Integrand size = 22, antiderivative size = 121 \[ \int x^{7/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2}{9} a^3 A x^{9/2}+\frac {2 a^2 (3 A b+a B) x^{\frac {9}{2}+n}}{9+2 n}+\frac {6 a b (A b+a B) x^{\frac {9}{2}+2 n}}{9+4 n}+\frac {2 b^2 (A b+3 a B) x^{\frac {9}{2}+3 n}}{3 (3+2 n)}+\frac {2 b^3 B x^{\frac {9}{2}+4 n}}{9+8 n} \] Output:
2/9*a^3*A*x^(9/2)+2*a^2*(3*A*b+B*a)*x^(9/2+n)/(9+2*n)+6*a*b*(A*b+B*a)*x^(9 /2+2*n)/(9+4*n)+2*b^2*(A*b+3*B*a)*x^(9/2+3*n)/(9+6*n)+2*b^3*B*x^(9/2+4*n)/ (9+8*n)
Time = 0.37 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98 \[ \int x^{7/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=2 \left (\frac {1}{9} a^3 A x^{9/2}+\frac {a^2 (3 A b+a B) x^{\frac {9}{2}+n}}{9+2 n}+\frac {3 a b (A b+a B) x^{\frac {9}{2}+2 n}}{9+4 n}+\frac {b^2 (A b+3 a B) x^{\frac {9}{2}+3 n}}{9+6 n}+\frac {b^3 B x^{\frac {9}{2}+4 n}}{9+8 n}\right ) \] Input:
Integrate[x^(7/2)*(a + b*x^n)^3*(A + B*x^n),x]
Output:
2*((a^3*A*x^(9/2))/9 + (a^2*(3*A*b + a*B)*x^(9/2 + n))/(9 + 2*n) + (3*a*b* (A*b + a*B)*x^(9/2 + 2*n))/(9 + 4*n) + (b^2*(A*b + 3*a*B)*x^(9/2 + 3*n))/( 9 + 6*n) + (b^3*B*x^(9/2 + 4*n))/(9 + 8*n))
Time = 0.46 (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 = {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.
\(\displaystyle \int x^{7/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (a^3 A x^{7/2}+a^2 x^{n+\frac {7}{2}} (a B+3 A b)+b^2 x^{3 n+\frac {7}{2}} (3 a B+A b)+3 a b x^{2 n+\frac {7}{2}} (a B+A b)+b^3 B x^{4 n+\frac {7}{2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{9} a^3 A x^{9/2}+\frac {2 a^2 x^{n+\frac {9}{2}} (a B+3 A b)}{2 n+9}+\frac {2 b^2 x^{3 n+\frac {9}{2}} (3 a B+A b)}{3 (2 n+3)}+\frac {6 a b x^{2 n+\frac {9}{2}} (a B+A b)}{4 n+9}+\frac {2 b^3 B x^{4 n+\frac {9}{2}}}{8 n+9}\) |
Input:
Int[x^(7/2)*(a + b*x^n)^3*(A + B*x^n),x]
Output:
(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(9/2 + n))/(9 + 2*n) + (6*a*b *(A*b + a*B)*x^(9/2 + 2*n))/(9 + 4*n) + (2*b^2*(A*b + 3*a*B)*x^(9/2 + 3*n) )/(3*(3 + 2*n)) + (2*b^3*B*x^(9/2 + 4*n))/(9 + 8*n)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1248\) vs. \(2(107)=214\).
Time = 1.03 (sec) , antiderivative size = 1249, normalized size of antiderivative = 10.32
Input:
int(x^(7/2)*(a+b*x^n)^3*(A+B*x^n),x,method=_RETURNVERBOSE)
Output:
2/27*x^(9/2)*(384*n^4+6400*n^3+27020*n^2+41600*n+21121)/(32*n^3+228*n^2+43 2*n+243)/(9+4*n)*(a+b*x^n)^3*(A+B*x^n)-40/27*x^2*(40*n^3+294*n^2+596*n+357 )/(32*n^3+228*n^2+432*n+243)/(9+4*n)*(7/2*x^(5/2)*(a+b*x^n)^3*(A+B*x^n)+3* x^(5/2)*(a+b*x^n)^2*(A+B*x^n)*b*x^n*n+x^(5/2)*(a+b*x^n)^3*B*x^n*n)+80/27*x ^3*(14*n^2+48*n+37)/(32*n^3+228*n^2+432*n+243)/(9+4*n)*(35/4*x^(3/2)*(a+b* x^n)^3*(A+B*x^n)+18*x^(3/2)*(a+b*x^n)^2*(A+B*x^n)*b*x^n*n+6*x^(3/2)*(a+b*x ^n)^3*B*x^n*n+6*x^(3/2)*(a+b*x^n)*(A+B*x^n)*b^2*(x^n)^2*n^2+6*x^(3/2)*(a+b *x^n)^2*B*(x^n)^2*n^2*b+3*x^(3/2)*(a+b*x^n)^2*(A+B*x^n)*b*x^n*n^2+x^(3/2)* (a+b*x^n)^3*B*x^n*n^2)-80/27*x^4*(5+4*n)/(128*n^4+1200*n^3+3780*n^2+4860*n +2187)*(105/8*x^(1/2)*(a+b*x^n)^3*(A+B*x^n)+213/4*x^(1/2)*(a+b*x^n)^2*(A+B *x^n)*b*x^n*n+71/4*x^(1/2)*(a+b*x^n)^3*B*x^n*n+45*x^(1/2)*(a+b*x^n)*(A+B*x ^n)*b^2*(x^n)^2*n^2+45*x^(1/2)*(a+b*x^n)^2*B*(x^n)^2*n^2*b+45/2*x^(1/2)*(a +b*x^n)^2*(A+B*x^n)*b*x^n*n^2+15/2*x^(1/2)*(a+b*x^n)^3*B*x^n*n^2+6*x^(1/2) *b^3*(x^n)^3*n^3*(A+B*x^n)+18*x^(1/2)*(a+b*x^n)*B*(x^n)^3*n^3*b^2+18*x^(1/ 2)*(a+b*x^n)*(A+B*x^n)*b^2*(x^n)^2*n^3+18*x^(1/2)*(a+b*x^n)^2*B*(x^n)^2*n^ 3*b+3*x^(1/2)*(a+b*x^n)^2*(A+B*x^n)*b*x^n*n^3+x^(1/2)*(a+b*x^n)^3*B*x^n*n^ 3)+32/27/(128*n^4+1200*n^3+3780*n^2+4860*n+2187)*x^5*(105/16*(a+b*x^n)^3*( A+B*x^n)/x^(1/2)+129/x^(1/2)*(a+b*x^n)*(A+B*x^n)*b^2*(x^n)^2*n^2+129/x^(1/ 2)*(a+b*x^n)^2*B*(x^n)^2*n^2*b+48/x^(1/2)*b^3*(x^n)^3*n^3*(A+B*x^n)+144/x^ (1/2)*(a+b*x^n)*B*(x^n)^3*n^3*b^2+144/x^(1/2)*(a+b*x^n)*(A+B*x^n)*b^2*(...
Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (107) = 214\).
Time = 0.13 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.80 \[ \int x^{7/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2 \, {\left (9 \, {\left (16 \, B b^{3} n^{3} + 132 \, B b^{3} n^{2} + 324 \, B b^{3} n + 243 \, B b^{3}\right )} x^{\frac {9}{2}} x^{4 \, n} + 3 \, {\left (2187 \, B a b^{2} + 729 \, A b^{3} + 64 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{3} + 504 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{2} + 1134 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n\right )} x^{\frac {9}{2}} x^{3 \, n} + 27 \, {\left (243 \, B a^{2} b + 243 \, A a b^{2} + 32 \, {\left (B a^{2} b + A a b^{2}\right )} n^{3} + 228 \, {\left (B a^{2} b + A a b^{2}\right )} n^{2} + 432 \, {\left (B a^{2} b + A a b^{2}\right )} n\right )} x^{\frac {9}{2}} x^{2 \, n} + 9 \, {\left (243 \, B a^{3} + 729 \, A a^{2} b + 64 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{3} + 312 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{2} + 486 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n\right )} x^{\frac {9}{2}} x^{n} + {\left (128 \, A a^{3} n^{4} + 1200 \, A a^{3} n^{3} + 3780 \, A a^{3} n^{2} + 4860 \, A a^{3} n + 2187 \, A a^{3}\right )} x^{\frac {9}{2}}\right )}}{9 \, {\left (128 \, n^{4} + 1200 \, n^{3} + 3780 \, n^{2} + 4860 \, n + 2187\right )}} \] Input:
integrate(x^(7/2)*(a+b*x^n)^3*(A+B*x^n),x, algorithm="fricas")
Output:
2/9*(9*(16*B*b^3*n^3 + 132*B*b^3*n^2 + 324*B*b^3*n + 243*B*b^3)*x^(9/2)*x^ (4*n) + 3*(2187*B*a*b^2 + 729*A*b^3 + 64*(3*B*a*b^2 + A*b^3)*n^3 + 504*(3* B*a*b^2 + A*b^3)*n^2 + 1134*(3*B*a*b^2 + A*b^3)*n)*x^(9/2)*x^(3*n) + 27*(2 43*B*a^2*b + 243*A*a*b^2 + 32*(B*a^2*b + A*a*b^2)*n^3 + 228*(B*a^2*b + A*a *b^2)*n^2 + 432*(B*a^2*b + A*a*b^2)*n)*x^(9/2)*x^(2*n) + 9*(243*B*a^3 + 72 9*A*a^2*b + 64*(B*a^3 + 3*A*a^2*b)*n^3 + 312*(B*a^3 + 3*A*a^2*b)*n^2 + 486 *(B*a^3 + 3*A*a^2*b)*n)*x^(9/2)*x^n + (128*A*a^3*n^4 + 1200*A*a^3*n^3 + 37 80*A*a^3*n^2 + 4860*A*a^3*n + 2187*A*a^3)*x^(9/2))/(128*n^4 + 1200*n^3 + 3 780*n^2 + 4860*n + 2187)
Timed out. \[ \int x^{7/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\text {Timed out} \] Input:
integrate(x**(7/2)*(a+b*x**n)**3*(A+B*x**n),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24 \[ \int x^{7/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2}{9} \, A a^{3} x^{\frac {9}{2}} + \frac {2 \, B b^{3} x^{4 \, n + \frac {9}{2}}}{8 \, n + 9} + \frac {2 \, B a b^{2} x^{3 \, n + \frac {9}{2}}}{2 \, n + 3} + \frac {2 \, A b^{3} x^{3 \, n + \frac {9}{2}}}{3 \, {\left (2 \, n + 3\right )}} + \frac {6 \, B a^{2} b x^{2 \, n + \frac {9}{2}}}{4 \, n + 9} + \frac {6 \, A a b^{2} x^{2 \, n + \frac {9}{2}}}{4 \, n + 9} + \frac {2 \, B a^{3} x^{n + \frac {9}{2}}}{2 \, n + 9} + \frac {6 \, A a^{2} b x^{n + \frac {9}{2}}}{2 \, n + 9} \] Input:
integrate(x^(7/2)*(a+b*x^n)^3*(A+B*x^n),x, algorithm="maxima")
Output:
2/9*A*a^3*x^(9/2) + 2*B*b^3*x^(4*n + 9/2)/(8*n + 9) + 2*B*a*b^2*x^(3*n + 9 /2)/(2*n + 3) + 2/3*A*b^3*x^(3*n + 9/2)/(2*n + 3) + 6*B*a^2*b*x^(2*n + 9/2 )/(4*n + 9) + 6*A*a*b^2*x^(2*n + 9/2)/(4*n + 9) + 2*B*a^3*x^(n + 9/2)/(2*n + 9) + 6*A*a^2*b*x^(n + 9/2)/(2*n + 9)
Time = 0.18 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.45 \[ \int x^{7/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2}{9} \, A a^{3} x^{\frac {9}{2}} + \frac {2 \, B b^{3} x^{\frac {9}{2}} \sqrt {x}^{8 \, n}}{8 \, n + 9} + \frac {2 \, B a b^{2} x^{\frac {9}{2}} \sqrt {x}^{6 \, n}}{2 \, n + 3} + \frac {2 \, A b^{3} x^{\frac {9}{2}} \sqrt {x}^{6 \, n}}{3 \, {\left (2 \, n + 3\right )}} + \frac {6 \, B a^{2} b x^{\frac {9}{2}} \sqrt {x}^{4 \, n}}{4 \, n + 9} + \frac {6 \, A a b^{2} x^{\frac {9}{2}} \sqrt {x}^{4 \, n}}{4 \, n + 9} + \frac {2 \, B a^{3} x^{\frac {9}{2}} \sqrt {x}^{2 \, n}}{2 \, n + 9} + \frac {6 \, A a^{2} b x^{\frac {9}{2}} \sqrt {x}^{2 \, n}}{2 \, n + 9} \] Input:
integrate(x^(7/2)*(a+b*x^n)^3*(A+B*x^n),x, algorithm="giac")
Output:
2/9*A*a^3*x^(9/2) + 2*B*b^3*x^(9/2)*sqrt(x)^(8*n)/(8*n + 9) + 2*B*a*b^2*x^ (9/2)*sqrt(x)^(6*n)/(2*n + 3) + 2/3*A*b^3*x^(9/2)*sqrt(x)^(6*n)/(2*n + 3) + 6*B*a^2*b*x^(9/2)*sqrt(x)^(4*n)/(4*n + 9) + 6*A*a*b^2*x^(9/2)*sqrt(x)^(4 *n)/(4*n + 9) + 2*B*a^3*x^(9/2)*sqrt(x)^(2*n)/(2*n + 9) + 6*A*a^2*b*x^(9/2 )*sqrt(x)^(2*n)/(2*n + 9)
Time = 4.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95 \[ \int x^{7/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2\,A\,a^3\,x^{9/2}}{9}+\frac {x^n\,x^{9/2}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )}{2\,n+9}+\frac {x^{3\,n}\,x^{9/2}\,\left (2\,A\,b^3+6\,B\,a\,b^2\right )}{6\,n+9}+\frac {2\,B\,b^3\,x^{4\,n}\,x^{9/2}}{8\,n+9}+\frac {6\,a\,b\,x^{2\,n}\,x^{9/2}\,\left (A\,b+B\,a\right )}{4\,n+9} \] Input:
int(x^(7/2)*(A + B*x^n)*(a + b*x^n)^3,x)
Output:
(2*A*a^3*x^(9/2))/9 + (x^n*x^(9/2)*(2*B*a^3 + 6*A*a^2*b))/(2*n + 9) + (x^( 3*n)*x^(9/2)*(2*A*b^3 + 6*B*a*b^2))/(6*n + 9) + (2*B*b^3*x^(4*n)*x^(9/2))/ (8*n + 9) + (6*a*b*x^(2*n)*x^(9/2)*(A*b + B*a))/(4*n + 9)
Time = 0.18 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.19 \[ \int x^{7/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2 \sqrt {x}\, x^{4} \left (144 x^{4 n} b^{4} n^{3}+1188 x^{4 n} b^{4} n^{2}+2916 x^{4 n} b^{4} n +2187 x^{4 n} b^{4}+768 x^{3 n} a \,b^{3} n^{3}+6048 x^{3 n} a \,b^{3} n^{2}+13608 x^{3 n} a \,b^{3} n +8748 x^{3 n} a \,b^{3}+1728 x^{2 n} a^{2} b^{2} n^{3}+12312 x^{2 n} a^{2} b^{2} n^{2}+23328 x^{2 n} a^{2} b^{2} n +13122 x^{2 n} a^{2} b^{2}+2304 x^{n} a^{3} b \,n^{3}+11232 x^{n} a^{3} b \,n^{2}+17496 x^{n} a^{3} b n +8748 x^{n} a^{3} b +128 a^{4} n^{4}+1200 a^{4} n^{3}+3780 a^{4} n^{2}+4860 a^{4} n +2187 a^{4}\right )}{1152 n^{4}+10800 n^{3}+34020 n^{2}+43740 n +19683} \] Input:
int(x^(7/2)*(a+b*x^n)^3*(A+B*x^n),x)
Output:
(2*sqrt(x)*x**4*(144*x**(4*n)*b**4*n**3 + 1188*x**(4*n)*b**4*n**2 + 2916*x **(4*n)*b**4*n + 2187*x**(4*n)*b**4 + 768*x**(3*n)*a*b**3*n**3 + 6048*x**( 3*n)*a*b**3*n**2 + 13608*x**(3*n)*a*b**3*n + 8748*x**(3*n)*a*b**3 + 1728*x **(2*n)*a**2*b**2*n**3 + 12312*x**(2*n)*a**2*b**2*n**2 + 23328*x**(2*n)*a* *2*b**2*n + 13122*x**(2*n)*a**2*b**2 + 2304*x**n*a**3*b*n**3 + 11232*x**n* a**3*b*n**2 + 17496*x**n*a**3*b*n + 8748*x**n*a**3*b + 128*a**4*n**4 + 120 0*a**4*n**3 + 3780*a**4*n**2 + 4860*a**4*n + 2187*a**4))/(9*(128*n**4 + 12 00*n**3 + 3780*n**2 + 4860*n + 2187))