Integrand size = 19, antiderivative size = 158 \[ \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx=a^2 d^3 x+\frac {a d^2 (2 b d+3 a e) x^{1+n}}{1+n}+\frac {d \left (b^2 d^2+6 a b d e+3 a^2 e^2\right ) x^{1+2 n}}{1+2 n}+\frac {e \left (3 b^2 d^2+6 a b d e+a^2 e^2\right ) x^{1+3 n}}{1+3 n}+\frac {b e^2 (3 b d+2 a e) x^{1+4 n}}{1+4 n}+\frac {b^2 e^3 x^{1+5 n}}{1+5 n} \] Output:
a^2*d^3*x+a*d^2*(3*a*e+2*b*d)*x^(1+n)/(1+n)+d*(3*a^2*e^2+6*a*b*d*e+b^2*d^2 )*x^(1+2*n)/(1+2*n)+e*(a^2*e^2+6*a*b*d*e+3*b^2*d^2)*x^(1+3*n)/(1+3*n)+b*e^ 2*(2*a*e+3*b*d)*x^(1+4*n)/(1+4*n)+b^2*e^3*x^(1+5*n)/(1+5*n)
Time = 0.82 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.94 \[ \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx=x \left (a^2 d^3+\frac {a d^2 (2 b d+3 a e) x^n}{1+n}+\frac {d \left (b^2 d^2+6 a b d e+3 a^2 e^2\right ) x^{2 n}}{1+2 n}+\frac {e \left (3 b^2 d^2+6 a b d e+a^2 e^2\right ) x^{3 n}}{1+3 n}+\frac {b e^2 (3 b d+2 a e) x^{4 n}}{1+4 n}+\frac {b^2 e^3 x^{5 n}}{1+5 n}\right ) \] Input:
Integrate[(a + b*x^n)^2*(d + e*x^n)^3,x]
Output:
x*(a^2*d^3 + (a*d^2*(2*b*d + 3*a*e)*x^n)/(1 + n) + (d*(b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^(2*n))/(1 + 2*n) + (e*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2)*x^ (3*n))/(1 + 3*n) + (b*e^2*(3*b*d + 2*a*e)*x^(4*n))/(1 + 4*n) + (b^2*e^3*x^ (5*n))/(1 + 5*n))
Time = 0.58 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {897, 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.
| \(\displaystyle \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx\) |
| \(\Big \downarrow \) 897 |
| \(\displaystyle \int \left (d x^{2 n} \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+e x^{3 n} \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+a^2 d^3+a d^2 x^n (3 a e+2 b d)+b e^2 x^{4 n} (2 a e+3 b d)+b^2 e^3 x^{5 n}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d x^{2 n+1} \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )}{2 n+1}+\frac {e x^{3 n+1} \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )}{3 n+1}+a^2 d^3 x+\frac {a d^2 x^{n+1} (3 a e+2 b d)}{n+1}+\frac {b e^2 x^{4 n+1} (2 a e+3 b d)}{4 n+1}+\frac {b^2 e^3 x^{5 n+1}}{5 n+1}\) |
Input:
Int[(a + b*x^n)^2*(d + e*x^n)^3,x]
Output:
a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^(1 + n))/(1 + n) + (d*(b^2*d^2 + 6*a* b*d*e + 3*a^2*e^2)*x^(1 + 2*n))/(1 + 2*n) + (e*(3*b^2*d^2 + 6*a*b*d*e + a^ 2*e^2)*x^(1 + 3*n))/(1 + 3*n) + (b*e^2*(3*b*d + 2*a*e)*x^(1 + 4*n))/(1 + 4 *n) + (b^2*e^3*x^(1 + 5*n))/(1 + 5*n)
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b , c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Time = 1.21 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(a^{2} d^{3} x +\frac {d \left (3 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x \,x^{2 n}}{1+2 n}+\frac {e \left (a^{2} e^{2}+6 a b d e +3 b^{2} d^{2}\right ) x \,x^{3 n}}{1+3 n}+\frac {e^{3} b^{2} x \,x^{5 n}}{5 n +1}+\frac {a \,d^{2} \left (3 a e +2 b d \right ) x \,x^{n}}{1+n}+\frac {b \,e^{2} \left (2 a e +3 b d \right ) x \,x^{4 n}}{1+4 n}\) | \(154\) |
| norman | \(a^{2} d^{3} x +\frac {d \left (3 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}+\frac {e \left (a^{2} e^{2}+6 a b d e +3 b^{2} d^{2}\right ) x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}+\frac {e^{3} b^{2} x \,{\mathrm e}^{5 n \ln \left (x \right )}}{5 n +1}+\frac {a \,d^{2} \left (3 a e +2 b d \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {b \,e^{2} \left (2 a e +3 b d \right ) x \,{\mathrm e}^{4 n \ln \left (x \right )}}{1+4 n}\) | \(164\) |
| parallelrisch | \(\frac {120 x \,a^{2} d^{3} n^{5}+274 x \,a^{2} d^{3} n^{4}+225 x \,a^{2} d^{3} n^{3}+85 x \,a^{2} d^{3} n^{2}+15 x \,a^{2} d^{3} n +462 x \,x^{n} a^{2} d^{2} e \,n^{3}+308 x \,x^{n} a b \,d^{3} n^{3}+213 x \,x^{n} a^{2} d^{2} e \,n^{2}+142 x \,x^{n} a b \,d^{3} n^{2}+42 x \,x^{n} a^{2} d^{2} e n +28 x \,x^{n} a b \,d^{3} n +360 x \,x^{n} a^{2} d^{2} e \,n^{4}+240 x \,x^{n} a b \,d^{3} n^{4}+a^{2} d^{3} x +x \,x^{3 n} a^{2} e^{3}+3 x \,x^{n} a^{2} d^{2} e +2 x \,x^{n} a b \,d^{3}+x \,x^{2 n} b^{2} d^{3}+50 x \,x^{5 n} b^{2} e^{3} n^{3}+35 x \,x^{5 n} b^{2} e^{3} n^{2}+40 x \,x^{3 n} a^{2} e^{3} n^{4}+10 x \,x^{5 n} b^{2} e^{3} n +78 x \,x^{3 n} a^{2} e^{3} n^{3}+60 x \,x^{2 n} b^{2} d^{3} n^{4}+49 x \,x^{3 n} a^{2} e^{3} n^{2}+24 x \,x^{5 n} b^{2} e^{3} n^{4}+3 x \,x^{3 n} b^{2} d^{2} e +13 x \,x^{2 n} b^{2} d^{3} n +3 x \,x^{2 n} a^{2} d \,e^{2}+107 x \,x^{2 n} b^{2} d^{3} n^{3}+2 x \,x^{4 n} a b \,e^{3}+3 x \,x^{4 n} b^{2} d \,e^{2}+12 x \,x^{3 n} a^{2} e^{3} n +59 x \,x^{2 n} b^{2} d^{3} n^{2}+e^{3} b^{2} x \,x^{5 n}+60 x \,x^{4 n} a b \,e^{3} n^{4}+240 x \,x^{3 n} a b d \,e^{2} n^{4}+468 x \,x^{3 n} a b d \,e^{2} n^{3}+354 x \,x^{2 n} a b \,d^{2} e \,n^{2}+78 x \,x^{2 n} a b \,d^{2} e n +294 x \,x^{3 n} a b d \,e^{2} n^{2}+360 x \,x^{2 n} a b \,d^{2} e \,n^{4}+72 x \,x^{3 n} a b d \,e^{2} n +642 x \,x^{2 n} a b \,d^{2} e \,n^{3}+120 x \,x^{3 n} b^{2} d^{2} e \,n^{4}+82 x \,x^{4 n} a b \,e^{3} n^{2}+123 x \,x^{4 n} b^{2} d \,e^{2} n^{2}+234 x \,x^{3 n} b^{2} d^{2} e \,n^{3}+180 x \,x^{2 n} a^{2} d \,e^{2} n^{4}+90 x \,x^{4 n} b^{2} d \,e^{2} n^{4}+122 x \,x^{4 n} a b \,e^{3} n^{3}+183 x \,x^{4 n} b^{2} d \,e^{2} n^{3}+6 x \,x^{3 n} a b d \,e^{2}+39 x \,x^{2 n} a^{2} d \,e^{2} n +6 x \,x^{2 n} a b \,d^{2} e +22 x \,x^{4 n} a b \,e^{3} n +33 x \,x^{4 n} b^{2} d \,e^{2} n +147 x \,x^{3 n} b^{2} d^{2} e \,n^{2}+321 x \,x^{2 n} a^{2} d \,e^{2} n^{3}+36 x \,x^{3 n} b^{2} d^{2} e n +177 x \,x^{2 n} a^{2} d \,e^{2} n^{2}}{\left (1+2 n \right ) \left (1+3 n \right ) \left (5 n +1\right ) \left (1+n \right ) \left (1+4 n \right )}\) | \(969\) |
| orering | \(\text {Expression too large to display}\) | \(2626\) |
Input:
int((a+b*x^n)^2*(d+e*x^n)^3,x,method=_RETURNVERBOSE)
Output:
a^2*d^3*x+d*(3*a^2*e^2+6*a*b*d*e+b^2*d^2)/(1+2*n)*x*(x^n)^2+e*(a^2*e^2+6*a *b*d*e+3*b^2*d^2)/(1+3*n)*x*(x^n)^3+e^3*b^2/(5*n+1)*x*(x^n)^5+a*d^2*(3*a*e +2*b*d)/(1+n)*x*x^n+b*e^2*(2*a*e+3*b*d)/(1+4*n)*x*(x^n)^4
Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (158) = 316\).
Time = 0.12 (sec) , antiderivative size = 667, normalized size of antiderivative = 4.22 \[ \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx=\frac {{\left (24 \, b^{2} e^{3} n^{4} + 50 \, b^{2} e^{3} n^{3} + 35 \, b^{2} e^{3} n^{2} + 10 \, b^{2} e^{3} n + b^{2} e^{3}\right )} x x^{5 \, n} + {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3} + 30 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n^{4} + 61 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n^{3} + 41 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n^{2} + 11 \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n\right )} x x^{4 \, n} + {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3} + 40 \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n^{4} + 78 \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n^{3} + 49 \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n^{2} + 12 \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n\right )} x x^{3 \, n} + {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + 60 \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n^{4} + 107 \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n^{3} + 59 \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n^{2} + 13 \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n\right )} x x^{2 \, n} + {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e + 120 \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n^{4} + 154 \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n^{3} + 71 \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n^{2} + 14 \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n\right )} x x^{n} + {\left (120 \, a^{2} d^{3} n^{5} + 274 \, a^{2} d^{3} n^{4} + 225 \, a^{2} d^{3} n^{3} + 85 \, a^{2} d^{3} n^{2} + 15 \, a^{2} d^{3} n + a^{2} d^{3}\right )} x}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1} \] Input:
integrate((a+b*x^n)^2*(d+e*x^n)^3,x, algorithm="fricas")
Output:
((24*b^2*e^3*n^4 + 50*b^2*e^3*n^3 + 35*b^2*e^3*n^2 + 10*b^2*e^3*n + b^2*e^ 3)*x*x^(5*n) + (3*b^2*d*e^2 + 2*a*b*e^3 + 30*(3*b^2*d*e^2 + 2*a*b*e^3)*n^4 + 61*(3*b^2*d*e^2 + 2*a*b*e^3)*n^3 + 41*(3*b^2*d*e^2 + 2*a*b*e^3)*n^2 + 1 1*(3*b^2*d*e^2 + 2*a*b*e^3)*n)*x*x^(4*n) + (3*b^2*d^2*e + 6*a*b*d*e^2 + a^ 2*e^3 + 40*(3*b^2*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*n^4 + 78*(3*b^2*d^2*e + 6 *a*b*d*e^2 + a^2*e^3)*n^3 + 49*(3*b^2*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*n^2 + 12*(3*b^2*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*n)*x*x^(3*n) + (b^2*d^3 + 6*a*b* d^2*e + 3*a^2*d*e^2 + 60*(b^2*d^3 + 6*a*b*d^2*e + 3*a^2*d*e^2)*n^4 + 107*( b^2*d^3 + 6*a*b*d^2*e + 3*a^2*d*e^2)*n^3 + 59*(b^2*d^3 + 6*a*b*d^2*e + 3*a ^2*d*e^2)*n^2 + 13*(b^2*d^3 + 6*a*b*d^2*e + 3*a^2*d*e^2)*n)*x*x^(2*n) + (2 *a*b*d^3 + 3*a^2*d^2*e + 120*(2*a*b*d^3 + 3*a^2*d^2*e)*n^4 + 154*(2*a*b*d^ 3 + 3*a^2*d^2*e)*n^3 + 71*(2*a*b*d^3 + 3*a^2*d^2*e)*n^2 + 14*(2*a*b*d^3 + 3*a^2*d^2*e)*n)*x*x^n + (120*a^2*d^3*n^5 + 274*a^2*d^3*n^4 + 225*a^2*d^3*n ^3 + 85*a^2*d^3*n^2 + 15*a^2*d^3*n + a^2*d^3)*x)/(120*n^5 + 274*n^4 + 225* n^3 + 85*n^2 + 15*n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 3376 vs. \(2 (151) = 302\).
Time = 7.94 (sec) , antiderivative size = 3376, normalized size of antiderivative = 21.37 \[ \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx=\text {Too large to display} \] Input:
integrate((a+b*x**n)**2*(d+e*x**n)**3,x)
Output:
Piecewise((a**2*d**3*x + 3*a**2*d**2*e*log(x) - 3*a**2*d*e**2/x - a**2*e** 3/(2*x**2) + 2*a*b*d**3*log(x) - 6*a*b*d**2*e/x - 3*a*b*d*e**2/x**2 - 2*a* b*e**3/(3*x**3) - b**2*d**3/x - 3*b**2*d**2*e/(2*x**2) - b**2*d*e**2/x**3 - b**2*e**3/(4*x**4), Eq(n, -1)), (a**2*d**3*x + 6*a**2*d**2*e*sqrt(x) + 3 *a**2*d*e**2*log(x) - 2*a**2*e**3/sqrt(x) + 4*a*b*d**3*sqrt(x) + 6*a*b*d** 2*e*log(x) - 12*a*b*d*e**2/sqrt(x) - 2*a*b*e**3/x + b**2*d**3*log(x) - 6*b **2*d**2*e/sqrt(x) - 3*b**2*d*e**2/x - 2*b**2*e**3/(3*x**(3/2)), Eq(n, -1/ 2)), (a**2*d**3*x + 9*a**2*d**2*e*x**(2/3)/2 + 9*a**2*d*e**2*x**(1/3) + a* *2*e**3*log(x) + 3*a*b*d**3*x**(2/3) + 18*a*b*d**2*e*x**(1/3) + 6*a*b*d*e* *2*log(x) - 6*a*b*e**3/x**(1/3) + 3*b**2*d**3*x**(1/3) + 3*b**2*d**2*e*log (x) - 9*b**2*d*e**2/x**(1/3) - 3*b**2*e**3/(2*x**(2/3)), Eq(n, -1/3)), (a* *2*d**3*x + 4*a*d**2*x**(3/4)*(3*a*e + 2*b*d)/3 - 4*b**2*e**3/x**(1/4) + 4 *b*e**2*(2*a*e + 3*b*d)*log(x**(1/4)) + 2*d*sqrt(x)*(3*a**2*e**2 + 6*a*b*d *e + b**2*d**2) + 4*e*x**(1/4)*(a**2*e**2 + 6*a*b*d*e + 3*b**2*d**2), Eq(n , -1/4)), (a**2*d**3*x + 5*a*d**2*x**(4/5)*(3*a*e + 2*b*d)/4 + 5*b**2*e**3 *log(x**(1/5)) + 5*b*e**2*x**(1/5)*(2*a*e + 3*b*d) + 5*d*x**(3/5)*(3*a**2* e**2 + 6*a*b*d*e + b**2*d**2)/3 + 5*e*x**(2/5)*(a**2*e**2 + 6*a*b*d*e + 3* b**2*d**2)/2, Eq(n, -1/5)), (120*a**2*d**3*n**5*x/(120*n**5 + 274*n**4 + 2 25*n**3 + 85*n**2 + 15*n + 1) + 274*a**2*d**3*n**4*x/(120*n**5 + 274*n**4 + 225*n**3 + 85*n**2 + 15*n + 1) + 225*a**2*d**3*n**3*x/(120*n**5 + 274...
Time = 0.03 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.53 \[ \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx=a^{2} d^{3} x + \frac {b^{2} e^{3} x^{5 \, n + 1}}{5 \, n + 1} + \frac {3 \, b^{2} d e^{2} x^{4 \, n + 1}}{4 \, n + 1} + \frac {2 \, a b e^{3} x^{4 \, n + 1}}{4 \, n + 1} + \frac {3 \, b^{2} d^{2} e x^{3 \, n + 1}}{3 \, n + 1} + \frac {6 \, a b d e^{2} x^{3 \, n + 1}}{3 \, n + 1} + \frac {a^{2} e^{3} x^{3 \, n + 1}}{3 \, n + 1} + \frac {b^{2} d^{3} x^{2 \, n + 1}}{2 \, n + 1} + \frac {6 \, a b d^{2} e x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, a^{2} d e^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {2 \, a b d^{3} x^{n + 1}}{n + 1} + \frac {3 \, a^{2} d^{2} e x^{n + 1}}{n + 1} \] Input:
integrate((a+b*x^n)^2*(d+e*x^n)^3,x, algorithm="maxima")
Output:
a^2*d^3*x + b^2*e^3*x^(5*n + 1)/(5*n + 1) + 3*b^2*d*e^2*x^(4*n + 1)/(4*n + 1) + 2*a*b*e^3*x^(4*n + 1)/(4*n + 1) + 3*b^2*d^2*e*x^(3*n + 1)/(3*n + 1) + 6*a*b*d*e^2*x^(3*n + 1)/(3*n + 1) + a^2*e^3*x^(3*n + 1)/(3*n + 1) + b^2* d^3*x^(2*n + 1)/(2*n + 1) + 6*a*b*d^2*e*x^(2*n + 1)/(2*n + 1) + 3*a^2*d*e^ 2*x^(2*n + 1)/(2*n + 1) + 2*a*b*d^3*x^(n + 1)/(n + 1) + 3*a^2*d^2*e*x^(n + 1)/(n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 962 vs. \(2 (158) = 316\).
Time = 0.14 (sec) , antiderivative size = 962, normalized size of antiderivative = 6.09 \[ \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx =\text {Too large to display} \] Input:
integrate((a+b*x^n)^2*(d+e*x^n)^3,x, algorithm="giac")
Output:
(120*a^2*d^3*n^5*x + 24*b^2*e^3*n^4*x*x^(5*n) + 90*b^2*d*e^2*n^4*x*x^(4*n) + 60*a*b*e^3*n^4*x*x^(4*n) + 120*b^2*d^2*e*n^4*x*x^(3*n) + 240*a*b*d*e^2* n^4*x*x^(3*n) + 40*a^2*e^3*n^4*x*x^(3*n) + 60*b^2*d^3*n^4*x*x^(2*n) + 360* a*b*d^2*e*n^4*x*x^(2*n) + 180*a^2*d*e^2*n^4*x*x^(2*n) + 240*a*b*d^3*n^4*x* x^n + 360*a^2*d^2*e*n^4*x*x^n + 274*a^2*d^3*n^4*x + 50*b^2*e^3*n^3*x*x^(5* n) + 183*b^2*d*e^2*n^3*x*x^(4*n) + 122*a*b*e^3*n^3*x*x^(4*n) + 234*b^2*d^2 *e*n^3*x*x^(3*n) + 468*a*b*d*e^2*n^3*x*x^(3*n) + 78*a^2*e^3*n^3*x*x^(3*n) + 107*b^2*d^3*n^3*x*x^(2*n) + 642*a*b*d^2*e*n^3*x*x^(2*n) + 321*a^2*d*e^2* n^3*x*x^(2*n) + 308*a*b*d^3*n^3*x*x^n + 462*a^2*d^2*e*n^3*x*x^n + 225*a^2* d^3*n^3*x + 35*b^2*e^3*n^2*x*x^(5*n) + 123*b^2*d*e^2*n^2*x*x^(4*n) + 82*a* b*e^3*n^2*x*x^(4*n) + 147*b^2*d^2*e*n^2*x*x^(3*n) + 294*a*b*d*e^2*n^2*x*x^ (3*n) + 49*a^2*e^3*n^2*x*x^(3*n) + 59*b^2*d^3*n^2*x*x^(2*n) + 354*a*b*d^2* e*n^2*x*x^(2*n) + 177*a^2*d*e^2*n^2*x*x^(2*n) + 142*a*b*d^3*n^2*x*x^n + 21 3*a^2*d^2*e*n^2*x*x^n + 85*a^2*d^3*n^2*x + 10*b^2*e^3*n*x*x^(5*n) + 33*b^2 *d*e^2*n*x*x^(4*n) + 22*a*b*e^3*n*x*x^(4*n) + 36*b^2*d^2*e*n*x*x^(3*n) + 7 2*a*b*d*e^2*n*x*x^(3*n) + 12*a^2*e^3*n*x*x^(3*n) + 13*b^2*d^3*n*x*x^(2*n) + 78*a*b*d^2*e*n*x*x^(2*n) + 39*a^2*d*e^2*n*x*x^(2*n) + 28*a*b*d^3*n*x*x^n + 42*a^2*d^2*e*n*x*x^n + 15*a^2*d^3*n*x + b^2*e^3*x*x^(5*n) + 3*b^2*d*e^2 *x*x^(4*n) + 2*a*b*e^3*x*x^(4*n) + 3*b^2*d^2*e*x*x^(3*n) + 6*a*b*d*e^2*x*x ^(3*n) + a^2*e^3*x*x^(3*n) + b^2*d^3*x*x^(2*n) + 6*a*b*d^2*e*x*x^(2*n) ...
Time = 0.94 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.99 \[ \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx=a^2\,d^3\,x+\frac {x\,x^{2\,n}\,\left (3\,a^2\,d\,e^2+6\,a\,b\,d^2\,e+b^2\,d^3\right )}{2\,n+1}+\frac {x\,x^{3\,n}\,\left (a^2\,e^3+6\,a\,b\,d\,e^2+3\,b^2\,d^2\,e\right )}{3\,n+1}+\frac {b^2\,e^3\,x\,x^{5\,n}}{5\,n+1}+\frac {a\,d^2\,x\,x^n\,\left (3\,a\,e+2\,b\,d\right )}{n+1}+\frac {b\,e^2\,x\,x^{4\,n}\,\left (2\,a\,e+3\,b\,d\right )}{4\,n+1} \] Input:
int((a + b*x^n)^2*(d + e*x^n)^3,x)
Output:
a^2*d^3*x + (x*x^(2*n)*(b^2*d^3 + 3*a^2*d*e^2 + 6*a*b*d^2*e))/(2*n + 1) + (x*x^(3*n)*(a^2*e^3 + 3*b^2*d^2*e + 6*a*b*d*e^2))/(3*n + 1) + (b^2*e^3*x*x ^(5*n))/(5*n + 1) + (a*d^2*x*x^n*(3*a*e + 2*b*d))/(n + 1) + (b*e^2*x*x^(4* n)*(2*a*e + 3*b*d))/(4*n + 1)
Time = 0.21 (sec) , antiderivative size = 902, normalized size of antiderivative = 5.71 \[ \int \left (a+b x^n\right )^2 \left (d+e x^n\right )^3 \, dx=\frac {x \left (240 x^{3 n} a b d \,e^{2} n^{4}+468 x^{3 n} a b d \,e^{2} n^{3}+294 x^{3 n} a b d \,e^{2} n^{2}+72 x^{3 n} a b d \,e^{2} n +360 x^{2 n} a b \,d^{2} e \,n^{4}+642 x^{2 n} a b \,d^{2} e \,n^{3}+354 x^{2 n} a b \,d^{2} e \,n^{2}+78 x^{2 n} a b \,d^{2} e n +120 a^{2} d^{3} n^{5}+274 a^{2} d^{3} n^{4}+225 a^{2} d^{3} n^{3}+85 a^{2} d^{3} n^{2}+15 a^{2} d^{3} n +x^{3 n} a^{2} e^{3}+x^{5 n} b^{2} e^{3}+x^{2 n} b^{2} d^{3}+24 x^{5 n} b^{2} e^{3} n^{4}+50 x^{5 n} b^{2} e^{3} n^{3}+35 x^{5 n} b^{2} e^{3} n^{2}+10 x^{5 n} b^{2} e^{3} n +2 x^{4 n} a b \,e^{3}+3 x^{4 n} b^{2} d \,e^{2}+40 x^{3 n} a^{2} e^{3} n^{4}+78 x^{3 n} a^{2} e^{3} n^{3}+49 x^{3 n} a^{2} e^{3} n^{2}+12 x^{3 n} a^{2} e^{3} n +3 x^{3 n} b^{2} d^{2} e +3 x^{2 n} a^{2} d \,e^{2}+60 x^{2 n} b^{2} d^{3} n^{4}+107 x^{2 n} b^{2} d^{3} n^{3}+59 x^{2 n} b^{2} d^{3} n^{2}+13 x^{2 n} b^{2} d^{3} n +3 x^{n} a^{2} d^{2} e +2 x^{n} a b \,d^{3}+a^{2} d^{3}+60 x^{4 n} a b \,e^{3} n^{4}+122 x^{4 n} a b \,e^{3} n^{3}+82 x^{4 n} a b \,e^{3} n^{2}+22 x^{4 n} a b \,e^{3} n +90 x^{4 n} b^{2} d \,e^{2} n^{4}+183 x^{4 n} b^{2} d \,e^{2} n^{3}+123 x^{4 n} b^{2} d \,e^{2} n^{2}+33 x^{4 n} b^{2} d \,e^{2} n +6 x^{3 n} a b d \,e^{2}+120 x^{3 n} b^{2} d^{2} e \,n^{4}+234 x^{3 n} b^{2} d^{2} e \,n^{3}+147 x^{3 n} b^{2} d^{2} e \,n^{2}+36 x^{3 n} b^{2} d^{2} e n +180 x^{2 n} a^{2} d \,e^{2} n^{4}+321 x^{2 n} a^{2} d \,e^{2} n^{3}+177 x^{2 n} a^{2} d \,e^{2} n^{2}+39 x^{2 n} a^{2} d \,e^{2} n +6 x^{2 n} a b \,d^{2} e +360 x^{n} a^{2} d^{2} e \,n^{4}+462 x^{n} a^{2} d^{2} e \,n^{3}+213 x^{n} a^{2} d^{2} e \,n^{2}+42 x^{n} a^{2} d^{2} e n +240 x^{n} a b \,d^{3} n^{4}+308 x^{n} a b \,d^{3} n^{3}+142 x^{n} a b \,d^{3} n^{2}+28 x^{n} a b \,d^{3} n \right )}{120 n^{5}+274 n^{4}+225 n^{3}+85 n^{2}+15 n +1} \] Input:
int((a+b*x^n)^2*(d+e*x^n)^3,x)
Output:
(x*(24*x**(5*n)*b**2*e**3*n**4 + 50*x**(5*n)*b**2*e**3*n**3 + 35*x**(5*n)* b**2*e**3*n**2 + 10*x**(5*n)*b**2*e**3*n + x**(5*n)*b**2*e**3 + 60*x**(4*n )*a*b*e**3*n**4 + 122*x**(4*n)*a*b*e**3*n**3 + 82*x**(4*n)*a*b*e**3*n**2 + 22*x**(4*n)*a*b*e**3*n + 2*x**(4*n)*a*b*e**3 + 90*x**(4*n)*b**2*d*e**2*n* *4 + 183*x**(4*n)*b**2*d*e**2*n**3 + 123*x**(4*n)*b**2*d*e**2*n**2 + 33*x* *(4*n)*b**2*d*e**2*n + 3*x**(4*n)*b**2*d*e**2 + 40*x**(3*n)*a**2*e**3*n**4 + 78*x**(3*n)*a**2*e**3*n**3 + 49*x**(3*n)*a**2*e**3*n**2 + 12*x**(3*n)*a **2*e**3*n + x**(3*n)*a**2*e**3 + 240*x**(3*n)*a*b*d*e**2*n**4 + 468*x**(3 *n)*a*b*d*e**2*n**3 + 294*x**(3*n)*a*b*d*e**2*n**2 + 72*x**(3*n)*a*b*d*e** 2*n + 6*x**(3*n)*a*b*d*e**2 + 120*x**(3*n)*b**2*d**2*e*n**4 + 234*x**(3*n) *b**2*d**2*e*n**3 + 147*x**(3*n)*b**2*d**2*e*n**2 + 36*x**(3*n)*b**2*d**2* e*n + 3*x**(3*n)*b**2*d**2*e + 180*x**(2*n)*a**2*d*e**2*n**4 + 321*x**(2*n )*a**2*d*e**2*n**3 + 177*x**(2*n)*a**2*d*e**2*n**2 + 39*x**(2*n)*a**2*d*e* *2*n + 3*x**(2*n)*a**2*d*e**2 + 360*x**(2*n)*a*b*d**2*e*n**4 + 642*x**(2*n )*a*b*d**2*e*n**3 + 354*x**(2*n)*a*b*d**2*e*n**2 + 78*x**(2*n)*a*b*d**2*e* n + 6*x**(2*n)*a*b*d**2*e + 60*x**(2*n)*b**2*d**3*n**4 + 107*x**(2*n)*b**2 *d**3*n**3 + 59*x**(2*n)*b**2*d**3*n**2 + 13*x**(2*n)*b**2*d**3*n + x**(2* n)*b**2*d**3 + 360*x**n*a**2*d**2*e*n**4 + 462*x**n*a**2*d**2*e*n**3 + 213 *x**n*a**2*d**2*e*n**2 + 42*x**n*a**2*d**2*e*n + 3*x**n*a**2*d**2*e + 240* x**n*a*b*d**3*n**4 + 308*x**n*a*b*d**3*n**3 + 142*x**n*a*b*d**3*n**2 + ...