Integrand size = 22, antiderivative size = 111 \[ \int (c x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {a^2 A (c x)^{1+m}}{c (1+m)}+\frac {a (2 A b+a B) x^n (c x)^{1+m}}{c (1+m+n)}+\frac {b (A b+2 a B) x^{2 n} (c x)^{1+m}}{c (1+m+2 n)}+\frac {b^2 B x^{3 n} (c x)^{1+m}}{c (1+m+3 n)} \] Output:
a^2*A*(c*x)^(1+m)/c/(1+m)+a*(2*A*b+B*a)*x^n*(c*x)^(1+m)/c/(1+m+n)+b*(A*b+2 *B*a)*x^(2*n)*(c*x)^(1+m)/c/(1+m+2*n)+b^2*B*x^(3*n)*(c*x)^(1+m)/c/(1+m+3*n )
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.70 \[ \int (c x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=x (c x)^m \left (\frac {a^2 A}{1+m}+\frac {a (2 A b+a B) x^n}{1+m+n}+\frac {b (A b+2 a B) x^{2 n}}{1+m+2 n}+\frac {b^2 B x^{3 n}}{1+m+3 n}\right ) \] Input:
Integrate[(c*x)^m*(a + b*x^n)^2*(A + B*x^n),x]
Output:
x*(c*x)^m*((a^2*A)/(1 + m) + (a*(2*A*b + a*B)*x^n)/(1 + m + n) + (b*(A*b + 2*a*B)*x^(2*n))/(1 + m + 2*n) + (b^2*B*x^(3*n))/(1 + m + 3*n))
Time = 0.42 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {950, 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.
| \(\displaystyle \int (c x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx\) |
| \(\Big \downarrow \) 950 |
| \(\displaystyle \int \left (a^2 A (c x)^m+b x^{2 n} (c x)^m (2 a B+A b)+a x^n (c x)^m (a B+2 A b)+b^2 B x^{3 n} (c x)^m\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 A (c x)^{m+1}}{c (m+1)}+\frac {a x^{n+1} (c x)^m (a B+2 A b)}{m+n+1}+\frac {b x^{2 n+1} (c x)^m (2 a B+A b)}{m+2 n+1}+\frac {b^2 B x^{3 n+1} (c x)^m}{m+3 n+1}\) |
Input:
Int[(c*x)^m*(a + b*x^n)^2*(A + B*x^n),x]
Output:
(a*(2*A*b + a*B)*x^(1 + n)*(c*x)^m)/(1 + m + n) + (b*(A*b + 2*a*B)*x^(1 + 2*n)*(c*x)^m)/(1 + m + 2*n) + (b^2*B*x^(1 + 3*n)*(c*x)^m)/(1 + m + 3*n) + (a^2*A*(c*x)^(1 + m))/(c*(1 + m))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^ n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGt Q[p, 0] && IGtQ[q, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.53 (sec) , antiderivative size = 699, normalized size of antiderivative = 6.30
| method | result | size |
| risch | \(\frac {x \left (3 B \,b^{2} m^{2} x^{3 n}+3 A \,b^{2} m^{2} x^{2 n}+3 m \,b^{2} B \,x^{3 n}+3 A \,b^{2} x^{2 n} m +B \,b^{2} m^{3} x^{3 n}+A \,b^{2} m^{3} x^{2 n}+a^{2} A +6 A a b \,m^{2} x^{n}+10 B \,a^{2} m n \,x^{n}+6 A a b \,x^{n} m +20 A a b m n \,x^{n}+10 A a b \,m^{2} n \,x^{n}+2 a b A \,x^{n}+6 B \,x^{n} a^{2} n^{2}+5 B \,x^{n} a^{2} n +6 A \,a^{2} m^{2} n +11 A \,a^{2} m \,n^{2}+12 A \,a^{2} m n +a^{2} B \,x^{n}+6 A \,a^{2} n^{3}+11 A \,a^{2} n^{2}+6 A \,a^{2} n +2 B \,x^{3 n} b^{2} n^{2}+3 A \,x^{2 n} b^{2} n^{2}+3 B \,x^{3 n} b^{2} n +4 A \,x^{2 n} b^{2} n +A \,a^{2} m^{3}+3 A \,a^{2} m^{2}+B \,a^{2} m^{3} x^{n}+3 B \,a^{2} x^{n} m +3 B \,a^{2} m^{2} x^{n}+b^{2} B \,x^{3 n}+A \,b^{2} x^{2 n}+3 a^{2} A m +2 A a b \,m^{3} x^{n}+5 B \,a^{2} m^{2} n \,x^{n}+6 B \,a^{2} m \,n^{2} x^{n}+12 A \,x^{n} a b \,n^{2}+12 A a b m \,n^{2} x^{n}+10 A \,x^{n} a b n +2 B a b \,x^{2 n}+6 B \,x^{2 n} a b \,n^{2}+8 B \,x^{2 n} a b n +3 B \,b^{2} m^{2} n \,x^{3 n}+2 B \,b^{2} m \,n^{2} x^{3 n}+6 B a b \,x^{2 n} m +8 A \,b^{2} m n \,x^{2 n}+6 B a b \,m^{2} x^{2 n}+4 A \,b^{2} m^{2} n \,x^{2 n}+3 A \,b^{2} m \,n^{2} x^{2 n}+2 B a b \,m^{3} x^{2 n}+6 B \,b^{2} m n \,x^{3 n}+16 B a b m n \,x^{2 n}+8 B a b \,m^{2} n \,x^{2 n}+6 B a b m \,n^{2} x^{2 n}\right ) c^{m} x^{m} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c x \right ) m \left (\operatorname {csgn}\left (i c x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i c x \right )+\operatorname {csgn}\left (i c \right )\right )}{2}}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right ) \left (1+m +3 n \right )}\) | \(699\) |
| parallelrisch | \(\frac {8 B x \,x^{2 n} \left (c x \right )^{m} a b n +3 B x \,x^{3 n} \left (c x \right )^{m} b^{2} m^{2} n +2 B x \,x^{3 n} \left (c x \right )^{m} b^{2} m \,n^{2}+4 A x \,x^{2 n} \left (c x \right )^{m} b^{2} m^{2} n +3 A x \,x^{2 n} \left (c x \right )^{m} b^{2} m \,n^{2}+6 B x \,x^{3 n} \left (c x \right )^{m} b^{2} m n +2 B x \,x^{2 n} \left (c x \right )^{m} a b \,m^{3}+8 A x \,x^{2 n} \left (c x \right )^{m} b^{2} m n +6 B x \,x^{2 n} \left (c x \right )^{m} a b \,m^{2}+6 B x \,x^{2 n} \left (c x \right )^{m} a b \,n^{2}+A x \left (c x \right )^{m} a^{2}+8 B x \,x^{2 n} \left (c x \right )^{m} a b \,m^{2} n +6 B x \,x^{2 n} \left (c x \right )^{m} a b m \,n^{2}+16 B x \,x^{2 n} \left (c x \right )^{m} a b m n +12 A x \,x^{n} \left (c x \right )^{m} a b \,n^{2}+10 B x \,x^{n} \left (c x \right )^{m} a^{2} m n +6 A x \,x^{n} \left (c x \right )^{m} a b m +10 A x \,x^{n} \left (c x \right )^{m} a b n +B x \,x^{n} \left (c x \right )^{m} a^{2} m^{3}+6 A x \left (c x \right )^{m} a^{2} m^{2} n +11 A x \left (c x \right )^{m} a^{2} m \,n^{2}+3 B x \,x^{n} \left (c x \right )^{m} a^{2} m^{2}+6 B x \,x^{n} \left (c x \right )^{m} a^{2} n^{2}+12 A x \left (c x \right )^{m} a^{2} m n +3 B x \,x^{n} \left (c x \right )^{m} a^{2} m +5 B x \,x^{n} \left (c x \right )^{m} a^{2} n +2 A x \,x^{n} \left (c x \right )^{m} a b +6 B x \,x^{2 n} \left (c x \right )^{m} a b m +2 A x \,x^{n} \left (c x \right )^{m} a b \,m^{3}+5 B x \,x^{n} \left (c x \right )^{m} a^{2} m^{2} n +6 B x \,x^{n} \left (c x \right )^{m} a^{2} m \,n^{2}+6 A x \,x^{n} \left (c x \right )^{m} a b \,m^{2}+10 A x \,x^{n} \left (c x \right )^{m} a b \,m^{2} n +12 A x \,x^{n} \left (c x \right )^{m} a b m \,n^{2}+20 A x \,x^{n} \left (c x \right )^{m} a b m n +A x \left (c x \right )^{m} a^{2} m^{3}+6 A x \left (c x \right )^{m} a^{2} n^{3}+3 A x \left (c x \right )^{m} a^{2} m^{2}+11 A x \left (c x \right )^{m} a^{2} n^{2}+3 A x \left (c x \right )^{m} a^{2} m +6 A x \left (c x \right )^{m} a^{2} n +B x \,x^{n} \left (c x \right )^{m} a^{2}+2 B x \,x^{3 n} \left (c x \right )^{m} b^{2} n^{2}+3 A x \,x^{2 n} \left (c x \right )^{m} b^{2} m^{2}+3 A x \,x^{2 n} \left (c x \right )^{m} b^{2} n^{2}+3 B x \,x^{3 n} \left (c x \right )^{m} b^{2} m +3 B x \,x^{3 n} \left (c x \right )^{m} b^{2} n +3 A x \,x^{2 n} \left (c x \right )^{m} b^{2} m +4 A x \,x^{2 n} \left (c x \right )^{m} b^{2} n +2 B x \,x^{2 n} \left (c x \right )^{m} a b +B x \,x^{3 n} \left (c x \right )^{m} b^{2} m^{3}+A x \,x^{2 n} \left (c x \right )^{m} b^{2} m^{3}+3 B x \,x^{3 n} \left (c x \right )^{m} b^{2} m^{2}+B x \,x^{3 n} \left (c x \right )^{m} b^{2}+A x \,x^{2 n} \left (c x \right )^{m} b^{2}}{\left (1+m \right ) \left (1+m +n \right ) \left (1+m +2 n \right ) \left (1+m +3 n \right )}\) | \(982\) |
| orering | \(\text {Expression too large to display}\) | \(1351\) |
Input:
int((c*x)^m*(a+b*x^n)^2*(A+B*x^n),x,method=_RETURNVERBOSE)
Output:
x*(3*B*b^2*m^2*n*(x^n)^3+2*B*b^2*m*n^2*(x^n)^3+a^2*A+6*A*a*b*m^2*x^n+10*B* a^2*m*n*x^n+6*B*a*b*(x^n)^2*m+6*A*a*b*x^n*m+16*B*a*b*m*n*(x^n)^2+20*A*a*b* m*n*x^n+8*B*a*b*m^2*n*(x^n)^2+6*B*a*b*m*n^2*(x^n)^2+10*A*a*b*m^2*n*x^n+2*B *a*b*(x^n)^2+2*a*b*A*x^n+2*B*(x^n)^3*b^2*n^2+3*A*(x^n)^2*b^2*n^2+3*B*(x^n) ^3*b^2*n+4*A*(x^n)^2*b^2*n+6*B*x^n*a^2*n^2+5*B*x^n*a^2*n+6*A*a^2*m^2*n+11* A*a^2*m*n^2+12*A*a^2*m*n+a^2*B*x^n+6*A*a^2*n^3+11*A*a^2*n^2+6*A*a^2*n+b^2* B*(x^n)^3+A*a^2*m^3+3*A*a^2*m^2+3*B*b^2*m^2*(x^n)^3+3*A*b^2*m^2*(x^n)^2+B* a^2*m^3*x^n+3*m*b^2*B*(x^n)^3+3*A*b^2*(x^n)^2*m+3*B*a^2*x^n*m+B*b^2*m^3*(x ^n)^3+A*b^2*m^3*(x^n)^2+3*B*a^2*m^2*x^n+A*b^2*(x^n)^2+3*a^2*A*m+2*A*a*b*m^ 3*x^n+8*A*b^2*m*n*(x^n)^2+5*B*a^2*m^2*n*x^n+6*B*a^2*m*n^2*x^n+6*B*a*b*m^2* (x^n)^2+6*B*(x^n)^2*a*b*n^2+12*A*x^n*a*b*n^2+12*A*a*b*m*n^2*x^n+8*B*(x^n)^ 2*a*b*n+10*A*x^n*a*b*n+4*A*b^2*m^2*n*(x^n)^2+3*A*b^2*m*n^2*(x^n)^2+2*B*a*b *m^3*(x^n)^2+6*B*b^2*m*n*(x^n)^3)/(1+m)/(1+m+n)/(1+m+2*n)/(1+m+3*n)*c^m*x^ m*exp(1/2*I*Pi*csgn(I*c*x)*m*(csgn(I*c*x)-csgn(I*x))*(-csgn(I*c*x)+csgn(I* c)))
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (111) = 222\).
Time = 0.10 (sec) , antiderivative size = 527, normalized size of antiderivative = 4.75 \[ \int (c x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {{\left (B b^{2} m^{3} + 3 \, B b^{2} m^{2} + 3 \, B b^{2} m + B b^{2} + 2 \, {\left (B b^{2} m + B b^{2}\right )} n^{2} + 3 \, {\left (B b^{2} m^{2} + 2 \, B b^{2} m + B b^{2}\right )} n\right )} x x^{3 \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + {\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 2 \, B a b + A b^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} m^{2} + 3 \, {\left (2 \, B a b + A b^{2} + {\left (2 \, B a b + A b^{2}\right )} m\right )} n^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} m + 4 \, {\left (2 \, B a b + A b^{2} + {\left (2 \, B a b + A b^{2}\right )} m^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} m\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + B a^{2} + 2 \, A a b + 3 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 6 \, {\left (B a^{2} + 2 \, A a b + {\left (B a^{2} + 2 \, A a b\right )} m\right )} n^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} m + 5 \, {\left (B a^{2} + 2 \, A a b + {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )} + {\left (A a^{2} m^{3} + 6 \, A a^{2} n^{3} + 3 \, A a^{2} m^{2} + 3 \, A a^{2} m + A a^{2} + 11 \, {\left (A a^{2} m + A a^{2}\right )} n^{2} + 6 \, {\left (A a^{2} m^{2} + 2 \, A a^{2} m + A a^{2}\right )} n\right )} x e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{m^{4} + 6 \, {\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \, {\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \, {\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \] Input:
integrate((c*x)^m*(a+b*x^n)^2*(A+B*x^n),x, algorithm="fricas")
Output:
((B*b^2*m^3 + 3*B*b^2*m^2 + 3*B*b^2*m + B*b^2 + 2*(B*b^2*m + B*b^2)*n^2 + 3*(B*b^2*m^2 + 2*B*b^2*m + B*b^2)*n)*x*x^(3*n)*e^(m*log(c) + m*log(x)) + ( (2*B*a*b + A*b^2)*m^3 + 2*B*a*b + A*b^2 + 3*(2*B*a*b + A*b^2)*m^2 + 3*(2*B *a*b + A*b^2 + (2*B*a*b + A*b^2)*m)*n^2 + 3*(2*B*a*b + A*b^2)*m + 4*(2*B*a *b + A*b^2 + (2*B*a*b + A*b^2)*m^2 + 2*(2*B*a*b + A*b^2)*m)*n)*x*x^(2*n)*e ^(m*log(c) + m*log(x)) + ((B*a^2 + 2*A*a*b)*m^3 + B*a^2 + 2*A*a*b + 3*(B*a ^2 + 2*A*a*b)*m^2 + 6*(B*a^2 + 2*A*a*b + (B*a^2 + 2*A*a*b)*m)*n^2 + 3*(B*a ^2 + 2*A*a*b)*m + 5*(B*a^2 + 2*A*a*b + (B*a^2 + 2*A*a*b)*m^2 + 2*(B*a^2 + 2*A*a*b)*m)*n)*x*x^n*e^(m*log(c) + m*log(x)) + (A*a^2*m^3 + 6*A*a^2*n^3 + 3*A*a^2*m^2 + 3*A*a^2*m + A*a^2 + 11*(A*a^2*m + A*a^2)*n^2 + 6*(A*a^2*m^2 + 2*A*a^2*m + A*a^2)*n)*x*e^(m*log(c) + m*log(x)))/(m^4 + 6*(m + 1)*n^3 + 4*m^3 + 11*(m^2 + 2*m + 1)*n^2 + 6*m^2 + 6*(m^3 + 3*m^2 + 3*m + 1)*n + 4*m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 5882 vs. \(2 (99) = 198\).
Time = 3.83 (sec) , antiderivative size = 5882, normalized size of antiderivative = 52.99 \[ \int (c x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\text {Too large to display} \] Input:
integrate((c*x)**m*(a+b*x**n)**2*(A+B*x**n),x)
Output:
Piecewise(((A + B)*(a + b)**2*log(x)/c, Eq(m, -1) & Eq(n, 0)), ((A*a**2*lo g(x) + 2*A*a*b*x**n/n + A*b**2*x**(2*n)/(2*n) + B*a**2*x**n/n + B*a*b*x**( 2*n)/n + B*b**2*x**(3*n)/(3*n))/c, Eq(m, -1)), (A*a**2*Piecewise((0**(-3*n - 1)*x, Eq(c, 0)), (Piecewise((-1/(3*n*(c*x)**(3*n)), Ne(n, 0)), (log(c*x ), True))/c, True)) + 2*A*a*b*Piecewise((-x*x**n*(c*x)**(-3*n - 1)/(2*n), Ne(n, 0)), (x*x**n*(c*x)**(-3*n - 1)*log(x), True)) + A*b**2*Piecewise((-x *x**(2*n)*(c*x)**(-3*n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(c*x)**(-3*n - 1)*lo g(x), True)) + B*a**2*Piecewise((-x*x**n*(c*x)**(-3*n - 1)/(2*n), Ne(n, 0) ), (x*x**n*(c*x)**(-3*n - 1)*log(x), True)) + 2*B*a*b*Piecewise((-x*x**(2* n)*(c*x)**(-3*n - 1)/n, Ne(n, 0)), (x*x**(2*n)*(c*x)**(-3*n - 1)*log(x), T rue)) + B*b**2*x*x**(3*n)*(c*x)**(-3*n - 1)*log(x), Eq(m, -3*n - 1)), (A*a **2*Piecewise((0**(-2*n - 1)*x, Eq(c, 0)), (Piecewise((-1/(2*n*(c*x)**(2*n )), Ne(n, 0)), (log(c*x), True))/c, True)) + 2*A*a*b*Piecewise((-x*x**n*(c *x)**(-2*n - 1)/n, Ne(n, 0)), (x*x**n*(c*x)**(-2*n - 1)*log(x), True)) + A *b**2*x*x**(2*n)*(c*x)**(-2*n - 1)*log(x) + B*a**2*Piecewise((-x*x**n*(c*x )**(-2*n - 1)/n, Ne(n, 0)), (x*x**n*(c*x)**(-2*n - 1)*log(x), True)) + 2*B *a*b*x*x**(2*n)*(c*x)**(-2*n - 1)*log(x) + B*b**2*Piecewise((x*x**(3*n)*(c *x)**(-2*n - 1)/n, Ne(n, 0)), (x*x**(3*n)*(c*x)**(-2*n - 1)*log(x), True)) , Eq(m, -2*n - 1)), (A*a**2*Piecewise((0**(-n - 1)*x, Eq(c, 0)), (Piecewis e((-1/(n*(c*x)**n), Ne(n, 0)), (log(c*x), True))/c, True)) + 2*A*a*b*x*...
Time = 0.04 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.40 \[ \int (c x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {B b^{2} c^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {2 \, B a b c^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {A b^{2} c^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B a^{2} c^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {2 \, A a b c^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (c x\right )^{m + 1} A a^{2}}{c {\left (m + 1\right )}} \] Input:
integrate((c*x)^m*(a+b*x^n)^2*(A+B*x^n),x, algorithm="maxima")
Output:
B*b^2*c^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 2*B*a*b*c^m*x*e^(m*l og(x) + 2*n*log(x))/(m + 2*n + 1) + A*b^2*c^m*x*e^(m*log(x) + 2*n*log(x))/ (m + 2*n + 1) + B*a^2*c^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + 2*A*a*b* c^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + (c*x)^(m + 1)*A*a^2/(c*(m + 1) )
Leaf count of result is larger than twice the leaf count of optimal. 2951 vs. \(2 (111) = 222\).
Time = 0.15 (sec) , antiderivative size = 2951, normalized size of antiderivative = 26.59 \[ \int (c x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\text {Too large to display} \] Input:
integrate((c*x)^m*(a+b*x^n)^2*(A+B*x^n),x, algorithm="giac")
Output:
(B*b^2*m^3*x*x^(3*n)*e^(m*log(c) + m*log(x)) + 3*B*b^2*m^2*n*x*x^(3*n)*e^( m*log(c) + m*log(x)) + 2*B*b^2*m*n^2*x*x^(3*n)*e^(m*log(c) + m*log(x)) + 2 *B*a*b*m^3*x*x^(2*n)*e^(m*log(c) + m*log(x)) + A*b^2*m^3*x*x^(2*n)*e^(m*lo g(c) + m*log(x)) + B*b^2*m^3*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 8*B*a*b*m ^2*n*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 4*A*b^2*m^2*n*x*x^(2*n)*e^(m*log( c) + m*log(x)) + 3*B*b^2*m^2*n*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 6*B*a*b *m*n^2*x*x^(2*n)*e^(m*log(c) + m*log(x)) + 3*A*b^2*m*n^2*x*x^(2*n)*e^(m*lo g(c) + m*log(x)) + 2*B*b^2*m*n^2*x*x^(2*n)*e^(m*log(c) + m*log(x)) + B*a^2 *m^3*x*x^n*e^(m*log(c) + m*log(x)) + 2*A*a*b*m^3*x*x^n*e^(m*log(c) + m*log (x)) + 2*B*a*b*m^3*x*x^n*e^(m*log(c) + m*log(x)) + A*b^2*m^3*x*x^n*e^(m*lo g(c) + m*log(x)) + B*b^2*m^3*x*x^n*e^(m*log(c) + m*log(x)) + 5*B*a^2*m^2*n *x*x^n*e^(m*log(c) + m*log(x)) + 10*A*a*b*m^2*n*x*x^n*e^(m*log(c) + m*log( x)) + 8*B*a*b*m^2*n*x*x^n*e^(m*log(c) + m*log(x)) + 4*A*b^2*m^2*n*x*x^n*e^ (m*log(c) + m*log(x)) + 3*B*b^2*m^2*n*x*x^n*e^(m*log(c) + m*log(x)) + 6*B* a^2*m*n^2*x*x^n*e^(m*log(c) + m*log(x)) + 12*A*a*b*m*n^2*x*x^n*e^(m*log(c) + m*log(x)) + 6*B*a*b*m*n^2*x*x^n*e^(m*log(c) + m*log(x)) + 3*A*b^2*m*n^2 *x*x^n*e^(m*log(c) + m*log(x)) + 2*B*b^2*m*n^2*x*x^n*e^(m*log(c) + m*log(x )) + A*a^2*m^3*x*e^(m*log(c) + m*log(x)) + B*a^2*m^3*x*e^(m*log(c) + m*log (x)) + 2*A*a*b*m^3*x*e^(m*log(c) + m*log(x)) + 2*B*a*b*m^3*x*e^(m*log(c) + m*log(x)) + A*b^2*m^3*x*e^(m*log(c) + m*log(x)) + B*b^2*m^3*x*e^(m*log...
Time = 3.87 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.39 \[ \int (c x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {A\,a^2\,x\,{\left (c\,x\right )}^m}{m+1}+\frac {a\,x\,x^n\,{\left (c\,x\right )}^m\,\left (2\,A\,b+B\,a\right )\,\left (m^2+5\,m\,n+2\,m+6\,n^2+5\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1}+\frac {b\,x\,x^{2\,n}\,{\left (c\,x\right )}^m\,\left (A\,b+2\,B\,a\right )\,\left (m^2+4\,m\,n+2\,m+3\,n^2+4\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1}+\frac {B\,b^2\,x\,x^{3\,n}\,{\left (c\,x\right )}^m\,\left (m^2+3\,m\,n+2\,m+2\,n^2+3\,n+1\right )}{m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1} \] Input:
int((c*x)^m*(A + B*x^n)*(a + b*x^n)^2,x)
Output:
(A*a^2*x*(c*x)^m)/(m + 1) + (a*x*x^n*(c*x)^m*(2*A*b + B*a)*(2*m + 5*n + 5* m*n + m^2 + 6*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2*n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1) + (b*x*x^(2*n)*(c*x)^m*(A*b + 2*B*a)*(2*m + 4*n + 4*m*n + m^2 + 3*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2*n + 3* m^2 + m^3 + 11*n^2 + 6*n^3 + 1) + (B*b^2*x*x^(3*n)*(c*x)^m*(2*m + 3*n + 3* m*n + m^2 + 2*n^2 + 1))/(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2*n + 3*m^2 + m^3 + 11*n^2 + 6*n^3 + 1)
Time = 0.19 (sec) , antiderivative size = 480, normalized size of antiderivative = 4.32 \[ \int (c x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {x^{m} c^{m} x \left (2 x^{3 n} b^{3} n^{2}+3 x^{3 n} b^{3} n +a^{3}+3 x^{2 n} a \,b^{2}+3 x^{n} a^{2} b +x^{3 n} b^{3}+6 a^{3} n^{3}+11 a^{3} n^{2}+6 a^{3} n +9 x^{2 n} a \,b^{2} n^{2}+12 x^{2 n} a \,b^{2} n +18 x^{n} a^{2} b \,n^{2}+15 x^{n} a^{2} b n +3 x^{3 n} b^{3} m^{2}+3 x^{3 n} b^{3} m +6 a^{3} m^{2} n +11 a^{3} m \,n^{2}+12 a^{3} m n +9 x^{n} a^{2} b m +a^{3} m^{3}+3 a^{3} m^{2}+3 a^{3} m +12 x^{2 n} a \,b^{2} m^{2} n +9 x^{2 n} a \,b^{2} m \,n^{2}+24 x^{2 n} a \,b^{2} m n +15 x^{n} a^{2} b \,m^{2} n +18 x^{n} a^{2} b m \,n^{2}+30 x^{n} a^{2} b m n +3 x^{3 n} b^{3} m^{2} n +x^{3 n} b^{3} m^{3}+2 x^{3 n} b^{3} m \,n^{2}+6 x^{3 n} b^{3} m n +3 x^{2 n} a \,b^{2} m^{3}+9 x^{2 n} a \,b^{2} m^{2}+9 x^{2 n} a \,b^{2} m +3 x^{n} a^{2} b \,m^{3}+9 x^{n} a^{2} b \,m^{2}\right )}{m^{4}+6 m^{3} n +11 m^{2} n^{2}+6 m \,n^{3}+4 m^{3}+18 m^{2} n +22 m \,n^{2}+6 n^{3}+6 m^{2}+18 m n +11 n^{2}+4 m +6 n +1} \] Input:
int((c*x)^m*(a+b*x^n)^2*(A+B*x^n),x)
Output:
(x**m*c**m*x*(x**(3*n)*b**3*m**3 + 3*x**(3*n)*b**3*m**2*n + 3*x**(3*n)*b** 3*m**2 + 2*x**(3*n)*b**3*m*n**2 + 6*x**(3*n)*b**3*m*n + 3*x**(3*n)*b**3*m + 2*x**(3*n)*b**3*n**2 + 3*x**(3*n)*b**3*n + x**(3*n)*b**3 + 3*x**(2*n)*a* b**2*m**3 + 12*x**(2*n)*a*b**2*m**2*n + 9*x**(2*n)*a*b**2*m**2 + 9*x**(2*n )*a*b**2*m*n**2 + 24*x**(2*n)*a*b**2*m*n + 9*x**(2*n)*a*b**2*m + 9*x**(2*n )*a*b**2*n**2 + 12*x**(2*n)*a*b**2*n + 3*x**(2*n)*a*b**2 + 3*x**n*a**2*b*m **3 + 15*x**n*a**2*b*m**2*n + 9*x**n*a**2*b*m**2 + 18*x**n*a**2*b*m*n**2 + 30*x**n*a**2*b*m*n + 9*x**n*a**2*b*m + 18*x**n*a**2*b*n**2 + 15*x**n*a**2 *b*n + 3*x**n*a**2*b + a**3*m**3 + 6*a**3*m**2*n + 3*a**3*m**2 + 11*a**3*m *n**2 + 12*a**3*m*n + 3*a**3*m + 6*a**3*n**3 + 11*a**3*n**2 + 6*a**3*n + a **3))/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m* n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1)