x**2/2 + D*a**2*log(x) + D*a*b*x**2 + D*b**2*x**4/4)/c**7 , Eq(m, -7)), ((-A*a**2/(4*x**4) - A*a*b/x**2 + A*b**2*log(x) - B*a**2/(2* x**2) + 2*B*a*b*log(x) + B*b**2*x**2/2 + C*a**2*log(x) + C*a*b*x**2 + C*b* *2*x**4/4 + D*a**2*x**2/2 + D*a*b*x**4/2 + D*b**2*x**6/6)/c**5, Eq(m, -5)) , ((-A*a**2/(2*x**2) + 2*A*a*b*log(x) + A*b**2*x**2/2 + B*a**2*log(x) + B* a*b*x**2 + B*b**2*x**4/4 + C*a**2*x**2/2 + C*a*b*x**4/2 + C*b**2*x**6/6 + D*a**2*x**4/4 + D*a*b*x**6/3 + D*b**2*x**8/8)/c**3, Eq(m, -3)), ((A*a**2*l og(x) + A*a*b*x**2 + A*b**2*x**4/4 + B*a**2*x**2/2 + B*a*b*x**4/2 + B*b**2 *x**6/6 + C*a**2*x**4/4 + C*a*b*x**6/3 + C*b**2*x**8/8 + D*a**2*x**6/6 + D *a*b*x**8/4 + D*b**2*x**10/10)/c, Eq(m, -1)), (A*a**2*m**5*x*(c*x)**m/(m** 6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 35*A* a**2*m**4*x*(c*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**...
Time = 0.06 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.48 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {D b^{2} c^{m} x^{11} x^{m}}{m + 11} + \frac {2 \, D a b c^{m} x^{9} x^{m}}{m + 9} + \frac {C b^{2} c^{m} x^{9} x^{m}}{m + 9} + \frac {D a^{2} c^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, C a b c^{m} x^{7} x^{m}}{m + 7} + \frac {B b^{2} c^{m} x^{7} x^{m}}{m + 7} + \frac {C a^{2} c^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, B a b c^{m} x^{5} x^{m}}{m + 5} + \frac {A b^{2} c^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{2} c^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a b c^{m} x^{3} x^{m}}{m + 3} + \frac {\left (c x\right )^{m + 1} A a^{2}}{c {\left (m + 1\right )}} \] Input:
integrate((c*x)^m*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x, algorithm="maxima")
Output:
D*b^2*c^m*x^11*x^m/(m + 11) + 2*D*a*b*c^m*x^9*x^m/(m + 9) + C*b^2*c^m*x^9* x^m/(m + 9) + D*a^2*c^m*x^7*x^m/(m + 7) + 2*C*a*b*c^m*x^7*x^m/(m + 7) + B* b^2*c^m*x^7*x^m/(m + 7) + C*a^2*c^m*x^5*x^m/(m + 5) + 2*B*a*b*c^m*x^5*x^m/ (m + 5) + A*b^2*c^m*x^5*x^m/(m + 5) + B*a^2*c^m*x^3*x^m/(m + 3) + 2*A*a*b* c^m*x^3*x^m/(m + 3) + (c*x)^(m + 1)*A*a^2/(c*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 1152 vs. \(2 (155) = 310\).
Time = 0.15 (sec) , antiderivative size = 1152, normalized size of antiderivative = 7.43 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\text {Too large to display} \] Input:
integrate((c*x)^m*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x, algorithm="giac")
Output:
((c*x)^m*D*b^2*m^5*x^11 + 25*(c*x)^m*D*b^2*m^4*x^11 + 2*(c*x)^m*D*a*b*m^5* x^9 + (c*x)^m*C*b^2*m^5*x^9 + 230*(c*x)^m*D*b^2*m^3*x^11 + 54*(c*x)^m*D*a* b*m^4*x^9 + 27*(c*x)^m*C*b^2*m^4*x^9 + 950*(c*x)^m*D*b^2*m^2*x^11 + (c*x)^ m*D*a^2*m^5*x^7 + 2*(c*x)^m*C*a*b*m^5*x^7 + (c*x)^m*B*b^2*m^5*x^7 + 524*(c *x)^m*D*a*b*m^3*x^9 + 262*(c*x)^m*C*b^2*m^3*x^9 + 1689*(c*x)^m*D*b^2*m*x^1 1 + 29*(c*x)^m*D*a^2*m^4*x^7 + 58*(c*x)^m*C*a*b*m^4*x^7 + 29*(c*x)^m*B*b^2 *m^4*x^7 + 2244*(c*x)^m*D*a*b*m^2*x^9 + 1122*(c*x)^m*C*b^2*m^2*x^9 + 945*( c*x)^m*D*b^2*x^11 + (c*x)^m*C*a^2*m^5*x^5 + 2*(c*x)^m*B*a*b*m^5*x^5 + (c*x )^m*A*b^2*m^5*x^5 + 302*(c*x)^m*D*a^2*m^3*x^7 + 604*(c*x)^m*C*a*b*m^3*x^7 + 302*(c*x)^m*B*b^2*m^3*x^7 + 4082*(c*x)^m*D*a*b*m*x^9 + 2041*(c*x)^m*C*b^ 2*m*x^9 + 31*(c*x)^m*C*a^2*m^4*x^5 + 62*(c*x)^m*B*a*b*m^4*x^5 + 31*(c*x)^m *A*b^2*m^4*x^5 + 1366*(c*x)^m*D*a^2*m^2*x^7 + 2732*(c*x)^m*C*a*b*m^2*x^7 + 1366*(c*x)^m*B*b^2*m^2*x^7 + 2310*(c*x)^m*D*a*b*x^9 + 1155*(c*x)^m*C*b^2* x^9 + (c*x)^m*B*a^2*m^5*x^3 + 2*(c*x)^m*A*a*b*m^5*x^3 + 350*(c*x)^m*C*a^2* m^3*x^5 + 700*(c*x)^m*B*a*b*m^3*x^5 + 350*(c*x)^m*A*b^2*m^3*x^5 + 2577*(c* x)^m*D*a^2*m*x^7 + 5154*(c*x)^m*C*a*b*m*x^7 + 2577*(c*x)^m*B*b^2*m*x^7 + 3 3*(c*x)^m*B*a^2*m^4*x^3 + 66*(c*x)^m*A*a*b*m^4*x^3 + 1730*(c*x)^m*C*a^2*m^ 2*x^5 + 3460*(c*x)^m*B*a*b*m^2*x^5 + 1730*(c*x)^m*A*b^2*m^2*x^5 + 1485*(c* x)^m*D*a^2*x^7 + 2970*(c*x)^m*C*a*b*x^7 + 1485*(c*x)^m*B*b^2*x^7 + (c*x)^m *A*a^2*m^5*x + 406*(c*x)^m*B*a^2*m^3*x^3 + 812*(c*x)^m*A*a*b*m^3*x^3 + ...
Timed out. \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\int {\left (c\,x\right )}^m\,{\left (b\,x^2+a\right )}^2\,\left (A+B\,x^2+C\,x^4+x^6\,D\right ) \,d x \] Input:
int((c*x)^m*(a + b*x^2)^2*(A + B*x^2 + C*x^4 + x^6*D),x)
Output:
int((c*x)^m*(a + b*x^2)^2*(A + B*x^2 + C*x^4 + x^6*D), x)
Time = 0.15 (sec) , antiderivative size = 661, normalized size of antiderivative = 4.26 \[ \int (c x)^m \left (a+b x^2\right )^2 \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {x^{m} c^{m} x \left (b^{2} d \,m^{5} x^{10}+25 b^{2} d \,m^{4} x^{10}+2 a b d \,m^{5} x^{8}+b^{2} c \,m^{5} x^{8}+230 b^{2} d \,m^{3} x^{10}+54 a b d \,m^{4} x^{8}+27 b^{2} c \,m^{4} x^{8}+950 b^{2} d \,m^{2} x^{10}+a^{2} d \,m^{5} x^{6}+2 a b c \,m^{5} x^{6}+524 a b d \,m^{3} x^{8}+b^{3} m^{5} x^{6}+262 b^{2} c \,m^{3} x^{8}+1689 b^{2} d m \,x^{10}+29 a^{2} d \,m^{4} x^{6}+58 a b c \,m^{4} x^{6}+2244 a b d \,m^{2} x^{8}+29 b^{3} m^{4} x^{6}+1122 b^{2} c \,m^{2} x^{8}+945 b^{2} d \,x^{10}+a^{2} c \,m^{5} x^{4}+302 a^{2} d \,m^{3} x^{6}+3 a \,b^{2} m^{5} x^{4}+604 a b c \,m^{3} x^{6}+4082 a b d m \,x^{8}+302 b^{3} m^{3} x^{6}+2041 b^{2} c m \,x^{8}+31 a^{2} c \,m^{4} x^{4}+1366 a^{2} d \,m^{2} x^{6}+93 a \,b^{2} m^{4} x^{4}+2732 a b c \,m^{2} x^{6}+2310 a b d \,x^{8}+1366 b^{3} m^{2} x^{6}+1155 b^{2} c \,x^{8}+3 a^{2} b \,m^{5} x^{2}+350 a^{2} c \,m^{3} x^{4}+2577 a^{2} d m \,x^{6}+1050 a \,b^{2} m^{3} x^{4}+5154 a b c m \,x^{6}+2577 b^{3} m \,x^{6}+99 a^{2} b \,m^{4} x^{2}+1730 a^{2} c \,m^{2} x^{4}+1485 a^{2} d \,x^{6}+5190 a \,b^{2} m^{2} x^{4}+2970 a b c \,x^{6}+1485 b^{3} x^{6}+a^{3} m^{5}+1218 a^{2} b \,m^{3} x^{2}+3489 a^{2} c m \,x^{4}+10467 a \,b^{2} m \,x^{4}+35 a^{3} m^{4}+6786 a^{2} b \,m^{2} x^{2}+2079 a^{2} c \,x^{4}+6237 a \,b^{2} x^{4}+470 a^{3} m^{3}+16059 a^{2} b m \,x^{2}+3010 a^{3} m^{2}+10395 a^{2} b \,x^{2}+9129 a^{3} m +10395 a^{3}\right )}{m^{6}+36 m^{5}+505 m^{4}+3480 m^{3}+12139 m^{2}+19524 m +10395} \] Input:
int((c*x)^m*(b*x^2+a)^2*(D*x^6+C*x^4+B*x^2+A),x)
Output:
(x**m*c**m*x*(a**3*m**5 + 35*a**3*m**4 + 470*a**3*m**3 + 3010*a**3*m**2 + 9129*a**3*m + 10395*a**3 + 3*a**2*b*m**5*x**2 + 99*a**2*b*m**4*x**2 + 1218 *a**2*b*m**3*x**2 + 6786*a**2*b*m**2*x**2 + 16059*a**2*b*m*x**2 + 10395*a* *2*b*x**2 + a**2*c*m**5*x**4 + 31*a**2*c*m**4*x**4 + 350*a**2*c*m**3*x**4 + 1730*a**2*c*m**2*x**4 + 3489*a**2*c*m*x**4 + 2079*a**2*c*x**4 + a**2*d*m **5*x**6 + 29*a**2*d*m**4*x**6 + 302*a**2*d*m**3*x**6 + 1366*a**2*d*m**2*x **6 + 2577*a**2*d*m*x**6 + 1485*a**2*d*x**6 + 3*a*b**2*m**5*x**4 + 93*a*b* *2*m**4*x**4 + 1050*a*b**2*m**3*x**4 + 5190*a*b**2*m**2*x**4 + 10467*a*b** 2*m*x**4 + 6237*a*b**2*x**4 + 2*a*b*c*m**5*x**6 + 58*a*b*c*m**4*x**6 + 604 *a*b*c*m**3*x**6 + 2732*a*b*c*m**2*x**6 + 5154*a*b*c*m*x**6 + 2970*a*b*c*x **6 + 2*a*b*d*m**5*x**8 + 54*a*b*d*m**4*x**8 + 524*a*b*d*m**3*x**8 + 2244* a*b*d*m**2*x**8 + 4082*a*b*d*m*x**8 + 2310*a*b*d*x**8 + b**3*m**5*x**6 + 2 9*b**3*m**4*x**6 + 302*b**3*m**3*x**6 + 1366*b**3*m**2*x**6 + 2577*b**3*m* x**6 + 1485*b**3*x**6 + b**2*c*m**5*x**8 + 27*b**2*c*m**4*x**8 + 262*b**2* c*m**3*x**8 + 1122*b**2*c*m**2*x**8 + 2041*b**2*c*m*x**8 + 1155*b**2*c*x** 8 + b**2*d*m**5*x**10 + 25*b**2*d*m**4*x**10 + 230*b**2*d*m**3*x**10 + 950 *b**2*d*m**2*x**10 + 1689*b**2*d*m*x**10 + 945*b**2*d*x**10))/(m**6 + 36*m **5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)