Integrand size = 22, antiderivative size = 88 \[ \int x^{5/2} \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {2}{7} a^2 A x^{7/2}+\frac {2 a (2 A b+a B) x^{\frac {7}{2}+n}}{7+2 n}+\frac {2 b (A b+2 a B) x^{\frac {7}{2}+2 n}}{7+4 n}+\frac {2 b^2 B x^{\frac {7}{2}+3 n}}{7+6 n} \] Output:
2/7*a^2*A*x^(7/2)+2*a*(2*A*b+B*a)*x^(7/2+n)/(7+2*n)+2*b*(A*b+2*B*a)*x^(7/2 +2*n)/(7+4*n)+2*b^2*B*x^(7/2+3*n)/(7+6*n)
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.99 \[ \int x^{5/2} \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=2 \left (\frac {1}{7} a^2 A x^{7/2}+\frac {a (2 A b+a B) x^{\frac {7}{2}+n}}{7+2 n}+\frac {b (A b+2 a B) x^{\frac {7}{2}+2 n}}{7+4 n}+\frac {b^2 B x^{\frac {7}{2}+3 n}}{7+6 n}\right ) \] Input:
Integrate[x^(5/2)*(a + b*x^n)^2*(A + B*x^n),x]
Output:
2*((a^2*A*x^(7/2))/7 + (a*(2*A*b + a*B)*x^(7/2 + n))/(7 + 2*n) + (b*(A*b + 2*a*B)*x^(7/2 + 2*n))/(7 + 4*n) + (b^2*B*x^(7/2 + 3*n))/(7 + 6*n))
Time = 0.39 (sec) , antiderivative size = 88, 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.
\(\displaystyle \int x^{5/2} \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (a^2 A x^{5/2}+a x^{n+\frac {5}{2}} (a B+2 A b)+b x^{2 n+\frac {5}{2}} (2 a B+A b)+b^2 B x^{3 n+\frac {5}{2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{7} a^2 A x^{7/2}+\frac {2 a x^{n+\frac {7}{2}} (a B+2 A b)}{2 n+7}+\frac {2 b x^{2 n+\frac {7}{2}} (2 a B+A b)}{4 n+7}+\frac {2 b^2 B x^{3 n+\frac {7}{2}}}{6 n+7}\) |
Input:
Int[x^(5/2)*(a + b*x^n)^2*(A + B*x^n),x]
Output:
(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(7/2 + n))/(7 + 2*n) + (2*b*(A* b + 2*a*B)*x^(7/2 + 2*n))/(7 + 4*n) + (2*b^2*B*x^(7/2 + 3*n))/(7 + 6*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. \(620\) vs. \(2(78)=156\).
Time = 0.45 (sec) , antiderivative size = 621, normalized size of antiderivative = 7.06
method | result | size |
orering | \(\frac {24 x^{\frac {7}{2}} \left (4 n^{3}+44 n^{2}+109 n +74\right ) \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{7 \left (48 n^{3}+308 n^{2}+588 n +343\right )}-\frac {8 x^{2} \left (22 n^{2}+90 n +77\right ) \left (\frac {5 x^{\frac {3}{2}} \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{2}+2 x^{\frac {3}{2}} \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n +x^{\frac {3}{2}} \left (a +b \,x^{n}\right )^{2} B \,x^{n} n \right )}{7 \left (48 n^{3}+308 n^{2}+588 n +343\right )}+\frac {32 x^{3} \left (4+3 n \right ) \left (\frac {15 \sqrt {x}\, \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{4}+8 \sqrt {x}\, \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n +4 \sqrt {x}\, \left (a +b \,x^{n}\right )^{2} B \,x^{n} n +2 \sqrt {x}\, b^{2} x^{2 n} n^{2} \left (A +B \,x^{n}\right )+4 \sqrt {x}\, \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{2} b +2 \sqrt {x}\, \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{2}+\sqrt {x}\, \left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{2}\right )}{7 \left (48 n^{3}+308 n^{2}+588 n +343\right )}-\frac {16 x^{4} \left (\frac {15 \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{8 \sqrt {x}}+\frac {23 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n}{2 \sqrt {x}}+\frac {23 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n}{4 \sqrt {x}}+\frac {9 b^{2} x^{2 n} n^{2} \left (A +B \,x^{n}\right )}{\sqrt {x}}+\frac {18 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{2} b}{\sqrt {x}}+\frac {9 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{2}}{\sqrt {x}}+\frac {9 \left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{2}}{2 \sqrt {x}}+\frac {6 b^{2} x^{2 n} n^{3} \left (A +B \,x^{n}\right )}{\sqrt {x}}+\frac {6 x^{3 n} b^{2} n^{3} B}{\sqrt {x}}+\frac {12 \left (a +b \,x^{n}\right ) B \,x^{2 n} n^{3} b}{\sqrt {x}}+\frac {2 \left (a +b \,x^{n}\right ) \left (A +B \,x^{n}\right ) b \,x^{n} n^{3}}{\sqrt {x}}+\frac {\left (a +b \,x^{n}\right )^{2} B \,x^{n} n^{3}}{\sqrt {x}}\right )}{7 \left (48 n^{3}+308 n^{2}+588 n +343\right )}\) | \(621\) |
Input:
int(x^(5/2)*(a+b*x^n)^2*(A+B*x^n),x,method=_RETURNVERBOSE)
Output:
24/7*x^(7/2)*(4*n^3+44*n^2+109*n+74)/(48*n^3+308*n^2+588*n+343)*(a+b*x^n)^ 2*(A+B*x^n)-8/7*x^2*(22*n^2+90*n+77)/(48*n^3+308*n^2+588*n+343)*(5/2*x^(3/ 2)*(a+b*x^n)^2*(A+B*x^n)+2*x^(3/2)*(a+b*x^n)*(A+B*x^n)*b*x^n*n+x^(3/2)*(a+ b*x^n)^2*B*x^n*n)+32/7*x^3*(4+3*n)/(48*n^3+308*n^2+588*n+343)*(15/4*x^(1/2 )*(a+b*x^n)^2*(A+B*x^n)+8*x^(1/2)*(a+b*x^n)*(A+B*x^n)*b*x^n*n+4*x^(1/2)*(a +b*x^n)^2*B*x^n*n+2*x^(1/2)*b^2*(x^n)^2*n^2*(A+B*x^n)+4*x^(1/2)*(a+b*x^n)* B*(x^n)^2*n^2*b+2*x^(1/2)*(a+b*x^n)*(A+B*x^n)*b*x^n*n^2+x^(1/2)*(a+b*x^n)^ 2*B*x^n*n^2)-16/7/(48*n^3+308*n^2+588*n+343)*x^4*(15/8*(a+b*x^n)^2*(A+B*x^ n)/x^(1/2)+23/2/x^(1/2)*(a+b*x^n)*(A+B*x^n)*b*x^n*n+23/4/x^(1/2)*(a+b*x^n) ^2*B*x^n*n+9/x^(1/2)*b^2*(x^n)^2*n^2*(A+B*x^n)+18/x^(1/2)*(a+b*x^n)*B*(x^n )^2*n^2*b+9/x^(1/2)*(a+b*x^n)*(A+B*x^n)*b*x^n*n^2+9/2/x^(1/2)*(a+b*x^n)^2* B*x^n*n^2+6/x^(1/2)*b^2*(x^n)^2*n^3*(A+B*x^n)+6*(x^n)^3*b^2*n^3*B/x^(1/2)+ 12/x^(1/2)*(a+b*x^n)*B*(x^n)^2*n^3*b+2/x^(1/2)*(a+b*x^n)*(A+B*x^n)*b*x^n*n ^3+1/x^(1/2)*(a+b*x^n)^2*B*x^n*n^3)
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (78) = 156\).
Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.17 \[ \int x^{5/2} \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {2 \, {\left (7 \, {\left (8 \, B b^{2} n^{2} + 42 \, B b^{2} n + 49 \, B b^{2}\right )} x^{\frac {7}{2}} x^{3 \, n} + 7 \, {\left (98 \, B a b + 49 \, A b^{2} + 12 \, {\left (2 \, B a b + A b^{2}\right )} n^{2} + 56 \, {\left (2 \, B a b + A b^{2}\right )} n\right )} x^{\frac {7}{2}} x^{2 \, n} + 7 \, {\left (49 \, B a^{2} + 98 \, A a b + 24 \, {\left (B a^{2} + 2 \, A a b\right )} n^{2} + 70 \, {\left (B a^{2} + 2 \, A a b\right )} n\right )} x^{\frac {7}{2}} x^{n} + {\left (48 \, A a^{2} n^{3} + 308 \, A a^{2} n^{2} + 588 \, A a^{2} n + 343 \, A a^{2}\right )} x^{\frac {7}{2}}\right )}}{7 \, {\left (48 \, n^{3} + 308 \, n^{2} + 588 \, n + 343\right )}} \] Input:
integrate(x^(5/2)*(a+b*x^n)^2*(A+B*x^n),x, algorithm="fricas")
Output:
2/7*(7*(8*B*b^2*n^2 + 42*B*b^2*n + 49*B*b^2)*x^(7/2)*x^(3*n) + 7*(98*B*a*b + 49*A*b^2 + 12*(2*B*a*b + A*b^2)*n^2 + 56*(2*B*a*b + A*b^2)*n)*x^(7/2)*x ^(2*n) + 7*(49*B*a^2 + 98*A*a*b + 24*(B*a^2 + 2*A*a*b)*n^2 + 70*(B*a^2 + 2 *A*a*b)*n)*x^(7/2)*x^n + (48*A*a^2*n^3 + 308*A*a^2*n^2 + 588*A*a^2*n + 343 *A*a^2)*x^(7/2))/(48*n^3 + 308*n^2 + 588*n + 343)
Timed out. \[ \int x^{5/2} \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\text {Timed out} \] Input:
integrate(x**(5/2)*(a+b*x**n)**2*(A+B*x**n),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.18 \[ \int x^{5/2} \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {2}{7} \, A a^{2} x^{\frac {7}{2}} + \frac {2 \, B b^{2} x^{3 \, n + \frac {7}{2}}}{6 \, n + 7} + \frac {4 \, B a b x^{2 \, n + \frac {7}{2}}}{4 \, n + 7} + \frac {2 \, A b^{2} x^{2 \, n + \frac {7}{2}}}{4 \, n + 7} + \frac {2 \, B a^{2} x^{n + \frac {7}{2}}}{2 \, n + 7} + \frac {4 \, A a b x^{n + \frac {7}{2}}}{2 \, n + 7} \] Input:
integrate(x^(5/2)*(a+b*x^n)^2*(A+B*x^n),x, algorithm="maxima")
Output:
2/7*A*a^2*x^(7/2) + 2*B*b^2*x^(3*n + 7/2)/(6*n + 7) + 4*B*a*b*x^(2*n + 7/2 )/(4*n + 7) + 2*A*b^2*x^(2*n + 7/2)/(4*n + 7) + 2*B*a^2*x^(n + 7/2)/(2*n + 7) + 4*A*a*b*x^(n + 7/2)/(2*n + 7)
Time = 0.14 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.40 \[ \int x^{5/2} \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {2}{7} \, A a^{2} x^{\frac {7}{2}} + \frac {2 \, B b^{2} x^{\frac {7}{2}} \sqrt {x}^{6 \, n}}{6 \, n + 7} + \frac {4 \, B a b x^{\frac {7}{2}} \sqrt {x}^{4 \, n}}{4 \, n + 7} + \frac {2 \, A b^{2} x^{\frac {7}{2}} \sqrt {x}^{4 \, n}}{4 \, n + 7} + \frac {2 \, B a^{2} x^{\frac {7}{2}} \sqrt {x}^{2 \, n}}{2 \, n + 7} + \frac {4 \, A a b x^{\frac {7}{2}} \sqrt {x}^{2 \, n}}{2 \, n + 7} \] Input:
integrate(x^(5/2)*(a+b*x^n)^2*(A+B*x^n),x, algorithm="giac")
Output:
2/7*A*a^2*x^(7/2) + 2*B*b^2*x^(7/2)*sqrt(x)^(6*n)/(6*n + 7) + 4*B*a*b*x^(7 /2)*sqrt(x)^(4*n)/(4*n + 7) + 2*A*b^2*x^(7/2)*sqrt(x)^(4*n)/(4*n + 7) + 2* B*a^2*x^(7/2)*sqrt(x)^(2*n)/(2*n + 7) + 4*A*a*b*x^(7/2)*sqrt(x)^(2*n)/(2*n + 7)
Time = 4.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int x^{5/2} \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {2\,A\,a^2\,x^{7/2}}{7}+\frac {x^n\,x^{7/2}\,\left (2\,B\,a^2+4\,A\,b\,a\right )}{2\,n+7}+\frac {x^{2\,n}\,x^{7/2}\,\left (2\,A\,b^2+4\,B\,a\,b\right )}{4\,n+7}+\frac {2\,B\,b^2\,x^{3\,n}\,x^{7/2}}{6\,n+7} \] Input:
int(x^(5/2)*(A + B*x^n)*(a + b*x^n)^2,x)
Output:
(2*A*a^2*x^(7/2))/7 + (x^n*x^(7/2)*(2*B*a^2 + 4*A*a*b))/(2*n + 7) + (x^(2* n)*x^(7/2)*(2*A*b^2 + 4*B*a*b))/(4*n + 7) + (2*B*b^2*x^(3*n)*x^(7/2))/(6*n + 7)
Time = 0.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.75 \[ \int x^{5/2} \left (a+b x^n\right )^2 \left (A+B x^n\right ) \, dx=\frac {2 \sqrt {x}\, x^{3} \left (56 x^{3 n} b^{3} n^{2}+294 x^{3 n} b^{3} n +343 x^{3 n} b^{3}+252 x^{2 n} a \,b^{2} n^{2}+1176 x^{2 n} a \,b^{2} n +1029 x^{2 n} a \,b^{2}+504 x^{n} a^{2} b \,n^{2}+1470 x^{n} a^{2} b n +1029 x^{n} a^{2} b +48 a^{3} n^{3}+308 a^{3} n^{2}+588 a^{3} n +343 a^{3}\right )}{336 n^{3}+2156 n^{2}+4116 n +2401} \] Input:
int(x^(5/2)*(a+b*x^n)^2*(A+B*x^n),x)
Output:
(2*sqrt(x)*x**3*(56*x**(3*n)*b**3*n**2 + 294*x**(3*n)*b**3*n + 343*x**(3*n )*b**3 + 252*x**(2*n)*a*b**2*n**2 + 1176*x**(2*n)*a*b**2*n + 1029*x**(2*n) *a*b**2 + 504*x**n*a**2*b*n**2 + 1470*x**n*a**2*b*n + 1029*x**n*a**2*b + 4 8*a**3*n**3 + 308*a**3*n**2 + 588*a**3*n + 343*a**3))/(7*(48*n**3 + 308*n* *2 + 588*n + 343))