Integrand size = 22, antiderivative size = 119 \[ \int x^{5/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2}{7} a^3 A x^{7/2}+\frac {2 a^2 (3 A b+a B) x^{\frac {7}{2}+n}}{7+2 n}+\frac {6 a b (A b+a B) x^{\frac {7}{2}+2 n}}{7+4 n}+\frac {2 b^2 (A b+3 a B) x^{\frac {7}{2}+3 n}}{7+6 n}+\frac {2 b^3 B x^{\frac {7}{2}+4 n}}{7+8 n} \] Output:
2/7*a^3*A*x^(7/2)+2*a^2*(3*A*b+B*a)*x^(7/2+n)/(7+2*n)+6*a*b*(A*b+B*a)*x^(7 /2+2*n)/(7+4*n)+2*b^2*(A*b+3*B*a)*x^(7/2+3*n)/(7+6*n)+2*b^3*B*x^(7/2+4*n)/ (7+8*n)
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99 \[ \int x^{5/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=2 \left (\frac {1}{7} a^3 A x^{7/2}+\frac {a^2 (3 A b+a B) x^{\frac {7}{2}+n}}{7+2 n}+\frac {3 a b (A b+a B) x^{\frac {7}{2}+2 n}}{7+4 n}+\frac {b^2 (A b+3 a B) x^{\frac {7}{2}+3 n}}{7+6 n}+\frac {b^3 B x^{\frac {7}{2}+4 n}}{7+8 n}\right ) \] Input:
Integrate[x^(5/2)*(a + b*x^n)^3*(A + B*x^n),x]
Output:
2*((a^3*A*x^(7/2))/7 + (a^2*(3*A*b + a*B)*x^(7/2 + n))/(7 + 2*n) + (3*a*b* (A*b + a*B)*x^(7/2 + 2*n))/(7 + 4*n) + (b^2*(A*b + 3*a*B)*x^(7/2 + 3*n))/( 7 + 6*n) + (b^3*B*x^(7/2 + 4*n))/(7 + 8*n))
Time = 0.42 (sec) , antiderivative size = 119, 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.
\(\displaystyle \int x^{5/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (a^3 A x^{5/2}+a^2 x^{n+\frac {5}{2}} (a B+3 A b)+b^2 x^{3 n+\frac {5}{2}} (3 a B+A b)+3 a b x^{2 n+\frac {5}{2}} (a B+A b)+b^3 B x^{4 n+\frac {5}{2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{7} a^3 A x^{7/2}+\frac {2 a^2 x^{n+\frac {7}{2}} (a B+3 A b)}{2 n+7}+\frac {2 b^2 x^{3 n+\frac {7}{2}} (3 a B+A b)}{6 n+7}+\frac {6 a b x^{2 n+\frac {7}{2}} (a B+A b)}{4 n+7}+\frac {2 b^3 B x^{4 n+\frac {7}{2}}}{8 n+7}\) |
Input:
Int[x^(5/2)*(a + b*x^n)^3*(A + B*x^n),x]
Output:
(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(7/2 + n))/(7 + 2*n) + (6*a*b *(A*b + a*B)*x^(7/2 + 2*n))/(7 + 4*n) + (2*b^2*(A*b + 3*a*B)*x^(7/2 + 3*n) )/(7 + 6*n) + (2*b^3*B*x^(7/2 + 4*n))/(7 + 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 = 1.04 (sec) , antiderivative size = 1249, normalized size of antiderivative = 10.50
Input:
int(x^(5/2)*(a+b*x^n)^3*(A+B*x^n),x,method=_RETURNVERBOSE)
Output:
2/7*x^(7/2)*(384*n^4+4800*n^3+15260*n^2+17760*n+6841)/(96*n^3+532*n^2+784* n+343)/(7+4*n)*(a+b*x^n)^3*(A+B*x^n)-40/7*x^2*(40*n^3+210*n^2+308*n+135)/( 96*n^3+532*n^2+784*n+343)/(7+4*n)*(5/2*x^(3/2)*(a+b*x^n)^3*(A+B*x^n)+3*x^( 3/2)*(a+b*x^n)^2*(A+B*x^n)*b*x^n*n+x^(3/2)*(a+b*x^n)^3*B*x^n*n)+80/7*x^3*( 14*n^2+32*n+17)/(96*n^3+532*n^2+784*n+343)/(7+4*n)*(15/4*x^(1/2)*(a+b*x^n) ^3*(A+B*x^n)+12*x^(1/2)*(a+b*x^n)^2*(A+B*x^n)*b*x^n*n+4*x^(1/2)*(a+b*x^n)^ 3*B*x^n*n+6*x^(1/2)*(a+b*x^n)*(A+B*x^n)*b^2*(x^n)^2*n^2+6*x^(1/2)*(a+b*x^n )^2*B*(x^n)^2*n^2*b+3*x^(1/2)*(a+b*x^n)^2*(A+B*x^n)*b*x^n*n^2+x^(1/2)*(a+b *x^n)^3*B*x^n*n^2)-80/7*x^4*(3+4*n)/(384*n^4+2800*n^3+6860*n^2+6860*n+2401 )*(15/8*(a+b*x^n)^3*(A+B*x^n)/x^(1/2)+69/4/x^(1/2)*(a+b*x^n)^2*(A+B*x^n)*b *x^n*n+23/4/x^(1/2)*(a+b*x^n)^3*B*x^n*n+27/x^(1/2)*(a+b*x^n)*(A+B*x^n)*b^2 *(x^n)^2*n^2+27/x^(1/2)*(a+b*x^n)^2*B*(x^n)^2*n^2*b+27/2/x^(1/2)*(a+b*x^n) ^2*(A+B*x^n)*b*x^n*n^2+9/2/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+1/x^(1/2)*(a+b*x^n)^3*B*x^n*n^3)+32/ 7/(384*n^4+2800*n^3+6860*n^2+6860*n+2401)*x^5*(24/x^(3/2)*b^3*(x^n)^3*n^3* (A+B*x^n)+72/x^(3/2)*(a+b*x^n)*B*(x^n)^3*n^3*b^2+72/x^(3/2)*(a+b*x^n)*(A+B *x^n)*b^2*(x^n)^2*n^3+72/x^(3/2)*(a+b*x^n)^2*B*(x^n)^2*n^3*b+108/x^(3/2)*( a+b*x^n)*B*(x^n)^3*n^4*b^2+42/x^(3/2)*(a+b*x^n)*(A+B*x^n)*b^2*(x^n)^2*n...
Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (107) = 214\).
Time = 0.14 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.85 \[ \int x^{5/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2 \, {\left (7 \, {\left (48 \, B b^{3} n^{3} + 308 \, B b^{3} n^{2} + 588 \, B b^{3} n + 343 \, B b^{3}\right )} x^{\frac {7}{2}} x^{4 \, n} + 7 \, {\left (1029 \, B a b^{2} + 343 \, A b^{3} + 64 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{3} + 392 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n^{2} + 686 \, {\left (3 \, B a b^{2} + A b^{3}\right )} n\right )} x^{\frac {7}{2}} x^{3 \, n} + 21 \, {\left (343 \, B a^{2} b + 343 \, A a b^{2} + 96 \, {\left (B a^{2} b + A a b^{2}\right )} n^{3} + 532 \, {\left (B a^{2} b + A a b^{2}\right )} n^{2} + 784 \, {\left (B a^{2} b + A a b^{2}\right )} n\right )} x^{\frac {7}{2}} x^{2 \, n} + 7 \, {\left (343 \, B a^{3} + 1029 \, A a^{2} b + 192 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{3} + 728 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n^{2} + 882 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} n\right )} x^{\frac {7}{2}} x^{n} + {\left (384 \, A a^{3} n^{4} + 2800 \, A a^{3} n^{3} + 6860 \, A a^{3} n^{2} + 6860 \, A a^{3} n + 2401 \, A a^{3}\right )} x^{\frac {7}{2}}\right )}}{7 \, {\left (384 \, n^{4} + 2800 \, n^{3} + 6860 \, n^{2} + 6860 \, n + 2401\right )}} \] Input:
integrate(x^(5/2)*(a+b*x^n)^3*(A+B*x^n),x, algorithm="fricas")
Output:
2/7*(7*(48*B*b^3*n^3 + 308*B*b^3*n^2 + 588*B*b^3*n + 343*B*b^3)*x^(7/2)*x^ (4*n) + 7*(1029*B*a*b^2 + 343*A*b^3 + 64*(3*B*a*b^2 + A*b^3)*n^3 + 392*(3* B*a*b^2 + A*b^3)*n^2 + 686*(3*B*a*b^2 + A*b^3)*n)*x^(7/2)*x^(3*n) + 21*(34 3*B*a^2*b + 343*A*a*b^2 + 96*(B*a^2*b + A*a*b^2)*n^3 + 532*(B*a^2*b + A*a* b^2)*n^2 + 784*(B*a^2*b + A*a*b^2)*n)*x^(7/2)*x^(2*n) + 7*(343*B*a^3 + 102 9*A*a^2*b + 192*(B*a^3 + 3*A*a^2*b)*n^3 + 728*(B*a^3 + 3*A*a^2*b)*n^2 + 88 2*(B*a^3 + 3*A*a^2*b)*n)*x^(7/2)*x^n + (384*A*a^3*n^4 + 2800*A*a^3*n^3 + 6 860*A*a^3*n^2 + 6860*A*a^3*n + 2401*A*a^3)*x^(7/2))/(384*n^4 + 2800*n^3 + 6860*n^2 + 6860*n + 2401)
Timed out. \[ \int x^{5/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\text {Timed out} \] Input:
integrate(x**(5/2)*(a+b*x**n)**3*(A+B*x**n),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.26 \[ \int x^{5/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2}{7} \, A a^{3} x^{\frac {7}{2}} + \frac {2 \, B b^{3} x^{4 \, n + \frac {7}{2}}}{8 \, n + 7} + \frac {6 \, B a b^{2} x^{3 \, n + \frac {7}{2}}}{6 \, n + 7} + \frac {2 \, A b^{3} x^{3 \, n + \frac {7}{2}}}{6 \, n + 7} + \frac {6 \, B a^{2} b x^{2 \, n + \frac {7}{2}}}{4 \, n + 7} + \frac {6 \, A a b^{2} x^{2 \, n + \frac {7}{2}}}{4 \, n + 7} + \frac {2 \, B a^{3} x^{n + \frac {7}{2}}}{2 \, n + 7} + \frac {6 \, A a^{2} b x^{n + \frac {7}{2}}}{2 \, n + 7} \] Input:
integrate(x^(5/2)*(a+b*x^n)^3*(A+B*x^n),x, algorithm="maxima")
Output:
2/7*A*a^3*x^(7/2) + 2*B*b^3*x^(4*n + 7/2)/(8*n + 7) + 6*B*a*b^2*x^(3*n + 7 /2)/(6*n + 7) + 2*A*b^3*x^(3*n + 7/2)/(6*n + 7) + 6*B*a^2*b*x^(2*n + 7/2)/ (4*n + 7) + 6*A*a*b^2*x^(2*n + 7/2)/(4*n + 7) + 2*B*a^3*x^(n + 7/2)/(2*n + 7) + 6*A*a^2*b*x^(n + 7/2)/(2*n + 7)
Time = 0.15 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.47 \[ \int x^{5/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2}{7} \, A a^{3} x^{\frac {7}{2}} + \frac {2 \, B b^{3} x^{\frac {7}{2}} \sqrt {x}^{8 \, n}}{8 \, n + 7} + \frac {6 \, B a b^{2} x^{\frac {7}{2}} \sqrt {x}^{6 \, n}}{6 \, n + 7} + \frac {2 \, A b^{3} x^{\frac {7}{2}} \sqrt {x}^{6 \, n}}{6 \, n + 7} + \frac {6 \, B a^{2} b x^{\frac {7}{2}} \sqrt {x}^{4 \, n}}{4 \, n + 7} + \frac {6 \, A a b^{2} x^{\frac {7}{2}} \sqrt {x}^{4 \, n}}{4 \, n + 7} + \frac {2 \, B a^{3} x^{\frac {7}{2}} \sqrt {x}^{2 \, n}}{2 \, n + 7} + \frac {6 \, A a^{2} b x^{\frac {7}{2}} \sqrt {x}^{2 \, n}}{2 \, n + 7} \] Input:
integrate(x^(5/2)*(a+b*x^n)^3*(A+B*x^n),x, algorithm="giac")
Output:
2/7*A*a^3*x^(7/2) + 2*B*b^3*x^(7/2)*sqrt(x)^(8*n)/(8*n + 7) + 6*B*a*b^2*x^ (7/2)*sqrt(x)^(6*n)/(6*n + 7) + 2*A*b^3*x^(7/2)*sqrt(x)^(6*n)/(6*n + 7) + 6*B*a^2*b*x^(7/2)*sqrt(x)^(4*n)/(4*n + 7) + 6*A*a*b^2*x^(7/2)*sqrt(x)^(4*n )/(4*n + 7) + 2*B*a^3*x^(7/2)*sqrt(x)^(2*n)/(2*n + 7) + 6*A*a^2*b*x^(7/2)* sqrt(x)^(2*n)/(2*n + 7)
Time = 4.42 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.97 \[ \int x^{5/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2\,A\,a^3\,x^{7/2}}{7}+\frac {x^n\,x^{7/2}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )}{2\,n+7}+\frac {x^{3\,n}\,x^{7/2}\,\left (2\,A\,b^3+6\,B\,a\,b^2\right )}{6\,n+7}+\frac {2\,B\,b^3\,x^{4\,n}\,x^{7/2}}{8\,n+7}+\frac {6\,a\,b\,x^{2\,n}\,x^{7/2}\,\left (A\,b+B\,a\right )}{4\,n+7} \] Input:
int(x^(5/2)*(A + B*x^n)*(a + b*x^n)^3,x)
Output:
(2*A*a^3*x^(7/2))/7 + (x^n*x^(7/2)*(2*B*a^3 + 6*A*a^2*b))/(2*n + 7) + (x^( 3*n)*x^(7/2)*(2*A*b^3 + 6*B*a*b^2))/(6*n + 7) + (2*B*b^3*x^(4*n)*x^(7/2))/ (8*n + 7) + (6*a*b*x^(2*n)*x^(7/2)*(A*b + B*a))/(4*n + 7)
Time = 0.19 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.23 \[ \int x^{5/2} \left (a+b x^n\right )^3 \left (A+B x^n\right ) \, dx=\frac {2 \sqrt {x}\, x^{3} \left (336 x^{4 n} b^{4} n^{3}+2156 x^{4 n} b^{4} n^{2}+4116 x^{4 n} b^{4} n +2401 x^{4 n} b^{4}+1792 x^{3 n} a \,b^{3} n^{3}+10976 x^{3 n} a \,b^{3} n^{2}+19208 x^{3 n} a \,b^{3} n +9604 x^{3 n} a \,b^{3}+4032 x^{2 n} a^{2} b^{2} n^{3}+22344 x^{2 n} a^{2} b^{2} n^{2}+32928 x^{2 n} a^{2} b^{2} n +14406 x^{2 n} a^{2} b^{2}+5376 x^{n} a^{3} b \,n^{3}+20384 x^{n} a^{3} b \,n^{2}+24696 x^{n} a^{3} b n +9604 x^{n} a^{3} b +384 a^{4} n^{4}+2800 a^{4} n^{3}+6860 a^{4} n^{2}+6860 a^{4} n +2401 a^{4}\right )}{2688 n^{4}+19600 n^{3}+48020 n^{2}+48020 n +16807} \] Input:
int(x^(5/2)*(a+b*x^n)^3*(A+B*x^n),x)
Output:
(2*sqrt(x)*x**3*(336*x**(4*n)*b**4*n**3 + 2156*x**(4*n)*b**4*n**2 + 4116*x **(4*n)*b**4*n + 2401*x**(4*n)*b**4 + 1792*x**(3*n)*a*b**3*n**3 + 10976*x* *(3*n)*a*b**3*n**2 + 19208*x**(3*n)*a*b**3*n + 9604*x**(3*n)*a*b**3 + 4032 *x**(2*n)*a**2*b**2*n**3 + 22344*x**(2*n)*a**2*b**2*n**2 + 32928*x**(2*n)* a**2*b**2*n + 14406*x**(2*n)*a**2*b**2 + 5376*x**n*a**3*b*n**3 + 20384*x** n*a**3*b*n**2 + 24696*x**n*a**3*b*n + 9604*x**n*a**3*b + 384*a**4*n**4 + 2 800*a**4*n**3 + 6860*a**4*n**2 + 6860*a**4*n + 2401*a**4))/(7*(384*n**4 + 2800*n**3 + 6860*n**2 + 6860*n + 2401))